xvEPA
EPA/600/R-17/236
June 2017
https://www.epa.gov/bmds
CATEGORICAL REGRESSION (CATREG)
USER GUIDE
VERSION 3.1.0.7
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Categorical Regression (CatReg) User Guide
TABLE OF CONTENTS
1.0 INTRODUCTION 5
1.1 About CatReg 6
1.2 CatReg's Two Models 6
1.3 CatReg and Toxicology Data 7
1.4 "Concentration" and "Dose" are Equivalent Terms in CatReg 8
2.0 INSTALLING CATREG 9
2.1 Step 1: Meet CatReg System Requirements 9
2.2 Step 2: Install CatReg 9
3.0 OVERVIEW OF A CATREG SESSION 10
4.0 INPUT FILE FORMATTING CHECKLIST 12
5.0 DATASET AND VARIABLES TAB 14
5.1 Loading an Input File 15
5.2 Input Files Must be in CSV Format 15
5.3 Model Variable Mapping 16
5.4 Managing Treatment Groups 16
5.4.1 CatReg's Default Grouping Algorithm 17
5.4.2 Custom Grouping 17
5.5 CatReg's Validation Criteria After Custom Grouping 18
5.6 Filtering Data Values by Variable 18
5.7 Minimum Data Requirements 18
5.8 Data Requirements: Outcomes 19
5.9 Data Requirements: Severity Levels 19
5.10 Data Issues that Trigger Error Messages 21
5.11 Input File Examples 22
5.11.1 chemx.csv 22
5.11.2 chemy.csv 24
5.11.3 chemz.csv 28
6.0 MODEL AND BMD TAB 30
6.1 Setting BMD Specifications for an Analysis 31
6.2 Setting Model Specifications for an Analysis 31
6.3 Running an Analysis 32
6.4 About the CatReg Output Files 32
6.5 Changing the Output File's Name and Location 32
6.6 Stratifying 33
6.7 Clustering 33
6.8 Link Function 34
6.9 Model Form 35
6.9.1 Understanding the Model Equations 36
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6.10 Assume Zero Background Response 36
6.11 Worst Case Analysis 37
7.0 PLOTS TAB 38
7.1 About the CatReg Plot Files: Locations, Formats, Names 38
7.2 Copying Plots from the Graphs Window 38
7.3 Plot Functions and Options 40
7.4 Jitter 40
7.5 Doses and Times for Designated ERC and Severity (catplot) 41
7.6 Doses and Times for Designated Probability and Severity of Strata (stratplot) 43
7.7 Extra Risk Concentration at Desired Time (confplot) 44
7.8 Probability Versus One Explanatory Variable (prplot) 45
7.9 Data Plotted by Stratum (dataplot) 46
7.10 Contribution to Deviance for Individual Datum (devplot) 47
7.11 Observed and Predicted Probability Versus Dose/Time combination (barplot) 50
8.0 HYPOTHESES TAB 51
8.1 Testing Parameters 52
9.0 ASSESSING MODEL FIT 54
10.0 MENUS, TOOL BARS, & STATUS BARS 60
10.1 File Menu 60
10.2 Preferences Menu 60
10.3 Help Menu 60
10.4 Text Window (Results) Menus 60
10.4.1 File 60
10.4.2 Edit 61
10.4.3 Preferences 61
10.5 Dataset Context Menu 61
10.6 Status Bar 61
11.0 KEEPING CATREG UP TO DATE 62
11.1 When an Update is Ready for Download 62
11.2 To Use the Newly Installed Software 63
11.3 Managing Your Existing Files After an Automatic Update 63
12.0 REFERENCES 64
13.0 GLOSSARY 65
APPENDIX A: DISTRIBUTION OF CONTINUOUS RESPONSE DATA OVER SEVERITY LEVELS 66
APPENDIX B: TECHNICAL DISCUSSION 69
B.1 Link Functions 69
B.2 Interval Censoring 70
B.3 Parameter Estimation 70
B.3.1 Maximum Likelihood Estimation 70
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B.3.2 Generalized Likelihood Estimation 71
B.4 Confidence Limit Calculations 72
APPENDIX C: TECHNICAL BACKGROUND: MODELS AND EXTRA RISK 73
C.1 Exposure-response Models 73
C.2 Extra Risk Concentration (ERC) 74
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INTRODUCTION
1.0 INTRODUCTION
The U.S. Environmental Protection Agency (EPA) developed the Categorical Regression
(CatReg) application as a tool to facilitate exposure-response analyses. The latest version can be
obtained from the EPA CatReg website.
In general, CatReg should be viewed as a statistical tool for examining exposure-response-time
relationships and addressing related questions. A thorough analysis may require numerous
executions of CatReg, ideally guided by both toxicological and statistical considerations. The
program's Windows interface and model options facilitate sensitivity analysis and produce
numerous plots in addition to hypothesis-testing results.
Key features of CatReg include:
• The use of multiple independent variables to explain the response
• The ability to predict exposures related to various levels of effect severity
• Extensive plot features for the analysis of model results
• A user-friendly hypothesis testing interface that assists the user in:
• Assessing and comparing how well models fit the data
• Determining the feasibility of combining data from multiple studies by testing for
differences across studies and the significance of covariates within single or pooled
studies
• Detecting outliers
CatReg's advantages over other dose-response analysis tools include its abilities to:
• Use multiple independent variables to explain the response. Most often, the independent
variables modeled in CatReg are exposure (or dose) and duration of exposure. The latter
allows for estimating risk at different exposure durations—even time points not included in the
dataset being modeled.
• Predict exposures related to various levels of effect severity. Using categorical
regression to evaluate different levels of severity is of particular use for developing
emergency planning guidelines, which are defined in terms of the exposure limits needed to
protect people from increasingly severe effects (e.g., protection from mild effects, severe
effects, and death).
• Combine multiple studies in a single analysis. The meta-analytical use of categorical
regression can be invaluable when individual studies provide only minimal useful dose-
response data. Classifying toxic effects into severity categories provides a way to put
different endpoints on a common scale for analysis and also allows the use of all types of
data in the analysis. For example, individual animal studies investigating liver lesions and
serum biochemistry can be combined with human studies reporting preclinical and clinical
liver effects to provide an overall estimation of the dose-response for liver toxicity.
Continuous, descriptive and categorical data can be used, in addition to incidence data, as
long as these effects can be classified into severity categories (Appendix A describes an
approach for converting continuous response information into severity categories).
• Test hypotheses. Finally, the CatReg 3.0 implementation of categorical regression offers
valuable and practical plotting and hypothesis testing tools, combined with a user-friendly
interface, to assist the user in determining whether data associated with variables that can
sometimes influence a dose-response (e.g., sex, species, diet, age, genetic differences)
should be evaluated separately or combined together into a more robust, data-rich multiple
study meta-analysis.
This documentation provides instruction on how to use CatReg. However, the documentation
does not address in detail CatReg concepts or guidance on CatReg methods. While the EPA
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Categorical Regression (CatReg) User Guide
INTRODUCTION
CatReg methods guidance has not been finalized at this time, every attempt has been made to
make this software consistent with the most recent working draft guidance and discussions of the
EPA Benchmark Dose Work group.
Until formal CatReg methods guidance is available, users of this software are strongly
encouraged to review existing background material such as the CatReg Software User Manual:
R-Version (EPA, 2006) before using this software.
Note: This user guide supersedes the 2006 User Manual with respect to software-specific
instructions such as installation procedures and user interface features and options.
The U.S. Environmental Protection Agency's (EPA's) National Center for Environmental
Assessment encourages the broad application of this software. In this document, however, EPA
has chosen to focus on the application of the software to the assessment of adverse effects
associated with acute inhalation exposure. The description of the CatReg input files for the
software reflect this application. The user is free to modify the input fields to support other
applications. Appendices B and C provide additional technical description and background for the
statistical methods and models used by the program.
1.1 About CatReg
CatReg is a computer program developed to support toxicologists, risk assessors, and health
scientists in conducting exposure-response or, when intake information is available, dose-
response analyses, most often for controlled animal experiments.
"Exposure" or "intake" has two components:
• Exposure or intake level, indicated by a concentration or dose of the agent of interest
• Exposure or intake duration (time), when the dose varies within the data
"Response" refers to occurrence of a detrimental health effect of a user-defined level of severity.
More specifically, effects observed in toxicological studies are assigned to ordinal severity
categories and associated with the exposure conditions (e.g., dose and time) under which the
effects occurred. "Ordinal" here means that the categories have a natural ordering in terms of
severity or strength of response, but the spacing between ordinal scores is not subject to direct
interpretation.
For example, response might have four levels of severity coded as:
• 0 = "no adverse effect"
• 1 = "mild adverse effect"
• 2 = "moderate/severe effect"
• 3 = "lethal effect"
An ordinal response of 2 is higher than a response of 1, but the difference is not necessarily the
same as the difference between 3 and 2. The simplest case is dichotomous response data, with
just two severity levels, such as: 0 = "no adverse effect", 1 = "adverse effect".
If data are reported on a continuous scale, such as mean and standard error of respiratory rate
depression, the user can distribute the total number of experimental subjects over the severity
levels using a method discussed in Section 5.11.3 and Appendix A.
1.2 CatReg's Two Models
CatReg provides two basic models, with variations to be explained, to relate the probabilities of
the different severity categories to exposure level and exposure duration, taking user-defined
covariates into account (e.g., species, gender, target organ, etc.).
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INTRODUCTION
The parameters in the models are an intercept term and coefficients of dose and time, either of
which may be log-transformed (to the base 10, denoted as "log," "Iog10," or"log10").
• Model 1, the cumulative odds model, allows the intercept term to vary with severity level, but
not the coefficients of dose and time.
• Model 2, the unrestricted cumulative model, allows any of the parameters to vary with
severity level.
The probability that a specified severity level or worse will occur increases as dose or time
increases. The user can choose for either Model 1 or Model 2 to conform to the logistic, normal,
or Gumbel cumulative probability distribution (see Appendix B and Appendix Q.
There is a function (called the link function) in each case that transforms the probability for each
severity level to a linear function of the unknown parameters, the format of a linear statistical
model. The link functions are the logit, probit, and cloglog (complementary log-log) functions for
the logistic, normal, and Gumbel cumulative probability distributions, respectively.
The parameter estimates and their statistical characteristics, including standard errors and
significance levels, are routinely output by CatReg, along with an analysis of deviance table to
assess model fit and a table of estimates of extra risk concentrations (ERC) (doses at which extra
risk is a user-specified value) for 1-, 4-, 8-, and 24-hour exposure durations.
1.3 CatReg and Toxicology Data
CatReg was developed for, but is not limited to, meta-analysis of toxicology data from controlled
animal studies. Meta-analysis refers to the analysis of data or results from multiple studies
simultaneously. Meta-analysis becomes valuable when individual experiments are too narrow to
address broad concerns.
For example, in acute inhalation risk assessment, it is important to investigate the combined
effects of dose and time of exposure, but few published experiments vary both the dose and the
time of exposure (Guth et al., 1997). By combining information from multiple studies, the
contribution of both dose and time to toxicity can be estimated. Moreover, the combined analysis
allows the analyst to investigate variation among experiments, an important benchmark for the
level of model uncertainty.
Different exposure-response experiments may consider the same or different toxicological
endpoints, and toxicological judgment is required to determine if, and when, two different
endpoints, or gradations of the same endpoint, are of comparable severity.
A relatively simple example is analysis of mortality studies, with two severity levels: 0 = "not
lethal", 1 = "lethal". The same endpoint is used for all studies and no intermediate degrees of
health gradation are addressed.
A little more complicated example might involve a single health effect, or mode of action, but with
more than one severity level corresponding to manifestations of progressive "stages" of
development.
Where studies address dissimilar endpoints that may be the consequence of different modes of
action, particular care needs to be exercised to decide if comparable severity levels can be
assigned across studies. It may not be reasonable to include all studies in the same analysis.
For example, two toxicology experiments might report stages of anesthesia while another reports
suppression of the shock-avoidance response. It might be the case that a toxicologist can
confidently assign endpoints of the first two studies to comparable severity levels, but not be able
to include the third study. In that case, one analysis could address the first two studies and a
second analysis the third study, since the studies cannot all be put on a "toxicologically
equivalent" severity scale for analysis together.
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INTRODUCTION
1.4 "Concentration" and "Dose" are Equivalent Terms in CatReg
You will see the terms "concentration," "dose," and "concentration or dose" used throughout this
User Guide. (In the CatReg user interface, "Dose" is used throughout.)
CatReg can be used on oral or inhalation toxicity data, and as such "dose" and "concentration"
are used interchangeably throughout this document.
Indeed, CatReg was originally developed for acute, inhalation data, hence the term "extra risk
concentration" (ERC) is used in the program. However, in practice, the ERC values are similar,
and can serve the same purpose as, benchmark dose (BMD) values traditionally used by EPA for
the derivation of reference values and cancer slope factors.
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INSTALLING CATREG
2.0 INSTALLING CATREG
2.1 Step 1: Meet CatReg System Requirements
• 16 Megabytes of RAM
• Microsoft Windows 7 or higher
• Microsoft .NET 4.5 or later. If your computer is set up to allow or be notified of downloads of
Microsoft software, then it is likely that that version of the .NET framework is already installed.
Otherwise, you can download and install Microsoft's latest version of .NET for client desktops.
• R for Windows version 3.4 or later (either 32- or 64-bit).
Note: CatReg relies on the open source package R for Windows to perform all statistical
calculations. For more information on R, refer to the R Project's web site.
2.2 Step 2: Install CatReg
You do not need to uninstall previous versions of CatReg. You can run this version of CatReg
alongside earlier versions.
1. Download the CatReg31 .msi Windows Installer file from the CatReg Web site.
2. Run the CatReg31 .msi file. Your system's antivirus program may warn you about the
file; if so, approve or allow the file so the install can continue.
3. Step through the install wizard. You can enter a different install directory or accept the
default.
4. Navigate to the install directory and select the CatReg.exe file to start the program.
When installing the CatReg application folder, you should:
• Place the folder in the simplest, shortest directory possible. For most EPA users, the
preferred path would be C:\Users\[EPA user's LAN ID]; for non-EPA users, this could be as
simple as C:\.
• Have administrative rights for the folder.
• Ensure there are no special characters or spaces in the folder names.
If you have problems or concerns, please contact the CatReg development team using our Help
Desk Form.
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Categorical Regression (CatReg) User Guide
OVERVIEW OF A CATREG SESSION
3.0 OVERVIEW OF A CATREG SESSION
The following steps illustrate one path through a CatReg session.
1. Open CatReg by selecting the CatReg.exe file.
2. From the menu bar, select File > New Analysis (for a new input file containing one or more
datasets) or Open Analysis (to continue work from a previous session). The Analysis Screen
opens, as shown in the following figure.
Figure 1.—Initial CatReg window.
oj Categorical Regress
on Version 3.0.1.1 Beta - [Analysis Screen]
1 = 10 U-a-l
| File Help
_ B X
Dataset and Variables | Model and BMP | Plots ~| Hypotheses
Dataset
Click here to load data
Model Variable Mapping
Incidence N
Filter Data Values by Variable
Filter 2 Filter 3 Filter 4
Summary of Run Options
Output File:
Filtered Out: I
Clustered:
Stratified:
Censoring:
Save Analysis As— ]
Right-dick on Dataset control for additional option(s).
^
3. In the Analysis Screen, select the Click here to load data button. CatReg displays an Open
dialog box; navigate to and select the CSV input file you want to open. CatReg displays the
loaded dataset(s) and enables the dropdown lists.
Note: A CatReg input file can contain one or more sets of dose-response data.
Note: At this time, CatReg can only read files in CSV format.
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OVERVIEW OF A CATREG SESSION
CatReg automatically displays the data sorted into treatment groups and displays the
dialog box shown in Figure 2. Click on Select Grouping... to group the data by other
criteria.
Figure 2.—Grouping message box, with grouped data.
Analysis Screen
Dataset and Variables pModel and BMP | Plots | Hypotheses]
Dataset: c:\usepa\catreg_3016_beta\data\chemx.csv
~ Grouping
r^wirs!
CatReg has grouped your data. If this grouping is
incorrect or you would like to change the grouping
criteria, click the Select Grouping button below.
This will not alter your orginal data file.
Model Variable Mapping
][ SevLo ~ [Not Used] ~
Filter Data Values by Variable
Filter 1 Filter 2 Filter 3
[[Not Used] ~[[ [Not Used] ¦¦•][[Not Used] ~
Summary of Run Options
A. innn-«\r-i+D«'
4.
In the Dataset and Variables tab, you must map the dose-response data variables to
CatReg's standardized model variable names. If you want to filter out specific values, select
them here. Note that the summary fields at the bottom of the screen record the selections you
make.
Note: Although you can select Run Analysis after specifying the model variables and any
filtering, it is highly recommended that you review the Model and BMD tab settings first.
