\>EPA
EPA/600/R-17/236
September 2018
https://www.epa.gov/bmds
Benchmark Dose Software (BMDS)
USER GUIDE
VERSION 3.0 (BETA]

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Contents
1.0 OVERVIEW	5
1.1	History of BMDS Development	5
1.2	How EPA Uses BMD Methods	5
1.3	Future of BMDS	6
2.0 USING BMDS 3.0	6
3.0 SETTING UP BMDS 3.0	8
3.1	System Requirements	8
3.2	Creating a BMDS Desktop Icon	8
3.3	Uninstalling Previous Versions of BMDS	8
4.0 TUTORIAL: ANALYZING MULTIPLE DATASETS IN BMDS 3.0	8
4.1	Step 1: Analysis Documentation	9
4.2	Step 2: Insert or Import Datasets	9
4.2.1	For Dichotomous Response Data	9
4.2.2	For Continuous Response Data	9
4.2.3	For Nested Dichotomous Data	10
4.3	Step 3: Select and Save Modeling Options	10
4.3.1	Continuous Response Models and Options	10
4.3.2	Dichotomous Response Models and Options	11
4.3.3	Dichotomous - Multi-tumor Models and Options	11
4.3.4	Dichotomous - Nested Models and Options	11
4.4	Step 4: Run Models, Review Results, and Prepare Summary Report(s)	12
5.0 MODEL OPTIONS	13
5.1	Dichotomous Model Options	13
5.1.1 Note about BMR and Graphs	14
5.2	Continuous Model Options	14
5.2.1	Unique Options for Exponential Models	15
5.2.2	Lognormal Response Option	15
5.2.3	Definition of BMR Types under Lognormal Distribution Assumption	16
5.3	Nested Model Options	16
6.0 MULTIPLE TUMOR ANALYSIS	17
6.1	Assumptions and Results	17
6.2	Obtaining the Combined BMD	18
6.3	Running an Analysis and Viewing Results	18
6.4	Troubleshooting a Tumor Analysis	18
6.5	Continuous Response Data With Negative Means	19
6.6	Test for Combining Two Datasets for the Same Endpoint	19
7.0 OUTPUT FROM INDIVIDUAL MODELS	20
7.1	Model Run Documentation	20
7.2	Parameter Estimates	20
7.3	Graphic Output from Models	20
7.4	Plot Error Bar Calculations	21
7.4.1	Continuous Models	21
7.4.2	Dichotomous Models	21
7.4.3	Nested Models	22
7.5	Outputs Specific to Continuous Models	22
7.5.1	Asymptotic Correlation Matrix of Parameter Estimates	22
7.5.2	Table of Data and Estimated Values of Interest	22
7.6	Outputs Specific to Frequentist Dichotomous Models	27
7.6.1	Asymptotic Correlation Matrix of Parameter Estimates	27
7.6.2	Analysis of Deviance Table	27

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7.6.3	Goodness of Fit	28
7.6.4	Cancer Slope Factor (CSF)—Multistage Model Only	28
7.7 Outputs Specific to Bayesian Dichotomous Models	29
8.1 Outputs Specific to Nested Models	29
8.1.1	Analysis of Deviance Table	29
8.1.2	Goodness of Fit Information—Litter Data and Grouped Data	29
9.0 MODEL DESCRIPTIONS	30
9.1	Models Included in BMDS 3.0	30
9.2	Model Types and Abbreviations	32
9.3	Optimization Algorithms Used in BMDS	32
9.4	Frequentist Continuous Model Descriptions	33
9.4.1	Special Considerations for Models for Continuous Endpoints in Simple Designs
	33
9.4.2	Likelihood Function	34
9.4.3	BMD Computation	35
9.4.4	BMDL Computation	35
9.4.5	Lognormal Distributions	36
9.4.6	Linear and Polynomial Continuous Frequentist Models	36
9.4.7	Power Continuous Model	37
9.4.8	Hill Continuous Model	38
9.4.9	Exponential Continuous Model	39
9.5	Frequentist Dichotomous Model Descriptions	40
9.5.1	Special Considerations for Models for Dichotomous Endpoints in Simple Designs
	40
9.5.2	Special Options for Models	40
9.5.3	Likelihood Function	40
9.5.4	BMD Computation	41
9.5.5	BMDL Computation	42
9.5.6	Gamma Model	42
9.5.7	Logistic Model	43
9.5.8	Log-Logistic Model	44
9.5.9	Multistage Model	44
9.5.10	Probit Model	45
9.5.11	Log-Probit Model	46
9.5.12	Quantal Linear and Weibull Models	47
9.6	Bayesian Dichotomous Model Descriptions	48
9.7	Frequentist Nested Model Descriptions	48
9.7.1	Special Considerations for Models for Nested Dichotomous Endpoints	48
9.7.2	Likelihood Function	48
9.7.3	Goodness of Fit Information—Litter Data	49
9.7.4	Logistic Nested Model	51
9.7.5	NCTR Model	52
9.8	Multitumor (MS_Combo) Model Description	53
10.0 TROUBLESHOOTING	55
10.1	Avoid Using Windows Reserved Characters in File and Path Names	55
10.2	Request Support with eTicket	55
APPENDIX A: VERSION HISTORY	56
A.1 BMDS 1.2	56
A.2 BMDS 1.2.1	56
A.3 BMDS 1.3	56
A.4 BMDS 1.3.1	56
A.5 BMDS 1.3.2	57
A.6 BMDS 1.4.1	57
A.7 BMDS 2.0 (beta)	57

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A.8	BMDS 2.0 (final)	58
A.9	BMDS 2.1 (beta)	58
A.10	BMDS 2.1 (Build 52)	59
A.11	BMDS 2.1.1 (Build 55)	59
A.12	BMDS 2.1.2 (Build 60)	59
A.13	BMDS 2.2 (Build 66)	61
A.14	BMDS 2.2 (Build 67)	61
A.15	BMDS 2.3 (Build 68)	61
A.16	BMDS 2.3.1 (Build 69)	63
A.17	BMDS 2.4 (Build 70)	63
A.18	BMDS 2.5 (Build 82)	64
A.19	BMDS 2.6	65
A.20	BMDS 2.6.0.1	67
A.21	BMDS 2.7	67
APPENDIX B: CITATION FORMAT AND ACKNOWLEDGEMENTS	68
Table of Tables
Table 1. Likelihood values and models	24
Table of Figures
Figure 1. BMDS 3.0 "Logic" worksheet with recommendation decision logic	12
Figure 2. Flow Chart of BMDS 3.0 model recommendation logic using EPA default logic assumptions
(Figure 1). Logic assumptions can be changed by the user	13

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Benchmark Dose Software (BMDS)
User Guide
1.0 Overview
The U.S. Environmental Protection Agency (EPA) Benchmark Dose Software (BMDS)
was developed as a tool to facilitate the application of benchmark dose (BMD) methods
to EPA hazardous pollutant risk assessments. This help file provides instruction on how
to use the BMDS, but is not intended to address or replace EPA BMD guidance.
However, every attempt has been made to make this software consistent with EPA
guidance, including the Risk Assessment Forum (RAF) Benchmark Dose Technical
Guidance Document (U.S. EPA, 2012).
1.1 History of BMDS Development
Research into model development for BMDS started in 1995 and the first BMDS
prototype was internally reviewed by EPA in 1997. After external and public reviews in
1998-1999, and extensive Quality Assurance testing in 1999-2000, the first public version
of BMDS, version 1.2, was released in April 2000.
A complete history of the versions of BMDS released by EPA is contained in the Version
History appendix. The models contained in the current version of BMDS are listed in the
Model Descriptions section of this document.
1.2 How EPA Uses BMD Methods
EPA uses BMD methods to derive risk estimates such as reference doses (RfDs),
reference concentrations (RfCs), and Cancer Slope Factors (CSF), which are used along
with other scientific information to set standards for human health effects.
Prior to the availability of tools such as BMDS, noncancer risk assessment benchmarks
such as RfDs and RfCs were determined from no-observed-adverse-effect levels
(NOAELs), which represent the highest experimental dose for which no adverse health
effects have been documented.
However, using the NOAEL in determining RfDs and RfCs has long been recognized as
having limitations:
•	It is limited to one of the doses in the study and is dependent on study design
•	It does not account for variability in the estimate of the dose-response
•	It does not account for the slope of the dose-response curve
•	It cannot be applied when there is no NOAEL, except through the application of an
uncertainty factor (Crump, 1984; Kimmel and Gaylor, 1988).
A goal of the BMD approach is to define a starting point of departure (POD) for the
computation of a reference value (RfD or RfC) or cancer slope factor (CSF) that is more
independent of study design. The EPA Risk Assessment Forum has published technical
guidance for the application of the BMD approach in cancer and non-cancer dose-
response assessments (U.S. EPA, 2012).
Using BMD methods involve fitting mathematical models to dose-response data and
using the different results to select a BMD that is associated with a predetermined
benchmark response (BMR), such as a 10% increase in the incidence of a particular
lesion or a 10% decrease in body weight gain.
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BMDS facilitates these operations by providing simple data-management tools and an
easy-to-use interface to run multiple models on the same dose-response dataset. Results
from all models include a reiteration of the model formula and model run options chosen
by the user, goodness-of-fit information, the BMD, and the estimate of the lower-bound
confidence limit on the BMD (BMDL). Model results are presented in textual and
graphical output files that can be printed or saved and incorporated into other documents.
1.3 Future of BMDS
EPA plans to continually improve and expand the BMDS system. Future plans include
the development of an online version of BMDS that will be integrated with the EPA Health
& Environmental Research Online (HERO) database and Health Assessment Workspace
Collaborative (HAWC) website and the incorporation of Bayesian modeling and model
averaging methods to alleviate existing issues and uncertainties associated with
bounding frequentist model parameters and selecting a "best" model (see Model
Descriptions section).
Use the BMDS web page at (http://epa.gov/ncea/bmds.htm) as your most up-to-date
source of information and updates pertaining to the BMDS. The entire BMDS system or
model updates can be downloaded from the web site. The source code files for the
models used in the BMDS system are also available via the BMDS web site to reviewers
and programmers who might be interested in performing an in-depth analysis of the
model algorithms and features. Hard copy documentation is available from EPA's BMDS
web site at http:/epa.qov/ncea/bmds.htm.
We welcome and encourage your comments on the BMDS software and the model
source code files. Please provide comments, recommendations, suggested revisions, or
corrections through our Help Desk Form. Once in the BMDS system, if you have
problems or concerns, please use the "Problem Report" feature listed in the BMDS
"Help" menu.
References
Crump, K. 1984. A new method for determining allowable daily intakes. Fund. Appl.
Toxicol. 4: 854-871.
Kimmel, C.; Gaylor, D. 1988. Issues in qualitative and quantitative risk analysis for
developmental toxicology. Risk Anal. 8: 15-21.
U.S. EPA. 2012. Benchmark Dose Technical Guidance. Risk Assessment Forum, U.S.
Environmental Protection Agency, Washington, DC 20460: EPA/100/R-12/001, June
2012.
2.0 Using BMDS 3.0
BMDS 3.0 is a major re-design of BMDS that contains substantial model code and
interface enhancements that reflect nearly two decades of experience and feedback on
the needs of risk assessors with respect to benchmark dose modeling. New Bayesian
dichotomous models have been added and pre-existing dichotomous and continuous
models have been recoded to stabilize and improve performance.
The new BMDS 3.0 interface has been designed to better facilitate performing and
tracking dose-response analyses of multiple dichotomous and continuous response
datasets. Depending on the needs of the risk assessment, users can focus a BMDS 3.0
analysis on datasets associated by study (e.g., for chemicals with a large database of
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Benchmark Dose Software (BMDS)
User Guide
studies), by chemical (e.g., for chemicals that are not well-studied), or by health outcome
(e.g., for chemicals with health outcomes that have been assessed in multiple studies
and/or by multiple response measures).
Options for a multiple dataset analysis can be saved and retrieved at any time prior to
modeling. When modeling is performed, outputs are recorded in separate Excel
workbooks for each dataset analyzed. All of the options used in the analysis are saved in
each Excel output workbook such that the analysis can be re-initiated in the main BMDS
3.0 interface from the Excel output workbook (e.g., to rerun the analysis using different
options).
BMDS 3.0 retains backwards compatibility with BMDS 2.7 and BMDS Wizard 1.11, but
also contains expanded capabilities for batch processing and revolutionary new methods
for Bayesian dose-response analysis and the averaging of dose-response modeling
results (currently only available for dichotomous data).
For experienced users, BMDS 3.0 resembles the pre-existing BMDS Wizard in these
ways:
•	Excel-based
•	Enables users to see and specify modeling options in a single worksheet
•	Includes auto-selection features for identifying the "best" results in accordance with
EPA recommended or user defined
•	Documents all inputs and outputs in a single Excel results file
•	Provides flexible print options for displaying results in Microsoft Word tables
formatted in a manner suitable for presentation in a risk assessment.
However, unlike the BMDS Wizard, BMDS 3.0 includes unique enhancements that
expand the efficient application of benchmark dose methods:
•	A separate installation of BMDS is not required: all calculations are handled within
the BMDS 3.0 Main workbook and results are backward-compatible with BMDS 2.7
•	Users can analyze multiple datasets using multiple BMDS dose-response models,
without needing to select from multiple Excel templates
•	Users can specify, from intuitive forms and picklists, traditional frequentist
dichotomous, continuous, nested dichotomous, and multitumor models available in
previous versions of BMDS
•	New to the BMDS model suite are Bayesian versions of all traditional frequentist
dichotomous models and a Bayesian model averaging feature (currently only
available for dichotomous data).
BMDS 3.0 contains all of the models and features that were available in BMDS 2.7 and
BMDS Wizard 1.11 except for the Toxicodiffusion and Ten Berge models. The Ten Berge
model is superseded by the latest version of EPA's categorical regression software
(CatRea 3.1.0.7). which has additional functionality.
Both the Toxicodiffusion and Ten Berge models can be run from BMDS 2.7, which will be
available from the BMDS website as an archive version of BMDS.
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3.0 Setting Up BMDS 3.0
3.1 System Requirements
•	BMDS 3.0 is distributed as a .zip file on the BMDS website download page, which
can be unzipped to any folder where the user has read/write privileges (Administrator
privileges are not required).
•	BMDS 3.0 requires Microsoft Excel 2016 or later with macros enabled (visit the
Microsoft support site for information on enabling Excel macros').
Note To keep your BMDS system current, check the BMDS website
(http://www.epa.gov/ncea/bmds') periodically for updates to the software or help
manual.
3.2 Creating a BMDS Desktop Icon
You may find it more convenient to run BMDS from a desktop shortcut icon. To do so:
1.	Delete any older BMDS shortcut icons on your desktop.
2.	In Windows Explorer, navigate to the newly installed BMDS application folder.
3.	Right-click the BMDSxxx.xIsm file (where "xxx" denotes the current BMDS version
number). A context menu appears.
4.	Click Send To. A submenu appears.
5.	Click "Desktop (Create Shortcut)". Windows creates a shortcut to the file on your
desktop.
3.3 Uninstalling Previous Versions of BMDS
It is not necessary to uninstall previous versions of BMDS to install and run BMDS 3.0.
4.0 Tutorial: Analyzing Multiple DataSets in BMDS 3.0
BMDS 3.0 has been designed to facilitate the process of analyzing, recording and
reporting dose-response analyses in a manner that is typically necessary and in
accordance with EPA recommendations and guidelines for the development of an EPA
chemical risk assessment. Once a dataset(s) has been entered and modeling options
defined the BMDS 3.0 analysis can be saved recalled at any time.
To perform and save a BMDS analysis a user must
1.	Document the analysis
2.	Insert or import datasets
3.	Select and save modeling options
4.	Run the models, review the results, and prepare summary report(s).
The following sections provide a simplified tutorial that overviews each step of the
process. The tutorial also references more detailed explanations located elsewhere in
this documentation.
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4.1 Step 1: Analysis Documentation
The first step in performing an analysis is to identify a directory where you want to store
Excel output files (modeling results) and Word report files (created from modeling
results). This is done in the "Main" worksheet of BMDS. Unless specified, all Excel result
and Word report files will be output to the same directory as the main BMDS program.
Separate Excel result and Word report files are created for each dataset analyzed. By
default, the name of the Excel results and Word report files will be the dataset name
provided on the "Data" worksheet of BMDS.
You can also name and describe your analysis in the "Main" worksheet. While not a
required step, such documentation is useful for analyses that you want to save for future
use or consideration. As stated previously, BMDS 3.0 has been designed such that the
saved "analysis" can be a collection of dose-response datasets from a single study, all
dose-response datasets available for a chemical (e.g., for chemicals that are not well-
studied), or all dose-response datasets related to a particular health outcome (e.g.,
multiple measures of cardiovascular effects).
4.2 Step 2: Insert or Import Datasets
After entering the analysis documentation information in the "Main" tab, the dose-
response data can be entered in the "Data" worksheet of BMDS. The user can enter or
import data multiple datasets associated with four response types, summarized
continuous (e.g., mean and SD), individual continuous (e.g., dose and response for each
test subject), dichotomous (e.g., cancer incidence) and nested dichotomous (e.g.,
developmental study) responses. In each case, dose-response data can be entered as
integers or integers with decimals (e.g., fractions). To insert new datasets, the user must
choose a response type and identify the number of rows required.
Users should enter a unique name for each dataset. This name will be used by BMDS to
reference the dataset on the "Main" worksheet (where users can select datasets to
include in a modeling analysis) and to name all Excel result and Word report files
generated from modeling the dataset. The user can also add more detailed notes to
describe the dataset.
4.2.1	For Dichotomous Response Data
The default column headers are "Dose," "N" and "Incidence" (blue row), but for reporting
purposes the user can enter replacement terms underneath these headers (grey row)
such as "mg/kg-day," "Subjects" and "Cases."
4.2.2	For Continuous Response Data
For summarized continuous response data the default column headers are "Dose," "N,"
"Mean" and "Std. Dev." (blue row).
For individual continuous response data the default column headers are "Dose,"
"Response."
Again, for reporting purposes the user can enter replacement terms underneath these
headers (grey row). The BMDS "Data" worksheet offers a tool for converting standard
errors to the required standard deviation metric. For continuous response data the user
can also choose to either allow BMDS to choose the adverse direction based on the
dose-response trend, or manually identify the dose-response direction as "Up" or "Down."
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This will impact the derivation of the benchmark response (BMR) for which the
benchmark dose (BMD) is estimated.
4.2.3 For Nested Dichotomous Data
The default column headers are "Dose," "Litter Size," "Incidence" and "Litter Specific
Covariate" (LSC) (blue row). Again, for reporting purposes the user can enter
replacement terms underneath these headers (grey row). There must be data in the LSC
row even if the modeling options do not call for the use of LSC.
4.3 Step 3: Select and Save Modeling Options
All models (see Model Descriptions) and modeling options (see Modeling Options)
available for use in an analysis can be selected in the "Main" worksheet of BMDS.
Options can be saved and reloaded at any time before or after running an analysis. An
analysis can involve the use of any or all of four Model Types, Continuous, Dichotomous,
Dichotomous - Multi-tumor and Dichotomous - Nested. Each model type offers a
different set of models and/or modeling options. Like previous versions of BMDS users
can choose to run multiple models in an analysis, but unlike previous versions of BMDS,
BMDS 3.0 allows users to run the selected models against multiple, user-defined
modeling "Option Sets" and multiple Datasets. BMDS 3.0 lists the datasets entered in the
"Data" worksheet in the "Main" worksheet and allows the user to choose datasets of the
appropriate Model Type to analyze using the selected Models and Option Sets. The
results for each Model-Option Set combination are recorded in separate worksheets
within dataset-specific Excel workbooks.
4.3.1 Continuous Response Models and Options
All of the traditional frequentist models and options that were available for analyzing
continuous response data in previous versions of BMDS are available in BMDS 3.0
(Bayesian models and Bayesian model averaging will be available for continuous
responses in a future version of BMDS). As in previous versions of BMDS, the user can
choose to run the Hill, Polynomial and Power models restricted or unrestricted, the Linear
model is not restricted and the Exponential models can only be run restricted (see
Models Descriptions). In the "Main" worksheet of BMDS 3.0, the user can define multiple
Option Sets to apply to multiple user-selected models and multiple user-selected
datasets in a single "batch" process. The adverse direction of each dataset can be
manually set to "Up" (increasing with dose) or "Down" (decreasing with dose) or "Auto-
detect" via trend testing (default). Continuous model Option Sets are user-defined with
respect to:
•	BMR Type: Standard Deviation, Relative Deviation, Absolute Deviation, Point, Hybrid
- Extra Risk (see Modeling Options for details)
•	BMRF: BMR factor (see Modeling Options for details)
•	Tail Probability: only applicable to Hybrid model (see Modeling Options for details)
•	Confidence Level: fraction between 0 and 1; 0.95 is recommended by EPA (2012)
•	Distribution: Normal or Log-normal (see Modeling Options for details)
•	Variance: Constant or Non-constant (see Modeling Options for details)
•	Background: Not currently specifiable for continuous models
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4.3.2	Dichotomous Response Models and Options
BMDS 3.0 offers the traditional frequentist dichotomous response models available in
previous versions of BMDS plus Bayesian versions of each model, as well as a Bayesian
model averaging feature (see Model Descriptions for details of the models and model
averaging feature). Most frequentist models can be run restricted or unrestricted (see
restriction details in Model Descriptions). The EPA default recommendation for initial runs
is to restrict the Gamma, Log-Logistic, Multistage and Weibull models and un-restrict the
Log-Probit and Dichotomous Hill model (the Logistic, Probit and Quantal Linear models
are not restricted). Dichotomous model Option Sets are user-defined with respect to:
•	BMR Type: Extra Risk or Added Risk (see Modeling Options for details)
•	BMR: a fraction between 0 and 1; EPA standard is 0.1
•	Confidence Level: fraction between 0 and 1; 0.95 is recommended by EPA (2012)
•	Background: Estimated, Zero or User-Specified; usually estimated unless strong
evidence for zero or specific value (see Modeling Options for details)
4.3.3	Dichotomous - Multi-tumor Models and Options
Like previous versions of BMDS, BMDS 3.0 allows users to run the EPA's Multi-tumor
(MS_Combo) model (see Model Descriptions) to determine the BMD, BMDL and BMDU
that is associated with a benchmark response (BMR) for the risk of experiencing any of
multiple tumor types. Unlike previous versions of BMDS, BMDS 3.0 provides users with
the option to manually select or allow BMDS to "Auto-select" the degree of Multistage
model to apply to a dataset. The auto-selection process follows the most recent EPA
technical guidance for selecting the Multistage model degree for the analysis of cancer
datasets. which differs from the model selection process described by EPA (2012) for
other modeling scenarios. Dichotomous model Option Sets are user-defined with respect
to:
•	BMR Type: Extra Risk or Added Risk (see Modeling Options for details)
•	BMR: a fraction between 0 and 1; EPA standard is 0.1
•	Confidence Level: fraction between 0 and 1; 0.95 is recommended by EPA (2012)
•	Background (dataset-specific): Estimated, Zero or User-Specified; usually estimated
unless strong evidence for zero or specific value
4.3.4	Dichotomous - Nested Models and Options
BMDS 3.0 allows users to run the EPA's Nested Logistic and NCTR nested dichotomous
models (see Model Descriptions). EPA no longer supports the Rai and Van Ryzin model
that was available in previous versions of BMDS. Unlike previous versions of BMDS,
BMDS 3.0 does not require the user to specify the model form, but rather automatically
runs all forms of the available nested models. Dichotomous nested model Option Sets
are user-defined with respect to:
•	BMR Type: Extra Risk or Added Risk (see Modeling Options for details)
•	BMR: a fraction between 0 and 1; EPA standard is 0.1 (see Modeling Options for
details)
•	Confidence Level: fraction between 0 and 1; 0.95 is recommended by EPA (2012)
•	Litter Specific Covariate: Overall or Control Group Mean (see Modeling Options for
details)
•	Background (dataset-specific): Estimated, Zero or User-Specified; usually estimated
unless strong evidence for zero or specific value (see Modeling Options for details)
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4.4 Step 4: Run Models, Review Results, and Prepare
Summary Report(s)
When modeling is performed, outputs are recorded in separate Excel workbooks for each
dataset analyzed. All of the options used in the analysis are saved in each Excel output
workbook such that the analysis can be re-initiated in the main BMDS 3.0 interface from
the Excel output workbook (e.g., to rerun the analysis using different options).
In the Report Options worksheet of BMDS 3.0, users can select the "User Inputs" and
"Analysis Results" to be output to the Excel results file generated from an analysis, as
well as "Word Report Default Options" that will be applied for the generation of tabular
documentation of modeling results in Microsoft Word. The "User Inputs" and "Analysis
Results" must be set separately for each of the different model/analysis types. The "Word
Report Default Options" will be applied to the Word reports that are generated from the
Excel result files.
Users familiar with the previous BMDS Wizard will note that BMDS 3.0 uses a similar
approach to analyzing modeling results and making automatic recommendations
regarding models selection that are consistent with the 2012 EPA Benchmark Dose
Technical Guidance. These criteria can be altered in the "Logic" worksheet of BMDS 3.0,
as presented below in Figure 1. Decision logic can be turned on or off, and specific
criteria can be enabled or disabled for different dataset types. Notice that the logic
depends on which type of data is entered in the BMDS Wizard (continuous, dichotomous,
cancer-dichotomous).
Figure 1. BMDS 3.0 "Logic" worksheet with recommendation decision logic.
Model Recommendatloni'BIn Placement Logic
Test Description
Test On/Off
Test
Threshold
(where
appropriate)
Bin Placement If
Test Is Failed
Notes to Show
Continuous
Dichotomous
Dichotomous
Cancer
Nested
BMD is riot calculated
On
On
On
On