5. In the Model and BMD tab, specify the options that CatReg will use to calculate an exposure-
response curve. All you need to do is select options from picklists, but understanding
clustering and stratification will help inform your selections.
6. Select the Run Analysis button. CatReg opens a separate window to display the text-based
results (.OTX file). By default, CatReg saves the .OTX file to the directory containing the input
file used for the current analysis.
7. On the Plots tab (which is enabled after an analysis is run), select the plots you want
generated for the analysis. Select the Run Plots button. CatReg opens a separate window to
display the plots. You can copy and paste the plots into other graphics programs, such as
PowerPoint or GnuPlot, to print or edit them. By default, CatReg saves all plots to the
directory containing the input file used for the current analysis.
8. On the Hypotheses tab (which is enabled after an analysis is run), you can test exposure-
response hypotheses for Intercept, Dose, and Time parameters. Select Run Tests. CatReg
will display a separate window to display the text-based analysis (.ANX file). By default, the
file is saved to CatReg's "OptionFiles" directory. Select the "Save Analysis As..." button to
save to another directory.
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INPUT FILE FORMATTING CHECKLIST
4.0 INPUT FILE FORMATTING CHECKLIST
A CatReg input file contains dose-response data from one or more studies.
CatReg input files that conform to the following guidelines should provide consistent operation
and results.
Note CatReg includes two sample input files in its Data subdirectory: Chemx.csv and
ChemxHu.csv.
~ The input file is a comma-delimited text file with a ".CSV" extension. Spaces are interpreted
as characters. For example,MU," is distinct from
~ The data and units must be capitalized correctly. Because R is case-sensitive, "mil," is
different from "Mu,".
~ The first row for each set of dose-response data contains variable names (columns).
Subsequent rows contain data for the variables.
~ Each dataset should contain the following columns that you will map to variables in CatReg.
The columns can be in any order. See Table 3 in Section 5.9, "Data Requirements: Severity
Levels" for more information.
o Exposure concentration or dose (required)
o Exposure time in hours (required unless times are equal; default=1)
o Number of subjects in the treatment group (required)
o Incidence of severity level or severity range for the record (required)
o Lowest severity level for the record (required)
o Highest severity level for the record (required for "censored" analysis if response
spans more than one severity level; see Section 5.9, "Data Requirements: Severity
Levels," for more information.)
~ The following conditions apply for severity levels in the input file. See Section 5.9, "Data
Requirements: Severity Levels," for more information.
o The minimum number of severity levels is two (severity levels coded as 0 and 1,
corresponding to absence or presence of an effect),
o Severity levels must be specified as small, consecutive ordinal values (0, 1,2,3, 4).
o It is not necessary to include rows that explicitly specify incidence=0.
o Individual studies in the input file can have incidence=0 for a severity level.
o Each severity level in the input file should have incidence > 0; that is, at least one
study in the input file must have incidence > 0 for a given severity level.,
o If a severity level has zero incidence (whether explicitly indicated by row entries or
not) across all studies in an input file, all severity levels in the input file must be
renumbered to eliminate the zero incidence. For example, an input file has severity
levels 0,1,2, and 3, but there is zero incidence for level 1. In this case, all the
severity levels should be renumbered to 0, 1 (formerly level 2), and 2 (formerly level
3).
o Filtering out all instances of a severity level is the same as having incidence=0 for
that severity level for the whole input file,
o For stratification, it is acceptable to have some strata where incidence=0 for a
severity level.
~ If both concentration/dose and time effects are to be modeled, then both C and T need to be
varying in the data. (If T is constant, then a reduced model is fit so that T is dropped from the
model.)
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INPUT FILE FORMATTING CHECKLIST
~ A treatment group's incidence must sum to the treatment group size (Nsub).
o If you use CatReg's default grouping method, then a treatment group's records must
be consecutive in the dataset.
o If you use CatReg's new grouping command to select column(s) that uniquely identify
treatment groups, then the treatment group records can be in any order. See Section
5.4, "Managing Treatment Groups," for more information.
o If a Group variable is present, CatReg will use it to sort and display treatment groups.
Otherwise, CatReg determines treatment groups by reading records until the values
of Incid sum to Nsub, then starting over with the next record.
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Categorical Regression (CatReg) User Guide
DATASET AND VARIABLES TAB
5.0 DATASET AND VARIABLES TAB
After either starting a new analysis or opening a previous analysis file, CatReg opens its Analysis
Screen.
Figure 3.—CatReg Analysis Screen with loaded dataset, striped treatment groups,
and mapped variables.
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DATASET AND VARIABLES TAB
Table 1: Selections Available on the Dataset and Variables tab
Specifications
Description
Notes
Variables
Variables as defined in the datasefs
column headers.
N/A
Dose
Specify the variable for exposure
dose.
Required.
Time
Specify the variable for duration.
Exposure time in hours. Required unless all times are equal.
(Default value = 1.)
Incidence
Specify the variable for incidence of
severity level or severity range.
Required.
N
Specify the variable for number of
subjects in a treatment group.
Required.
SevLo
Specify the variable for the lowest
severity level.
Required.
SevHi
Specify the variable for the highest
severity level.
Required if response spans more than one severity level.
SevHi is required if the severity level for one or more records
is entered as a range, but not otherwise.
For example, if the severity level for a record is a range such
as level 1 to level 2, then SevLo is entered as 1 and SevHi is
entered as 2. Data records for which SevLo * SevHi are
referred as censored data. If only one severity level applies,
e.g. level 1, then SevLo = SevHi = 1
Filter
Check to display a popup window
showing values for the selected
variable.
Check the option boxes beside a
value to remove (filter) it from the
analysis.
The summary fields at the bottom of
the screen display the filtered values
in the Filtered Out text box.
Use this option to fit the exposure-response curve to a subset
of the data. One reason to filter data is to investigate how the
fit changes when certain observations are excluded.
For instance, a particular study may be suspect, and it may be
desirable to compare the parameter estimates with and
without the suspect study.
5.1 Loading an Input File
• If no dataset is loaded: Select File > New Analysis, and select the Click here to load data
button.
• If a dataset is already loaded: right-click on the Dataset and Variables window and select
Load Dataset... from the context menu.
5.2 Input Files Must be in CSV Format
Microsoft Excel spreadsheets can be used to construct an input file, but the file must be saved as
a comma-delimited file with a ".CSV" extension, rather than as an Excel file with a ".XLS"
extension.
Because CatReg assumes that data are separated by commas rather than by blank spaces,
spaces are interpreted as characters and should be avoided unless intended to be part of the
data. For instance,", MU," is distinct from ",MU,". Because R is case-sensitive, "mU," is different
from "Mu,".
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DATASET AND VARIABLES TAB
5.3 Model Variable Mapping
For the Model Variable Mapping, select from the picklists the dose-response variables that
correspond to the CatReg variables of Dose, Time, Incidence, etc. For Time and Sev Hi, [Not
Used] is a valid choice.
CatReg automatically maps variables based on the loaded input file's column names.
a. If default mapping fails for required variables, then the error message shown in Figure 4
appears:
Figure 4.—Error message when CatReg cannot map required variables.
Mapping data to variables | S3 |
Dos^, Incidence, and N are the minimum required variables to validate data.
( OK 1
In addition, CatReg does not stripe the data into groups, as shown in Figure 5. For this
case, you must manually map the required variables so CatReg can perform its default
validation.
Figure 5.—Unstriped treatment groups mean CatReg cannot validate the input file data.
Dataset an
Variables Model
nd BMD 1 Plots 1 Hypotheses
beta\data\chemxhu
Exp.
Group
Species
Target
tisIsTheDOseColum
SevLo Nsub
Incid
-
~ 1
1
MU
12S9
1.25
0
10
10
J
2
i
2
MU
1259
1.6
0
10
9
3
i
2
MU
1259
1.6
1
10
1
4
i
3
MU
1259
2
0
10
4
5
i
3
MU
1259
2
1
10
6
6
i
4
MU
1259
2.5
0
10
1
7
i
4
MU
1259
2.5
1
10
9
8
i
5
MU
1585
1.2S
0
10
7
9
i
5
MU
1585
1.25
1
10
3
10
i
6
MU
1585
1.6
0
10
3
11
i
6
MU
1585
1.6
1
10
7
-
Model Variable Mapping
Filter Data Values by Variable
Dose Time Incidence N
Sev L
Sev H
Fitt
r1 Filter
Filter =!
Filter 4
| [Required] -»|| Hours
• linen
~ Nsub
~ |[sevLo
*¦ [Not Used]
[Not Used] w || [Not Used]
' II [Not Used]
¦'II [Not Used]
If the validation succeeds, the data will be striped into treatment groups, as shown in
Figure 3. If you prefer a different grouping, right-click on the Dataset window and select
"Regroup Dataset..." from the menu. See the next section for more information on
managing treatment groups in CatReg.
b. If default mapping is successful, then a Grouping message box (see Figure 2) will appear.
Each time you change the Model Variable Mapping, and if all required variables are mapped,
then CatReg performs its validation methods (see Section 5.5).
5.4 Managing Treatment Groups
CatReg attempts to automatically sort treatment groups in the loaded input file based on column
names. Although CatReg displays sorted treatment groups in the Dataset and Variables tab,
CatReg does not change or rewrite the input file on disk.
To save the grouped data to disk, right-click on the Analysis Screen and select Save Dataset
As... You can choose to overwrite the existing input file or save the displayed grouped data to a
new file.
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5.4.1 CatReg's Default Grouping Algorithm
CatReg performs the following treatment-group checks on loaded dose-response groups to
ensure that the required variables dose, incidence, and number of subjects are mapped:
1. The data is read row by row in the order it is provided (no sorting), and the sum of the
incidence column is calculated.
2. The end of a group is determined if the sum of the incidences = number of subjects, or if the
dose value or the time (optionally, if specified) value changes.
3. The grouping validation fails if for any group the incidence does not equal the number of
subjects.
5.4.2 Custom Grouping
You can choose to proceed with CatReg's default grouping or you can specify a custom grouping.
To specify a custom grouping:
1. Right-click in the Dataset window and select Regroup Dataset... from the menu. The
Grouping dialog box appears (Figure 6).
Figure 6.—Grouping dialog box.
~ Grouping ] m || E || £3 |
Selectcolumns that uniquely identify
a group
~ Exp.
~ Group
Q Species
^ Target
| ] mg/m3
^ Hours
~ SevLo
^ Nsub
Incid
[ Cancel
Group
I [ Use Default
2. Select any number of column(s) that will uniquely identify groups in the dataset.
3. Select the Group button and CatReg will sort the groups based on the selected columns.
Behind the scenes, CatReg sorts the data provided (without altering the original .CSV file) based
on the selected columns.
• The ordering of items in the "Grouping" dialog list determines the sorting order. For example,
if "Exp" and "Group" are selected, then the data is sorted by "Exp" and then by "Group" for
each value of "Exp". If all the values in a column are numeric, then sorting is done
numerically. If the columns contain any non-numeric values, then sorting (including numbers)
is done alphabetically
• CatReg reads the data row by row in the sorted order and calculates the sum of the incidence
column.
• CatReg determines the end of a group if any values in the selected grouping columns
change.
• Grouping validation fails if, for any group, the incidence total does not equal the number of
subjects.
To save the regrouped data to disk, right-click on the Analysis Screen and select Save Dataset
As... You can choose to overwrite the existing input file or save the regrouped data to a new file.
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5.5 CatReg's Validation Criteria After Custom Grouping
CatReg uses the following criteria to validate the data if the grouping method is changed:
• Grouping validation. The data is checked according to the grouping algorithm used. See
Section 5.4.1, "CatReg's Default Grouping Algorithm," for more information.
• Numeric validation. Certain variables are required to be numeric. CatReg validates that the
mapped columns in the data corresponding to these variables do not contain any non-
numeric values.
If validation fails:
• A separate dialog box appears for each failed validation method (grouping or numeric
validation) with a list of the row numbers that failed validation (see Figure 7).
Figure 7.—Example of failed validation error message.
Data Error S3
Incidence does not sum to N on line 31,68,104,138
OK
• A red "X" icon appears on the tool strip at the bottom left of the Analysis Screen with the
message "Validation Failed" (see Figure 8). Click the icon to redisplay any failed validation
messages.
Figure 8.—Validation Failed error.
5.6 Filtering Data Values by Variable
From the dropdown lists, select the dose-response group data variables whose records you want
to be removed (filtered) from the analysis, without removing them from the input file. If no variable
is selected, "[Not Used]" appears. (See Figure 3.)
This option is used to fit the exposure-response curve to a subset of the data. One reason to filter
data is to investigate how the fit changes when certain observations are excluded. For instance, a
particular study may be suspect, and it may be desirable to compare the parameter estimates
with and without the suspect study.
To filter data, select the variable of interest; CatReg will display all the values for that variable.
Tick the box for the value you want to be removed, and select OK.
This option is also useful for scanning a particular field to see what values have been observed.
Unexpected values may indicate a problem with the input file (e.g., errors in data entry).
5.7 Minimum Data Requirements
Certain minimal data requirements need to be satisfied to estimate the categorical regression
model:
• There needs to be at least one response in each severity category. For example, if the input
file contains a severity 4, then severity categories 3, 2, and 1 should each contain responses.
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If any categories in the input file contain no non-zero response, then categories will need to
be combined. Refer to Section 5.9, "Data Requirements: Severity Levels," for more
information.
• If both concentration/dose and time effects are to be modeled, then both C and 7 need to be
varying in the data. If 7 is constant, then a reduced model is fit so that 7 is dropped from the
model.
There are additional technical limitations on the complexity of the model relative to the data:
• The most obvious is that the model cannot include more parameters than the number of
independent observations.
• Less obvious problems sometimes occur if certain variables are redundant. CatReg will
display R error messages if there are redundant variables in the model. This means that at
least one variable in the model can be expressed in terms of the others. Reducing the
number of stratification variables usually will solve the problem.
• Sometimes, R will return a "failure to converge" message. This is an indication that there are
too many variables in the model, and that the model needs to be simplified. This problem
relates to the number of variables needed to completely isolate the different severity
categories. The solution is to remove one or more variables from the model.
5.8 Data Requirements: Outcomes
When the source document for an experiment does not report the outcome for individual subjects,
or otherwise report the incidence of different health effects, the data may not be suitable for
CatReg. For example, a report of "mild" pathology for a treatment group might mean that a few or
many in the group manifested that response or that the "mild" response was the most common,
with both lesser and more severe effects also present in the group. In either case, there is not
sufficient information to divide a treatment group into incidence of severity categories.
It is sometimes reasonable to represent a health outcome measured on a continuous scale as
categorical data. Continuous data from acute studies, such as enzyme activities, tidal volume,
respiratory rate, blood pressure, etc., often are reported as a mean value, with a measure of
dispersion, such as the standard error or standard deviation, for each treatment group. To convert
these data to severity levels for CatReg, each severity level needs to be equated to an interval of
values on the continuous scale.
For example, if the full range of responses is 0 to 100, the user might decide to classify outcomes
0 to 20 as "no effect", 21 to 40 as a "mild adverse effect", 41 to 65 a "moderate adverse effect",
and 66 to 100 "severe effect". The mean for a treatment group falls into a single severity level, but
some of the individual responses of subjects in the group may have been dispersed over adjacent
severity levels.
Knowing the mean and standard deviation (or standard error that can be converted to a standard
deviation by multiplying by the square root of the size of the treatment group) and assuming a
distribution for the continuous data (e.g., normal), an estimate can be made of the incidence at
each severity level (see Appendix A for details). The estimated incidence figures need not be
whole numbers, but must still sum to the total group size. Incidence estimation is not possible if
the mean is reported without a measure of dispersion.
5.9 Data Requirements: Severity Levels
The same category system of severity levels must be used for all data in an input file.
Considerable toxicological judgment may be required for classification of various health effects
into severity levels and for achieving comparability across experiments. When that cannot be
done for all the studies of interest, it may be necessary to group the studies into more than one
input file. Classification judgments must be made systematically according to documented
criteria.
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The minimum number of severity levels is two (severity levels coded as 0 and 1, corresponding to
absence or presence of an effect).
Suggested severity categories for a three-category classification are "no adverse effect", "adverse
effect", and "lethal effect", coded as severity levels 0,1,2, respectively.
A four-category scheme might be "no adverse effect", "mild adverse effect", "moderate/severe
effect", and "lethal effect", coded as 0, 1, 2, 3, respectively.
In some toxicology studies, it may not be possible to score all response data completely.
Consider a four-category scoring system in which 0 = "no observable effect," 1 = "mild effect," 2 =
"moderate effect," and 3 = "severe effect." Published data from an animal mortality study may not
include nonlethal outcomes; therefore, the response score for an animal that survives is
uncertain, or "censored." That score is known to be less than 3, but it is not known whether the
score should be 0, 1, or 2. Such an observation is said to be "interval censored." Another situation
where the response score may be interval censored is in combining data from experiments with
different endpoints. For some endpoints, it may not be clear from the toxicology whether a
specific response should be considered "mild" or "moderate." An interval censored analysis
simply could report that the response is either 1 or 2, but the specific score is not known.
The ability to include partial information about the ordinal scores is one of the important features
of CatReg. CatReg incorporates this type of partial information in an interval-censored analysis.
In general, interval censoring occurs if the response is known only to lie in an interval of potential
values. Such intervals are specified in CatReg by supplying the lower and upper limits of the
known range for each observation. Suggested codes to indicate species and sex in an input file
are provided in Table 2.
Table 2: Recommended Codes for Species and Sex
Species
Code
Sex
Code
Human
HU
Female
F
Rat
RT
Male
M
Mouse
MU
Both sexes
B
Rabbit
RB
N/A
N/A
Guinea pig
GP
N/A
N/A
Each column of the user input file is referred to as a data field, with the first record (row) being
variable names and all subsequent rows containing data for the variables. CatReg requires
information for four- six data fields, depending on the data. The names of these data fields and
the corresponding default variable names that CatReg looks for in the user input file are shown in
Table 3. For example, "cone" refers to a data field for exposure concentration and CatReg looks
for the variable name "mg/m3" to identify that field, unless the variable name has been changed
from the default. The default variable name for "cone" might be changed, for example, if the
concentration or dose used in experiments is different from milligrams per cubic meter.
Using the default variable names in Table 3 as an example, the user input file must include data
in each record (beyond the first that contains variable names) for variables "mg/m3", "Nsub",
"Incid", and "SevLo". Data are also required for the variable "Hours" unless all exposure duration
times are equal, in which case it can be omitted (CatReg uses 1 as the default value in that case)
The variable SevHi is required if the severity level for one or more records is entered as a range,
but not otherwise. For example, if the severity level for a record is a range such as level 1 to level
2, then SevLo is entered as 1 and SevHi is entered as 2. Data records for which SevLo £ SevHi
are referred as censored data. If only one severity level applies, e.g. level 1, then SevLo = SevHi
= 1.
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Table 3: Data Fields That May Be Required
Data Field
Default Variable
Name
Description
Cone
mg/m3
Exposure concentration or dose. Use of the human equivalent concentration
or dose is recommended. Always required.
Time
Hours
Exposure time in hours. Required unless all times are equal. (Default value
= 1.)
Nsub
Nsub
Number of subjects in a treatment group. Always required.
Incid
Incid
Incidence of severity level (or severity range) for the record. Always
required.
Loscore
SevLo
Lowest severity level for the record. Always required.
Hiscore
SevHi
Highest severity level for the record. Required if response spans more than
one severity level.
A separate record is entered for each severity level (or range of severity levels in the case of
censored data) observed in a treatment group. For example, if the user determines three severity
classifications for health effects, denoted as 0, 1, and 2, then the outcome for a treatment group
is represented as an incidence for each severity level that is observed. To illustrate, a treatment
group of size 10 might result in 3 subjects being classified at severity level 0, 4 at severity level 1,
and 3 at severity level 2, which would require three consecutive records (consecutive rows of
data) in the input file. The records for the treatment group must not only be consecutive but their
incidence (values of Incid) must sum to the treatment group size (Nsub).
A severity level with no observations need not be entered as a separate record. For example, if a
treatment group of size 10 had 6 subjects classified at severity level 0 and 4 at severity level 1,
then only two records would be required to enter the data. The value of Nsub would be 10 in both
records; the value of Incid would be 6 in one record and 4 in the other.
Variables in addition to those in Table 3 may be added to the input file at the user's discretion,
either for use in execution of CatReg or to facilitate organizing and keeping track of the data. The
user can refer to them in the same manner as variables in the required fields when using CatReg
options. For example, one might want to add "strain" as a variable to distinguish between two
strains of mice and have CatReg test whether their exposure-response curves are significantly
different in some respect, or add "Ref.id" to record the source of the data, even if it is not used
during the execution of CatReg.
5.10 Data Issues that Trigger Error Messages
CatReg attempts to display any R error messages in plain language.
Sometimes "NaN" (not a number) is given for some of the model deviance iterations. It is
displayed when CatReg is searching for the solutions to the parameter estimates in early
iterations of the program. CatReg tries to compute the deviances for those solutions and finds
they are undefined (NaN). Once a solution is determined for runs like this, the following warning
message may appear: nas produced in: log (likes) (Na means "not available").
The message Warning : Gamma hit its maximum bound! ! ! may OCCur when the
parameter y (gamma) is being estimated. The smallest positive value of concentration or dose in
an input file is used as a practical boundary on gamma. The estimate shown for gamma, and the
other parameters in the summary table of estimates, are not maximum likelihood in that case, and
the user is advised to consider the setup option that assumes the background risk is zero.
An error message occurs if a coefficient of concentration/dose or time is negative, e.g., Error:
TIME is negative! Estimates of coefficient parameters do not satisfy
non-negativity constraint on the parameters. A negative estimate is
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evidence of no effect. This run will terminate. The user needs to modify the
run. The data may be indicating there is no effect or there may just be too many parameters in
the model.
Similarly, the estimates of severity intercepts may violate the order constraint, resulting in a
message Such as Sev 1 < Sev 2! Estimates of severity parameters do not satisfy
constraint on order of parameters. Incorrectly ordered severity
estimates is evidence of too many severity levels in the data. This run
will terminate. The user needs to modify the dose-response group data by, for example,
combining severity levels.
Most error messages at this point in a CatReg run result from an attempt to calculate too many
parameters with too little data. This may generate error and /or warning messages from the
optimization routine (used to find maximum likelihood estimates), such as: Error in
optim(transcoefs, fn = hfdeviance, gr = hfdeviance.grad, method =
"BFGS",: initial value in vmmin is not finite
In general, when errors occur during the calculation of parameter estimates, the user should
change the link function if the cloglog link function was used or simplify the model and run it
again. The model can be simplified by reducing or omitting stratification of parameters. If Model 2
(unrestricted cumulative model) is in use, the user can consider switching to Model 1 (cumulative
odds model). Additionally, one can try different options for the scale of the explanatory variables
Dose and Time. Dumped means that the CatReg analysis has been terminated because a
solution could not be computed.
5.11 Input File Examples
Three varied examples of input files follow, described as experimental results for hypothetical
chemicals named chemx, chemy, and chemz.
The data were generated by simulation, except the data for what is being called chemy that were
constructed from actual rodent experiments.
• The input file for chemx is an example of four experiments, one each on the four
combinations of species (RT and MU) and target organs (C and L).
• The input file for chemy has a more complicated structure and illustrates how a toxicologist
might determine the severity levels.
• The input file for chemz is an illustration of converting a continuous response to severity
categories for use in CatReg.
5.11.1 chemx.csv
Table 4 displays the first part of the input file for chemx.csv. Four experiments were conducted
under identical exposure conditions, each consisting of 10 observations at each combination of
four concentrations/dosages (mg/m3) and four times (Hours), for a total of 64 treatment groups.
The dosages are 1259, 1585, 2000, and 2512 mg/m3; the times are 1.25, 1.6, 2.0, and 2.5 hours.
There are three severity levels: no adverse effect (SevLo = 0), mild adverse effect (SevLo = 1),
moderate/severe effect (SevLo = 2). Two experiments are on mice (Species = MU) and two are
on rats (Species = RT), with one of the two experiments on each species reporting effects on the
central nervous system (Target = C) and the other reporting effects on the liver (Target = L).
"Exp" denotes an experiment number, "Group" the treatment group within the experiment, "Nsub"
the number of subjects in the treatment group, and "Incid" the incidence in the treatment group of
the severity level (SevLo) shown in the record (row of data).
The names of data fields are user-specified, but the data fields associated with the Table 4
variable names mg/m3, SevLo, Nsub, and Incid are required. CatReg displays an error message
(see Figure 4 in Section 5.3, "Model Variable Mapping") if any of them is missing. In this example,
the exposure times vary so the variable Hours is included. SevHi would have been included as a
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variable if the severity level had been censored for one or more records (spanned more than one
severity level). For example, to make the 10 subjects in the first record (row of data) classified as
severity level 0-1, the variable SevHi would be added to the input file and the first record would
remain unchanged except for SevHi = 1. In that case, subsequent records that are not censored
would be given the same value for SevLo and SevHi. For example, the second record indicates
that 9 subjects were classified at severity level 0. If SevHi were included as a variable, then SevHi
would be set to 9 for that record, making SevLo = SevHi = 9.
It may be noted that variables Exp., Group, Species, and Target are names created by the user.
Species and Target were included in this case to be able to distinguish between species and
target organ in the data analysis, but other names could be used in their place. Exp. and Group
were added by the user to facilitate record keeping. The variable Group is not required by CatReg
but records for the same treatment group must be together in the input file, all with the common
value of Nsub and values of Incid that sum to Nsub. Adding a variable such as Group provides a
convenient check of the data for the user. CatReg determines treatment groups by reading
records until the values of Incid sum to Nsub, then starting over with the next record.
Table 4: Part of the Input File CHEMX.CSV
Exp.
Group
Species
Target
mg/m3
Hours
SevLo
Nsub
Incid
1
1
MU
C
1259
1.25
0
10
10
1
2
MU
C
1259
1.6
0
10
9
1
2
MU
c
1259
1.6
1
10
1
1
3
MU
c
1259
2
0
10
4
1
3
MU
c
1259
2
1
10
6
1
4
MU
c
1259
2.5
0
10
1
1
4
MU
c
1259
2.5
1
10
9
1
5
MU
c
1585
1.25
0
10
7
1
5
MU
c
1585
1.25
1
10
3
1
6
MU
c
1585
1.6
0
10
3
1
6
MU
c
1585
1.6
1
10
7
1
7
MU
c
1585
2
0
10
1
1
7
MU
c
1585
2
1
10
7
1
7
MU
c
1585
2
2
10
2
1
8
MU
c
1585
2.5
1
10
4
1
8
MU
c
1585
2.5
2
10
6
1
9
MU
c
2000
1.25
0
10
7
1
9
MU
c
2000
1.25
1
10
3
1
10
MU
c
2000
1.6
0
10
2
1
10
MU
c
2000
1.6
1
10
5
1
10
MU
c
2000
1.6
2
10
3
1
11
MU
c
2000
2
1
10
6
1
11
MU
c
2000
2
2
10
4
1
12
MU
c
2000
2.5
1
10
4
1
12
MU
c
2000
2.5
2
10
6
1
13
MU
c
2512
1.25
1
10
9
1
13
MU
c
2512
1.25
2
10
1
1
14
MU
c
2512
1.6
1
10
5
1
14
MU
c
2512
1.6
2
10
5
1
15
MU
c
2512
2
1
10
2
1
15
MU
c
2512
2
2
10
8
1
16
MU
c
2512
2.5
2
10
10
2
1
RT
c
1259
1.25
0
10
10
2
2
RT
c
1259
1.6
0
10
9
2
2
RT
c
1259
1.6
1
10
1
2
3
RT
c
1259
2
0
10
9
2
3
RT
c
1259
2
1
10
1
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5.11.2 chemy.csv
Table 5 is part of a larger input file that illustrates a more elaborate coding system and some
other features not included in the preceding example, e.g. censoring, and provides a realistic
example for discussion of toxicological judgment in severity classification. The available studies
varied on the organ sites and endpoints examined and a four-category system of severity levels
was implemented: no adverse effect (0), mild adverse effect (1), moderate/severe effect (2), and
lethal effect (3).
Again, CatReg expects to find variables mg/m3, Nsub, Incid, and SevLo, at a minimum, and
Hours as well, if exposure time varies, and SevHi if there are any censored data.
Also (again) notice that a separate record is required for each different severity (or range of
severity in the case of censored data) in a treatment group, and that Incid sums to Nsub for
records in the same treatment group.
"Ref.id" (reference identification) is a number assigned by the user to the source of the
information in the record. In the example, Ref.id = 20938 is the source of the material used to
construct the first 10 data records shown.
"Exp." identifies experiments within each Ref.id, numbered sequentially from 1; "Group" numbers
treatment groups within each experiment (i.e., subjects alike with respect to all methods and
materials variables); and "Nsub" is the number of subjects in each group.
A separate record (row) is entered for each severity level (or range of severity levels) in a
treatment group. "Marker" just numbers the data records. The severity levels are entered under
"SevLo", the lowest possible severity level for the record, and "SevHi", the highest severity level
for the record.
When SevLo £ SevHi, "y" is entered under "Censored"; "n" when SevLo = SevHi. In Table 5, the
variable Censored has been added by the user to readily distinguish between records with a
single severity level and those with a range of severity levels; it is not required by CatReg. The
variable BestNum also has been added by the user to indicate the most likely severity level when
SevLo £ SevHi; otherwise, the common value for SevLo and SevHi is entered for BestNum. To
use the scores in the BestNum column, the default variable names SevLo and SevHi must both
be changed to BestNum by editing the names in the input file.
The first reference (Ref.id = 20938) reported two experiments, one with mice (Species = MU) and
one with rats (Species = RT), with both sexes in both experiments (Sex = B).
There is a user-defined variable coded to indicate the target organ (e.g., Target = Resp) and the
primary endpoint (Endpoint = Lethality), both of which were the same for both experiments.
The first group (Group = 1) under the first experiment (Exp = 1) is for rats (Species = RT) of both
sexes (Sex = B) exposed at a concentration or dose of 330 mg/m3 (mg/m3 = 330) for 6 h (Hours
= 6). There were 26 rats at risk (Nsub = 26). The user-defined severity classification was the
same for all 26: severity level 0 to 2 (SevLo = 0, SevHi = 2), with severity level 1 the best guess
for a single severity level (BestNum = 1). Because the results variables were the same for all 26
subjects, only one record is needed to record the data for the whole treatment group.
The sixth and seventh records (Markers 6 and 7) are for the first treatment group (Group = 1) of
the second experiment (Exp = 2) in Ref.id = 20938. Mice (Species = MU) of both sexes (Sex = B)
were exposed to 360 mg/m3 (mg/m3 = 360) for 6 h (Hours = 6). Two records are required
because there are two distinct severity classifications: one for 23 subjects (Incid = 23) with
severity level 1 to 2 (SevLo = 1, SevHi = 2) and severity level 2 the best guess (BestNum = 2)
and the other with three subjects at severity level 3 (SevLo = 3, SevHi = 3) and BestNum = 3. Six
records in the example have different values for SevLo and SevHi, as indicated by Censored = y.
In Ref.id = 20938 of Table 5, the effect severity for dosages at which no subjects died were
censored 0 to 2 (e.g., Marker 1) for no adverse effects to severe adverse effects, because the
effects were unknown. Survivors from groups in which some subjects died (e.g., Marker 2) were
assumed to have suffered adverse effects and were censored 1 to 2 because the effects could
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have been mild to severe. Survivors from groups in which most of the subjects died (e.g., Marker
5) were assumed to have suffered severe effects.
For the 360-, 390-, 420-, and 460-mg/m3 ChemY exposures, there is one record for the number
of subjects exhibiting lethal effects, and another for the number of subjects exhibiting nonlethal
effects. Only one record was made for exposure to 330 mg/m3, because all subjects were
assumed to exhibit effects of severity 0 to 2, and one record was made for exposure to 500
mg/m3, because all subjects died (severity 3).
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Table 5: Part of the Input File CHEMY.CSV
Marker
Ref.id
Exp
Group
Species
Sex
mg/m3
Hours
Target
Endpoint
Nsub
Incid
BestNum
SevLo
SevHi
Censored
1
20938
1
1
RT
B
330
6
Resp
Lethality
26
26
1
0
2
y
2
20938
1
2
RT
B
390
6
Resp
Lethality
26
20
2
1
2
y
3
20938
1
2
RT
B
390
6
Resp
Lethality
26
6
3
3
3
n
4
20938
1
3
RT
B
460
6
Resp
Lethality
26
23
3
3
3
n
5
20938
1
3
RT
B
460
6
Resp
Lethality
26
3
2
2
2
n
6
20938
2
1
MU
B
360
6
Resp
Lethality
26
23
2
1
2
y
7
20938
2
1
MU
B
360
6
Resp
Lethality
26
3
3
3
3
n
8
20938
2
2
MU
B
420
6
Resp
Lethality
26
13
3
3
3
n
9
20938
2
2
MU
B
420
6
Resp
Lethality
26
13
2
1
2
y
10
20938
2
3
MU
B
500
6
Resp
Lethality
26
26
3
3
3
n
11
61831
1
1
RT
M
14
4
Resp
N lavage
12
12
0
0
0
n
12
61831
1
2
RT
M
278
4
Resp
N lavage
12
12
1
0
1
y
13
61831
1
3
RT
M
556
4
Resp
N lavage
12
12
2
1
2
y
Notes:
Marker - Record number.