Unusable Bin
BMD not calculated
BMDL is not calculated
On
On
On
On

Unusable Bin
BMDL not calculated
BMDU is not calculated
On
On
On
On

Unusable Bin
BMDU not calculated
AIC is not calculated
On
On
On
On

Unusablo Bin
AIC not calculated
Constant Variance
Off



0 1
Unusable Bin
Vananco not wo# modeled (Tost 2 p valuo > 0,1)
Variable Variance
Off



0 1
Unusablo Bin
Variance not wo# modeled (Tost 3 p value <0.1)
Goodness of fit p test
Off



01
Questionable Bin
Goodness of fit p value <0.1
Goodness of fit p test (cancer)
Off



005
Questionable Bin
Goodness of fit p value < 0.05
Ratio of BMD'BMDL (serious)
On
On
On
On
20
Questionable Bin
BMD 'BMDL ratio > 20
Ratio of BMD/BMDL (caution)
On
On
On
On
5
No Bin Change (Warning)
BMD/BMDL ratio > 5
Abs(Residual of interest) too large
Off
Off
On
On
2
Questionable Bin
|Rosidual for Dose Group Near BMD] > 2
BMDS Model Warnings
Off
Off
On
On

No Bin Change (Warning)
BMDG output file included warning
BMD higher than highest dose
On
On
On
On
1
No Bin Change (Warning)
BMD higher than maximum dose
BMDL higher than highest dose
On
On
On
On
1
No Bin Change (Warning)
BMDL higher than maximum dose
BMD lower than lowest dose (warning)
On
On
On
On
3
No Bin Change (Warning)
BMD 3x tower than lowest nan zero dose
BMDL lower than lowest dose (warning)
On
On
On
On
3
No Bin Change (Warning)
BMDL 3x lower than lowest non-zero dose
BMD tower than lowest dose (senous)
On
On
On
On
10
Questionable Bin
BMD lOx lower than towest non-zero dose
BMDL lower than towest dose (serious)
On
On
On
On
10
Questionable Bin
BMDL 10x tower lhan lowest non-zero dose
Abs(Residual al control) too large
Of!



2
No Bin Change (Warning)
| Residual al controi| > 2
Poor corrirol dose ski cterv
Off



1 5
No Bin Change (Warning)
Modeled conliol response sld dev >|1 5| actual response sld dev
d f equals 0
Off