Ref. id - Source identifier.
Exp - Experiment number within a source.
Group - Treatment group number (within an experiment)
Species - Species.
Sex - Sex.
mg/m3 - Exposure concentration or dose.
Hours - Exposure time.
Target - Target organ.
Page 26 of 75
Endpoint - Toxic endpoint.
Nsub - Number of subjects in treatment group.
Incid - Number of animals responding.
BestNum - Analyst's best estimate of severity category for censored
data; same as SevLo and SevHi for noncensored data.
SevLo - Lowest applicable severity level.
SevHi - Highest applicable severity level.
Censored - Severity level reported as a range if "y".
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Categorical Regression (CatReg) User Guide
DATASET AND VARIABLES TAB
Ref.id = 61831 of the example illustrates a case where health effects are described for each
treatment group, for which the severity levels can be decided, but incidence of the severity levels
is unknown. These data are not suitable for CatReg because correct values for Incid are
unknown. Nevertheless, one might want to test the sensitivity of the results of CatReg to different
assumptions regarding the incidence values and it provides a useful example for discussion of
deciding severity levels. It is included here primarily as an example of deciding severity levels
from toxicological data. The data are from a study in which groups of 12 rats were exposed to 0,
14, 278, and 556 mg/m3 ChemY for 4 h. Incid = 12 is used in the example for illustration, which is
equivalent to assuming that all 12 rats in each treatment group have the same severity
classification shown for the group as a whole.
Data from this study were assigned to severity levels as follows. Groups of four rats were killed at
1, 20, and 44 h after 4-h exposure for the examination of biochemical indicators of injury and
inflammatory response in the respiratory tract. Nasal lavage fluid was examined for lactate
dehydrogenase (LDH), alkaline phosphatase (ALP), protein, and number of nucleated cells.
Bronchoalveolar lavage (BAL) fluid was examined for activities of LDH, ALP, and D-glutamyl
transpeptidase. All measurements were reported as means, plus or minus standard deviations.
No changes in any parameters were noted among rats exposed to 14 mg/m3 ChemY. The only
parameter significantly different from controls in nasal lavage fluid at all post-exposure time
periods was increased cellularity, which was significant at the 556-mg/m3 exposure. In BAL fluid,
LDH activity was elevated at 44 h post-exposure, and ALP activity was significantly decreased at
20 and 44 h after the 278-mg/m3 exposure. At all post-exposure times for the 556-mg/m3
exposure, protein concentration and LDH activity were elevated, but ALP was decreased.
Because a number of biochemical and cellular parameters were measured at several post-
exposure periods, significant changes were considered to be adverse only if the changes were
still significant at the last post-exposure measurement. In other words, reversible changes were
classified as no-observed-adverse effects. Table 6 shows how these effects were categorized
using the four-category severity scheme. Because no changes were noted after exposure to
14mg/m3 ChemY, effects were coded 0 for no adverse effect. Effects at the 278-mg/m3 ChemY
exposure were estimated to range from severity 0 to 1 (no adverse effect to mild adverse effect)
because significant changes in two biochemical parameters were observed. The adversity of
those changes was uncertain but assumed to be less than severe. Effects caused by exposure to
556 mg/m3 ChemY were estimated to range from severity 1 to 2 (mild to severe adverse effect)
because of changes in the activities of several enzymes and nasal cytopathology. No deaths
occurred during this study, so category 3 (lethality) was not used.
Table 6: Example of Severity Categorization for Nonlethal Effects
Exposure
Concentration or
Dose
(mg/m3 ChemY)
Statistically Significant
Effects Reported
Severity
Score
Censored
14
None
0
No
2748
tLDH, lALP
0-1
Yes
556
TProtein, tLDH, 4-ALP,
tcellularity, nasal cytopathology
1-2
Yes
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5.11.3 chemz.csv
An artificial example of how continuous data might be coded is displayed in Table 7. It is
assumed that the standard error was included along with the mean for each treatment group. In
this case, the incidence of each severity in a group was estimated from the mean and standard
error, which produces fractional subjects. The method to estimate incidence at each severity from
group means and a measure of variability is described in Appendix A.
Male rats in treatment groups of 10 each were exposed to a toxicant at various dosages and
times. Adverse effects occurred in the respiratory tract, with severity indicated by lung weight. For
illustration, a four-category severity classification is used, with lung weight 0 to 20 classified as
"no effect", 21 to 40 as "mild adverse effect", 41 to 65 as "moderate adverse effect", and 66 to
100 as "severe effect". The mean for a treatment group falls into a single severity level, but some
of the individual responses of subjects in the group may have been dispersed over adjacent
severity levels.
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Table 7: Part of the Input File CHEMZ.CSV
Marker
Ref.id
Exp
Group
Species
Sex
mg/m3
Hours
Target
Endpoint
Nsub
Incid
BestNum
SevLo
SevHi
Censored
1
1
1
1
RT
M
330
2
Resp
Lung wt
10
1
0
0
0
n
2
1
1
2
RT
M
360
2
Resp
Lung wt
10
1
1
1
1
n
3
1
1
3
RT
M
390
2
Resp
Lung wt
10
1
1
1
2
y
4
1
1
4
RT
M
410
2
Resp
Lung wt
10
1
2
2
2
n
5
1
1
5
RT
M
460
2
Resp
Lung wt
10
1
2
2
2
n
6
1
2
1
RT
M
460
1
Resp
Lung wt
10
1
1
1
1
n
7
1
2
2
RT
M
510
1
Resp
Lung wt
10
1
1
1
1
n
8
1
2
3
RT
M
560
1
Resp
Lung wt
10
1
2
2
2
n
9
1
2
4
RT
M
610
1
Resp
Lung wt
10
1
2
2
2
n
10
1
3
1
RT
M
560
0.5
Resp
Lung wt
10
1
1
1
1
n
11
1
3
2
RT
M
610
0.5
Resp
Lung wt
10
1
1
1
2
y
12
1
3
3
RT
M
660
0.5
Resp
Lung wt
10
1
2
2
2
n
13
1
3
4
RT
M
710
0.5
Resp
Lung wt
10
1
2
2
2
n
14
2
1
1
RT
M
330
2
Resp
Lung wt
10
7.1
0
0
0
n
15
2
1
1
RT
M
330
2
Resp
Lung wt
10
2.9
0
1
1
n
16
2
1
2
RT
M
360
2
Resp
Lung wt
10
2.9
1
0
0
n
17
2
1
2
RT
M
360
2
Resp
Lung wt
10
7.1
1
1
1
n
18
2
1
3
RT
M
390
2
Resp
Lung wt
10
4.9
1
1
1
n
19
2
1
3
RT
M
390
2
Resp
Lung wt
10
5.1
1
2
2
n
20
2
1
4
RT
M
410
2
Resp
Lung wt
10
1.2
1
1
1
n
21
2
1
4
RT
M
410
2
Resp
Lung wt
10
8.8
2
2
2
n
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Categorical Regression (CatReg) User Guide
MODEL AND BMD TAB
6.0 MODEL AND BMD TAB
Figure 9.—Model and BMD tab.
o-' Categorical Regression Version 3.0.1.1 Beta - [Analysis Screen]
U^Uajwaall
Dataset and Variables] Model and BMD [ Plots j Hypotheses j
BMD Specifications
Risk: Extra
BMR: 0.1
Confidence Level (%): 95
Model Specifications
| Cumulative Odds ' |
Assume Zero Background Response:
Worst Case Analysis:
Stratification
Model variables can be stratified by one
Stratify... Intercept Dose
by
then by
then by '
et variables.
Clustering
Selected variables will be used to define groups of coirelated data.
Summary of Run Options
Output File: C:\usepa\CatReg\Data\Oiemx.otx
filtered Out:
Clustered:
Stratified:
Censoring:
"Q
Run Analysis
Right-click on Dataset control for additional option(s).
The Model and BMD tab provides further opportunity to specify the options that CatReg will use
to calculate an exposure-response curve. All you need to do is select or deselect option boxes or
buttons, but having an understanding of clustering and stratification will help inform your
selections.
See also Appendix C. for technical background discussions on exposure-response models and
extra risk concentration (ERC).
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MODEL AND BMD TAB
6.1 Setting BMD Specifications for an Analysis
Table 8: BMD Specification Options on the Model and BMD Tab
Specifications
Description
Risk
Extra is the default risk type. 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).
BMR
The response, generally expressed as in excess of background, at which a benchmark dose or
concentration is desired.
User input value (or default of .1000). must be a number >0 and <1.
Confidence Level (%)
The confidence level (default 0.95) associated with the statistical lower bound of BMD (BMDL)
calculation.
Confidence level must be a number >0 and <1.
Time
Exposure duration, in hours
6.2 Setting Model Specifications for an Analysis
Table 9: Model Specification Options on the Model and BMD Tab
Specifications
Available Options
Notes
Link Function
Options are:
• Logit (default)
• Probit
• Cloglog
For more information, see page 34 .
Model Form
Options are:
• Cumulative Odds (default)
• Unrestricted Odds
For more information, see page 35.
Zero Background Response
Options are:
• Yes (default)
• No
For more information, see page 36.
Dose
Options are:
• Log10 (default)
• Linear
N/A
Time
Options are:
• Log10 (default)
• Linear
N/A
Worst Case Analysis
Options are:
• Yes
• No (default)
For more information, see page 37.
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6.3 Running an Analysis
After specifying the parameters and options, select the Run Analysis button. CatReg will execute
the analysis according to your selections.
CatReg opens a new window to display the output file specified in the Output File box at the
bottom of the Analysis Screen. In the following screenshot, the output file's name is
ChemxHu.otx.
Figure 10.—CatReg Output File.
~ CatReg v3.0.1.8 - Beta
File Preferences Help
Dataset and Variables Model
Dataset c:\users\mbrown06\(
mg/m3
C:\Users\m b rown06\C atReg_3018Beta\Data\Ch emxH u. otx
File Edit Preferences
CatReg Version 3.0.1.5
Run Date: Sunday, 21 May 2017 at 22:24:21
Source data file: c:/users/mbrown06/catreg_3018beta/data/chemxhu.c
Type of analysis: Censored
Input file : c:/users/mbrown06/catreg_3018beta/data/chemxhu.csv
Model Variable Mapping:
Dose: mg/m3
Time: Hours
Incidence: Incid
N: Nsub
Sev Lo: SevLo
Sev Hi: [Not Used]
Model Specifications:
Model: cumulative odds model
Link: logit
Assume Zero Background Response: Y
Worst Case Analysis: n
Scale for Concentration: loglO( mg/m3 )
Scale for Duration: loglO( Hours )
BMD Specifications:
Risk: extra risk
BMR: 0.1
Confidence Level(%): 95
Time: 0.5 12 4 8
HDI0E
Summary of Run Options
Output Rle: | C:\UsersVr
Rltered Out: 1
Clustered: |
Filtered data:
Clustering :
Filter 4
[Not Used]
JLZJ
6.4 About the CatReg Output Files
CatReg's output file provides summary information such as the name of the input file, the setup
options used, the table of coefficient estimates, an analysis of deviance table, and extra risk
concentrations (ERCs) with upper and lower confidence bounds at exposure times of 1, 4, 8, and
24 hours.
CatReg's output file has the extension *.otx. Figure 24 in Section 9.0 contains a sample output
file.
The .otx output file is a simple text file that CatReg automatically saves to the directory holding
the currently loaded input file. From the text file window, you can save, print, edit, and otherwise
manipulate the file's text. You can also set the file's font and line wrapping preferences.
CatReg by default suggests a name for the output file based on the currently loaded input file's
name. For example, if the input file is named "ChemZ.csv," then CatReg will save the output file
as "ChemZ.otx".
CatReg also saves the output results in .csv format to the directory holding the currently loaded
input file. The .csv file name uses the input file's name and appends "_summary".
6.5 Changing the Output File's Name and Location
To change the name and location of the output file, select the ... button beside the Output File
field in the Analysis Screen's Summary of Run Options area. CatReg displays a Save As dialog
box. Use this dialog to define a new location and filename.
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By default, CatReg saves output files to the directory holding the currently loaded input file.
6.6 Stratifying
On the Model and BMD tab, select the variable(s) from the picklists to stratify by Intercept, Dose,
or Time.
Stratification is a way of allowing one or more of the regression parameters (intercept, coefficient
for concentration or dose ("mg/m3"), and coefficient of time ("Hours") to change when a specified
variable changes value. For example, instead of assuming a common intercept parameter for
three different species, stratification of intercept on the variable Species adds two more intercept
parameters so there is one for each species. Stratification of the intercept on three species
defines three subgroups (strata) of data, one for each species. The same parameter can be
stratified on more than one variable. For example, the intercept might be stratified on both
Species and Target. If there are two target organs for each of three species, then there are six
strata, each corresponding to a distinct combination of species and target. CatReg will provide six
intercept estimates, one for each species-target combination. To stratify the intercept on Species
and Target, enter those two variable names in response to CatReg's query on whether to stratify
the intercept (enter both on the same line, separated by a space, or on separate lines). In the
same way, CatReg queries for variables on which to stratify the coefficients of dose and time.
Stratification is often conducted to test if the value of a regression parameter (e.g., the intercept)
is the same for two or more values of a variable (e.g., Species). The user typically wants to
produce an exposure-response curve that is suitably "accurate" by taking account of different
parameter values that may occur between species, endpoints, etc., but that achieves that
objective as simply as possible (i.e., with the minimal number of parameters). Stratification can be
a way of implementing toxicological considerations, as the following example illustrates.
Suppose the data contain mortality results from experiments using rats and mice. The basic
explanatory variables are atmospheric toxicant dose and time of exposure. The response score
equals 0 for surviving animals and 1 for animals that died. In this case, the maximum severity
score is S = 1. Assuming C and T enter the model logarithmically, the basic model has the form
L[Pr(Y = 1 |C,T)] = on +/3Hogio(C)+/32*logio(T),
where L is the link function (the inverse function of H in Appendix C. Eq. 1a). Logarithmic scaling
is typical when the explanatory variables range over two or more factors of 10. Because of
different rates of respiration and metabolism among rats and mice, it may be reasonable to
assume the internal dose for rats should be rescaled compared to that of mice. One possibility is
to assume that a dose of C for rats is equivalent to a dose of kC for mice, where k is common to
all mice in the study. Then, for the mice,
L[Pr(Y = 1 |C,T)] = on + /3r logio (kC) + /32* logio (7).
= [en + /3r log-io (k)] + /3r log-io (C) + /32* log-io (T).
This shows that the mice, in effect, have a different intercept than do the rats, namely criMU = criRT
+ /3r logio(Ar), where MU and RT refer to mouse and rat parameters, respectively. By stratifying
the intercept parameter, the data are allowed to determine the estimate of the conversion factor k.
Whether k is significant would be determined by testing criMU = criRT. Refer to Section 7.1 for an
explanation of how to perform hypothesis testing in CatReg.1
6.7 Clustering
On the Model and BMD tab, select the variable(s) from the list boxes that are part of a cluster.
The list boxes display the variables for selection.
1 Note in particular the discussion at the end of that section on the testing of the intercept parameter
estimate, which is done slightly differently from other parameters because the intercept parameter
estimates are reported in CatReg as incremental changes from a reference intercept parameter estimate.
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MODEL AND BMD TAB
An input file may consist of subsamples of data from common sources that causes them to be
more similar to each other than to observations from another source.
Example 1: Suppose there are reports from three different "identical" experiments conducted at
three different laboratories. The data from each laboratory may be considered a cluster because
of the following reasons:
• There are likely some differences among laboratories in the way subjects were fed, their
animal suppliers, the age of the subjects, the conditions under which the subjects have been
taken off the study, the protocol for histopathology (or just different histopathologists), etc.
• The differences can be viewed as random effects by thinking of the specific laboratories as a
random sample among a population of laboratories.
• The magnitude of the differences among the specific laboratories may vary.
• The laboratories are reasonably homogeneous (i.e., there is not one or more of them that is
unreliable or consistently different in some way from the others).
Example 2: Suppose that an experiment is conducted wherein pregnant female rats are exposed
to a toxic substance. Each rat gives birth to a litter and the pups are examined for specific health
effects. Each litter could be considered a cluster sample.
Clustering is necessary whenever there is reason to suspect that batches of data are correlated
(i.e., when the design of the study involves cluster sampling). The cluster variable should uniquely
identify each batch of correlated data uniquely.
Cluster labels might be text identifiers, identification numbers, combinations of variables, etc.
The only requirements are that observations from the same cluster have the same cluster label,
and those from different clusters have different cluster labels. If no cluster variables are specified,
the program treats all data as being independent.
CatReg assumes that responses from the same cluster are correlated, whereas observations
from different clusters are independent. It adjusts for the cluster sampling effect using the method
of generalized estimating equations (GEE). The cluster adjustment affects standard errors,
confidence limits and hypothesis tests (p-values), but it does not affect parameter estimates or
the deviance (a statistic used to measure the fit of exposure-response curve to the data). For
technical background on GEE, see Simpson et al. (1996b) and Diggle et al. (1994). It also may be
noted that cluster sampling invalidates the large sample F distribution of the generalized F-
statistic. However, it is common practice to compute F as a rough guideline (see Venables and
Ripley, 1994, p. 187). In any case, the R2 statistic gives an idea of how much variation in the
response is accounted for by the explanatory variables (see Section 7 for information on how F
and R2 are computed). Ignoring clusters of observations typically leads to underestimation of
variability in estimates and confidence bounds that are inappropriately narrow.
6.8 Link Function
A link function is a function applied to the exposure-response curve to transform it to a simple
linear relationship in dose and time. By also transforming the observed responses, the link
function reduces the mathematical complexity of estimating the parameters. The parameter
estimates then are substituted into the (untransformed) exposure-response curve.
CatReg provides three different link functions for the exposure-response curves.
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MODEL AND BMD TAB
Table 10: Link Functions
The Link Function...
Corresponds to Probability Function...
Logit
Logistic
Probit
Normal
Cloglog (complementary log-log)
Gumbel
A comparison of how well the different link functions fit the data may be assessed using the AIC
(Akaike Information Criteria, Akaike (1974)). A link with a smaller AIC provides a closer fit to the
data. The AIC for different link functions may be compared only if there are no changes in the
data or in use of the data filtering and stratification options. Otherwise, the differences in AIC may
result from the changes.
The parameter estimates and their statistical characteristics, including standard errors and
significance levels, are routinely output by CatReg, along with an analysis of deviance table to
assess model fit and a table of estimates of extra risk concentrations (ERC) (concentrations at
which extra risk is a user-specified value) for 1-, 4-, 8-, and 24-hour exposure times.
6.9 Model Form
There are two choices of models, Model 1 (cumulative odds model) and Model 2 (unrestricted
cumulative model).
CatReg provides a choice of two models, the cumulative odds model and the unrestricted
cumulative model.
The Cumulative Odds Model is described by the following equation:
Pr(Y> s\C,T) = H[as + P1 */1(C) + P2 * f2(T)\ Model 1
The Unrestricted Cumulative Model is described by the following equation:
Pr(Y> s|C, T) = /-/[as + P1 s * /1 (C) + P2s * /2(7)] Model 2
CatReg refers to any model of the form of Model 1 as a cumulative odds model because the
model is expressed in terms of the cumulative probabilities, or odds, for Y > s.
Note that Model 1 is a special case of Model 2 wherein parameters pis and P2s do not depend
on s (which denotes severity level). That is, Model 1 is a simplification of Model 2 in which only
the intercept term can vary across severity levels, not the coefficients of dose or time (a restriction
called parallelism).
In other words, the cumulative odds model (Model 1) states that:
• the probability that a severity level s or higher will occur at a given dose or concentration (C)
and time (T) is given by the exposure-response curve (logistic, normal, or Gumbel,
determined by the choice of link function), and
• the intercept parameters may differ by severity level (i.e., a different intercept for each
severity level), but the coefficients for C and Tdo not differ by severity.
A primary use of fitting Model 2 is to test whether the simpler Model 1 is adequate. Model 2,
although more general than Model 1, has the undesirable feature that the regression lines for
different severity levels may cross. Often the crossing is well outside the range of values of
interest, so the model can be used to make empirical risk estimates. The user has the option to
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MODEL AND BMD TAB
add an additional parameter, y, which represents a hypothetical background concentration or
dose, in some circumstances.
6.9.1 Understanding the Model Equations
The left-hand side of the Model 1 and Model 2 equations above is read as follows: the probability
that a response of severity level s or greater occurs, given that concentration or dose is C and
time is T (time refers to exposure duration). No expression for s = 0 is included because this is
the minimal category, and Y is always greater than or equal to 0 (i.e., Pr(Y> 0|C,T) = 1).
The right-hand side is described as follows:
• H is a probability function taking values between 0 and 1, for which the user has three
choices: logistic, normal, and Gumbel.
• The parameter as is the intercept for severity level s, s = 2,...,S (to be called the intercept or
severity parameters). The severity parameters are ordered as ai3 a,23...3as. This constraint is
a consequence of the requirement that the probability of exceeding a lower score is larger
than the probability of exceeding a higher score for any fixed levels of C and T.
• In Model 1, the parameter pi determines the dependence of the response on concentration or
dose (I), whereas P2 determines the dependence on time (T). In Model 2, the parameters are
also indexed by s because they may change values with severity level s.
• Current choices for/1 and /2 are "untransformed" and "base-10 logarithm." Other
transformations of C and T may be obtained by transforming the input data.
• Parameters are as, Pis (to be called the coefficient of concentration or dose), P2s (to be called
the coefficient of time or duration), for severity levels s running from 1 to S. All parameters
may be stratified on variables, as discussed in Section 6.6.
6.10 Assume Zero Background Response
The option to assume zero background response is unavailable and permanently set to "No"
unless the option to Log10 transform the concentration (Dose) parameter is chosen. In all other
cases, the background response is not assumed to be zero and is estimated. When the
concentration (Dose) term is Log10 transformed, then you can choose Yes or No for the Assume
Zero Background Response option.
• If you choose Yes, then the implied probability of an adverse response at zero concentration
or dose is zero and observations at zero concentration are uninformative (treated the same
as if they were filtered out). Warning: This option should only be chosen if it is not
reasonable to expect severity >_1 responses at zero concentration or dose (e.g., when
evaluating mortality from acute exposures) and when the observed data do not have severity
> 1 responses at zero concentration or dose.
• If you choose No, then CatReg adds a hypothetical background concentration or dose to the
administered doses, denoted as the parameter y (gamma) that is estimated by maximum
likelihood and displayed in the summary table of parameter estimates.
For example, If Assume Zero Background Response is set to No, an experimental concentration
or dose of 50 mg/m3 is treated as an observation at dose (50 + y) mg/m3, where y is estimated
from the data simultaneously with the other model parameters. If the logarithmic option is chosen
for time (Hours), the implied probability of an adverse response at zero time is zero and
observations at zero time are uninformative (treated the same as if they were filtered out).
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6.11 Worst Case Analysis
CatReg provides an option to do a worst-case analysis when there is at least one record in the
input file that contains censored data:
In a worst case analysis, censored responses are treated as occurring at their highest possible
(worst) severities.
Although the graphical presentation of censored points will not change in a worst-case analysis,
higher estimates of risk will be produced than those for the corresponding censored analysis.
Comparison of risk estimates from the two methods provides an indication of the sensitivity of the
results to the severity scoring.
When the worst case option is selected, the output file contains the line: Type of analysis: Worst-
case.
When the worst-case option is not selected, the output line is: Type of analysis: Censored.
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7.0 PLOTS TAB
CatReg includes functions for making plots after an exposure-response curve has been fit to the
data.
Figure 11.—Plots tab.
To specify a plot:
1. Select the Severity Level from the drop down list. The choices are 0, 1, 2 (default), and All.
2. Select one or more plot functions. The thumbnail illustrations indicate the type of graph that
CatReg will generate and display. Note that some plots (such as prplot and barplot) require
additional inputs.
3. Select Run Plots. The plots will be displayed in their own window (see Figure 12).
7.1 About the CatReg Plot Files: Locations, Formats, Names
CatReg saves any plots you've selected to the directory that contains the currently loaded input
file.
CatReg names the files based on the input file's name and the select plot type, plus any
options/parameters for filtering, stratification, etc. that were selected. Examples:
ChemxHu_barplot-SEV2-MU.png, Chemx_catplot-SEV2.emf.
Note If you re-run an analysis without changing any parameters and generate a plot, CatReg
will overwrite the previous plot file.
The plots are saved in either .PNG format (barplot) or .EMF format (all other plots). You can
import these files into PowerPoint, GnuPlot, or other graphics application for printing.
For the EMF-formatted plots only, you can use PowerPoint, GnuPlot, or another graphics
program to manipulate or edit the plot and its components. For example, import an EMF plot into
PowerPoint, right-click on the graphic, and select the Group>Ungroup command to edit each
element of the plot (text, points, lines, etc.).
7.2 Copying Plots from the Graphs Window
Right-click on any plot in the window and select Copy to Clipboard. You can then paste the plot
into PowerPoint, GnuPlot, or other graphics application for printing.
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Figure 12.—Plots generated by CatReg.
1 Graphs for->Analysis Screen
ERC16 Une |SEV2:C}wttnss% too-6Kfea contxtence bounds.
A um =K>git
•' effec: a saiercy 1 ¦ SewertyZ x Cemetea
ERC16 L!n«witn9e%two-4»09d confluence bounds
B Link = JoflTt
" Ms effec i 5«ertv 1 ¦ Several x C
Al Strata: ERC16 ones «t JEV=2. Lin* = K>grt
Q ERC10 |$EV2[wtth30%tivo-6Htea to rimsnee bounds
Vertcai lines are w thout stratltcatHxi
S 1 258-53 [-
Data pot(an SEV points( w itn strata
200 250 300 350 400 450 500
OoseC T.~ie (Hous> - 10. _s* - cor. j
100 120 140
3 v*. Predicted Tor Strata « C
o effect Severity 1 Severity 2
Observed vs. Predicted for Strata ¦ L
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PLOTS TAB
7.3 Plot Functions and Options
The following tables describe the plot functions and specific options available within CatReg.
Table 11: Plot Functions and Options
Function
Description
prplot(gp=x)
Probability of exceeding a specified severity level on y-axis, dose or time on x-
axis. Plots the probability curve as a function of dose, keeping time fixed, or as a
function of time with dose fixed.
Use the dropdown list to set the time or dose level to be used.
catplotO
Dose on y-axis, time on x-axis. Plot of ERC line with confidence interval for a
single severity level and stratum, and response data for all severity levels.
stratplotO
Dose on y-axis, time on x-axis. Plot of ERC lines and response data for all strata
for a single severity level.
confplot(Time)
Strata for a single severity level displayed on y-axis, dose on x-axis. Plot of ERC
with confidence interval for the unstratified model, and for individual strata of the
model, for a single exposure time.
dataplot()
Dose on y-axis, time on x-axis. Plots response data for all severity levels
combined by stratum.
devplotO
Generalized deviance residuals on y-axis and data observation number, log-dose,
or log-time on x-axis.
barplotQ
Bar plot of probability of actual incidences of data, with calculated probability as a
point, and error bars for each.
7.4 Jitter
CatReg's Jitter setting (which can be toggled from Preferences>Jitter) can be useful to get a
sense of data density in cases where observations pile on top of one another.
The Jitter function adds "noise" - small random offsets to coincident observations. Although
CatReg bounds the amount of offset, the randomness of the offset causes each jitter of the data
to produce a slightly different plot.
You can see the effects in the following figure. In plot a., it is impossible to discern the density of
data as it appears to be a grid of points. But with jitter enabled in plot b, we can see the data
distribution more easily. While plot b would not be useful for analyzing the data (CatReg provides
text results for that), the jittered graph provides a quick visual look at the underlying data.
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Figure 13.—Plots without jitter (a.) and with jitter (b.)
ERC10 Line (SEV2:C) with 90% two-sided confidence bounds,
Link = logit
No effect A Seventy 1 ¦ Seventy 2 x Censored
ERC10 Line (SEV2:C) with 90% two-sided confidence bounds,
Link = logit
No effect A Seventy 1 ¦ Seventy 2 x Censored
~~I 1 1—
16 18 2 0
Time (Hours)
—I r~
18 20
Time (Hours)
After you toggle the Jitter preferences setting, you will need to select the Run Plots button to
display plots with the new setting.
7.5 Doses and Times for Designated ERC and Severity (catplot)
The catplotQ plots the response data and graphs the extra risk concentration (ERC) with
confidence interval, for a single severity level and stratum.
Exposure concentration or dose is on the y-axis and exposure duration (time) on the x-axis.
The current ERC settings are used for ERC percentile and severity level.
The confidence interval percentile is 90% (two-sided) and cannot be overridden.
By default, dose and time are graphed on a log-linear scale. This type of graph is useful for
showing how extra risk changes with dose or time. Be aware that the weighting of individual
points is not shown.
Each call to catplot generates a new graphics window. Thus, repeated use of catplot allows
comparison of results across strata.
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Figure 14.—Illustration of catplotQ.
ERC10 Line (SEV2:L) with 90% two-sided confidence bounds,
Link = logit
° No effect A Severity 1 ¦ Severity 2 -< Censored
Time {Hours)
- These 90% two-sided confidence bounds are equivalent to 95% one-sided confidence bounds
in each direction.
The x-y locations of symbols on the graph indicate the exposure doses and times of observations
on mice with the central nervous system as the target organ. The symbol itself indicates the
severity category, as shown in the legend. The lines on the graph are the estimated ERC10 (solid
line) and upper and lower 95% one-sided confidence bounds (dashed lines) (equivalently, two
sided 90% confidence bounds). Slicing the graph vertically at a specific time gives the confidence
interval that confplot would graph for that time. As expected, longer times require lower doses to
achieve the estimated 10% level of extra risk.
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7.6 Doses and Times for Designated Probability and Severity of Strata (stratplot)
The command stratplotO plots the response data by stratum for all severity levels and graphs the
extra risk concentration (ERC) (without confidence interval) by stratum for a single severity level.
Exposure concentration or Dose is on the y-axis and exposure duration or Time on the x-axis.
The current ERC settings are used (for the ERC percentile and ERC severity level).
Response to a pop-up menu determines whether to include a legend.
By default, Dose and Time are graphed on a log-linear scale.
Be aware that the weighting of individual points is not shown. The number of points in each
severity category will be displayed in the R command window, along with the number of hidden
points.
This graph provides comparison of the ERC curves for different strata by plotting them on the
same graph.
Figure 15.— Illustration of stratplotO.
All Strata: ERC 10 lines at SEV=2, Link = logit
—m— 1 - - 2" ^7" 3 4 5
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7.7 Extra Risk Concentration at Desired Time (confplot)
The command confplot(Time) displays the extra risk concentration (ERC) with confidence interval
for the unstratified model and for individual strata of the stratified model, for a single severity level
at the exposure duration set in the "Time =" field. The default value is 10, for 10 hours exposure.
The time argument is required unless time is not included in the data as an explanatory variable.
The current ERC settings are used for the ERC percentile and severity level.
The confidence interval percentile is 90% (two-sided).
This graph is useful for comparing ERC estimates and confidence intervals among strata, and
comparing individual strata with the unstratified model.
Figure 16.—Illustration of confplotQ.
ERC 10 (SEV2) with 90% two-sided confidence bounds
Vertical lines are without stratification
>
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364.47
200
300
400
Dose (Time (Hours) = 10, Link = logil.)
500
-These 90% two-sided confidence bounds are equivalent to 95% one-sided confidence bounds
in each direction.
The above figure was produced with Time=10 hours. Dose is on the x-axis. The central dot for
each stratum is the estimated ERC10 for severity level 1.
The vertical solid line is the estimated ERC10 for the unstratified exposure-response curve, and
the vertical dashed lines are the associated confidence intervals (90% for two-sided bounds,
determined from the ERC settings).
To display the confidence interval for another duration, repeat the confplot command and reset
the Time field.
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7.8 Probability Versus One Explanatory Variable (prplot)
The command prplot(gp=x) displays the probability curve (i.e., the exposure-response curve) as a
function of concentration or dose, keeping time fixed, or as a function of time with dose fixed.
If either of these variables is constant in the data, then prplot graphs the probability against the
non-constant variable.
If both dose and time vary in the data , then you need to tell prplot which variable to hold
constant. You can specify either Time or Dose from the "gp=" dropdown list:
• To plot the response probability versus time at a fixed dose, select gp=Dose.
• To plot the response probability versus dose at a fixed time, select gp=Time.
The severity level is determined by the current ERC settings. The choice of stratum and whether
to include a legend are determined by response to pop-up menus.
The function prplot can help assess whether the fitted probability curve is consistent with the
data, and for representing the risk over a range of exposure levels.
Figure 17 was produced with gp=Time and x=1.25. The curve plots the probability of the
occurrence of a severity level 1 response or greater for the liver target organ in mice (species
MU) as a function of concentration or dose, with time fixed at 1.25 h. The value used for time
need not be an exposure time in the data. When that occurs, the probability curve is displayed but
there are no data to display.
Figure 17.—Illustration of prplotQ.
Time ( Hours ) = 1.25 Stratum = SEV1:3
0 No effect A Severity 1 ¦ Severity 2 x Censored
O
O
o
o
o
o
1400
1600 1 800 2000 2200 2400
Dose (mg/m3)
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In these graphs, the vertical location of a symbol represents whether the response at a particular
dose was equal to or greater than the severity of interest (adverse, severity 1, for the above
figure).