Questionable Bin
d 1 ~0. saturated model (Goodness ol lit lesl cannot tie calculated)
Based on the decision logic entered by the user as described above, BMDS will attempt
to select a "recommended" model. A user must ultimately select a model and may
choose to disagree with the BMDS auto-determination. Full justification should be
provided for why a model was selected. The "Notes" column in the "Results" tab provides
a space for entering such documentation.
The BMDS Wizard places each model into one of three different bins:
•	Viable (highest quality model, no serious deficiencies found based on user-defined
logic but may contain warnings)
•	Questionable (serious deficiencies with model based on user-defined decision logic)
•	Unusable (required outputs such as BMD or BMDL not calculated)
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After all models with the same BMR have been placed into one of three different quality
bins, a model is recommended from the highest quality bin based on BMDL or AIC
criteria defined in the 2012 EPA Benchmark Dose Technical Guidance. The default
setting for "sufficiently close" BMDLs is a 3-fold range. Figure 2 reflects the BMDS 3.0
model recommendation logic using default assumptions shown in Figure 1.
Figure 2. Flow Chart of BMDS 3.0 model recommendation logic using EPA default logic assumptions (Figure 1).
Logic assumptions can be changed by the user.
Unusable bin
No bin change
{warnings included)
Questionable bin
Imtiolly assume models are in the ...
Viable bin
... and then begin testing
' Invalid BMD
•	Invalid BMDL
Invalid AIC
Continuous datasets only:
•	Wrong variance model (based on o-test 2)
Variance poorly modeled (p-test 3 < 0.1)
All dataset
•	8MD/BMDL ratio >20
•	| Scaled residual of interest| > 2
•	BMDL higher than maximum dose
• Goodness of fit p-test >0.1
Dichotomous-cancer datasets:
¦ Goodness of fit p-test > 0.05
ies:
BMD/BMDL ratio >5
BMDS output file included warning
BMD higher than maximum dose
BMD 3x lower than lowest non-zero dose
8M0L 3x lower than lowest non-zero dose
| Scaled residual at control I > 2
ontinuous datasets only:
Modeled response standard deviation > 1.5x
actual response standard deviation at control
If any models exist in the "Viable"
bin, a model is recommended.
A "questionable" or "unusable"
model is never recommended.
Otherwise, recommend lowest
BMDL.
If range of BMDLs in "Viable" bin <
3, recommend lowest AIC.
Model Recommendation
Criteria:
When is a Model
Recommended?
5.0 Model Options
5.1 Dichotomous Model Options
+Risk Type
Choices are "Extra" (Default) or "Added."
Added risk is the additional proportion of total animals that respond in the presence of the
dose, or the predicted probability of response at dose d, P(d), minus the predicted
probability of response in the absence of exposure, P(0).
Extra risk is the additional risk divided by the predicted proportion of animals that will not
respond in the absence of exposure, 1 - P(0). The BMRF for all dichotomous models
must be between 0 and 1 (not inclusive).
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5.1.1 Note about BMR and Graphs
The response associated with the BMR that is displayed in the graphical model output
will only be the same as the BMR when P(0) = 0. This is because to obtain the actual
response value one must solve for P(d) in the equation for added or extra risk discussed
above.
5.2 Continuous Model Options
Constant Variance
When selected (Default), the model assumes a constant variance across all dose groups.
If not selected, then the model assumes that the variance can be different for each dose
group, and varies as a power function of the mean response (see Continuous Model
Descriptions for more details).
Adverse Direction
Choices for the Adverse Direction option are "Automatic," (Default), "Up" or "Down." This
option refers to whether adversity increases as the dose-response curve rises "up" or
falls "down." If automatic is chosen, the software chooses the adverse direction based on
the shape of the dose-response curve. Manually choose the adverse direction if you
know the direction of adversity for the endpoint being studied. This selection only impacts
how the user-designated BMR is used in conjunction with model results to obtain the
BMD.
BMR Type
The BMR type is the method of choice for defining the response level used to derive the
benchmark dose (BMD). The choices allowed are "Rel. Dev." (Default), "Abs. Dev.," "Std.
Dev.," "Point" and "Extra" (Hill model only).
•	Rel. Dev. (Relative Deviation) means the response associated with the BMR will be
the background estimate plus or minus (depending on the Adverse Direction) the
product of the background estimate times the BMRF entered by the user.
•	Abs. Dev. (Absolute Deviation) means the response associated with the BMR will
be the background estimate plus or minus the BMRF.
•	Std. Dev. (Standard Deviation) means the response associated with the BMR will
be the background estimate plus or minus the product of the BMRF times the
standard deviation for the control group data.
•	Point means the response associated with the BMR will be the BMRF value itself.
•	Extra (Hill only) means the response associated with the BMR will be the background
estimate plus or minus the product of the BMRF times the difference between the
background estimate and the model estimate of the maximum/minimum response.
"Extra" is similar to Extra risk for dichotomous data, except that the maximum (or
minimum) achievable response is not 1, but is estimated from the model.
Rel. Dev.Response = m(0) + (BMRF* m(0)) (Default)
Abs. Dev.Response = m(0) + BMRF
Std. Dev.Response = m(0) + (BMRF*STD)
Point Response = BMRF
Extra (Hill and some exponential models only)
for "up" Response = m(0) + (BMRF*(mmax- m(0)))
for "down" Response = m(0) - (BMRF*( m(0) - mmin))
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where m(0) is the mean response when exposure equals zero, STD is the standard
deviation when exposure equals zero, mmax is the maximum predicted mean from the
Hill or exponential model, and mmin is the minimum predicted mean from the Hill or
exponential model.
Note When response data is lognormally distributed, the BMR Types acquire different
meanings. As of BMDS 3.0, only continuous exponential models can assume
lognormal distribution. For more information, refer to Unique Options for
Exponential Models.
5.2.1	Unique Options for Exponential Models
The exponential model choice actually allows the user to run up to four models that have
exponential-dose terms. These models are referred to as exponential Models 2-5
(following a designation by Dr. Wout Slob, wherein the restricted (flat) model was model
1). Refer to the section on Exponential Continuous Model Description for additional
details. All of the exponential models are automatically run in BMDS 3.0.Options unique
to the exponential model option screen are as follows:
•	Distribution: The user may choose to assume that the data are normally or
lognormally distributed around the dose-group-specific means. The choice of the
distribution affects that type of MLE solution that may be obtained (see next option).
Moreover, when a lognormal distribution is assumed, only constant (log-scale)
variance models will be fit to the data; such models correspond to an assumption of a
constant coefficient of variation.
•	Solution: BMDS 3.0 provides an exact MLE solutions when possible. When the data
are assumed to be lognormally distributed and the data are presented in terms of
group-specific means and standard deviations, then the exact MLE solution cannot
be obtained. In that case, the "Solution" is "Approximate" and the means and
standard deviations of the log-transformed data are estimated as follows:
log-scale mean = In(mean) - ln(1+(std/mean)2)/2
log-scale std = sqrt[ln(1+(std/mean)2)]
5.2.2	Lognormal Response Option
When modeling continuous response data, the standard assumption for the BMDS
continuous models is that the underlying distributions (one for each dose group) are
Normal, with a mean given by the dose-response model and a variance as specified by
the user (constant or a function of the mean response). An alternative assumption is that
the responses are Lognormally distributed.
Currently, only the Exponential models allow the user to choose between Normal and
Lognormal distribution assumptions. If the user has access to the individual response
data, those data can be log-transformed prior to analysis. If the user suspects that the
responses are Lognormally distributed, the best practice for now is to only use the
Exponential models, with the Lognormal option for underlying distribution. The set of
Exponential models covers a wide range of dose-response shapes and will be adequate
in many modeling contexts.
Using log-transformed responses in the analysis is not recommended, for the
following reasons:
•	If you choose to log-transform the data prior to analysis, then the interpretation of the
BMD and BMDL estimates would have to be considered carefully (and perhaps in
consultation with a statistician). Data interpretation when using log-transformed
responses will not be the same as when using the natural-scale response values.
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Indeed, the models—when "transformed back" to the natural scale—will not
correspond to any of the standard BMDS models.
For example, if using the power model on log-transformed responses, the user is
actually implicitly modeling the medians (on the natural-scale) with the function
exp{background + slope*dosepower} which is not a standard BMDS model and whose
characteristics (e.g., exponential increases in response) may not be those desired by
the user.
•	Similarly, the interpretation of the BMD will not correspond to simple expressions
(e.g., if the BMR is set equal to a relative deviation of 10%, that relative deviation will
be assessed on the log-scale and so will not yield BMD or BMDL estimates that
correspond to a 10% change in the original mean responses).
For these reasons, log-transforming the response values is not considered a "best
practice" and, as stated, should only be applied and interpreted with supporting statistical
expertise.
5.2.3 Definition of BMR Types under Lognormal Distribution Assumption
The Exponential models allow the user to assume that the response data are lognormally
distributed, with median values defined by the dose-response function and a constant
log-scale variance. Under such an assumption the BMR types are defined and
implemented so that they are calculated by the program to return BMDs as follows
(where BMRF is the numerical value, specified by the user, indicating the response, or
change in response, of interest):
•	Relative Deviation: The natural scale median value at the BMD, m(BMD), differs
from the natural scale median at 0 dose, m(0), such that |m(BMD) - m(0)|/m(0) =
BMRF.
•	Absolute Deviation: The natural scale median value at the BMD, m(BMD), differs
from the natural scale median at 0 dose, m(0), such that |m(BMD) - m(0)| = BMRF.
•	Standard deviation: The log-scale mean at the BMD, ln(m(BMD)), differs from the
log-scale mean at 0 dose, ln(m(0)), such that |ln(m(BMD)) - ln(m(0))|/D(0) = BMRF,
where a(0) is the log-scale standard deviation at 0 dose. Recall that D(0) =
ln(GSD(0)). This definition allows the user to use BMRF's typical of an analysis
where a normal distribution of responses is assumed (e.g., the EPA default of 1
standard deviation) and still maintain the logic and rationale for such choices, since
the log-transformed response values under the lognormal assumption would
themselves be normally distributed.
•	Point: The natural scale median value at the BMD, m(BMD), equals the BMRF, i.e.,
m(BMD) = BMRF.
5.3 Nested Model Options
Risk Type
Choices are "Extra" or "Added." Additional risk is the additional proportion of total animals
that respond in the presence of the dose, or the probability of response at dose d, P(d),
minus the probability of response in the absence of exposure, P(0). Extra risk is the
additional risk divided by the proportion of animals that will not respond in the absence of
exposure, 1 - P(0). Thus, extra and additional risk are equal when background rate is
zero.
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The following were options in previous versions of BMDS. BMDS 3.0 does not require the
user to specify these options anymore, but simply runs and provides output for all of the
possible option combinations.
Use a Litter Specific Covariate
Optionally, enables the user to account for inter-litter variability by using a litter specific
covariate (LSC). If the box is checked (the default), Theta values are estimated. If the box
is unchecked, the Theta values are set to zero. Do not use LSC if the corresponding
metric is affected by dose or if its use does not sufficiently improve model fit, as indicated
by a lower AIC value.
Fixed LSC Value
Choices are "Control Group Mean" (Default) or "Overall Mean." See Nested Model
Descriptions for an explanation as to why this option is necessary, and which choice
would be preferred for your given dataset. Basically, the Overall Mean should be used
under most circumstances. If the Litter Specific Covariate differs from dose to dose
(without any apparent consistent trend with respect to dose), consider using the Control
Group Mean.
Intralitter Correlations
Provides user with the option to allow the models to attempt to estimate intralitter
correlations or assume they are zero. If "Estimate Intralitter Correlations" is selected
(Default), all of the Phi values are estimated (one for each dose group). If "Assume
Intralitter Correlations Zero" is chosen, all of the Phi values are set to zero.
6.0 Multiple Tumor Analysis
6.1 Assumptions and Results
The analyses of multiple tumors have the following assumptions and results.
1.	The tumors are independent of one another.
2.	A multistage model is an appropriate model for each of the tumors separately. (The
individual multistage-cancer models fit to the individual tumors need not have the
same degree, however.)
3.	The user is interested in estimating the risk of getting one or more of the tumors
being analyzed; the results indicate the BMD and BMDL associated with the user-
defined benchmark response (BMR) level, where the BMD and BMDL are the
maximum likelihood and lower bound estimates of the dose that is estimated to give
an extra risk equal to the BMR for the "combination" (getting one or more of the
tumors).
A Multiple Tumor Analysis will always run the Multistage-Cancer model (at present, only
the Multistage-Cancer model is set up to analyze more than one dichotomous endpoint
and combine the model fits to obtain a combined BMD and BMDL as described above).
The user should ensure that the degree of the polynomial and other selections for
modeling parameters are as desired for the particular dataset(s) being analyzed.
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6.2 Obtaining the Combined BMD
Because of the form of the multistage-cancer model, the combined BMD is obtained in a
relatively straightforward manner from the maximum likelihood parameter estimates from
the models fit to the individual tumors.
The combined maximum log-likelihood is the sum of the individual maximized likelihoods
(summed over the individual tumor analyses).
The combined BMD is the dose that is estimated to yield an extra risk of getting one or
more of the tumors, where the extra risk is equal to the BMR.
The calculation of the combined BMDL is a more complicated computation based on the
profile-likelihood approach.
As such, it gives the lowest value of the dose that satisfies the following conditions:
•	There is a combination of parameters (across all models) for which the value of the
BMDL gives a combined extra risk equal to the BMR and, using those parameter
values,
•	The combined log-likelihood is greater than or equal to a minimum log-likelihood
defined by the maximum log-likelihood and the confidence level specified by the user
(i.e., the parameters that give the desired extra risk when the dose is equal to the
BMDL give a combined log-likelihood that is "close enough" to the maximum
combined log-likelihood).
6.3 Running an Analysis and Viewing Results
That output file will include analysis results for each individual tumor considered by itself
plus, at the end of the output, a short section that shows the estimated BMD and BMDL
for the BMR and confidence levels specified by the user. No graphical output is
produced; the plots for the individual multistage model runs can be obtained by running
the individual tumors in a usual data analysis or from within a regular dose-response
session. It is basically presumed here that the user has already run analyses of the
individual tumors to identify the endpoints of interest (and to determine the best form of
the multistage-cancer model for those selected tumors) that then will be used for the
combined analysis. Hence, no additional graphics are produced.
6.4 Troubleshooting a Tumor Analysis
If one or more of the tumors is estimated to have a BMD greater than three times the
highest dose tested (for that tumor), then the multiple tumor analysis will stop at an
intermediate point, i.e., after the fitting has been done for the tumor in question and the
magnitude of that BMD has been determined. No tumors listed below that tumor will be
analyzed and no combination will be completed.
It is probably the case that the tumor in question will not add substantially to the
estimation of a BMD for the combinations of tumors, assuming other tumors have BMDs
less than three times the highest dose; that is because the magnitude of response for the
tumor in question has not even reached the benchmark response level for such a high
exposure and so its individual contribution to the risk of getting one or more of the tumors
being analyzed will be small in comparison to that for the other tumors. The user might
attempt a combination that does not include the tumor in question.
The input file (.d file) for a multiple tumor analysis need not be edited by the user; the
BMDS GUI automatically creates such a file to reflect the user specifications in the
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Analysis Session screen (see above). If such a file is examined, the user will see that it
consists of a concatenation oft Multistage Dichotomous .d files (starting with the line that
specifies the number of dose groups and the desired degree of the model), all specifying
that the beta parameters are restricted to be greater than or equal to zero), where t is the
number of tumors being combined. Preceding that concatenation will be three lines of
text (which have no bearing on the model selections) and then a line that specifies the
value oft.
See also:
Multistage Model
6.5 Continuous Response Data With Negative Means
Data with negative means should only be modeled with a constant variance model.
It may occasionally be the case that, when modeling transformed data, you will need to
model negative data. In this case, the transformation used should be a variance-
stabilizing transformation so that a constant-variance model would be appropriate.
If a standard deviation-based BMR is used to define the BMD calculations, then a
constant can be added to all the observations (or means) to make the values (means)
positive. That will not change the standard deviations of the observations and would allow
you to model the variance.
6.6 Test for Combining Two Datasets for the Same Endpoint
At this time, BMDS does not include a formal test for similarity of dose response across
covariate values (e.g., across class variables like species or sex). EPA's categorical
regression software (CatReg, https://www.epa.gov/bmds/catreq) has that capability.
However, the following procedure can be used in BMDS if you have dose-response data
for two experiments that you are considering combining (e.g., for the two sexes within a
species, or two species, etc.).
1.	Choose a single model to consider for both runs.
2.	Model the two runs separately. For each run, record the following:
•	Maximum log-likelihood for each run. Add the numbers from each run to get the
summed log-likelihood.
•	The number of unconstrained parameters for each run. Add the numbers from
each run to get the summed unconstrained parameters.
3.	Combine the data from the two experiments and model them together. Record the
following:
•	The maximum log-likelihood for the combined run. This will be the combined
log-likelihood.
•	The number of unconstrained parameters for the combined run. This will be the
combined unconstrained parameters.
4.	Subtract the combined log-likelihood from from the summed log-likelihood. Then,
multiply the difference by 2.
5.	Compare the value from Step 4 to a chi-squared distribution. The degrees of freedom
for that chi-squared distribution will be the difference between the summed
unconstrained parameters (Step 2) and the combined unconstrained parameters
(Step 3).
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If the value from Step 4 is in the tail (say, greater than the 95th percentile) of the chi-
squared distribution in question, then reject the null hypothesis that the two sets have
the same dose-response relationship. If rejection occurs, then infer that it is not
proper to combine the two data sets.
7.0 Output From Individual Models
Dataset-specific Excel result workbooks generated by BMDS 3.0 contain separate
worksheets for each Model-Option Set combination that consist of tabular and graphical
summaries of the modeling inputs and results. The purpose of these worksheets is to
provide the user with goodness-of-fit criteria and model results to aid in determining the
appropriateness of the Model and Option Set to the benchmark dose derivation. This
section describing BMDS model outputs that are common to all models types is followed
by sections describing model-specific outputs.
7.1 Model Run Documentation
The output pages also give the user a quick verification of the options they had selected
on the model option screen.
For instance, when two users may be comparing results and obtained different answers,
they may consult the output pages to make sure the settings were the same or if they had
used the same (or most current) version of the software/models.
The Excel output worksheets for each Model-Option Set starts with tabular information
that the user can reference quickly to check/verify the version number, the date and time
of run, the input dataset used, that all the correct options were set, which model was
used, the explicit form of the dose response function for the model run, and review basic
data summaries (number of dose levels, etc).
7.2 Parameter Estimates
The parameter estimates are the actual estimates the program has found for the
particular model run. This table includes both the estimates for the true parameter values
as well as their estimated standard errors. The standard errors are given for two reasons:
1.	If standard errors are extraordinarily high, then the user may suspect that the
probability function may not have reached a maximum, and they may want to use
different starting points. There is not a guarantee if these are high that the function
has not, in fact, been maximized. The user should use this in conjunction with other
output to make a decision.
2.	To make inferences about the population parameters themselves. Under certain
assumptions, the user may be able to formulate tests for the true value of the
parameter.
7.3 Graphic Output from Models
The graphic output plot should display in the summary and individual model worksheets
of the Excel results workbook along with the tabular results.
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Bayesian Logistic Model with BMR of 10% Extra Risk for
the BMD and 0.95 Lower Confidence Limit for the BMDL
i
09
0.8
0.7
Estimated Probability
Response at BMD
BMDL
0 50 100 150 200 250 300 350 400 450 500
DOSE
•	The BMD and BMDL are indicated by the green and yellow vertical lines,
respectively, and are associated with the user-selected benchmark response (BMR),
the horizontal grey line.
•	The BMD curve estimated by the model is represented by a blue line.
•	Data points are shown as orange circles with their individual group confidence
intervals (see the section Error Bar Calculations for more information).
•	The graphic display features can be modified using Excel edit features.
7.4 Plot Error Bar Calculations
7.4.1 Continuous Models
BMDS uses a single error bar plotting routine for all continuous models.
1.	The plotting routine calculates the standard error of the mean (SEM) for each group.
The routine divides the group-specific observed variance (obs standard deviation
squared) by the group-specific sample size.
2.	The routine then multiplies the SEM by the Student-T percentiles (2.5th percentile or
97.5th percentile for the lower and upper bound, respectively) appropriate for the
group-specific sample size (i.e., having degrees of freedom one less than that
sample size). The routine adds the products to the observed means to define the
lower and upper ends of the error bar.
7.4.2 Dichotomous Models
The error bars shown on the plots of dichotomous data are derived using a method
discussed in J. Fleiss, "Statistical Methods for Rates and Proportions" (Third Edition,
1973), pp. 26-29. That method is a modification of the Wilson interval (based on the
score statistic) but with a continuity correction. For the upper bound, the calculation finds
the proportion, pi, such that
(|p - pi| -1/(2n)) / sqrt(pi * (1 - pi) / n) = z
where
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•	p is the observed proportion
•	n is the total number in the group in question
•	z = Z(1 -alpha/2) is the inverse standard normal cumulative distribution function
evaluated at 1 -alpha/2
This leads to equations for the lower and upper bounds of:
•	LL = {(2np + z2 - 1) - z * sqrt[ z2 - (2 + 1/n) + 4p(nq + 1) ]} / [ 2 (n + z2) ]
•	UL = {(2np + z2 + 1) + z * sqrt[ z2 + (2 - 1/n) + 4p(nq - 1) ]} / [ 2 (n + z2) ]
where q = 1-p.
The error bars shown in BMDS plots use alpha = 0.05 and so represent the 95%
confidence intervals on the observed proportions (independent of model).
7.4.3 Nested Models
The error bars shown for the plots of nested data are calculated in the same way as
those fordichotomous data. However, a Rao-Scott transformation is applied prior to the
calculations in order to express the observations in terms of an "effective" number of
affected divided by the total number in each group (the format required for the confidence
intervals of simple dichotomous responses).
7.5 Outputs Specific to Continuous Models
7.5.1 Asymptotic Correlation Matrix of Parameter Estimates
This table provides the user with a matrix of correlation estimates between each of the
parameters. Again, if these values seem to be high (in this case, very close to 1, in
absolute value), there may have been a problem in the maximization. However, as stated
before, high correlation does not confirm that the process of maximization did, in fact, fail.
Note The parameter standard errors and the correlation matrix elements are based on
a variance-covariance (VCV) matrix obtained by inverting the negative of the
Hessian matrix (the Fisher-observed information matrix). That matrix is made up
of second partial derivatives of the log-likelihood, with respect to the model
parameters. For all the continuous models, the partials are derived using a finite
difference approximation to those derivatives.
7.5.2 Table of Data and Estimated Values of Interest
This table gives a listing of the data as well as estimated means and standard deviations
from the model. This is a good place for the user to look, along with the Tests of Fit and
Maximum Likelihood below, to judge the appropriateness of the model. If a model fits
well, the observed and estimated means should be relatively close. The scaled residual
values printed in the final column of the table are defined as follows:
(Obs. Mean - Predicted Mean)/SE,
where the Predicted Mean is from the model and SE equals the estimated standard
deviation (square root of the estimated variance) divided by the square root of the sample
size.
The overall model should be called into question if the scaled residual value for any
individual dose group, particularly a dose group close to the BMD estimate, is greater
than 2 or less than -2.
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7.5.2.1 Continuous Model Maximum Likelihood
The BMDS uses likelihood theory to estimate function parameters and ultimately to make
inferences based on risk assessment data. Maximum likelihood is the process of
estimating the model parameters; the likelihood function is as large as possible
(maximized) given the form of the model under consideration and the data.
In other words, parameter values are "chosen" such that the subject model (e.g.,
polynomial or power) obtains the best possible fit to the data, given the constraints of the
model's parameter structure.
For example, suppose one wishes to fit a second degree polynomial model with a
constant variance to a dataset. The particular form of this model would be:
Y = bO + b1 * X + b2 * XA2
The parameters we wish to estimate in this case would be bO, b1, and b2, as well as the
constant variance parameter, call it SigmaA2. To estimate these parameters, BMDS uses
maximum likelihood procedures, the end result being a vector of parameters that
maximizes the likelihood function for the model specified.
The "Log(likelihood)" value given on the BMDS output page is the maximum value of the
natural logarithm of the likelihood function.
Also note that there are an associated number of parameters for each likelihood
calculated. The number of parameters reported for the model under consideration is the
total number possible for the model minus any parameter estimates that have values on
the bounds set for their estimation (either bounds specified by the user or those inherent
to the model).
In the example above, if all 4 parameters were estimated, and did not equal a bound
(e.g., did not equal 0 for the b parameters), the number of parameters reported for the
fitted model likelihood is 4.
The Akaike's Information Criterion (AIC) (Akaike, 1973; Linhart and Zucchini, 1986;
Stone, 1998) value given on the BMDS output page is -2L + 2p, where L is the log-
likelihood at the maximum likelihood estimates for the parameters, and p is the number of
model parameters estimated (and not on a boundary; see above). It can be used to
compare different types of models which use a similar fitting method, as do all
dichotomous, continuous and nested model types within BMDS. The model with the
lowest AIC would be presumed to be the better model under this method. Although such
methods are not exact, they can provide useful guidance in model selection.
The BMDS output file gives five likelihood and AIC values that may be of interest to the
user. These values are later used in asymptotic Chi-Square tests of fit. Each of these
likelihood values represents a model a user may consider in the analysis of the data. The
five models are summarized in the following table.
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Table 1. Likelihood values and models
Model
Description
A1: "Full" Constant Variance Model
Yij = Mu(i) + e(ij),
Var{e(ij)} = SigmaA2
A2: "Fullest" Model
Yij = Mu(i) + e(ij),
Var{e(ij)} = Sigma(i)A2
A3: "Full" Model with variance structure specified
by the user
Yij = Mu(i) + e(ij),
Var{e(ij)} = alpha*(Mu(i))Arho
R: "Reduced" Model
Yi = Mu + e(i),
Var{e(i)} = SigmaA2
Fitted Model
The user specified model
Model A1 estimates separate and independent means for the observed dose groups (it is
"full" or "saturated" in that respect) but posits a constant variance over those groups.
Model A2 is the "fullest" model in that it estimates separate and independent means for
the observed dose groups (as in Model A1) and it also estimates separate and
independent variances for those groups. There is no assumed functional relationship
among the means or among the variances across dose groups. This model is often
referred to as the "saturated" model (it has as many mean and variance parameters as
there are dose groups). The log-likelihood obtained for this model is the maximum
attainable, for the data under consideration.
Model A3 is similar to model A2, and may only differ with respect to its variance
parameters. Model A2 estimates separate and independent means for the observed dose
groups (like A1). If the user specifies a constant variance for the fitted model, then model
A3 will also assume that and it becomes identical to Model A1. If the user assumes a
non-constant variance for the fitted model, then Model A3 will also assume the same
functional form for the variance.
The reduced model (R) is the model that implies no difference in mean or variance over
the dose levels. In other words, it posits a constant mean response level with the same
variance around that mean at every dose level.
The last model, the fitted model, is the user-specified model (e.g., power or polynomial,
among others). A user may have reason to believe that a certain model may describe the
data well, and thus uses it to calculate the BMD and BMDL.
See the next section, "Tests of Fit," for a description of how these models are used to test
certain hypotheses about the data.
7.5.2.2 Tests of Fit
The BMDS software provides four different Tests of Fit that the user may use to
determine an appropriate model for fitting their data. These Tests of Fit are based on
asymptotic theories of the likelihood ratio. Without getting too technical, the likelihood
ratio is just the ratio of two likelihood values, many of which are given in the BMDS
output. Statistical theory proves that -2*log(likelihood ratio) converges to a Chi-Square
random variable as the sample size gets large and the number of dose levels gets large.
These values can in turn be used to obtain approximate probabilities to make inferences
about model fit. Chi-Square tables can be found in almost any statistical book.
Each of the five models described in the previous section on likelihood has a likelihood
value. The BMDS program uses these values to create ratios from two models that form
a meaningful test. Suppose the user wishes to test two models, A and B, for fit. One
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assumption that is made for these tests is that model A is "nested within" Model B, i.e.,
that Model B can be simplified (via restriction of some parameters in Model B) in such a
way that the simplified model is Model A. This implies that Model A has fewer varying
parameters. As an example, consider that the linear model is a "simpler" or "nested"
model relative to the power model because the linear model has the power parameter
restricted to be equal to 1.
Note The model with a higher number of parameters is always in the denominator of
this ratio.
Now, using the theory, -2*log{ L(A)/L(B)} approaches a Chi-Square random variable. This
can be simplified by using the fact that the log of a ratio is equal to the difference of the
logs, or simply put,
-2*log{ L(A)/L(B)} = -2*( log{L(A)} - log{L(B)}) = 2*log{L(B)} - 2*log{L(A)}.
The likelihood values given by BMDS are in fact the log-likelihoods, log{L(B)} and
log{L(A)}, so this likelihood ratio calculation becomes just a subtraction problem. This
value can then in turn be compared to a Chi-Square random variable with a specified
number of degrees of freedom.
As mentioned in the section on likelihood, each log likelihood value has an associated
number of parameters. The number of degrees of freedom for the Chi-Square test
statistic is merely the difference between the two model parameter counts. In the mini-
example above, suppose Model A has 5 fitted parameters, and that Model B has 8. In
this case, the Chi-Square value you would compare this to would be a Chi-Square with 8
-5 = 3 degrees of freedom.
In the A vs B example, what is exactly being tested? In terms of hypotheses, it would be:
HO: A models the data as well as B
H1: B models the data better than A
Keeping these tests in mind, suppose 2*log{L(B)} - 2*log{L(A)} = 4.89 based on 3
degrees of freedom. Also, suppose the rejection criteria is a Chi-Square probability of
less than .05. Looking on a Chi-Square table, 4.89 has a p-value somewhere between
.10 and .25. In this case, HO would not be rejected, and it would seem to be appropriate
to model the data using Model A. BMDS automatically does the "table look-up" for the
user, and provides the p-value associated with the calculated log-likelihood ratio having
degrees of freedom as described above.
The BMDS software provides four default tests. BMDS provides interpretation of the test
results, based on p-values that have been selected by EPA. However, the computed p-
values are presented so that the user is free to use any rejection criteria they want. Each
of the four default tests provided for any of the continuous models is discussed in some
detail below.
Test 1 (A2 vs R): Tests the null hypothesis that responses and variances don't
differ among dose levels. If this test fails to reject the null hypothesis, there may
not be a dose-response.
This test compares Model R (the simpler model) to Model A2. Model R is a simpler A2
(or nested within A2) since R can be obtained from A2 by restricting all the mean
parameters to be equal to one another and restricting all the variance parameters to be
equal to one another. If this test fails to reject the null hypothesis, then there may not be a
dose-response, as the inference would be that the simpler model (R) is not much worse
than the saturated model. The default p-value for the test (as reported in the Tests of
Interest section of the output) is 0.05. A p-value less than 0.05 is associated with the
statement that "There appears to be a difference between response and/or variances
among the dose levels. It seems appropriate to model the data." A p-value greater than
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0.05 is associated with the statement that the data may not be suitable for dose-response
modeling.
Test 2 (A1 vs A2): Tests the null hypothesis that variances are homogeneous. If
this test fails to reject the null hypothesis, the simpler constant variance model
may be appropriate.
This test compares A1 (the simpler model) to Model A2. Model A1 is a simpler A2 (or
nested within A2) since A1 can be obtained from A2 by restricting all the variance
parameters to be equal to one another. If this test rejects the null hypothesis, the
inference is that the constant variance assumption is incorrect and a modeled variance is
necessary to adequately represent the data. The default p-value for the test (as reported
in the Tests of Interest section of the output) is 0.1. A p-value less than 0.1 is associated
with the statement that the user should "Consider running a non-homogeneous variance
model. A p-value greater than 0.1 is associated with the statement that a constant
variance assumption is suitable for the dose-response modeling.
Test 3 (A3 vs A2): Tests the null hypothesis that the variances are adequately
modeled. If this test fails to reject the null hypothesis, it may be inferred that the
variances have been modeled appropriately.
Here, the test is one to see whether or not the user-specified variance model, is
appropriate. If the user-specified variance model is "constant variance," then Models A1
and A3 are identical; this test is the same as Test 2, with the same interpretation. If the
user-specified variance model is nonconstant (Sigma(i)A2 = alpha*Mu(i)Arho), this test
determines if that particular equation appears adequate to describe the variance across
dose groups. Model A3 is the simpler version of Model A2 obtained by constraining the
variances to fit the nonconstant variance equation. The default p-value for the test (as
reported in the Tests of Interest section of the output) is 0.1. A p-value less than 0.1 is
associated with the statement that "You may want to consider a different variance
model." [Unfortunately BMDS has no further way to model variance. Look for different
variance models in future releases of BMDS.] A p-value greater than 0.1 is associated
with the statement that the modeled variance appears to be suitable for the dose-
response modeling.
Test 4 (Fitted vs A3): Tests the null hypothesis that the model for the mean fits the
data. If this tests fails to reject the null hypothesis, the user has support for the
selected model.
This test compares the Fitted Model to Model A3. The Fitted Model is as simpler Model
A3 (or nested within Model A3) because it can be obtained by restricting the means
(unrestricted in A3) to be described by the dose-response function under consideration. If
this test fails to reject the null hypothesis, the inference is that the fitted model is
adequate to describe the dose-related changes in the means (conditional on the form of
the variance model; the form of the variance model is the same for the Fitted Model and
Model A3). Failure to reject the null hypothesis is associated with the inference that the
restriction of the means to the shape of the dose-response function under consideration
is adequate. The default p-value for the test (as reported in the Tests of Interest section
of the output) is 0.1. A p-value less than 0.1 is associated with the statement that "You
may want to try a different model." I.e., the fit of the Fitted Model is not good enough. A
p-value greater than 0.1 is associated with a statement that the Fitted Model appears to
be suitable for dose-response modeling.
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7.6 Outputs Specific to Frequentist Dichotomous Models
7.6.1	Asymptotic Correlation Matrix of Parameter Estimates
This table provides the user with a matrix of correlation estimates between each of the
parameters. Again, if these values seem to be high (in this case, very close to 1), there
may have been a problem in the maximization. Also, as stated before, high correlation
does not confirm that the problem of maximization in fact failed. The Weibull model, for
instance, tends to give high correlation between the slope and power parameters, even
when the likelihood was maximized.
Note The parameter standard errors and the correlation matrix elements are based on
a variance-covariance (VCV) matrix obtained by inverting the negative of the
Hessian matrix (the Fisher-observed information matrix). That matrix is made up
of second partial derivatives of the log-likelihood, with respect to the model
parameters.
For all the dichotomous models, except for the multistage model, the partials are
derived using a finite difference approximation to those derivatives.
For the multistage model, the partial derivatives are computed analytically (i.e.,
without approximating their values through the finite-difference method).
7.6.2	Analysis of Deviance Table
The analysis of deviance table lists three maximum likelihood values. The first is the "full
model". The full model would be any model that would perfectly fit all the positive
response proportions at the dose levels specified by the user. The second model is the
"fitted model" maximum likelihood value. This is the value of the maximum likelihood
function for the particular model selected and using the estimated parameter values. The
last likelihood value is the "reduced model" value, which would be the value of the
likelihood function if all data points where assumed to come from the same population
with the same population parameter. That is, for each dose level, the actual probability of
an adverse effect would be the same. These values are just the likelihood functions
evaluated according to the assumptions made at each step (i.e., the model assumption
for the fitted model).
Next to the likelihood values there are three values: Deviance, degrees of freedom (DF),
and P-value. The Deviance is the difference between the fitted or reduced model and the
full model likelihood values. This deviance measures whether or not the smaller model
(i.e., the fitted or reduced model) describe the data as well as the full model does. This
deviance is then used to formulate a Chi-Square random variable that tests exactly that.
The user may choose a rejection level (.05 is common) to test whether or not the model
fit is appropriate. The p-value for testing whether or not the fitted model adequately
describes the data is given next to the fitted model likelihood, and the user can reject or
not reject a hypothesis according to the p-value given. The reduced model p-value would
be used in the same way, but here the user would be testing whether or not there is in
fact a dose/response relationship where the true population proportion is a function of
dose, as opposed to a single population with one parameter (the proportion of affected
animals).
It will often happen that several models provide an adequate fit to a given dataset. These
models may be essentially unrelated to each other (for example a logistic model and a
probit model often do about as well at fitting dichotomous data) or they may be related to
each other in the sense that they are members of the same family that differ in which
parameters are fixed at some default value. One can consider the log-logistic, the log-
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logistic with non-zero background, and the log-logistic with threshold and non-zero
background to all be members of the same family of models. Generally, within a family of
models, as additional parameters are introduced the fit will appear to improve. Goodness-
of fit statistics presented in the main body of the Analysis of Deviance Table can be used
to compare such related models, but are not designed to compare unrelated models.
Alternative approaches are need for selecting between models that are not related (not in
the same family).
The Akaike's Information Criterion (AIC; Akaike, 1973; Linhart and Zucchini, 1986; Stone,
1998) is defined as -2L + 2p, where L is the log-likelihood at the maximum likelihood
estimates for the parameters, and p is the number of model parameters estimated. The
AlCsforthe model run are provided at the bottom of the Analysis of Deviance Table.
They can be used to compare different types of models which use a similar fitting method
(for example, least squares or a binomial maximum likelihood), as do all dichotomous,
continuous and nested model types within BMDS. The model with the lowest AIC would
be presumed to be the better model under this method. Although such methods are not
exact, they can provide useful guidance in model selection.
7.6.3	Goodness of Fit
This table gives both a listing of the data as well as residual and overall Chi-Square
Goodness of Fit tests. This is a good place for the user to look outside of the Analysis of
Deviance table to judge the appropriateness of the model. The table lists estimated
probabilities, the expected and observed number of affected animals and scaled
residuals for each dose group. If a model fits well, the observed and expected number of
affected animals should be relatively close. The overall scaled residual value, and it
corresponding p-value are indications of that "closeness". If the p-value is larger than
some predetermined critical p-value, then the user may be able to conclude that the
model is appropriate to model the data.
The scaled residual values printed at the end of the table are defined as follows:
(Obs. - Expected)/SE
where "Expected" is the predicted number of responders from the model and SE equals
the estimated standard error of that predicted number. For these models, the estimated
standard error is equal to sqrt[n*p*(1-p)], where n is the sample size and p is the model-
predicted probability of response."
n is the sample (litter) size, and
p is the model-predicted probability of response.
The overall model should be called into question if the scaled residual value for any dose
group, particularly a low dose group, is greater than 2 or less than -2.
7.6.4	Cancer Slope Factor (CSF)—Multistage Model Only
Some additional assessment tools are imparted by the Multistage model for use with
cancer response data. The output page for the Multistage model includes an estimate of
the cancer slope factor (CSF), defined by EPA as the linear slope between the extra risk
at the BMDL(10) and the extra risk at background (generally 0 dose). The Multistage
model plot also includes a dashed line representing this linear slope.
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7.7 Outputs Specific to Bayesian Dichotomous Models
Tk - [Insert explanation of difference between Bayesian model "BIC equivalent" and how
it is similar or different from the frequentist model AIC values]
7.8 Outputs Specific to Nested Models
7.8.1	Analysis of Deviance Table
The analysis of deviance table lists three maximum likelihood values.
•	The first is the "full model". The full model would be any model that would perfectly fit
all the positive response proportions at the dose levels specified by the user.
•	The second model is the "fitted model" maximum likelihood value. This is the value of
the maximum likelihood function for the particular model selected and using the
estimated parameter values.
•	The last likelihood value is the "reduced model" value, which would be the value of
the likelihood function if all data points where assumed to come from the same
population with the same population parameter. That is, for each dose level, the
actual probability of an adverse effect would be the same.
These values are just the likelihood functions evaluated according to the assumptions
made at each step (i.e., the model assumption for the fitted model).
Next to the likelihood values there are three values: Deviance, degrees of freedom (DF),
and P-value. These are asymptotic Chi-Square tests that investigate the appropriateness
of the model fit, as well the reduced model.
•	The Deviance is the difference between the fitted or reduced model and the full
model likelihood values. This deviance measures whether or not the smaller model
(i.e., the fitted or reduced model) describe the data as well as the full model does.
This deviance is then used to formulate a Chi-Square random variable that tests
exactly that. The user may choose a rejection level (.05 is common) to test whether
or not the model fit is appropriate.
•	The p-value for testing whether or not the fitted model adequately describes the data
is given next to the fitted model likelihood, and the user can reject or not reject a
hypothesis according to the p-value given .
•	The reduced model p-value would be used in the same way, but here the user would
be testing whether or not there is in fact a dose/response relationship where the true
population proportion is a function of dose, as opposed to a single population with
one parameter (the proportion of affected animals).
7.8.2	Goodness of Fit Information—Litter Data and Grouped Data
Both of these tables provide a listing of the data, expected and observed responses and
scaled residuals (observed - expected).
The "Litter Data" table contains this information for each litter.
To obtain the "Group Data" table, the Litter Data were sorted on Dose (first), and by Litter
Specific Covariate within Dose. Within dose, litters adjacent to each other with respect to
Litter Specific Covariate were grouped together until the expected number of affected
pups was at least one. This grouping was done prior to the estimation of an overall Chi-
Square and p-value to improve the validity of the Chi-Square approximation for the
goodness of fit statistic. Goodness of Fit statistics. Both tables list estimated probabilities,
the expected and observed number of affected animals and scaled residuals for each
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dose group. If a model fits well, the observed and expected number of affected animals
should be relatively close. The overall Chi-Square value, and it corresponding p-value are
an indication of that "closeness". If the p-value is larger than some predetermined critical
p-value, then the user may be able to conclude that the model is appropriate to model the
data.
The scaled residual values printed at the end of the table are defined as follows:
(Obs. - Expected)/SE
where "Expected" is the predicted number of responders from the model and SE equals
the estimated standard error of that predicted number. For these models, the estimated
standard error is equal to sqrt[n*p*(1-p)*(D*(n-1)+1)], where
•	n is the sample (litter) size,
•	p is the model-predicted probability of response, and
•	0 is the model-predicted intra-litter correlation coefficient.
The overall model should be called into question if the scaled residual value for any
individual dose and litter-specific covariate combination, particularly for a low dose group,
is greater than 2 or less than -2.
8.0 Model Descriptions
8.1 Models Included in BMDS 3.0
The following list details the models and version numbers contained in this BMDS
version.
Frequentist Dichotomous Models
Version (Date)
Dichotomous Hill
1.0 (09/30/2018)
Gamma Model
1.0 (09/30/2018)
Logistic and Log-Logistic Models
1.0 (09/30/2018)
Probit and Log Probit Models
1.0 (09/30/2018)
Multistage Model
1.0 (09/30/2018)
Weibull and Quantal Linear Models
1.0 (09/30/2018)
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Bayesian Dichotomous Models
Version (Date)
Dichotomous Hill
1.0 (09/30/2018)
Gamma Model
1.0 (09/30/2018)
Logistic and Log-Logistic Models
1.0 (09/30/2018)
Probit and Log Probit Models
1.0 (09/30/2018)
Multistage Model
1.0 (09/30/2018)
Weibull and Quantal Linear Models
1.0 (09/30/2018)