No effect responses are plotted at Pr = 0.
Adverse effects (severity 1) and severe effects (severity 2) are both plotted at Pr = 1.
There are no censored observations. If the data had contained some observations censored as
[0,1], they would have been graphed as at Y = 0.5 on the severity >1 graph, and as at Y =
0 on the severity >2 graph. This is because, in the first case, it is not known whether the censored
observation meets the severity threshold (i.e., whether severity >1), whereas in the second case
the severity is known to be less than 2.
7.9 Data Plotted by Stratum (dataplot)
The command dataplotO plots the response data for all severity levels by stratum, without a
response probability curve, as shown in the following figure.
Exposure concentration or dose is on the y-axis and exposure time on the x-axis. Notice that this
figure is very similar to the stratplotO figure, except that the dataplotO figure does not show the
ERC10 lines.
Figure 18.—Illustration of dataplotO.
Data plot(all SEV points) with strata
c 4 L
i
*
1 f* 1 ^ r-
1.2 1.4 1.6 1.8 2.0 2.2
Time (Hours)
2.4
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7.10 Contribution to Deviance for Individual Datum (devplot)
How well are individual observations explained by the exposure-response curve?
In linear regression, residual plots help to address that question, that is, plots of the differences
between observed and predicted responses.
Here the response is binary or ordinal, so there is a scaling problem in defining a "residual."
Various possibilities for resolving this problem exist.
CatReg uses the individual components of the deviance statistic to measure how well individual
observations are explained by the exposure-response curve. Using deviance residuals is
common practice in generalized linear modeling (see Venables and Ripley [1994, p. 188-189]).
For interval-censored ordinal data, other types of residuals are undefined, whereas the deviance
residuals are defined in the usual way. The observations that contribute to any lack of fit of the
exposure-response curve can be identified by examination of the individual contributions to the
deviance.
The devplotO function produces a diagnostic plot of generalized deviance residuals versus
observation number (excluding filtered data), dose (or log-dose), or time (or log-time), depending
on the user selection from a pop-up menu.
If the model is stratified, the points are labeled by strata.
Generally, plotting deviances versus observation number (Obs.Number) is a good choice. It
provides a representation of the relative effectiveness of the model in fitting the different
observations or strata. Devplot does not require jittering because each observation is represented
uniquely under the Obs.Number option. The fit is suspect if one or a few observations have much
larger deviance residuals than the remaining observations because the fit may be unduly
influenced by these observations. If one stratum has large deviances, the model may be
inadequate for this stratum. Rerunning the model without this stratum would allow one to
determine whether the results for the other groups are heavily influenced by the poorly fit subset.
Plots of deviance versus dose (or log-dose) or time (or log-time) are useful in studying the
adequacy of the functional form of the regression relationship. Trends in the deviances would
suggest a problem with the functional form. One should be aware that differences in the density
of the data at different dosages or times will affect the perception. Regions of the plot with more
data will tend to have more spread in the deviances because of random variation, even if the
model is adequate for the data.
Figure 19 was produced by selecting Observation # from the pop-up menu.
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Figure 19.—Illustration of devplotQ — Example 1.
Deviance plot
O 1:C • 2:C a 3:L " 4:L « 5:L
0 20 40 60 80 100 120 140
Obs.Number
Figure 20 is a devplotO example for a different set of dose-response group data. The input and
output files are not shown. The targets are C= central nervous system and L=liver. The species
are HU (human), MU (mice), and RT (rats). Intercepts were stratified on "Species" and "Target".
The notable feature of the figure is that the liver data for mice are relatively poorly described by
the curve (i.e., residual deviances are relatively high) in comparison to most of the other data.
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These data may or may not be influential on the fitted exposure-response curve. To examine
whether they are, the curve could be refit after filtering out liver data for mice.
Figure 20.—Illustration of devplotQ — Example 2.
Deviance plot
O
CO
IO
CMI
CD
CM
CD
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7.11 Observed and Predicted Probability Versus Dose/Time combination (barplot)
The barplotQ displays a stacked bar plot showing the observed probability of each severity level
for a given dose/time combination.
The plot overlays each bar with predicted probability points for each dose/time value at each
severity level (except the "no effect" level) along with error bars to represent the prediction's
standard error.
The barplotO offers the following X-axis options for Dose/Time combination:
• Dose/Time - sorts by dose first, and then time for each dose value
• Time/Dose - sorts by time first, and then dose for each time value
• Dose*Time - sorts by the product of dose*time
The plot also features a Grayscale option (checked by default) to output the plots in grayscale
rather than color.
Figure 21 plots the Observed and Predicted Probability vs. Dose/Time for the central nervous
system target organ in mice (species MU)
Figure 21.—Illustration of barplotQ
Observed vs. Predicted for Strata = 4
• No effect ¦ Severity 1 ~ Severity 2
Dose ( mg/m3 ) | Time ( Hours )
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HYPOTHESES TAB
8.0 HYPOTHESES TAB
Figure 22.—Hypotheses tab.
Dataset and Variables Model and BMD Plots Hypotheses
Automatic Tests
Intercept Term Equality...
Removal of Time
Removal of Dose
Coefficients tested for removal: None
Coefficients tested for equality:
Group 1 : SEV1 SEV2
Test statistics:
Chisquare df p-value
274.7552 1 1e-05
The P Value of the equality test is < =
0.05. This is generally considered
significant, indicating that all the
User-defined Tests
Coefficients tested for removal:
LG10TIME
Coefficients tested for equality:
None
Test statistics:
Chisquare df p-value
232.272 1 1e-05
The P Value of the removal test is < =
0.05. This is generally considered
Intercept
Available
Parameters
SEV1
SEV2
2: INTERCEPT
3: INTERCEPT
4: INTERCEPT
5: INTERCEPT
Value of 0
LG10CONC
Value of 0
Test for
Equality of
Parameters
Coefficients tested for removal:
LG10CONC
Coefficients tested for equality:
None
Test statistics:
Chisquare df p-value
240.8585 1 1e-05
The P Value of the removal test is < =
0.05. This is generally considered
LG10TIME
Value of 0
From the Hypotheses tab, you can test exposure-response hypotheses. The Hypotheses tab is
disabled until you have run an analysis. After running an analysis, the controls are enabled and
eligible parameters from the analysis are loaded into the Intercept, Dose, and Time parameter
lists.
• CatReg automatically runs parameter equality tests, based on the data. Those results are
displayed in the Automatic Tests section.
• Simply select and drag parameters from the Available Parameters box to the appropriate
Test for Equality boxes. For example, under Intercept, you can drag SEV1 to the left Test for
Equality box and SEV2 to the right box.
• Select Clear Tests to clear the Test for Equality boxes and start over.
• Select Run Tests to run the specified hypotheses tests.
• After running the tests, CatReg displays a new window showing the text-based results. The
window's title bar indicates the file name and location (in CatReg's Data\OptionFiles
directory).
• The hypotheses results are saved as a text file with the extension *.ANX.
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HYPOTHESES TAB
Figure 23.—Hypotheses results.
Hypotheses Result For; C:\Users\mbrown06\Documents\CatReg\CatReg_31Beta3c\Dat... ~ '| 0 ||
8.1 Testing Parameters
The output file includes tabular results for a simple test of the hypothesis that a parameter is zero.
Dividing the estimated coefficient by the standard error provides a Z statistic for the
corresponding parameter. This statistic provides a one-degree-of-freedom test of the null
hypothesis that the parameter equals zero.
Under the null hypothesis, the Z statistic has an approximate standard normal distribution. The
larger the sample, the better the approximation is.
The p-value in the table gives the significance level of the test. A parameter that is not
significantly different from zero may be considered a candidate for removal and simplification of
the exposure-response curve. The p-values apply to individual parameters considered singly,
however, and further testing is needed to test the joint hypothesis that more than one parameter
is zero, that two or more parameters are equal, or a combination of the two.
CatReg lists all the parameters in the Parameters list box. You can select a parameter and then
select the appropriate buttons to test the parameter(s) for removal or to select a group of
parameters to test for equality (the default is none in both cases).
The idea is to express the hypothesis to be tested as a set of constraints on the model
coefficients.
A test is then conducted of the hypothesis or joint hypotheses, if more than one was entered. The
test is a (generalized Wald-type) chi-square test of the null hypothesis that all of the specified
constraints hold. The distribution of the test statistic is derived from the sampling distribution of
the estimated model coefficients, and it takes into account any cluster sampling.
One may want to test for a gender difference, or whether there are interspecies differences in the
exposure-response.
For such a test, the null hypothesis is that the specified parameters are equal. A p-value less than
0.05 is usually taken as evidence that the hypothesis should be rejected (i.e., the specified
parameters are not equal).
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HYPOTHESES TAB
The parameters to be tested as equal must, of course, be included in the exposure-response
curve; this is accomplished by stratifying. For example, to test that there is no difference between
species in the coefficients of dose, the user would stratify the Dose parameter on species on the
Dataset and Variables tab.
When CatReg queries which coefficients to test for equality, the parameters to be tested (if any)
are entered from the Parameters list on the Hypotheses tab.
To test for no difference between species in the coefficients of dose, you would enter the
coefficient for each species. Some care needs to be used when tests involve stratified intercept
parameters because stratified intercept parameters are represented in CatReg as increments
relative to a reference. CatReg assigns the lowest alphabetical term to be the reference.
For illustration, consider the example provided in Section 6.6, "Stratifying." The intercept is
stratified on species (rat and mouse) in this example. In this case, the MU:INTERCEPT
parameter estimate will be the reference as it alphabetically precedes RT:INTERCEPT. Thus, the
SEV1 intercept value reported by CatReg will apply to mice. The reported rat intercept value will
be an incremental difference from the mouse reference value. To test whether the mouse and rat
SEV1 intercept estimates are equal, one would perform a Z-test of the null hypothesis that
RT:INTERCEPT is zero. To perform this test in the CatReg Hypothesis tab, simply select and
drag the RT:INTERCEPT parameter to one side and "Value of 0" to the other side of the Intercept
"Test for Equality of Parameters" boxes, then select "Run Tests."
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ASSESSING MODEL FIT
9.0 ASSESSING MODEL FIT
This section focuses on the fit of the exposure-response curve rather than on its parameters.
In linear regression analysis, it is common practice to consider the proportion of variation
accounted for by the model, the so-called R2 statistic, as a measure of the model's explanatory
power. This statistic, which ranges in value between 0 and 1, is the ratio of the model and total
sums of squares. These sums of squares, along with degrees of freedom, F-tests, and so on
usually are reported in the form of an analysis of variance table. Standard texts such as Weisberg
(1985) describe the use and interpretation of these statistics.
CatReg provides generalized analysis of variance and R2 statistics for assessing the explanatory
capacity of the exposure-response curve. These are derived from deviance statistics for
hierarchical models. Following McCullagh and Nelder(1989) and Venables and Ripley (1994),
this type of analysis is called the analysis of deviance. The analysis of deviance statistics and R2
statistic are in a table in the output file.
After running CatReg, the fitted curve is available for further analysis. As an example of the
analysis of deviance, consider the output file in Figure 24.
The summary output, which shows the coefficient estimates, indicates that seven parameters
have been estimated. There are 64 treatment groups (2 species x 2 targets x 4 doses x 4 times),
so total degrees of freedom (df) is 64 x (3 severity levels -1) = 128 (unadjusted for SEV1 and
SEV2). The estimates of the SEV1 and SEV2 intercepts provide no information about how well
the curve fits the data, so it is customary to adjust the total degrees of freedom for them in the
analysis of deviance table, leaving 128 - 2 = 126 degrees of freedom. The model has 5
parameters, aside from SEV1 and SEV2, which are MU:L:INTERCEPT, RT:C:INTERCEPT,
RT:L:INTERCEPT, LG10C0NC, and LG10TIME. This leaves 126 - 5 = 121 degrees of freedom
for the residual deviance.
The analysis of deviance table for the example shown in Figure 24 partitions the total deviance
into the sum of two components: the "model" deviance and the "residual" deviance. The total
deviance is the deviance when the only parameters in the exposure-response curve (or "model"
in the terminology being used here to show the comparison with the analysis of variance) are the
intercepts, SEV1 and SEV2. This is referred to as the null model because it contains no
explanatory variables. The residual deviance is simply the deviance of the fitted model, which
includes the 5 parameters, aside from SEV1 and SEV2, i.e., MU:L:INTERCEPT,
RT:C:INTERCEPT, RT:L:INTERCEPT, LG10C0NC, and LG10TIME, as explanatory variables.
The model deviance is that part of the total deviance that is explained by the model (the
proportion is the generalized R2). In this example, 46.4% of the total deviance is explained by the
model. R2 is a general measure of the proportion of the variation in the response that is
accounted for by the explanatory variables.
The mean deviance entries, labeled as "Mean.Dev," are computed as deviance divided by
degrees of freedom. An approximate F-test of the model is obtained as the ratio of model to
residual mean deviations. This is an approximate F-statistic in large samples under the ordinal
regression model with independent responses. This statistic tests the null hypothesis that all
explanatory variables can be dropped from the model.
Generally, the F-test will reject the null hypothesis unless the sample size is very small or the
model fits poorly. It merely verifies that there is some relationship between the response and the
explanatory variables. The generalized R2 statistic often will be of more direct interest as a
measure of the explanatory value of the variables in the model.
Cluster sampling invalidates the large sample F distribution of the generalized F-statistic.
However, it is common practice to compute F as a rough guideline; see Venables and Ripley
(1994, p. 187). In any case, the R2 statistic gives an idea of how much variation in the response is
accounted for by the explanatory variables.
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ASSESSING MODEL FIT
Figure 24.—Output file from CatReg. Input file:chemx.csv. Model 1. logit. Scales: Iog10.
Stratification: intercept on species and target. Assumed Zero Background Risk.
CatReg Version 3.1
Run Date: Thursday, 15 Jun 2017 at 12:45:6
Source data file: c:/users/mbrown06/documents/catreg/catreg_31beta3c/data/chemx.csv
Type of analysis: Censored
Input file : c:/users/mbrownO6/documents/catreg/catreg_31beta3c/data/chemx.csv
Model Variable Mapping:
Dose: mg/m3
Time: Hours
Incidence: Incid
N: Nsub
Sev Lo: SevLo
Sev Hi: [Not Used]
Model Specifications:
Model: cumulative odds model
Link: logit
Assume Zero Background Response: Y
Worst Case Analysis: n
Scale for Dose(Concentration): loglO( mg/m3 )
Scale for Time(Duration): loglO( Hours )
BMD Specifications:
Risk: extra risk
BMR: 0.1
Confidence Level(%): 95
Time: 0.5 1 2 4 8
Filtered data: none
Clustering : none
Stratification:
Intercept : Species Target
Dose(Concentration):
Time(Duration) :
Message
Iterations
Deviance
Residual DF
AIC
39 14
741.0266
121
755.0266
Coefficients:
Estimate
Std. Error
Z-Test=0
p-value
SEVl
-66.2375249
4.2947663
-15.422847
0.00001
SEV2
-70.3128027
4.4629521
-15 .754774
0.00001
MU:C:INTERCEPT
0. 0000000
0.0000000
NA
NA
MU:L:INTERCEPT
1.5443003
0.2803179
5 .509103
0.00001
RT:C:INTERCEPT
-0. 8580588
0.2701988
-3.175658
0.00149
RT:L:INTERCEPT
0.7735915
0.2703681
2.861253
0.00422
LG10CONC
19.6220725
1.2858838
15 .259600
0.00001
LG10TIME
18.0738765
1.2265874
14 .735091
0.00001
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ASSESSING MODEL FIT
Variance:
SEV1
SEV2
MU:L:INTERCEPT
RT:C:INTERCEPT
RT:L:INTERCEPT
LGIOCONC
LG10TIME
SEV1
SEV2
MU:L:INTERCEPT
RT:C:INTERCEPT
RT:L:INTERCEPT
LGIOCONC
LG10TIME
SEV1
18.4450174 19.
19.1496806 19.
-0.3256623 -0.
0.1304789 0.
-0.1730734 -0.
-5.5095904 -5.
-3.4384286 -3.
RT:L:INTERCEPT
-0.17307335
-0.18143003
0. 03822081
0. 03365877
0. 07309891
0. 04053521
0.04091319
SEV2 MU:L
1496806
9179410
3429295
1402984
1814300
7215951
6292401
LGIOCONC
-5.50959042
-5.72159505
0. 08581225
-0.04903025
0. 04053521
1.65349721
0.97163480
:INTERCEPT RT:
0.32566226
0.34292950
0.07857815
0. 03192034
0. 03822081
0.08581225
0. 08062509
LG10TIME
-3.43842860
-3.62924010
0. 08062509
-0.04554744
0. 04091319
0.97163480
1.50451662
C:INTERCEPT
0.13047885
0.14029845
0.03192034
0.07300738
0.03365877
-0.04903025
-0.04554744
Analysis of Deviance Statistics:
Generalized R-squared: 0.464
DF Deviance Mean.Dev Gen.F pvalue
Model 5 642.5451 128.509 20.984 0
Residual 121 741.0266 6.124
Total 126 1383.5717
#########################################################################
Note: About 4 6.4 % of the variation in the response is accounted
for by the explanatory variables in the current model fit.