Frequentist Continuous Models
Version (Date)
Exponential Model
1.0 (09/30/2018)
Hill Model
1.0 (09/30/2018)
Polynomial and Linear Models
1.0 (09/30/2018)
Power Model
1.0 (09/30/2018)

Nested Dichotomous Models
Version (Date)
NCTR Model
2.13 (04/27/2015)
Nlogistic Model
2.20 (04/27/2015)

Bayesian Model Averaging
Version (Date)
Bayesian Model Averaging
1.0 (09/30/2018)

Multi-Tumor (MSCombo) Model
Version (Date)
Multiple tumor analysis; combining
multistage-cancer model runs over
different tumors
1.8 (04/30/2014)
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8.2 Model Types and Abbreviations
BMDS uses the following naming conventions for model abbreviations.
Model Type
Model
Abbreviation
Continuous
Exponential
exp

Hill
hil

Linear
lin

Polynomial
ply

Power
pow
Dichotomous
Gamma
gam

Logistic
log

LogLogistic
Inl

LogProbit
Inp

Multistage
mst

Probit
pro

Weibull
wei

Quantal Linear
qln

Dichotomous Hill
dhl
Nested
Nested Logistic
nln

NCTR
net
Bayesian Model Averaging
ToxicoDiffusion
bma
Multitumor
MS_Combo
multi
8.3 Optimization Algorithms Used in BMDS
The NLopt optimization library is used for BMDS 3.0.
Several optimization algorithms available in the library are used to ensure reliability of the
estimation:
• For global optimization involving the maximum likelihood or maximum a-posteriori
estimation, the L-BFGS method is attempted first. If it fails to converge, gradient free
algorithms "subplex" and "BOBYQA" algorithms are then attempted.
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•	For profiling, when only non-linear inequality constraints are needed, the COBYLA
and MMA approaches are used and compared. In the case the methods return
different optimum, the values producing the larger of the two is used.
•	For equality-constrained optimization, the augmented Lagrangian algorithm is used
and either the L-BFGS, BOBYQA, or the "subplex" algorithm is used in the local
optimization step. When two approaches produce different results, the values
producing the larger optimum is used.
NLopt 2.4.1 was used when developing the BMDS 3.0 code. This version is available for
download from the NLopt GitHub site.
For more information regarding the algorithms, refer to the NLopt documentation site.
8.4 Frequentist Continuous Model Descriptions
8.4.1 Special Considerations for Models for Continuous Endpoints in
Simple Designs
Models in this section are for continuous endpoints, such as weight or enzyme activity
measures, in simple experimental designs that do not involve nesting or other
complications. The models predict the mean value of the response, ~ (dose), expected
for a given dose.
Models for continuous endpoints require consideration of more details than do those for
dichotomous endpoints in similar designs. While for dichotomous models, we normally
model the incidence of adversely affected individuals, and so expect the response to
increase with increasing dose, in continuous models the change in a measure is modeled
without regard for "adversity", and the response may increase or decrease. Thus, just
what constitutes an adverse change, and how to specify it, must be made explicit. The
models in BMDS allow that specification to be made in several ways, which will be
described below (BMD Computation).
Another important contrast with dichotomous models is the nature of the probability
distribution of response. In dichotomous models, the nature of the experimental design
guarantees that the binomial probability distribution is appropriate. There are many more
options for continuous distributions, however. In the current version of BMDS, the
distribution of continuous measures is assumed to be normal, with the exception of the
Exponential Models, for which the user may assume either a normal or a lognormal
distribution (see the section on the Lognormal Distribution below). Moreover, for all
models and normally distributed data, one may assume either a constant variance (that
is, the variance is the same regardless of dose group), or a variance that changes as a
power function of the mean value:
which is the modeled variance for the ith dose group, the expression A (c/ose/) is the
observed mean (from the model) for the ith dose group, and a (alpha) and p (Rho) are
estimated parameters. This formulation allows for several commonly encountered
situations. For example, if p = 2, then the coefficient of variation is constant, a common
situation especially for biochemical measures; if p = 1, then the variance is proportional
to the mean, which is sometimes appropriate for large counts (especially if the constant
of proportionality, k, is 1.0). When a lognormal distribution is assumed, the Exponential
Models assume a constant (log-scale) variance, equivalent to a constant coefficient of
variation.
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8.4.2 Likelihood Function
Suppose there are g doses,
with Ni subjects per dose group, and that y,y is the measurement for the jm subject in the
ith dose group. The form of the log-likelihood function depends upon whether the
variance is assumed to be constant, or to vary among doses.
For constant variance, the log-likelihood function is:
where
is the sample variance for the ith dose group,
is the sample mean for the ith dose group, g is the number of doses, Ni is the number of
subjects in the ith dose group, and a2 the variance which is same in all dose groups.
Generally, a2 and the parameters hidden here in A() are to be estimated.
If the variance is allowed to be a power function of the mean, the log-likelihood function
is:
where
with
The upper bound for the power parameter in the Hill and power models has been
(somewhat arbitrarily) set to 18. That value was selected because it represents a very
high degree of curvature that should accommodate almost every dataset, even ones with
very (or absolutely) flat dose-response at low doses followed by a very steep dose-
response at higher doses.
If the power parameter for the Hill or power model is reported equal to 18 and the
warning "... hit a bound... " appears, the parameter estimates are maximum likelihood
estimates only in the restricted sense that the power parameter has been assigned a
value and the other parameters are MLEs conditional on that assigned value. Such
model results are not strictly comparable with others in terms of AIC. In such a case, the
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BMD and BMDL could depend on the choice of power parameter; thus, sensitivity
analysis is indicated if one intends to rely on the reported BMD or BMDL.
8.4.3 BMD Computation
In the continuous models, the benchmark dose is always the dose that results in a
prespecified change in the mean response. The change can be expressed in several
ways:
•	an absolute change in the mean (Abs. Dev.);
•	a change in the mean equal to a specified number of control standard deviations
(Std. Dev);
•	a specified fraction of the control group mean (Rel. Dev.);
•	a specified value for the mean at the BMD (i.e., not a change, but a fixed value)
(Point);
•	a change equal to a specified fraction of the range of the response, applicable only
when the dose-response has an asymptote at high doses (Extra) [Hill and some
Exponential models only].
Symbolically, these are (where 8 represents the BMRF designated by the user):
8.4.4 BMDL Computation
BMDS currently only calculates one-sided confidence intervals, in accordance with
current BMD practice. The general approach to computing the confidence limit for the
BMD (called the BMDL here) is the same for all the models in BMDS, and is based on the
asymptotic distribution of the likelihood ratio (Crump and Howe, 1985). Two different
approaches are followed in these models. In one, the equations that define the
benchmark response in terms of the benchmark dose and the dose-response model are
solved for one of the model parameters. The resulting expression is substituted back into
the model equations, with the effect of reparameterizing the model so that BMD appears
explicitly as a parameter. A value for BMD is then found such that, when the remaining
parameters are varied to maximize the likelihood, the resulting log-likelihood is less than
that at the maximum likelihood estimates by exactly
In the polynomial exponential models, it is impractical or impossible to explicitly
reparameterize the dose-response model function to allow BMD to appear as an explicit
parameter. For this model, the BMR equation is used as a non-linear constraint, and the
minimum value of BMD is determined such that the log-likelihood is equal to the log-
likelihood at the maximum likelihood estimates less
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Occasionally, the following error message may appear for a model: "BMDL computation
is at best imprecise for these data." This is a flag that convergence for the BMDL was not
"successful" in the sense that the required level of convergence (< 1 e-3 relative change
in the target function by the time the optimizer terminates) has not been achieved.
8.4.5	Lognormal Distributions
In previous versions of BMDS, continuous data were always assumed to be normally
distributed. In the current version of BMDS, for the exponential models only, the user has
the option of specifying that the continuous data being analyzed are lognormally
distributed. Lognormal distributions are appropriate only for data that are strictly positive
and may be preferable for such data (since the normal distribution allows, in theory, both
positive and negative values, no matter what the mean and standard deviation). When a
lognormal distribution is specified, the models assume a constant log-scale variance,
which is equivalent to an assumption of a constant coefficient of variation (CV).
The likelihood function shown above is then correct for data on the log scale (log-
transformed) and is the basis for fitting the log-transformed version of the model in
question. That is, if jjL(dose) is the log-scale mean as a function of dose, the model being
fit is jjL(dose) = ln{m(dose)}, where m(dose) is the specified model (e.g., one of the
exponential models parameterized as shown in the section on Exponential Models).
Therefore, m(dose) will then be a description of the change in the median response as a
function of dose since the anti-log of the log-scale mean is the median.
When the input data are summarized in terms of the sample mean and sample standard
deviation (or standard error or variance), the exact likelihood of the data cannot be
determined if the data are lognormally distributed. In such cases, BMDS gives an
approximate MLE solution by estimating the log-scale sample mean and log-scale
sample standard deviation for each dose group as follows:
estimated log-scale sample standard deviation (sL): sqrt{ln[1 + s2/m2]}
estimated log-scale sample mean (mL): ln[m] - sL2/2
where m and s are the reported sample mean and sample standard deviation. When
individual responses are available, the user may input those values (where the input dax
file will have two columns reporting the dose and the response for each experimental
unit) and may request that the exact MLE solution be obtained (which the software does
by first log-transforming the individual responses) or that the approximate solution using
the estimates shown above be obtained (which the software does by first computing
sample means and sample standard deviations). This option allows the user to compare
estimates and determine the impact of the approximation or to provide consistency
across data sets if some data sets have individual responses while others do not.
8.4.6	Linear and Polynomial Continuous Frequentist Models
Model Form
The formula for the polynomial model is
Here n is the degree of the polynomial (labeled "Degree Poly." on the model option
screen), and is specified by the user. The degree must be a positive integer (typically less
than the number of dose groups).
The linear model is a special case of the polynomial model, with n restricted to 1.
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Parameters
Alpha is a from the variance model
Rho is p from the variance model
Betao... Betan is pi ... polynomial coefficients.
Special Options
Degree Poly
Degree of polynomial (maximum = 21).
Restriction
One of "None'", "Non-Positive", "Non-Negative". Determines restrictions on the
polynomial coefficients. Restricting them to be either non-positive or non-negative
guarantees that the resulting function will be strictly decreasing, strictly increasing, or
perfectly flat (when all the coefficients are zero). If the coefficients are unrestricted,
more complicated shapes are possible, and, particularly as the degree of the
polynomial approaches the number of dose groups minus one, the polynomial will
often be quite "wavy". When the coefficients are unrestricted and the degree is one
less than the number of dose groups (for example, if there are four dose groups,
including control, if the degree of the polynomial is three), then the model will exactly
reproduce the means of the dose groups.
BMD Computation
The appropriate relationship for the BMR is solved (see "BMD Computation" in Section
9.4.3) using numerical methods.
BMDL Computation
The BMR equation (see "BMD Computation" in Section 9.4.3) is used as a non-linear
constraint, and the minimum value of BMD is determined such that the log-likelihood is
equal to the log-likelihood at the maximum likelihood estimates less
8.4.7 Power Continuous Model
Model Form
The form of the Power model is:
M(dose) = y + p . (dose)5
Here, 00, and 18 > 8 > 0 with an option to restrict 8 > 1.
Parameters
Alpha is a from the variance model (see section 9.4)
Rho is p from the variance model (see section 9.4)
Control = y
Slope = (3
Power = 8 (The Power parameter must be a positive number < 18. If Power is
restricted, the number must be > 1.)
Special Options
Restrict power > 1. Restrict 8 > 1. If 8 < 1, then the slope of the dose-response
curve becomes infinite at the control dose. This is biologically unrealistic, and can
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lead to numerical problems when computing confidence limits, so several authors
have recommended restricting 8 > 1.
BMD Computation
The appropriate relationship for the BMR is solved (see "BMD Computation" in Section
9.4.3) analytically.
BMDL Computation
The equations that define the benchmark response in terms of the benchmark dose and
the dose-response model are solved for the slope. The resulting expression is substituted
back into the model equations, with the effect of reparameterizing the model so that BMD
appears explicitly as a parameter. A value for BMD is then found such that, when the
remaining parameters are varied to maximize the likelihood, the resulting log-likelihood is
less than that at the maximum likelihood estimates by exactly
8.4.8 Hill Continuous Model
Model Form
The form of the Hill model is:
Parameters
•	Intercept (Control) = y
•	Dose with half-maximal change = k (must be positive number)
•	Power = n (must be a positive number < 18. If n is restricted, the number must be >
1-)
•	Maximum change = v
Special Options
•	Restriction: When the "Restrict n > 1" box is checked, the power parameter will be
estimated to be greater than or equal to 1.
BMD Computation
The appropriate relationship for the BMR is solved (see "BMD Computation" in Section
9.4.3) analytically.
BMDL Computation
The BMR equation is used as a non-linear constraint, and the minimum value of BMD is
determined such that the log-likelihood is equal to the log-likelihood at the maximum
likelihood estimates less
Warning BMDL estimates from models that have an asymptote parameter (including
the Hill model) can be unstable when a wide range of parameter values can
give nearly identical likelihoods. One indicator of that problem is that the
estimated asymptotic response is far outside the range of the observed
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responses. The user should consult a statistician if this behavior is seen or
suspected.
8.4.9 Exponential Continuous Model
Dr. Wout Slob of RIVM in The Netherlands has proposed a set of nested models known
as the exponential models. Currently, these models should be fit only to data having
positive (mean) values.
There are four exponential models fit by BMDS and they are defined and labeled as
follows.
Model Form
•	Model 2: m(dose) = a*exp{sign*b*dose}
•	Model 3: m(dose) = a*exp{sign*(b*dose)d}
•	Model 4: m(dose) = a*(c - (c-1)*exp{-1*b*dose})
•	Model 5: m(dose) = a*(c - (c-1)*exp{-1*(b*dose)d})
[Model 1, as defined by Dr. Slob, is the constant-mean model, called R in BMDS, which is
estimated for every continuous data run.]
The parameter "sign" is the indicator of the direction of change: +1 for data trending up, -
1 for data trending down. It is very important that the user correctly specify the direction
of change in the data - for the Exponential Models the "automatic" choice of adverse
direction has not been included. Some indicators that the wrong direction has been used
for any given run include the observation that one or more models result in a flat curve fit,
that optimal solutions for MLE parameters or BMDLs have not been obtained, and/or that
the likelihoods associated with the models are much worse than models A1 to A3 (and
are more like model R).
Parameters
For all the exponential models the following restrictions apply:
•	Background Response: a (> 0)
•	Slope: b (> 0)
•	Asymptote Parameter: c [Models 4 and 5 only]
•	c >1 for increasing data
•	0 < c < 1 for decreasing data
Power: d (> 1) [Models 3 and 5 only]
Restrictions
There are no restrictions beyond the parameter constraints shown above for each model.
BMD Computation
The appropriate relationship for the BMR is solved (see "BMD Computation" in Section
9.4.3) analytically.
BMDL Computation
The BMR equation is used as a non-linear constraint, and the minimum value of BMD is
determined such that the log-likelihood is equal to the log-likelihood at the maximum
likelihood estimates less
Warning BMDL estimates from models that have an asymptote parameter (including
Exponential models 4 and 5) can be unstable when a wide range of parameter
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values can give nearly identical likelihoods. One indicator of that problem is
that the estimated asymptotic response is far outside the range of the
observed responses. The user should consult a statistician if this behavior is
seen or suspected.
8.5 Frequentist Dichotomous Model Descriptions
8.5.1	Special Considerations for Models for Dichotomous Endpoints in
Simple Designs
BMDS includes in this category models for dichotomous endpoints in which the
observations are independent of each other. In these models, the dose-response model
provides the probability that an animal will have an adverse response at a given dose.
The actual number of animals that have an adverse response is assumed to be
binomially distributed.
An example of such a dataset is a study in which adult animals are exposed to different
concentrations of a toxicant and then evaluated for the presence of liver toxicity. For
models for dichotomous endpoints in which the responses are nested (for example, pups
in litters, and litters nested within doses), see the section on Nested Model Descriptions.
BMDS contains ten models for dichotomous endpoints (the Probit, Log-Probit, Logistic,
Log-Logistic, Weibull, Quantal Linear, Gamma, Multistage, and Multistage-Cancer
models). They may all be written in the form:
Prob{response} = y + (1 - y)F(dose; a, (3, . . .)
Here F(dose; a, (3, . . .) is a cumulative distribution function and y, a, (3, . . ., are
parameters to be estimated using maximum likelihood methods. Sometimes
Prob{response} is written as P[cfose,y, a, (3, . . .] to indicate the relationship between the
response probability and the dose as well as parameters. When the function F(dose; a, (3,
. . .) approaches zero as dose approach zero, the parameter / represents the
background incidence. In the Logistic and Probit models, F(0) is not zero, unlike in the
Log-Logistic and Log-Probit models. In these models, y is set to 0.
8.5.2	Special Options for Models
In addition to the options that are available to all dichotomous models, there may be
model-specific options. Generally, these are options to restrict the legal range of a
parameter or set of parameters. The range of a parameter may be restricted for two
reasons:
•	The slope of the dose-response curve becomes infinite at a dose of 0 if the
parameter falls below 1, so that the default is to restrain that parameter to be at least
1, or
•	The quantal polynomial dose-response curve can become non-monotonic if the
coefficients are allowed to be negative, often resulting in the curve looking "wavy",
so the default is to restrict the coefficients to be non-negative.
The applicable special options are listed in the sections for the specific models.
8.5.3	Likelihood Function
All models in the current version of BMDS are fit using maximum likelihood methods. This
section describes the likelihood function used to fit the dichotomous models.
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Suppose we employ k doses:
and the total number of s and number of responding s in each dose group are
and, respectively,
The distribution of ni is assumed to be binomial with probability
where © is a vector of parameters. Then the log-likelihood function L can be written as
where
The upper bound for the power parameter in the gamma and Weibull models, and the
slope parameter for the log-probit and log-logistic models, has been (somewhat
arbitrarily) set to 18. That value was selected because it represents a very high degree
of curvature that should accommodate almost every data set, even ones with very (or
absolutely) flat dose-response at low doses followed by a very steep dose-response at
higher doses.
If the power parameter for the gamma or Weibull model, or the slope parameter for the
log-probit or log-logistic model, is reported equal to 18 and the warning "... hit a bound ...
" appears, the parameter estimates are maximum likelihood estimates only in the
restricted sense that the power parameter has been assigned a value and the other
parameters are MLEs conditional on that assigned value. Such model results are not
strictly comparable with others in terms of AIC. In such a case, the BMD and BMDL could
depend on the choice of power parameter; thus, sensitivity analysis is indicated if one
intends to rely on the reported BMD or BMDL.
8.5.4 BMD Computation
The BMD is computed as a function of the parameters of the model, which must have
already been estimated. The BMDs for dichotomous models are expressed as the dose
that would give an (estimated) increase in incidence of x% above the control incidence
(where x is often in the range of 1 to 10 percent). This increase in incidence is referred to
here as the "BMRF", for benchmark response factor. Note that, although the word
"response" is used here, we are really talking about an increase of the incidence over
the control incidence (added risk). The actual response associated with the BMR, will
only be the same as the BMR when P(0) = 0. This is because to obtain the actual
response associated with the BMR one must solve for P(d) in the equation for added or
extra risk.
Two formulations for computing the excess over background are in common use, the
extra risk model and the additional risk model. In the extra risk model,
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while in the additional risk model,
The equation appropriate to the risk type formulation that the user requests is solved to
get the BMD for a specific model and dataset. Details of this computation are included in
the descriptions of the models.
8.5.5 BMDL Computation
BMDS currently calculates one-sided confidence intervals, in accordance with current
BMD practice.
Note The Multistage and Multistage-Cancer models also calculate one-sided upper
confidence limits.
The general approach to computing the confidence limit for the BMD (called the BMDL
here) is the same for all the models in BMDS, and is based on the asymptotic distribution
of the likelihood ratio (Crump and Howe, 1985). Two different approaches are followed in
these models. In one, the equations that define the benchmark response in terms of the
benchmark dose and the dose-response model are solved for one of the model
parameters. The resulting expression is substituted back into the model equations, with
the effect of reparameterizing the model so that BMD appears explicitly as a parameter.
A value for BMD is then found such that, when the remaining parameters are varied to
maximize the likelihood, the resulting log-likelihood is less than that at the maximum
likelihood estimates by exactly
In a few models, it is impractical or impossible to explicitly reparameterize the dose-
response model function to allow BMD to appear as an explicit parameter. For these
models, the BMR equation is used as a non-linear constraint, and the minimum value of
BMD is determined such that the log-likelihood is equal to the log-likelihood at the
maximum likelihood estimates less
8.5.6 Gamma Model
Model Form
The Gamma Model formula is:
Here, 0 < y < 1, p > 0, and 18 > a > 0 with an option to restrict a > 1.
Parameters
•	"background" is y (If specified or initialized, the background parameter must be a
number > 0 and < 1.)
•	"power" is a (If specified or initialized, the Power parameter must be a positive
number < 18. If Power is restricted, the number must be > 1.)
•	"slope" is (3 (If specified or initialized, the Slope parameter must be a number > 0. If
Slope is restricted, the number must be > 1.)
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Restrict power > 1. Restrict a > 1. If a < 1, then the slope of the dose-response curve
becomes infinite at the control dose. This is biologically unrealistic, and can lead to
numerical problems when computing confidence limits, so several authors have
recommended restricting a > 1.
BMD Computation
Let
be the incomplete Gamma function and
be its inverse function. Then
BMDL Computation
To calculate the BMDL, the defining equations for the BMD are solved for the slope
parameter (3, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.5.7 Logistic Model
Model Form
The form of the probability function for the Logistic model is
Parameters
•	intercept is a
•	slope is (3 (If specified or initialized, the Slope parameter must be a number > 0. If
Slope is restricted, the number must be > 1.)
Special Options
•	None
BMD Computation
The BMD estimate for the Logistic model is defined implicitly by the following equation.
An iterative numerical method is used to determine the value of BMD:
where
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BMDL Computation
To calculate the BMDL, the defining equations for the BMD are solved for the intercept
parameter a, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.5.8 Log-Logistic Model
Model Form
The form of the probability function for the Log-Logistic model is
if dose > 0:
and if dose = 0:
Here, 0 < y < 1, P ^ 0, and 18 > a > 0 (with an option to restrict a > 1).
Parameters
•	background is y (If specified or initialized, the background parameter must be a
number > 0 and < 1.)
•	intercept is a
•	slope is (3 (If specified or initialized, the Slope parameter must be a number > 0. If
Slope is restricted, the number must be > 1.)
Special Options
•	Restrict slope > 1: If the slope is allowed to be less than 1, the slope of the dose-
response curve is infinite at zero dose.
BMD Computation
The BMD estimate for the Log-Logistic model is:
BMDL Computation
To calculate the BMDL, the defining equations for the BMD are solved for the intercept
parameter a, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.5.9 Multistage Model
Model Form
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The Multistage Model formula is:
Here, 0 < y < 1, and there is an option to restrict p, > 0 for all is. The degree of the
polynomial is n. The Multistage-Cancer model is the same as the Multistage model
except that the (3 parameters are always restricted to be positive (the Multistage model
allows them to be positive or negative).
Parameters
•	Background is y (If specified or initialized, the background parameter must be a
number > 0 and < 1.)
•	Dose Coefficients (Betai ... Betan) are (3>sub>1 ... pn
Special Options
•	Degree Poly: The maximum degree polynomial to fit (maximum = 23). The degree
must be a positive integer (typically less than the number of dose groups).
•	Restrict Betas > 0: When this box is checked, the polynomial coefficients are
restricted to be non-negative. This guarantees that the dose-response function will
either be perfectly flat or always increasing, with no "bumps". This restriction option
is not available for the Multistage-Cancer model because it is always implemented for
that model.
BMD Computation
There is no general analytic form for the BMD in terms of the BMR and the estimated
model parameters for the multistage model. Instead, BMD is the root of the equation
where
BMDL Computation
The BMR equation is used as a non-linear constraint, and the minimum value of BMD is
determined such that the log-likelihood is equal to the log-likelihood at the maximum
likelihood estimates less
8.5.10 Probit Model
Model Form
The form of the probability function for the Probit model is
Prob{response} = p(dose; ~, ~) = ~(~ + dose),
where
and
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(that is, 0 is the standard normal density function, and $ is the normal distribution
function), 18 > (3 > 0 (with an option to restrict (3 > 1).
Parameters
•	intercept is a
•	slope is (3 (If specified or initialized, the Slope parameter must be a number > 0. If
Slope is restricted, the number must be > 1.)
Special Options
•	Log of Dose: This results in the Log-Probit model.
•	Restrict slope > 1 (Log-Probit only): if the slope is allowed to be less than 1, the
slope of the dose-response curve is infinite at zero dose.
BMD Computation
The BMD estimate for the Probit model is defined implicitly by the following equation. An
iterative numerical method is used to determine the value of BMD:
BMDL Computation
To calculate the BMDL, the defining equations for the BMD are solved for the intercept
parameter a, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.5.11 Log-Probit Model
Model Form
The form of the probability function for the Log-Probit model is, if dose > 0:
and if dose = 0:
where
	and	
(that is, cp is the standard normal density function, and $ is the normal distribution
function), 0p>0 (with an option to restrict (3 > 1).
Parameters
•	background is y (If specified or initialized, the background parameter must be a
number > 0 and < 1.)
•	intercept is a
•	slope is (3 (If specified or initialized, the Slope parameter must be a number > 0. If
Slope is restricted, the number must be > 1.)
Special Options
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For the log-probit model, the slope of the model will approach zero as dose approaches
zero. However, depending on the data and parameter estimates, the slope for the log-
probit model, for some relatively low doses perhaps less than those corresponding to the
BMR, the slope can be quite steep, which may be manifested in terms of a relatively low
value for the BMDL (or perhaps an "NA" result for the BMDL if this causes convergence
problems because the steepness entails BMDL estimates that get very small).
BMD Computation
The BMD estimate for the Log-Probit model is:
BMDL Computation
To calculate the BMDL, the defining equations for the BMD are solved for the intercept
parameter a, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.5.12 Quantal Linear andWeibull Models
Model Form
The Weibull Model formula is:
Here, 0 < y < 1, p > 0, and 18 > a > 0 with an option to restrict a > 1.
Note The Quantal Linear model results from setting a equal to 1 in the Weibull Model.
Parameters
•	Background is y (If specified or initialized, the background parameter must be a
number > 0 and < 1.)
•	Slope is (3 (If specified or initialized, the Slope parameter must be a number > 0.)
•	Power is a (If specified or initialized, the Power parameter must be a non-negative
number < 18. If Power is restricted, the number must be > 1. If Power is unrestricted,
the number must be greater than or equal to the value entered for the Lower Bound
on Power (strictly greater than that bound if it is 0).
Special Options
•	Restrict power > 1: Restrict a > 1. If a < 1, then the slope of the dose-response
curve becomes infinite at the control dose. This is biologically unrealistic, and can
lead to numerical problems when computing confidence limits, so several authors
have recommended restricting a > 1.
BMD Computation
The BMD estimate for the Weibull model is:
BMDL Computation
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To calculate the BMDL, the defining equations for the BMD are solved for the slope
parameter (3, which is then replaced in the original model equations. This makes BMD
appear in the model equations as a parameter. See the section on Dichotomous Models'
BMDL Computation for further details.
8.6 Bayesian Dichotomous Model Descriptions
tk
8.7 Frequentist Nested Model Descriptions
8.7.1	Special Considerations for Models for Nested Dichotomous
Endpoints
The most common application of the models in this section will be to developmental
toxicology studies of organisms that have multiple offspring per litter, as do rodents. In
these study designs, pregnant females ("dams") are given one or several doses of a
toxicant, and the fetuses, embryos, or term offspring ("pups") are examined for signs of
abnormal development. In such studies, it is usual for the responses of pups in the same
litter to be more similar to each other than to the responses of pups in different litters
("intra-litter correlation", or "litter-effect"). Another way to describe the same phenomenon
is that the variance among the proportion of pups affected in litters is greater than would
be expected if the pups were responding completely independently of each other.
The models in this section make available two approaches to this feature of
developmental toxicology studies: they use a probability model that provides for extra
inter-litter variance of the proportion of pups affected (the beta-binomial probability model:
see the "Likelihood Function" section below); and they incorporate a litter-specific
covariate that is expected to account for at least some of the extra inter-litter variance.
This latter approach was introduced by Rai and Van Ryzin (1985), who reasoned that a
covariate that took into account the condition of the dam before dosing might explain
much of the observed litter effect. Those authors suggested that litter size would be an
appropriate covariate. For the reasoning to apply strictly, the measure of litter size should
not be affected by treatment; thus, in a study in which dosing begins after implantation,
the number of implantation sites would seem to be an appropriate measure. On the other
hand, the number of live fetuses in the litter at term would not be an appropriate measure
if there is any dose-related prenatal death or resorption (this has apparently been ignored
in most of the literature).
Carr and Portier (1991), in a simulation study, warn that in situations in which there is no
effect of litter size, statistical models that incorporate a litter size parameter, as do the
models in BMDS, will often erroneously indicate that there is a litter size effect. Thus, the
user should use litter size parameters with caution. Unfortunately, there are currently no
good diagnostics for determining whether a litter size effect actually exists.
8.7.2	Likelihood Function
Let g represent the number of dose groups. For the ith group, there are n, pregnant
females administered dose c/ose/. In the jth litter of the ith dose group there are s,y
fetuses, Xij affected fetuses, and, potentially, a litter-specific covariate nj which will often
be a measure of potential litter size, such as number of implantation sites, though this is
not a requirement of the models. In what follows, the dose-response model, which gives
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the probability that a fetus in the Jth litter of the ith dose group will be affected is
represented by
The beta-binomial distribution can be thought of as resulting from sampling in two stages.
First, each litter is assigned a probability, P,yfrom a beta distribution (beta distributions
represent a two-parameter family of probability distributions defined on the interval (0,1)).
The parameters of the beta distribution are determined by the administered dose, the
litter specific covariate nj and the degree of intra-litter correlation, v/. Note that the intra-
litter correlation parameter varies among doses. It is well known (Williams et al., 1988)
that when the true intra-litter correlation differs among doses, unbiased estimates of the
other parameters in a dose-response model can only be obtained if dose-specific intra-
litter correlation parameters are estimated. As a special case, if v/=0, then this part of the
process is completely deterministic, and
This allows for the possibility of no litter effect at all.
In the second stage of sampling, sy fetuses are assigned to the litter, and the number of
affected fetuses, xij is sampled from a binomial distribution with parameters P,yand s,y.
The log-likelihood function that results from this process is:
where
and
if a > b by convention.
8.7.3 Goodness of Fit Information—Litter Data
The "Litter Data" table provides a listing of the data, expected and observed responses
and scaled residuals, for each litter.
The scaled residual values printed in the last column of the table are defined as follows:
(Obs. - Expected)/SE
where "Expected" is the predicted number of responders from the model and SE equals
the estimated standard error of that predicted number. For these models, the estimated
standard error is equal to sqrt[n*p*(1 -p)*(*0(n-1)+1)], where
• n is the sample (litter) size,
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•	p is the model-predicted probability of response, and
•	f is the model-predicted intra-litter correlation coefficient.
The overall model should be called into question if the scaled residual values for several
individual dose and litter-specific covariate combinations, particularly for a dose group
near the BMD and for litter-specific covariate values close to the overall mean, are
greater than 2 or less than -2.
The goodness-of-fit p-values are calculated using a bootstrap approach.
1.	The MLE parameter values are used to generate B pseudo-datasets having the
same design features (number of doses and number of litters per dose), litter-sizes,
and, if necessary, litter-specific covariate values, as the original dataset. What varies
from pseudo-dataset to pseudo-dataset are the number of responding "units" within
litters, and those are generated, at random, as dictated by the values of the ML
estimates.
2.	Once the B bootstrap iterations are generated, a statistic referred to as chi-square is
calculated for each. The chi-square statistic is the sum of the squares of the scaled
residuals for each litter, as described above. Higher values of that statistic are
indicative of poorer match between the model predictions and the data.
Note The chi-square statistic is so called here because, in traditional testing situations,
that statistic would be approximated by a chi-squared random variable having a
certain degrees of freedom, and its "significance" (p-value) would be determined
from the appropriate chi-squared distribution function.
3.	The chi-square statistic from the original data is computed and compared to the
values from the B bootstrap iterations. The p-value is the proportion of chi-square
values from the iterations that are greater than the original chi-square value.
High p-values are indicative of adequate fit (i.e., there was a high proportion of chi-
square values associated with pseudo-datasets obtained from data known to be
consistent with the model and the ML estimates of the model parameters).
That calculation is repeated three times, and various percentiles of the generated chi-
square statistic are presented. This allows the user to determine if enough bootstrap
iterations (B) have been specified. The default for B is 1000 and should probably not be
reduced. The user may wish to increase the default if the percentiles for chi-square differ
markedly across the three runs (in particular the median and lower percentiles), or if the
p-values calculated from the three runs differ markedly. This may only be an issue when
the p-value is close to the value (e.g., 0.05 or 0.10) used as a critical value for deciding
whether or not the fit of the model to the data is adequate. If there is some variability in
the p-values, but they are all greater than 0.20, for example, then one probably need not
worry about increasing the value for B.
BMD Computation
BMD computation is similar to that for dichotomous models with the added wrinkle that a
value for a litter-specific covariate (LSC) may be used, in addition to dose, to describe
changes in the endpoint. It therefore affects the BMD calculation. If an LSC is included in
the model, the user can choose to plot results and compute BMDs for one of two specific
values of the LSC, either the overall mean (across all dose groups) or the control group
mean. Typically, the overall mean is the preferred choice, but the control group mean
might be appropriate in certain situations.
For example, suppose the LSC value varies enough from group to group to be
"interesting," but it goes up for some dose groups and down for others in a manner that
contraindicates a dose effect. In this case, you might decide to use the control group
mean LSC when the BMD is close to the background dose (i.e., basically deciding that
the LSC of interest in that region is more likely to be the average observed for the control
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group as opposed to the average across all the groups). If a covariate is found to be
affected by dose, i.e., if its value appears to have a consistent trend with respect to dose,
its use is discouraged.
BMDL Computation
BMDS currently only calculates one-sided confidence intervals, in accordance with
current BMD practice. The general approach to computing the confidence limit for the
BMD (called the BMDL here) is the same for all the models in BMDS, and is based on the
asymptotic distribution of the likelihood ratio (Crump and Howe, 1985).
The approach used for all the nested dichotomous models is the same. The equations
that define the benchmark response in terms of the benchmark dose and the dose-
response model are solved for one of the model parameters, using either the control
group mean or the overall mean of the litter-specific covariate. The resulting expression is
substituted back into the model equations, with the effect of reparameterizing the model
so that BMD appears explicitly as a parameter. A value for BMD is then found such that,
when the remaining parameters are varied to maximize the likelihood, the resulting log-
likelihood is less than that at the maximum likelihood estimates by exactly
8.7.4 Logistic Nested Model
Model Form
The Nested Logistic Model is the log-logistic model, modified to include a litter-specific
covariate. The model form for the Nested Logistic Model is:
Prob{response} = a + 0i ry + (1 - a - 0i ry) / (1 + exp[-p-02 ry - p*Ln(dose)])
if dose > 0, and a + 0inj if dose = 0.
In the above equation, nj is the litter-specific covariate for the jth litter in the ith dose
group; a > 0, p > 0, p > 0 with an option to restrict p > 0; and 1 > a + p > 0i ry > 0 for every
nj.
In addition there are g intra-litter correlation coefficients, 0 < 0/< 1 (i = 1, ..., g).
Parameters
•	Intercept = a
•	Power = p
•	Slope = p
•	First coefficient for litter-specific covariate = 01
•	Second coefficient for liter-specific covariate = 02
•	Intralitter correlation coefficients = 01 . . . 0g (If 0 is specified or initialized, 0 must be a
number > 0 and < 1.)
Special Options
•	Restriction: Power parameter p (rho) can be restricted to be >1 (Default)
Risk Type
Choices are "Extra" or "Added." Additional risk is the additional expected proportion of
total animals that respond in the presence of the dose, or the mean probability of
response at dose d, P(d), minus the mean probability of response in the absence of
exposure, P(0). Extra risk is the additional risk divided by the expected proportion of
animals that will not respond in the absence of exposure, 1 - P(0). Thus, extra and
additional risk are equal when the expected background rate is zero.
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BMD Computation
If rm represents either the control mean value for the litter-specific covariate or its overall
mean, then the BMD is computed as:
BMD	= Exp{[ln(A/(1 -A)) - (3 - 02 rm ] / p}
where
A	= BMRF	for extra risk
= BMRF/(1 - a - 01 rm)	for added risk
BMDL Computation
The parameter p is replaced with an expression derived from the BMD definition (above)
in the dose-response function, with the effect of reparameterizing the model so that BMD
appears explicitly as a parameter. A value for BMD is then found such that, when the
remaining parameters are varied to maximize the likelihood, the resulting log-likelihood is
less than that at the maximum likelihood estimates by exactly
8.7.5 NCTR Model
Model Form
The NCTR model is a Weibull model modified to include a litter-specific covariate. The
model form is:
Prob{response} = 1 - Exp{-(a+ 01 (rjj - rm ) - (P+ 02 0"ij - rm ) dosep }
where rjj is the litter-specific covariate for the jth litter in the ith dose group, rm is the
overall mean for the litter-specific covariate, a > 0, p>0, p > 0 with an option to restrict p
>1,and
01 (rij - rm ) > 0 and 02 (rjj - rm ) > 0
for every nj.
In addition there are g intra-litter correlation coefficients, 0 < < 1 (i = 1, ..., g).
Parameters
•	Intercept = a
•	Power = p
•	Slope = p
•	First coefficient for litter-specific covariate = 01
•	Second coefficient for liter-specific covariate = 02
•	Intralitter correlation coefficients = 01 . . . 0g (If 0 is specified or initialized, 0 must be a
number > 0 and < 1.)
Special Options