The p-value of the model fit is <= 0.05. This is generally considered
significant, indicating that the current model fit is acceptable.
#########################################################################
< ERC10 at specific time point(s) >
: One-sided 95% lower bound and one-sided 95i
* Risk Type : extra risk
upper bound
* ERC10 at SEV2:MU:C
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 5606.0094 4972.9013 6319.7196 408.4255
1.0 Hours 2960.5699 2769.3104 3165.0386 120.2035
2.0 Hours 1563.4961 1493.3890 1636.8944 43.6072
4.0 Hours 825.6924 758.0997 899.3118 42.8733
8.0 Hours 436.0535 378.5090 502.3466 37.5186
* ERC10 at SEV2:MU:L
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5
Hours
4676.8304
4178.5047
5234.5862
320.3478
1.0
Hours
2469.8644
2322.1674
2626.9555
92.5903
2.0
Hours
1304.3514
1241.2888
1370.6178
39.2971
4.0
Hours
688.8364
627.6448
755.9938
38.9591
8.0
Hours
363.7790
313.2137
422.5075
33.0993
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ASSESSING MODEL FIT
* ERC10 at SEV2:RT:C
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 6199.8772 5475.8851 7019.5918 468.0497
1.0 Hours 3274.1953 3051.3295 3513.3390 140.3247
2.0 Hours 1729.1238 1652.3140 1809.5041 47.7660
4.0 Hours 913.1615 841.3680 991.0809 45.4587
8.0 Hours 482.2465 420.3186 553.2985 40.2960
* ERC10 at SEV2:RT:L
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 5119.5203 4559.8071 5747.9379 360.3605
1.0 Hours 2703.6518 2536.5080 2881.8097 104.8931
2.0 Hours 1427.8160 1361.1725 1497.7224 41.4924
4.0 Hours 754.0389 689.3606 824.7856 41.1111
8.0 Hours 398.2128 344.0843 460.8564 35.3702
Note:
1. These One-sided 95% lower bound and one-sided 95% upper bound
confidence intervals are equivalent to
the lower bound and upper bound of two-sided 90% confidence intervals.
< ERC10 at specific time point(s) >
: One-sided 95% lower bound and one-sided 95% upper bound
* Risk Type : extra risk
* ERC10 at SEV1:MU:C
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 3475.0846 3129.8727 3858.3718 221.0445
1.0 Hours 1835.2147 1729.5769 1947.3045 66.1458
2.0 Hours 969.1887 912.5364 1029.3582 35.4899
4.0 Hours 511.8349 460.2481 569.2037 33.0579
8.0 Hours 270.3033 229.7032 318.0794 26.7462
* ERC10 at SEV1:MU:L
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 2899.0999 2623.0928 3204.1490 176.3342
1.0 Hours 1531.0334 1442.0397 1625.5193 55.7405
2.0 Hours 808.5487 754.9296 865.9760 33.7292
4.0 Hours 426.9998 380.3650 479.3523 30.0230
8.0 Hours 225.5014 189.8653 267.8260 23.5819
* ERC10 at SEV1:RT:C
ERC10 Lower Bound of ERC10 Upper Bound of ERC10 Standard Error
0.5 Hours 3843.2146 3450.7259 4280.3455 251.6993
1.0 Hours 2029.6265 1911.0637 2155.5451 74.2722
2.0 Hours 1071.8589 1013.0276 1134.1068 36.7859
4.0 Hours 566.0556 511.4375 626.5066 34.9186
8.0 Hours 298.9376 255.2595 350.0897 28.7068
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ASSESSING MODEL FIT
ERC10 at SEV1:RT:L
ERC10 Lower
Bound of ERC10 Upper
Bound of ERC10
Standard Error
0.5
Hours
3173.5169
2866.2209
3513.7591
196.4977
1.0
Hours
1675.9548
1579.5678
1778.2236
60.3518
2.0
Hours
885.0826
829.6099
944.2646
34.8282
4.0
Hours
467.4179
418.1326
522.5124
31.6636
8.0
Hours
246.8464
208.6956
291.9714
25.1955
Note:
1. These One-sided 95% lower bound and one-sided 95% upper bound
confidence intervals are equivalent to
the lower bound and upper bound of two-sided 90% confidence intervals.
•*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*••*• tomated Test (s) ':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r
Hypothesis Test Results
Coefficients tested for removal: None
Coefficients tested for equality:
Group 1 : SEV1 SEV2
Test statistics:
Chisquare df p-value
261.1419 1 le-05
##############################################################################
The P Value of the equality test is <= 0.05. This is qenerally considered
siqnificant, indicatinq that all the tested parameters should be retained
in the model.
##############################################################################
Removal of Time':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r':*r
Hypothesis Test Results
Coefficients tested for removal: LG10TIME
Coefficients tested for equality: None
Test statistics:
Chisquare df p-value
217.1229 1 le-05
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ASSESSING MODEL FIT
##############################################################################
The P Value of the removal test is <= 0.05. This is generally considered
significant, indicating that the tested parameters should not be removed
from the model.
##############################################################################
R0m.oVeil of
Hypothesis Test Results
Coefficients tested for removal: LGIOCONC
Coefficients tested for eguality: None
Test statistics:
Chisguare df p-value
232.8554 1 le-05
##############################################################################
The P Value of the removal test is <= 0.05. This is generally considered
significant, indicating that the tested parameters should not be removed
from the model.
##############################################################################
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Categorical Regression (CatReg) User Guide
MENUS, TOOL BARS, & STATUS BARS
10.0 MENUS, TOOL BARS, & STATUS BARS
At all times, CatReg displays a menu bar at the top of the CatReg application window, and a
status bar at the bottom of the window.
This section describes the different options provided by the CatReg menus and status bars.
10.1 File Menu
Command
Function
New Analysis
Opens a new Analysis Screen window.
Open Analysis
Displays an Open dialog box, where you can navigate to and select a previously saved analysis file
f.anx).
Exit
Closes the CatReg application. CatReg will prompt you to save any unsaved changes.
10.2 Preferences Menu
Command
Function
Jitter
(Plots tab) Adds small random offsets to coincident observations to produce a plot that better shows
density of the data.
10.3 Help Menu
Command
Function
About
Window describing the CatReg Sponsors and Credits, CatReg program version.
User Guide
Displays the CatReg User Guide PDF, located in the CatReg program directory
Check for
Updates...
Displays Update Preferences dialog for CatReg to periodically check for new versions or for the user to
manually check for updates.
10.4 Text Window (Results) Menus
10.4.1 File
Command
Function
Save
Saves changes to the selected file.
Save As...
Displays the Windows Save As dialog box so the file can be saved under a new filename.
Print
Prints the contents of the selected window.
Print Setup
Displays the Windows Print Setup dialog box, where you can select such options as orientation, margins,
paper size, and so on.
Print Preview
Displays a window showing how the file will look when printed.
Close
Closes the selected window. CatReg will prompt you to save any unsaved changes.
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Categorical Regression (CatReg) User Guide
MENUS, TOOL BARS, & STATUS BARS
10.4.2 Edit
Command
Function
Undo (Ctrl+Z)
Undo the most recent change.
Cut (Ctrl+X)
Selected data is cut from an active output file.
Copy (Ctrl+C)
Selected data is copied to the clipboard.
Paste (Ctrl+V)
Cut/copied data is pasted into output file at cursor location.
Select All (Ctrl+A)
Selects all text in current active window.
10.4.3 Preferences
Command
Function
Word Wrap
Toggle word wrap.
Font
Display font selection dialog.
10.5 Dataset Context Menu
With data loaded in the Analysis Screen, right-click to display the following options.
Command
Function
Load Dataset...
Displays an Open dialog box, where you can navigate to and select an input file f.csv).
Regroup Dataset...
Displays the Grouping dialog box.
Save Dataset As...
Displays a Save dialog box, where you can choose to overwrite an existing input file or save the displayed
data with a new name.
10.6 Status Bar
Each window in CatReg has its own status bar and communicates different information.
Status Bar
Location
Functions
CatReg application
window
Displays the results of actions executed within session and data windows, such as when rows or columns
are inserted, a session is saved, or when a parameter options file is opened for editing.
Analysis Screen
Displays such messages as "Right-click on Dataset control for additional option(s). or "Done processing!"
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Categorical Regression (CatReg) User Guide
KEEPING CATREG UP TO DATE
11.0 KEEPING CATREG UP TO DATE
After you start CatReg, and assuming your computer has an Internet connection, the software will
check the CatReg server for any updates. You can change the frequency of this check by
selecting Help>Check for Updates.
Figure 25—Update Preferences dialog box.
Update Preferences
When would you like to checkfor updates?
© Startup
Weekly
© Monthly
Never
Check for updates now
OK
Cancel
From this dialog box, you can select whether CatReg checks for a new update:
• On startup
• Weekly (the default)
• Monthly
Selecting Never means you will have to manually check for updates by selecting the Check for
updates now button.
Note that CatReg must be running so it can check for and notify you of an update.
11.1 When an Update is Ready for Download
When CatReg detects that an update is ready to download, it will display an update notification
dialog.
Figure 26.—Update Notification dialog box.
NotifyUpdate
i@UB-n
A new version of CatReg is available: CatReg 3.0.1.2
Do you want to download the new version?
® Download Now
© Download and Install Now
Q Do Not Download
Update File Location
Install Directory
C:\usepa\CatReg3012
OK
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Categorical Regression (CatReg) User Guide
KEEPING CATREG UP TO DATE
You can choose to:
• Download the update and install it yourself later (the default)
• Have CatReg download and automatically install the update. The dialog shows the Update
File Location where CatReg will download the new update file and the Install Directory where
the new CatReg directory will be created if automatically installed. These fields are read-only
and cannot be edited; CatReg enforces this information to ensure a stable installation
process.
• Not download the update
If you choose to download and install the software yourself, you can follow the instructions
described in the CatReg readme file included with the install package.
If you choose to have CatReg automatically download and install the update, then CatReg
displays an information box after installation confirming the update.
Note that this automatic install process is done under your user account and does not require a
system administrator.
11.2 To Use the Newly Installed Software
Following an automatic installation, the older CatReg version does not shut down; it remains
open.
To use the newly installed CatReg software:
1. Close the old CatReg version.
2. Review the readme file to learn about new features, bug fixes, or other need-to-know
items regarding the new software.
3. Follow the instructions in the readme file to:
• Create a shortcut for the new CatReg version on your desktop
• Determine that the new CatReg version works properly on your computer
• Fix possible issues with displaying the help file
11.3 Managing Your Existing Files After an Automatic Update
The automatic install does not replace or displace your existing data and results files. Your old
version of CatReg and related input filesw remain in their current locations. Input files do not have
to be moved from their current folder locations to work with the new version of CatReg. Follow
your organization's guidelines for locating and maintaining data and files.
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Categorical Regression (CatReg) User Guide
REFERENCES
12.0 REFERENCES
Akaike, M. (1974) A new look at statistical model identification. IEEE Transactions on Automatic
Control AU-19: 716-722.
Diggle, P. J.; Liang, K.-Y.; Zeger, S. L. (1994) Analysis of longitudinal data. New York, NY:
Clarendon Press.
Huber, P. J. (1967) The behavior of maximum likelihood estimates under nonstandard conditions.
In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability.
Volume 1. Berkeley, CA: University of California Press.
Liang, K.-Y.; Zeger, S. L. (1986) Longitudinal data analysis using generalized linear models.
Biometrika 73: 13-22.
McCullagh, P.; Nelder, J. A. (1989) Generalized linear models. 2nd ed. London, United Kingdom:
Chapman and Hall.
Morgan, B. J. T. (1992) Analysis of quantal response data. London, United Kingdom: Chapman &
Hall.
Simpson, D. G.; Carroll, R. J.; Xie, M.; Guth, D. J. (1996a) Weighted logistic regression and
robust analysis of diverse toxicology data. Commun. Stat. 25: 2615-2632.
Simpson, D. G.; Carroll, R. J.; Zhou, H.; Guth, D. J. (1996b) Interval censoring and marginal
analysis in ordinal regression. J. Agric. Biol. Environ. Stat. 1: 354-376.
U.S. Environmental Protection Agency. (2000) CatReg software documentation. Research
Triangle Park, NC: Office of Research and Development, National Center for Environmental
Assessment; report no. EPA/600/R-98/053.
U.S. Environmental Protection Agency. (2006) CatReg Software User Manual (R-Version).
Research Triangle Park, NC: Office of Research and Development, National Center for
Environmental Assessment; report no. EPA/600/R-04/006. Downloadable from the EPA Web
page, "CATREG SOFTWARE FOR CATEGORICAL REGRESSION ANALYSIS ."
Venables, W. N.; Ripley, B. D. (1994) Modern applied statistics with S-Plus®. New York, NY:
Springer-Verlag.
Weisberg, S. (1985) Applied linear regression. 2nd ed. New York, NY: Wiley.
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GLOSSARY
13.0 GLOSSARY
Akaike Information Criteria (AIC) - The deviance (-2 times the log of the maximized value of the
likelihood function) + 2 times the number of parameters in the model.
Categorical regression - A model expressing the probabilities of different response categories as
functions of explanatory variables.
Cluster sample - A data set comprised of subsamples of data from common sources. For
example, a data set may contain several data records per laboratory from several different
laboratories. The subgroup of data records from each individual laboratory would represent a
cluster sample.
Cumulative odds regression - Ordinal regression model for directly modeling the probabilities of
exceeding different severity levels.
Deviance - The minimized value of twice the negative logarithm of the likelihood function.
Extra risk concentration (ERC) - Concentration or dose at which extra risk is a user-specified
value.
ERC settings - Three numbers: ERC percentile, ERC severity level, and ERC percentile for
confidence intervals.
Filtering - Exclusion of selected data records from the analysis. This capability "filters out"
selected data without altering the data input file.
Generalized estimating equation - An equation depending on the data and the parameter values,
such that solving for the parameter values yields consistent estimates.
Hierarchical models - An ordered series of models, such that each model is a special case of the
next one in the series.
Inf- Infinite value
Interval censored data - Data for which the response is known only to lie in an interval of values.
Likelihood function - For categorical response data, a model for the joint probability of the
observed data values, expressed as a function of the model parameters.
Link function - A function applied to the categorical response probability to transform the
categorical regression model to linear units.
Meta-analysis - The analysis of data from multiple studies to determine overall trends and
increase power.
NaN - Not a number
Ordinal data - Data reported as ordered categories. The order is meaningful, but the numerical
difference between ordered categories is not.
Parallelism - The coefficients of dose and time in the exposure-response model (i.e., the
probability function) do not change with severity level (parallelism applies to Model 1, the
cumulative odds model, but not to Model 2, the unrestricted cumulative model).
Probability function - The function of the explanatory variables that gives the probability of
exceeding a given severity level.
Proportional odds regression - Ordinal regression in which the log-odds of exceeding different
severity levels are parallel across severity categories. It is a special case of cumulative odds
regression with the logistic link function.
Stratification - To create subsets of data by allowing the model parameters to vary by subset.
Covariate information such as species, sex, and target organ may be used as a basis for creating
the sub sets.
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Categorical Regression (CatReg) User Guide DISTRIBUTION OF CONTINUOUS RESPONSE
DATA OVER SEVERITY LEVELS
APPENDIX A: DISTRIBUTION OF CONTINUOUS RESPONSE DATA OVER
SEVERITY LEVELS
Response data are sometimes measured on a continuous scale, with the mean and standard
error reported. Section 5.11.3 contains a hypothetical example for lung weight in rats in which a
mean and standard error were reported for each treatment group. The lung weight data were
distributed over the severity levels shown for "Ref.id" = 2 in Table 7 for analysis by CatReg. The
following description explains how that was done and provides an example calculation for
reference. A technical explanation is provided at the end of the appendix.
The distribution of lung weights in healthy, unexposed rats is needed, either estimated from
control animals in the experiment or "known" from other sources. It is assumed here that the
distribution of lung weights is normally distributed with mean 1.0 g and standard deviation 0.05 g.
(Note: The normal distribution is assumed simply for illustration. The same idea could be applied
to other distributions). The user then determines weight intervals for severity levels to be used.
For this purpose, it may be helpful to estimate first the highest weight that might be considered in
the "normal" range for unexposed animals. The weight 1.15 g, which is three standard deviations
above the mean, is an upper bound on virtually all lung weights in unexposed animals (i.e., a
weight above 1.15 g is above the normal range of lung weights). The following correspondence
was made between severity levels and lung weights for the example: SevO (<1.15 g), Sev1
(1.15 to 1.50 g), and Sev2 (>1.50 g).