•	Restriction: Power parameter p (rho) can be restricted to be > 1 (Default)
Risk Type
Choices are "Extra" or "Added." Additional risk is the additional proportion of total animals
that respond in the presence of the dose, or the probability of response at dose d, P(d),
minus the probability of response in the absence of exposure, P(0). Extra risk is the
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additional risk divided by the proportion of animals that will not respond in the absence of
exposure, 1 - P(0). Thus, extra and additional risk are equal when background rate is
zero.
BMD Computation
BMD = [-(Ln(1-A)) / (P + 02 §r )](1/p)
where 8r is the average of (rjj - rm ) over either the control group or over all
observations, depending upon the option selected for "Fixed Litter Size" (when using the
overall mean, 8r is always 0), and
A = BMRF for extra risk
= BMRF/(1-a- 01 Sr) for added risk
BMDL Computation
The parameter p is replaced with an expression derived from the BMD definition (above)
in the dose-response function, with the effect of reparameterizing the model so that BMD
appears explicitly as a parameter. A value for BMD is then found such that, when the
remaining parameters are varied to maximize the likelihood, the resulting log-likelihood is
less than that at the maximum likelihood estimates by exactly
8.8 Multitumor (MS_Combo) Model Description
The purpose of the MS_Combo program in BMDS is to allow the user to calculate BMDs
and BMDLs for a combination of tumors (corresponding to a defined risk of getting one or
more of those tumors) when the individual tumor dose-responses have been modeled
using a Multistage-Cancer model.
Thus, the output of an MS_Combo run will present the results of fitting each individual
tumor (including the BMD and BMDL for that tumor) plus the combined log-likelihood,
BMD and BMDL for the combination of specified tumor responses.
In practice, the user should investigate each tumor individually and determine which
degree of the Multistage-Cancer model is most appropriate for each individual tumor.
That determination will involve all the usual considerations of fit, AIC, etc.
Once a specific form of the Multistage-Cancer model is chosen for each of the tumors of
interest (they need not have the same degree across all of the tumors in question), the
user should specify those choices in the MS_Combo run.
Note The following descriptions are valid only when the tumors are assumed to be
independent of one another (conditional on dose level).
Because of the form of the multistage model, the MLE estimates for the combined risk
are a function of the parameter values obtained for the individual tumor multistage model
fits. In fact, the combined probability function has a multistage model form:
P(d) = 1 - exp{-(po + pid + P2d2 + ...)}
and the terms of the combined probability function (po, pi, ...) are specified as follows
P0 = spoi
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P1 =spii
P2 = sp2i
etc.
where the sums are over i = 1, ..., t, with
t being the number of tumors under consideration, and
pXj being the xth parameter (0, 1, ...) for tumor j.
The pXj values are available directly from the Multistage-Cancer runs performed on the
individual tumors, but MS_Combo performs the calculations for the user, completing the
summations of the individual terms and computing the BMD based on the combined
parameter values and the user-specified BMR.
A profile-likelihood approach is used to derive the BMDL.
1.	Given the BMD and the log-likelihood associated with the MLE solution, a target
likelihood is defined based on the user-specified confidence level (e.g., 95%).
2.	That target likelihood is derived by computing the percentile of a chi-square (1 degree
of freedom) corresponding to the confidence level specified by the user (actually, the
alpha associated with the confidence level, times 2).
3.	That percentile is divided by 2 and subtracted from the maximum log-likelihood.
4.	That derivation is based on a likelihood ratio test with one degree of freedom; it can
be shown that estimating the BMDL corresponds to losing one degree of freedom,
regardless of the number of tumors being combined.
The BMDL for the combined response (one or more of the tumors of interest) is defined
as the smallest dose, D, for which the following two conditions are satisfied:
1.	There is a set of parameters such that the combined log-likelihood using D and those
parameters is greater than or equal to the target likelihood), and
2.	For that set of parameters, the risk at D is equal to the user-specified BMR.
Note that the combined log-likelihood is a function of the fits of the individual tumors (the
sum of the individual log-likelihoods), obtained using their tumor-specific (3 values. Thus,
the search for the parameters of the combined Multistage-Cancer model varies the
individual-tumor (3 values in such a way that the individual log-likelihoods add up to a
combined likelihood within the range desired (greater than or equal to the target).
However, to satisfy the second constraint, the sums of the individual-tumor parameters
(shown above to be the parameters of the combined probability function) are used to
evaluate the risk for any proposed BMDL, D.
Note that the individual tumors need not be modeled with the same degree of the
Multistage-Cancer model. Any terms not included for an individual tumor are assumed to
be zero (and will remain at zero during BMDL optimization) in the summations shown
above. The optimizer DONLP2 is used for the combined BMDL estimation.
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9.0 Troubleshooting
9.1	Avoid Using Windows Reserved Characters in File and
Path Names
BMDS allows any character, except for Windows reserved characters, to be used when
naming files or directories that BMDS will access.
However, the following Windows reserved characters are still disallowed and cannot be
used:
< (less than)
> (greater than)
| (vertical bar or pipe)
? (question mark)
* (asterisk)
" (double quote)
Note The backslash (\) should be used when specifying network drive paths.
9.2	Request Support with eTicket
For any technical problem related to running BMDS, please select Help>BMDS Support.
This will display the eTicket site in your default web browser, where you can request help,
ask a question, or check on the status of an existing issue.
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Appendix A: Version History
The following sections document new, changed, or updated BMDS features as
documented in each version's respective readme file. The information is included here for
historical and reference purposes.
A.1 BMDS 1.2
September 2, 2000 - A new user interface (BMDS0900.exe) was distributed to fix some
problems with installation of BMDS on certain Windows 98 configurations. If you
successfully installed BMDS version 1.2 using a previous installation procedure you do
not need this upgrade. This upgrade merely simplifies the installation process and
corrects some problems that did not allow BMDS to install to certain computer
hardware/software configurations. (This version of the software is no longer being made
available as there are newer versions now available which fix problems that were being
encountered on newer operating systems.)
A.2 BMDS 1.2.1
October 25, 2000 - A new version of BMDS, version 1.2.1, is being distributed at this
time. This version contains new versions of the continuous Polynomial (version 2.1) and
Hill (version 2.1) models. If you do not want to completely reinstall BMDS, you can
download the the new model executables (see Latest Versions of BMDS Help and Model
Files) and run them separately or under the BMDS version 1.2 interface. These new
versions of the polynomial and Hill models fix problems associated with running the
model on Windows NT/2000 operating systems, provide improved model fit for certain
unique data sets and improve upon the rate of convergence on a BMD and BMDL.
A.3 BMDS 1.3
March 22, 2001 - Version 1.3 of BMDS is now available! This latest version of BMDS,
version 1.3, contains new continuous Polynomial (v2.1), Power (v2.1) and Hill (v2.1)
models, new dichotomous Multistage (v2.1), Weibull (v2.1) and Gamma (v2.2) models,
and an improved user interface. The new models are more compact and stable (will
converge on BMD and BMDL solutions more often). The user interface upgrades are
described in the new help manual (PDF format) for version 1.3 and the readme.txt file
that is distributed with the upgrade.
A.4 BMDS 1.3.1
January 22, 2002 - Version 1.3.1 of BMDS is now available! Version 1.3.1 contains a
revised help manual and user interface, including a revision to the interface that allows
the Multistage model to calculate BMD and BMDL values for very low (below E-5)
benchmark response (BMR) levels.
November 13, 2002 - A new polynomial model (Version 2.2) is now available that fixes
the previous incompatibility with Windows 2000. Download it to your main bmds directory
(same directory as the bmds.exe file).
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A.5 BMDS 1.3.2
May 23, 2003 - Version 1.3.2 of BMDS is now available! Version 1.3.2 contains revised
polynomial (poly.exe) and nested logistic (nlogist.exe) models that are compatible with
Windows 2000. If you are using a Windows 98 or older operating system, you may need
to update your msvcrt.dll driver. We suggest that you obtain the latest msvcrt.dll driver
from Microsoft or download this version of the msvcrt.dll driver and copy it to the
c:\windows\system directory of your computer (you may have to exit Windows and do this
in DOS mode).
A.6 BMDS 1.4.1
February 5, 2007 - Version 1.4.1 is now available! All models have been recompiled to
improve speed, stability and compatibility with the latest Windows operating systems.
Improvements have been made to the model output format for all models. A Multistage-
Cancer model has been added that calculates and reports a cancer slope factor and plots
the linear extrapolation from the BMDL to the background response estimate per EPA's
2005 cancer guidelines. Unlike the Multistage model, the Multistage Cancer model does
not estimate added risk, nor does it allow beta coefficients to be unrestricted. The
Quantal-Quadratic model was removed from Dichotomous model choices (note: the user
can still run this model by specifying the power term of the Weibull model to be 2, but this
model is not retained in the BMDS dichotomous model listings)
Issues in the continuous models that caused occasional errors in degrees of freedom
assignments which impacted continuous model test results have been resolved.
Acceptance criteria for Tests 2, 3 and 4 was changed from p>=0.05 to p>=0.1 and default
risk type changed to "Std. Dev." for all continuous models to be consistent with EPA's
draft BMD technical guidance (EPA, 2000). Issues with the Hill model have been fixed,
including memory problems which were causing some operating systems to crash.
Parameter standard error estimates and Chi-squared residual calculations in all the
continuous models were checked and corrected if in error. Model A3 of the continuous
model testing procedures has been modified so that it always uses the user-specified
value for the parameter rho, including the constant-variance case where rho = 0. When
rho = 0, model A3 is the same as model A1, and it is reported explicitly in the constant-
variance runs. As a consequence, all model runs report the entire set of models (A1, A2,
A3, R and the fitted model) and all four hypothesis tests.
Issues in the Nested models that caused occasional errors in degrees of freedom
assignments have been resolved. Memory problems which were causing problems for
some NCTR model runs have been fixed.
August 29, 2007 - BMDS Version 1.4.1 b has been added to replace version 1.4.1. This
version contains an update to the BMDS help file.
November 9, 2007 - BMDS version 1.4.1 c is now available. This version updates
dichotomous models that were already included on BMDS version 1.4.1 b. The updates
primarily improve the handling of parameter specifications, particularly in situations where
the user may wish to specify the background parameter to be zero.
A.7 BMDS 2.0 (beta)
September 28, 2007 - BMDS Version 2.0 beta is now available for inspection and testing
(NOTE: this is a beta test version, provided only for your examination and testing - BMDS
1.4.1b should be used for definitive risk assessment calculations). BMDS 2.0 beta
employs a new graphical user interface and makes it easy to run a number of models for
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one data set and compare the results. BMDS 2.0 beta also has a new set of quantal
models with alternative background parameters (i.e., background additive to dose). We
welcome comments and suggestions on the functioning of the interface and its new
features, and on the new models.
October 10, 2007 - BMDS 2.0 beta - Build 19 released on October 10, 2007 replaces the
first BMDS 2.0 beta release of September 28, 2007 (Build 13). The new Build 19 has
important changes and enhancements as a result of additional testing and user exposure
and should be downloaded and used instead of Build 13. Enhancements include the
ability to better run a number of the BMD models and also added flexibility and fixes for
user interface features. Changes include the designation of the new Dichotomous models
as Alternate Dichotomous to better reflect their production status. Please refer to the
readme.txt file included with the software installation for more details on the BMDS 2.0
beta.
A.8 BMDS 2.0 (final)
July 10, 2008 - BMDS Version 2.0 final is now available. Released on July 10, 2008, it
replaces BMDS 1,4.1c as the official BMDS software. BMDS 2.0 is a rewrite of the user
interface and risk assessment modeling framework, with a markedly improved
functionality and enhanced multi-model processing capabilities. It uses the same
underlying source code for the models in BMDS 1.4.1 software, with minor corrections
and some important additions. For details on the new user interface, go to the BMDS 2.0
Help menu option in the installed software. BMDS 2.0 also has a new set of quantal
models with alternative background (i.e., background additive to dose) and asymptote
(i.e., Hill model) parameters, as well as a Beta Exponential set of models.
A.9 BMDS 2.1 (beta)
September 30, 2008 - EPA is making version 2.1 of BMDS available at this time for
public beta testing. Version 2.1 includes a beta (external peer review) version of a new
time-dependent toxicodiffusion model for continuous outcomes (Zhu et al., 2005),
incorporates graphical plots for the continuous exponential models and allows for the use
of individual animal continuous response data. The BMDS toxicodiffusion model was
developed by the USEPA National Center for Environmental Assessment (NCEA),
through partnerships with the USEPA Neurotoxicology Division (NTD) and the University
of South Florida, to characterize toxic effects (e.g., neurotoxicity) that potentially evolve
along critical time points. It does this by:
•	modeling a dose-response along a time-course of repeated response measures; and
•	computing benchmark doses and their confidence limits along the time course.
Documentation for the toxicodiffusion model can also be downloaded. The
documentation contains a full description of the model, input requirements, model run
options and sample runs.
In addition, EPA is distributing an external review (beta) version of a concentration-time
(CxT) model originally programmed by Wil ten Berge. The EPA ten Berge model
implements an approach to evaluating the CxT relationships for effects associated with
chemical exposures. The EPA's version 1.0 implementation of this model is being
distributed along with associated documentation and comments on the model received
from external peer reviewers. EPA plans to respond to external review comments and
incorporate the ten Berge model into a future version of BMDS.
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A.10 BMDS 2.1 (Build 52)
July 30, 2009 - EPA is now distributing the final release of Version 2.1 (Build 52) of the
Benchmark Dose Software (BMDS). BMDS 2.1 (Build 52) contains user interface
enhancements as well as several additions/enhancements to the suite of models
available for modeling dose-response data, including new features for the continuous
exponential models and a new interface for the ten Berge concentration-time model. For
details on the changes to the user interface, go to the BMDS 2.1 Help menu option in the
installed software. The Readme.rtf file distributed with BMDS describes the
improvements made in version 2.1 (Build 52), installation requirements, and known
problems.
The exponential models contained in this version of BMDS have been developed in
conjunction with the Netherlands' National Institute for Public Health and the Environment
(RIVM) to be consistent with the exponential models contained in the RIVM's PROAST
software . The USEPA and RIVM are working together to achieve consistency between
the BMDS and PROAST software and methods.
A.11 BMDS 2.1.1 (Build 55)
November 9, 2009 - EPA is now distributing Version 2.1.1 (Build 55) of the Benchmark
Dose Software (BMDS). BMDS 2.1.1 (Build 55) contains a flexible new feature that
allows users to export select BMDS summary report data and plots to Excel. It also
contains a comprehensive set of sample session and model option files to assist users in
running batch operations, and several improvements to the ten Berge model that were
not available in version 2.1. The Readme.rtf file distributed with BMDS provides details
on the improvements made in Version 2.1.1 (Build 55), installation requirements, and
known problems.
A.12 BMDS 2.1.2 (Build 60)
June 11, 2010 - BMDS 2.1.2 (Build 60) contains user interface enhancements to the
"Summary Report" feature, new sample session and model option files, and
improvements to the ten Berge model that were not available in version 2.1.1:
•	BMDS 2.1.2 can access folders or files with embedded space(s) in them.
Combinations of the path and file name must be less than 256 characters.
•	The default location for searching for certain files (i.e. "Session," "Option," "Data,"
"Plot," and "Output" files) is now the last location (folder) to which the user saved that
type of file or from which he accessed that type of file. There is a separate "memory"
for file location for each of the above-listed file types.
•	Improvements have been made to the "Export to Excel" feature associated with the
"Summary Report" of a session run.. This feature allows the user to select which
variables will be exported to Excel by checking/un-checking, in the "Export to Excel"
column, the boxes corresponding to the variable rows the user wishes to export. The
plots are exported as well, in a separate Excel "Plots" sheet.
•	Additional session and option files have been added to the "SessionFiles" and
"OptionFiles" folders. These folders contain sample sessions that allow the user to
quickly run a specific set of models and model options for a selected dataset.
•	The dichotomous Weibull model can accept a lower bound on power specified by the
user. The EPA default choice (power restricted to be greater than or equal to 1) can
still be checked, but if the user wishes not to restrict the power in that way, s/he may
specify a value greater than or equal to zero; zero was the only other option in
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previous versions and that could cause biologically unrealistic fitted curves and
numerical problems.
•	The Toxicodiffusion model no longer needs the R(D)com Active-X control and will run
with any version of R, 2.6.2 or later.
•	Plots created by BMDS can be viewed using capabilities within BMDS or using
GnuPlot. Those two options are accessed under the "Tools" menu item, the "View
Plot" option. Using GnuPlot, the user can edit the plots to choose colors, fonts, line
styles, etc.
•	The option screen for the ten Berge CxT (concentration-time) model BMDS 2.1.2 had
been improved to - address some issues and to allow saving the output file with a
name of the user's choosing. More importantly, CxT results from the ten Berge
model are automatically exported to an Excel template file containing two
customizable plots, one that shows the contour of concentration and time
combinations for a user-specified probability and a second that shows the probability
of response as a function of either concentration or time. In both plots, the user
specifies the values of the other variables in the model.
Like BMDS 2.1.1, BMDS 2.1.2 contains the following additions/enhancements over
BMDS 2.0:
•	The ten Berge CxT model alluded to in item 5 above, allows for fitting of dichotomous
response data sets having two or more explanatory variables (as in acute inhalation
toxicity experiments). The explanatory variables can be entered as main effects or in
interaction (cross-product) terms. The user can request the value (and its bounds) of
one explanatory variable when a response rate is specified (fixing the other
explanatory variables at some user-specified values) and/or conversely, the value
(and its bounds) of the response rate, given specification of all explanatory variables.
•	The executable for the set of models known as the exponential models, proposed by
Dr. Wout Slob of RIVM in The Netherlands has been expanded to allow the
assumption of log-normally distributed data (the previous versions of the exponential
models and all other continuous models in BMDS assume that the data are normally
distributed). The four exponential models fit by BMDS are defined and labeled as
follows:
•	Model 2: m(dose) = a*exp{sign*b*dose}
•	Model 3: m(dose) = a*exp{sign*(b*dose)d}
•	Model 4: m(dose) = a*(c-(c-1)*exp{-1*b*dose})
•	Model 5: m(dose) = a*(c-(c-1)*exp{-1*(b*dose)d})
where "sign" indicates the direction of change in the responses (sign=+1 for
increasing responses; sign=-1 for decreasing responses).
•	A version of a new ToxicoDiffusion model for continuous outcomes (Zhu et al., 2005)
allows for the analysis of repeated-measures data. The BMDS Toxicodiffusion model
is able to characterize toxic effects (e.g., neurotoxicity) that potentially evolve with
time points by
•	Modeling a dose-response along a time-course of repeated response
measurements;
•	Computing benchmark doses and their confidence limits along the time course.
The ToxicoDiffusion model includes graphical outputs showing the observed and
model-predicted time-course data, residuals, and a summary of the bootstrap-based
BMDL calculations.
For further detail see Zhu, Y., Jia, Z., Wang, W., Gift, J., Moser, V.C., and B.J. Pierre-
Louis (2005), Data Analysis of Neurobehavioral Screening Data: Benchmark Dose
Estimation. Regulatory Toxicology and Pharmacology, pp 190-201.
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A.13 BMDS 2.2 (Build 66)
September 6, 2011 - EPA is now distributing the final release of Version 2.2 (Build 66) of
the Benchmark Dose Software (BMDS). Key enhancements in BMDS 2.2 (Build 66)
include:
•	Multiple Tumor Analysis - BMDS 2.2 adds the capability to perform a combined
analysis of multiple tumors. If the user is willing to assume that those tumors are
independent and are well described by a multistage-cancer model, then the Multiple
Tumor Analysis capability (accessed through the File/New or File/Open tool-bar
choices) allows the user to estimate BMDs and BMDLs for the combined incidence of
the tumors in question (i.e., BMDs and BMDLs for the likelihood of getting one or
more of those tumors).
•	Trend Test for Dichotomous Data - Another major addition is the new capability to
perform a trend test on dichotomous data sets. This is the first in a series of trend test
to be added to BMDS (future versions will also include trend test for continuous and
nested data). The trend testing feature can be found on the dataset screens,
accessible once a dataset has been identified by the user as containing dichotomous
response data. The test performed is the Cochran-Armitage trend test described by
Haseman (1984).
•	The Dichotomous Hill model has been modified - Changes to the parameter
initialization section of the Dichotomous Hill code have improved the convergence
features of this model.
•	Automatic Transfer of Variable Name Changes to Other Option Files in a
Session - When working within a session, variable name changes (e.g., for dose,
sample size, response, mean, or standard deviation variables) made in one option
file (i.e., for one model) can be "transferred" to other option files included in that
session (i.e., those for other models). The user will be prompted to determine if
variable name assignment changes made in one option file should be made in all
other option files included in that session. Thus, users can change variable name
assignments once in a session, without having to make those changes separately in
every option file.
•	Default Column Headers for New Datasets - Note also that newly created
dichotomous, continuous or nested model data files will start with default column
headers, in a particular order, as appropriate for the type of data (e.g., Dose, N, and
Effect for dichotomous datasets; Dose, N, Mean, and Std for summarized continuous
datasets). The user may change those default headers, but will be warned that doing
so may affect the running of BMDS-supplied sessions that look for those default
names.
A.14 BMDS 2.2 (Build 67)
December 8, 2011 - EPA is now distributing BMDS 2.2 (Build 67), which include minor
modifications to the user interface.
A.15 BMDS 2.3 (Build 68)
September 12, 2012 - BMDS 2.3 introduces more flexible error-trapping functionality,
enabling users to store essential comments, documentation, notations, etc. on their data
in the datasets and spreadsheets without triggering a data validation error.
Previously, BMDS ran validation checks on the dataset each time the file was opened.
BMDS would interpret notes, comments, extraneous text, etc. contained in the dataset as
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data errors and would not open the file. This would require opening the file in a separate
program and purging any comments or text.
In BMDS 2.3, the application instead checks for data errors when the data is sent to the
model. BMDS bases its validation on the dataset's column assignments from the Option
Screen.
After you open the dataset, select the Model and Model Type, and then select Proceed,
the Option Screen displays. In the Column Assignments section of the Option Screen,
specify the data labels corresponding to the dataset variables.
When you click Run, BMDS will validate and error-check the data for only those assigned
columns. BMDS will then display a message box describing any errors related to invalid
values or blank cells found in those columns.
BMDS will also flag any duplicate column headers if they conflict with the Column
Assignments specifications. Note that BMDS is case-sensitive, so BMDS considers dose,
Dose, and DOSE to be separate variable names.
BMDS 2.3 also features the following fixes and enhancements:
•	Fixes a problem reported by users of Microsoft Office in Windows 7, in which
clipboard errors would crash both BMDS and Microsoft Office.
•	Warns the user when saving a file using a name that exceeds the Windows limit of
256 characters. To correct the problem, rename the file or move the file to a directory
higher in the hierarchy.
•	Enhances Option screen validation to provide more thorough parameter constraint
checking and to display pop-up messages describing errors.
•	Fixes bug that prevented BMDS from adding a new/existing data set to the Session
Grid.
•	Fixes bug related to dynamically drawn dialog boxes.
•	An issue was discovered in the computation of the A3 model log-likelihood for the
continuous models when the user specified the variance parameter alpha. While this
issue is being investigated and resolved, the option to specify alpha has been
disabled for those models.
•	Fixes intermittent bug where occasionally, when running the exponential model,
BMDS failed to display a summary graph and report, even when output files were
produced.
•	Edits and updates to several Help topics:
•	Updated all Model Description topics to reflect upper parameter limit of 18 for
some models.
•	Added the following topics under "Other Data or Analysis Types": Data With
Negative Means, Test for Combining Two Data Sets for the Same Endpoint
•	Added the following Troubleshooting topics: Help File Does Not Display, Decimal
Separator Should Be a Period (see Section 3, "Usage Tips," in this file for more
information)
•	Added description of multitumor model (.d) file to the Multiple Tumor Analysis
topic.
•	Added text to the Dichotomous Hill and Logistic models descriptions to address
occasional error messages that appear for these models.
•	Under the "Graphic Output from Models" topic, added the subtopic "Error Bar
Calculations," explaining how BMDS generates error bars for various model
types.
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A.16 BMDS 2.3.1 (Build 69)
September 28, 2012 - BMDS maintenance release Version 2.3.1 (Build 69) replaces
Version 2.3 (Build 68). It simplifies how data validation errors for certain models are
reported to the user. BMDS 2.3 also introduced user interface improvements.
None of the actual dose-response models in BMDS were modified for versions 2.3 and
2.3.1.
A.17 BMDS 2.4 (Build 70)
BMDS 2.4 includes several enhancements to improve usability and ensure accurate and
reliable results, such as more informative plot titles and the ability to change an option file
on the fly within a session. The help file includes a new section on "BMDS Best
Practices" containing valuable information and guidance on such topics as optimization
criteria, alternative models, re-initializing parameters, and lognormal response option,
among others.
Also, in this release, the BMDS install package includes ICF International's BMDS
Wizard, an Excel-based tool that facilitates the preparation and organization of, and
enhances the reporting capabilities of, BMDS modeling sessions.
BMDS 2.4 adds the following new or enhanced functionality:
•	BMDS plots now provide more informative titles, such as "Weibull Model, with BMR
of 10% Extra Risk and 0.95 Lower Confidence Limit for the BMD (BMDL)."
•	BMDS now includes the cancer slope factor for the MS_Combo output.
•	When an option file is modified via the Session Grid and saved under a new name,
BMDS now automatically links that file to the current session.
•	In a modeling session, after changes are made to a model option file, BMDS lists the
changes for the user and asks whether they should be applied to the other model
option files in the session. A warning is given to remind the user that changes to
option files will affect other sessions that use those option files.
BMDS 2.4 also features the following fixes or changes:
•	Gnuplot files now remain on the screen until the user closes them.
•	Fixes a problem where exporting rows to Excel would shift column values to different
headers on the Dichotomous Format and Continuous Format worksheets.
•	Fixes a problem where BMDS dichotomous models treated "%Positive" as Incidence
data.
•	Fixes a problem where cut and paste operations on session data fields worked
erratically on Windows 7 systems.
•	Fixes a problem where BMDS could not open datasets saved with a capital .DAX
extension.
•	Fixes a problem where selecting a column in the data grid for a Log10 transformation
returned incorrect results.
•	Removes "Exact" as an available solution in the exponential model for summary
data.
•	Removes the "Extra" option for all continuous models but Hill.
•	Removes the Optimization section from the options screens for all models that are
not affected by optimization settings.
•	Changes the default model iterations for all models from 250 to 500.
•	Harmonizes the output for all continuous models so that when the "Rel Dev." BMR
type is selected in the option screen, the "Risk Type" reported in the output file says
"Relative deviation" rather than "Relative risk."
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•	Fixes a problem in the Multitumor Model (ms_combo) where BMDS failed to return a
valid combined BMDL if the largest dose in the first dataset listed in the tumor
analysis was less than the maximum dose in the other tumor analysis datasets.
•	Edits and updates several Help topics, including:
•	A new section, "BMDS Best Practices for Obtaining Optimal Model Convergence."
•	A new topic under Troubleshooting: "Avoid using special characters in filenames."
•	A new topic documenting the model files (with their version numbers) used in BMDS
2.4.
•	A new appendix for BMDS version history information that originally appeared in
those versions' readme files.
•	A new note on log transformations for the topic "Data Transformation Types."
ICF INTERNATIONAL'S BMDS WIZARD
The BMDS 2.4 install package includes ICF International's BMDS Wizard, an Excel-
based tool that facilitates the preparation and organization of and enhances the reporting
capabilities of BMDS modeling sessions. This install includes multiple copies of the ICF
BMDS Wizard files that are preformatted for continuous, dichotomous, and dichotomous-
cancer datasets.
BMDS "power users" employ ICF BMDS Wizard as a shell to simplify the BMD modeling
process by streamlining data entry, model selection, option file development, output file
reporting, and model comparisons.
ICF BMDS Wizard 1.7 can also export Microsoft Word-formatted reports that employ the
latest EPA-approved reporting format (as of February 15, 2013). It can only export
reports for continuous, dichotomous, and dichotomous-cancer models.
To run the ICF BMDS Wizard, go to the BMDS 2.4 program directory and locate the
"BMDS Wizard 1.7" subdirectory. ICF has included a readme and quickstart guide to the
software. Please refer to that documentation for details on running the tool.
More information on the ICF BMDS Wizard can be found at ICF's site:
http://www.icfi.com/insiqhts/products-and-tools/bmds-wizard.
Please note that ICF BMDS Wizard is not endorsed or approved by EPA. Please contact
ICF International at wizard@icfi.com for support.
A.18 BMDS 2.5 (Build 82)
BMDS 2.5 provides updates to ICF International's BMDS Wizard, which includes support
for 32-bit and 64-bit versions of Microsoft Office on Windows 7, as well as a new
MS_Combo (multi-tumor) template.
In addition, BMDS 2.5 contains various improvements to model stability and reliability.
Resolved Issues
BMDS 2.5 features the following enhancement and fixes:
•	The BMDS Tools>View Plot menu entry has been simplified to make it easier to
generate a plot from a previously created .pit file.
•	The Multistage, Multistage Cancer and MS_Combo models now provide accurate
results when non-integer input data values for Incidence and Number of Subjects
used.
•	The Multistage, Multistage Cancer and MS_Combo models now work correctly when
beta parameters are specified by the user. Previously, the models would fail to
calculate BMDL.
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•	The Power model now honors the direction of adversity specified by the user.
Previously, the model always determined the direction automatically.
•	An intermittent crash in the MS_Combo model has been resolved.
•	Several Help topics have been revised, including:
•	Specified in the Troubleshooting section that path+filenames should not exceed 127
characters.
•	Added a topic documenting the MS_Combo model input file.
•	Added Appendix B, which documents, for each model options screen, all of that
screen's fields, including any field constraints.
•	Updated the following topics to reflect BMDS .out files content: Continuous Model
Text Output, Continuous Model Maximum Likelihood, Tests of Fit.
A.19 BMDS 2.6
BMDS 2.6 includes several enhancements to improve usability and ensure accurate and
reliable results.
BMDS 2.6 also introduces a significant new feature: the ability for BMDS to automatically
detect and, optionally, install software updates. This feature will ensure BMDS users
have access to the latest version of the software with up-to-date fixes and
enhancements.
The BMDS install package includes ICF International's BMDS Wizard, an Excel-based
tool that facilitates the preparation and organization of, and enhances the reporting
capabilities of, BMDS modeling sessions. See the next section on the ICF BMDS Wizard
for more information.
BMDS 2.6 features the following enhancement and fixes:
•	Resolved the following issues related to the nested models (NLogistic, NCTR, and
Rai and van Ryzin):
•	The approach to evaluating goodness-of-fit for the nested models has been
changed. Now, a bootstrap-based approach (using the fitted model and the
underlying beta-binomial distribution of the observations) is used to evaluate the
lack of fit. This obviates the need to group the litter-specific observations across
litters and avoids asymptotic approximations.
•	Fixed a calculation error that occurred when the litter-size covariate (LSC) was
larger than the litter size. (Although the reliability of the Rai and van Ryzin model
has been improved, there is a remaining issue where the model erroneously
reports the same value for the BMD and BMDL under certain circumstances.)
•	Added scaled residual of interest calculations to the results report. Output
includes min, average, and max scaled residual of interest, and number of litters
used for the calculation.
•	BMDS now handles file and path names more robustly across all models. Specific
changes include:
•	Full path length (folder plus file name) is now 255 characters rather than 127.
•	Spaces and ampersands (&) are now permitted.
•	Fixed several issues that occurred when the user specified output and session
names in some models.
•	BMDS now supports UNC (network) path names structured as
\\ComputerName\ShareName\Path (e.g.,
\\FileSrv1\Users\JDoe\USEPA\BMDS260). The total number of characters
cannot exceed 255.
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•	The ToxicoDiffusion and MS_Combo models are now out of "beta" status. Detailed
technical documentation of these models can be viewed or downloaded from the
BMDS Download page.
•	The ToxicoDiffusion model can now run when R has been installed on a per-user
basis.
•	Resolved an issue with the Linear and Polynomial models where incorrect MLE's
were produced for non-constant variance in some situations.
•	Resolved an issue so that BMDS now correctly calculates and displays Standard
Error Estimates for all models in the "Parameter Estimates" table, whereas previously
most models displayed
•	Changed the color scheme for BMDS plots so they are easier to read.
•	Included the most recent version of gnuplot (version 4.6.3).
•	Fixed the Export to Excel function so it can export results for sessions containing any
number of models. (Previously, sessions containing more than 24 models produced
an error during export.)
•	Fixed an issue so that BMDS now supports regional settings for the decimal
separator in the user interface and in spreadsheets created by the Export to Excel
function. However:
•	Files generated by BMDS (such as the .out files and .d files) will still show as
the decimal.
•	No "thousands separator" (regardless of regional setting) can be used in the
data; that is, one thousand can only be written as 1000 rather than as 1,000.
•	Added the following items to the summary table in the BMDS user interface: "Scaled
Residual for Control Group" and "Scaled Residual for Dose Group Nearest the BMD".
The relevant values are already printed out in the "Goodness of Fit" table.
•	Improved the e-Ticket system for users to request support and guidance. The new
URL is http://bmds.epa.gov/eticket.
•	Fixed an issue in the Power model where the specified power parameter output value
was displayed incorrectly in the output text.
•	Fixed an error in the restricted linear model when variance is non-constant.
•	Fixed the alpha parameter initialization feature in the model options screen.
•	New data grid windows now default to display 1000 rows. The previous default was
100.
•	In addition to minor changes to accommodate the fixes and enhancements described
above, the following Help file topics have been significantly revised or added:
•	Added: Troubleshooting>No Thousands Separator Can Be Used in the Data
•	Added: Troubleshooting>Requesting Support using eTicket
•	Added: Using BMDS 2.6.0>Keeping BMDS Up to Date
•	Added: Model Descriptions>Multitumor (MS_Combo) Model Description
•	Added a note to the Output From Models>Text Output from
Models>Dichotomous Model Text Output topic, describing how standard error
methods are calculated for parameters for multistage models vs. other
dichotomous models. Also added text to the Continuous Model Text Output and
Nested Model Text Output topics describing how their parameter standard errors
are obtained.
•	Revised text on standard error calculations for the continuous model topics.
•	Updated the Model Descriptions>Nested Model Descriptions topic to describe
how the goodness-of-fit p-values are calculated using a bootstrap approach. The
Nested Model Description subtopics also include revised formula descriptions.
•	Updated the topic "Models Included in BMDS 2.6.0" to reflect updated versioning
information.
•	Updated several topics in "Appendix A: Model Input File Format Descriptions" to
reflect user interface changes. Also added input file details for the Dichotomous
Hill and Quantal with Background Dose models.
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• Updated several topics in "Appendix B: Model Options Screen Fields Reference"
to reflect user interface changes.
A.20 BMDS 2.6.0.1
BMDS 2.6.0.1 contains the following fixes and enhancements:
•	Resolves an issue in BMDS 2.6 that prevented users from specifying parameter
values for Continuous models. (BMDS still prevents users from specifying the
Continuous model Alpha parameter to work around an open issue where BMDS
sometimes reports a non-optimal A3 log likelihood when the Alpha parameter is
specified.)
•	Resolves an issue with previous versions where BMDS occasionally reported
incorrect scaled residuals when a session referenced multiple data files with differing
dose values.
•	Enhances the Nested models' text results to display the minimum, maximum, and
mean of the absolute value of the scaled residuals, as well as the minimum,
maximum, and mean of the scaled residuals.
•	Resolves an issue where BMDS 2.6 could not display plots generated by previous
BMDS versions.
•	Implements a fix for an intermittent issue where the BMDS Wizard did not import
plots generated by BMDS 2.4 and later.
•	Correctly validates the v parameter for the Continuous Hill model, an issue since
BMDS 2.3. Although the v parameter must be 0 < v <1 for Dichotomous Hill, it should
be unconstrained for the Continuous Hill model.
•	Adds an Update button to the About BMDS box (accessible from the Help>About
menu item). Click the Update button to have BMDS check for and optionally install a
BMDS update.
A.21 BMDS 2.7
August 18, 2017 - EPA distributed BMDS 2.7, which features several enhancements and
fixes, including:
•	The benchmark dose upper confidence limit (BMDU) is now included with the BMDL
for most BMDS models (primarily continuous and dichotomous).
•	BMDS installation now uses a Windows Installer msi-based file, rather than the zip
files of previous releases. The .msi-based release is simpler, more robust, and
minimizes problems related to Windows 10's increased security protocols.
•	The auto-update feature (introduced in BMDS 2.6) works in a wider range of
connection scenarios so users can be assured they are running the most recent
version.
The Excel-based BMDS Wizard v1.11 is also included as part of the install package.
BMDS Wizard remains unchanged except for minor fixes needed for compatibility with
BMDS v2.7.
See the BMDS 2.7 readme file for details on this version's enhancements and fixes.
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Appendix B: Citation Format and Acknowledgements
Citation Format
U.S. EPA (Environmental Protection Agency), 2018. Benchmark Dose Software (BMDS)
Version 3.0. National Center for Environmental Assessment. Available from:
https://www.epa.gov/bmds/download-benchmark-dose-software-bmds (Accessed
mm/dd/yyyy).
Sponsors
EPA's National Center for Environmental Assessment with technical support from EPA's
National Health & Environmental Effects Research Laboratory.
Contractor
General Dynamics Information Technology (GDIT) - BMDS user interface with a risk
assessment modeling framework
Credits
•	Dr. Matthew Wheeler, The National Institute for Occupational Safety and Health
(NIOSH)
•	RIVM (National Institute for Public Health and the Environment of the Netherlands,
part of the Dutch administration), is a recognized leading center of expertise in the
fields of health, nutrition, medicines, consumer safety and environmental protection
working mainly for the Dutch government. RIVM is not a part of the US
Environmental Protection Agency, but has given US EPA a non-exclusive, limited
and revocable permission to use its logo only in recognition of RIVM's contribution to
the development of certain BMDS models. The use of this logo here is neither an
endorsement nor an advertisement for RIVM. RIVM does not accept any
responsibility or liability for the activities of—or failures to act by—US Environmental
Protection Agency.
•	The following contractor(s) contributed to the models included in BMDS:
• tk
We gratefully acknowledge the following software used in BMDS:
•	Microsoft Excel 2016
•	R
Note
Obtain the latest version of this program from the EPA BMDS web site
(https://www.epa.gov/bmds/download-benchmark-dose-software-bmds). The features
and models in the BMDS program that are downloadable from this web site have
received at least an internal EPA review. To allow for outside input towards improvement
of the individual models, the source code files for the latest versions of the models are
separately available from the BMDS web site for continuous review and comment.
Disclaimer
This software has been reviewed in accordance with U.S. Environmental Protection
Agency policy and approved for use. Mention of trade names or commercial products
does not constitute endorsement.
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