Suppose that the average lung weights and standard errors shown in Table A-1 were reported for
treatment groups in "Ref.id" = 2, Table 7.
Note: If SE is the standard error from a treatment group of size n, the estimate of the standard
deviation is SE.
There is a separate mean and standard error reported for each treatment group, denoted by ,
and SEi, respectively for index i, with i = 1 ,...,13. The ith treatment group is assumed to be a
sample from a normal distribution with unknown mean and standard deviation, denoted by jji and
ai, respectively. The treatment groups are rather small (ni = 10 for all i), so it was assumed that
the standard deviation was the same for treatment groups with similar estimates of the standard
deviation (i.e., SEtJni). The standard deviation was assumed equal for indices 1, 2, 6, and 10 (to
be called Group A), 3, 7, and 11 (Group B), 4, 8, and 12 (Group C), and 5, 9, and 13 (Group D) in
Table A-1. An estimate of the common standard deviation in Group A, cta, is calculated from SEi,
SE2, SEe, and SE10 as follows (estimation of the standard deviations for the other three groups is
similar).
Let oa2 be the common variance for Group A. The variance estimate from Sample 1 is S12 =
niSEi2 = 10 (0.03)2. The estimate of cm2, denoted by Sa2 is the sum of the estimates of a a2 from
samples 1, 2, 6, and 10, weighted by their degrees of freedom (df) (9 for each sample). This gives
Sa2 = 1/36 [9(10)(0.03)2 + 9(10)(0.025)2 + 9(10)(0.025)2 + 9(10)(0.025)2] = 0.00694. The
proportion of Sample 1 with lung weights less than 1.15 g is estimated as follows. IfXis a new
observation for Sample 1, then
has a t-distribution with nA - 4 df, where n1 is the number of observations in Sample 1 (i.e., n1 =
10), and nA is the number of observations in Group A (i.e., nA = 40). Pr (X < 1.15 g) is estimated
by Pr(T < (1.15 - 1,1)/[0.00694 (11/10)0.5] = 0.715, shown in Table A-2 under SevO for "Index" =
1. To estimate Pr (1.15 < X < 1.5), first estimate Pr (X < 1.5) as above, except with 1.15 g
replaced by 1.5 g, and then subtract the estimate of Pr(X < 1.15). Similarly, the relationship Pr(X
> 1.5 = 1 — Pr(X < 1.5) is used to estimate Pr (X >1.5). The estimated proportions of each sample
with lung weights in the intervals <1.15, 1.15 to 1.5, and >1.5 are displayed in Table A-2.
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Categorical Regression (CatReg) User Guide DISTRIBUTION OF CONTINUOUS RESPONSE
DATA OVER SEVERITY LEVELS
Table A-1: Summary Data for "Ref.ld" = 2 in Table 7
Index
Exp.
Group
mg/m3
Hours
Average Lung Weight (X)
Standard Error
1
1
1
330
2
1.1
0.03
2
1
2
360
2
1.2
0.025
3
1
3
390
2
1.5
0.04
4
1
4
410
2
1.8
0.08
5
1
5
460
2
1.7
0.1
6
2
1
460
1
1.2
0.025
7
2
2
510
1
1.3
0.04
8
2
3
560
1
1.6
0.08
9
2
4
610
1
1.8
0.1
10
3
1
560
0.5
1.3
0.025
11
3
2
610
0.5
1.5
0.04
12
3
3
660
0.5
1.6
0.08
13
3
4
710
0.5
1.8
0.1
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Table A-2: Estimated Proportions in Severity Levels
Group
Index
5
%
n + 1
X
1.15 -X
df
SevO
Pr(X<1.15)
1.5-X
Sev1
Pr(1.15 < X < 1.5)
Sev2
Pr(X> 1.5)
n
r n+1
r n+1
5V n
5V n
A
1
0.0874
1.1
0.572
36
0.715
4.577
0.285
0
A
2
0.0874
1.2
-0.572
36
0.285
3.432
0.714
0
B
3
0.1327
1.5
-2.638
27
0.007
0
0.493
0.500
C
4
0.2653
1.8
-2.450
27
0.011
-1.131
0.123
0.866
D
5
0.3317
1.7
-1.1658
27
0.127
-0.603
0.149
0.724
A
6
0.0874
1.2
-0.572
36
0.285
3.432
0.714
0.001
B
7
0.1327
1.3
-1.130
27
0.134
1.507
0.794
0.072
C
8
0.2653
1.6
-4.146
27
0
-0.377
0.354
0.645
D
9
0.3317
1.8
-1.960
27
0.030
-0.904
0.157
0.813
A
10
0.0874
1.3
-1.716
36
0.047
2.288
0.939
0.014
B
11
0.1327
1.5
-2.638
27
0.007
0
0.493
0.5
C
12
0.2653
1.6
-4.146
27
0
-0.377
0.354
0.645
D
13
0.3317
1.8
-1.960
27
0.030
-0.904
0.157
0.813
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Categorical Regression (CatReg) User Guide
TECHNICAL DISCUSSION
APPENDIX B: TECHNICAL DISCUSSION
B.1 Link Functions
Without loss of generality, link functions will be discussed in the context of Model 1, described by
Eq. 1a in Appendix C. specifically,
Pr(T > s\C,T) = H[as + ft * A(C) + ft * f2(T)]
CatReg currently supports three forms for H:
Logistic H(x) = ex I (1 + ex),
1 X ~z2
Normal H(x) = -== f e 2 dzand
v ' y[2n J-°°
Gumbel H(x) = 1 - exp(-ex).
The inverse of H, which is denoted by L, is called the link function in the statistical literature. The
link functions corresponding to the probability functions given above are
Log it L(p) = log[p/(1-p)],
Probit L(p) = 100 pth percentile of normal (0,1), and
Cloglog L(p) = log[-log(1 - p)],
where p is any number between 0 and 1. The link function and probability function are inverse to
each other in the sense that H[L(p)] = p and L[H(x)] = x. Applying the link function to both sides of
Model 1 gives
L[Pr(T > s\C,T)] = as+ ft *A(C) + ft * f2(T),
s = 1,2
This expression shows that the link-transformed probability follows a linear model.
The use of a link function is essential here. Without it, the linear model becomes unbounded, and
one is led to absurd estimates of probabilities for extreme values of C and T, namely negative
probabilities or probabilities greater than 1. Moreover, link functions may be derived from a basic
assumption that the ordinal severity score corresponds to exceeding an underlying toxic response
threshold (see discussion below). Under this approach, the ordinal response score is called a
quantal response because it is a quantization of the underlying response (see, e.g., Morgan,
1992). Any ordinal regression model of the type given in Model 1 may be interpreted as a quantal
response model. When pi and p2 are constant across severity categories data from one severity
category adds information to the modeling of another category.
A toxicological interpretation of the link function follows from quantal response analysis. In
particular, let Z denote a particular measure of health for a randomly selected subject given
exposure to concentration or dose C for time T. Larger values of Z correspond to a healthier
individual. Suppose that the health variable Z is distributed in the population as
Pr(Z < x) = H(-—-)
a
where x is a possible value for an individual's health, |j determines the median health for the
given level of exposure, and a is a measure of the population variation in health. In particular, if H
is Gaussian, then the health levels in the population are distributed normally with mean |j and
standard deviation a. The larger |j is, the healthier the population is at the given level of
exposure.
To model the situation where health deteriorates with increasing exposure, suppose that
f = -/Wi(0- ft */2 (70-
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Further, suppose that a toxic reaction of severity Y> s occurs if the health measure Z is below a
threshold aas, where a is specific to the health measure. Then, under exposure (C, T), the
probability of toxic severity of category s or higher is
Pr(7 < s) = Pr(Z < aas) = H[as + ft * A(C) + ft * f2(T)]
This is precisely the ordinal regression model given in Model 1. It is apparent that the link function
is a reflection of the underlying distribution of Z, which cannot be measured directly, but its
distribution, in particular the dependence on C and T, can be estimated indirectly from the
toxicological response data.
B.2 Interval Censoring
Although censored responses do not provide as much information as fully scored responses, they
do provide some information about the model. This information is used in the maximum likelihood
estimation and the generalized maximum likelihood estimation described in Section B.3. In fitting
the model by maximum likelihood, it is necessary to compute the probability of the observed
response as a function of the model parameters. Table B-1 shows how these probabilities are
computed in a three-category scoring system with interval censoring, using Model 1.
Table B-1: Interval Probabilities for Model 1 with Three Severity Categories
Interval
Probability
(0,0)
1 -H[a1+/31*f1(C)+ /32*f2(T)\
(0,1)
l-ff[a2+/Wi(0+ ft *f2(T)\
(1,1)
HK + ft * ACQ + ft * f2(T)\ - H[a2 + ft * A(C) + ft * f2{T)\
(1,2)
ffK+ft */i(C)+ ft *f2(T)\
(2,2)
H[a2 +ft *A(C) + ft * f2(T)]
B.3 Parameter Estimation
B.3.1 Maximum Likelihood Estimation
The likelihood function is defined to be the joint probability density of the data, viewed as a
function of the parameters. In categorical regression, the response variables are discrete, so the
likelihood may be interpreted as the probability that an investigation would result in the particular
values that were observed. This probability depends on the unknown parameters. Maximum
likelihood estimates the unknown parameters by the values that maximize the likelihood of the
observed data. It is often more convenient to work with the logarithm of the likelihood. For this
purpose, it is common to define the deviance function:
Deviance = -2 * log(likelihood).
The deviance is a nonnegative measure of model fit. Maximizing the likelihood is mathematically
equivalent to minimizing the deviance. The factor 2 is included because it is the correct multiplier
for certain likelihood-based, goodness-of-fit tests. Smaller deviances (larger likelihoods)
correspond to a closer fit of the model to the data. A deviance of 0 would indicate a perfect fit,
that is, a "saturated" model. Generally, a deviance of zero would indicate a model that is too
complicated. The deviance value shown in CatReg summary statistics is the "residual deviance"
discussed in Section 9.0: "ASSESSING MODEL FIT."
If all data are independent, the likelihood function for interval-censored ordinal regression has a
simple form. For/' = 1,2,..., n, let V, denote the ordinal response, and let C,and T,denote the dose
and time of exposure for the /th experimental subject, respectively. Yi may be known only to lie in
an interval. To account for this, let L, and U, denote the lower and upper endpoints of the known
interval for V,, respectively. If it is known that V, = k, then L, = U, = k. For convenient reference,
denote the probability of severity s or greater by
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Pi(s) = ]H[as + ftA(C) + p2f2{Tdl
(. 0,
if s = 0;
if s — 1,
if s > S.
Then the deviance for interval-censored ordinal regression is given by
Deviance = -2log[Pr(Lf < Y, < Ut\Ct,Tt)]
-ZS^loglPiM-Pim + l)]
Parameter estimates are computed by iteratively minimizing the deviance. CatReg uses the R
function optim() to perform this optimization.
B.3.2 Generalized Likelihood Estimation
Weighted ordinal regression analysis corresponds to a modified likelihood in which the probability
associated with the /th observation is raised to a positive power w,. This results in a modified
likelihood with a weighted deviance:
Deviance = -2£f=1Wj log[P;(L;) - P;({/; + 1)]
If the weights do not correspond to incidences, then this likelihood corresponds to a nonstandard
ordinal regression model. In this situation, it is more common to interpret the deviance as a
generalized criterion and to assume that the usual ordinal regression model holds. Under this
assumption, the generalized deviance still leads to consistent estimates of the parameters, but it
does not correspond to the likelihood of the data. Instead, the estimator is defined by a
generalized estimating equation, which provides the basis for computing valid large-sample
confidence intervals and test statistics.
A further modification arises when the data are cluster sampled. The likelihood for cluster-
sampled data does not have the simple form given above. Rather, it involves a product of multiple
integrals of conditional likelihoods. Such likelihoods are computationally challenging and the
results may be sensitive to the specification of the correlation structure. An alternative approach
is to assume the ordinal regression model holds in a population-average sense. Consistent
estimates then may be obtained quite generally, without making extensive distributional
assumptions about the correlation structure. To achieve this, CatReg takes the expression
derived above as a "working deviance" criterion. Minimizing it leads to consistent estimates under
the population-average model. The main impact on the analysis compared to a standard
likelihood analysis is the use of generalized estimating equation methods for making statistical
inferences. In particular, rather than reporting the inverse information matrix as the estimated
parameter covariance, the well-known sandwich formula is used.
Most applications of CatReg will involve cluster-sampled data and possible weighting of
observations. As noted above, CatReg uses the weighted independence criterion as an
estimating criterion, but computes confidence intervals and hypothesis tests without assuming the
criterion is the likelihood. This approach has a long history in the statistical literature. Huber
(1967) derived the large-sample theory of "maximum likelihood" estimators when the working
likelihood is different from the actual likelihood of the data. In the literature on robust statistics,
this type of estimator is called an "M-estimator" because it generalizes maximum likelihood. Liang
and Zeger (1986) extended the method to the analysis of correlated data, based on a "working"
correlation structure, without assuming the working correlation structure was correct. This general
approach is widely used in biometry, econometrics, and survey sampling.
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B.4 Confidence Limit Calculations
CatReg uses the method of generalized estimating equations, which is well accepted in the
literature (see Diggle et al., 1994), for the calculation of confidence limits for cluster-sampled
data. The classical likelihood ratio inferences do not apply to cluster-correlated data because the
likelihood ratio test assumes independent responses. The application of generalized likelihood
ratio tests for correlated data is, however, an area of active research that may produce usable
results in the future.
Confidence intervals and hypothesis tests about the parameters rely on a large-sample normal
approximation to the joint distribution of the parameter estimates. The main steps in the derivation
of this approximation are as follows. First, assuming the data are cluster sampled, write the
generalized deviance as
ZN % irij
) wylog[Py(z,y)-Py([/y+ 1)],
where -2 wylog[Py(iy) - Py(t/y + 1)] is the contribution of the yth individual from cluster i,j =
1,..., rii, i = 1,..., N. Let B denote the vector of all model parameters. The GD estimate of B solves
a generalized estimating equation
iN
0 *
ZN r—% Til
> Vij.
i=l^—'/=!
where the summand „¦((/„ + 1)
and dPij(t) denotes the vector of derivatives of Pij(t), with respect to the components of B,
evaluated at the estimated parameters. Expanding the estimating equation in a Taylor series
leads to a large-sample normal approximation. The estimated parameter vector, B, is
approximately multivariate normal. The mean of the approximating normal distribution equals the
true value of B, and the covariance matrix is given by the sandwich formula
Est.Cov(B) = J^CJ'1
where J is given by
ZN x irij
> dVij,
i=l£—tj=l
and C is a covariance estimate for the total score, given by
If the working likelihood were the actual likelihood of the data, then J and C would estimate the
same matrices, and the usual inverse information, J-\ would estimate the covariance of B.
Further details are given in Simpson et al. (1996a).
Standard errors of individual parameter estimates are obtained as the square roots of the
diagonal elements of the estimated covariance. Confidence intervals and hypothesis tests derive
from the normal approximation for B.
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TECHNICAL BACKGROUND: MODELS
AND EXTRA RISK
APPENDIX C: TECHNICAL BACKGROUND: MODELS AND EXTRA RISK
C.1 Exposure-response Models
Let Ydenote a dependent variable that represents the severity or intensity of the response.
Assume Y is an ordinal score taking one of the values (0,1 A score of 0 corresponds to the
lowest severity (e.g., no adverse effect), and a score of S corresponds to the highest severity
(e.g., lethal, in a toxicological context). Categorical regression is a method for modeling the
probability distribution of Y as a function of the explanatory variables, concentration or dose (C)
and time (T). It employs a generalized linear model (McCullagh and Nelder, 1989) for the
dependence of the probabilities of different severity categories on the explanatory variables.
CatReg provides a choice of two models (with variations on both):
• Model 1 (the cumulative odds model) described by Eq. 1a, and
• Model 2 (the unrestricted cumulative model) described by Eq. 1b.
CatReg refers to any model of the form of Model 1 as a cumulative odds model because the
model is expressed in terms of the cumulative probabilities, or odds, for Y> s. Note that Model 1
is a special case of Model 2 wherein parameters pls and /?2s do not depend on s (which denotes
severity level, as discussed below). A primary use of fitting Model 2 is to test whether the simpler
Model 1 is adequate.
Model 2, although more general than Model 1, has the undesirable feature that the regression
lines for different severity levels may cross. Often the crossing is well outside the range of values
of interest, so the model can be used to make empirical risk estimates. The user has the option to
add an additional parameter, y, which represents a hypothetical background concentration or
dose, in some circumstances (see Section 6.10: "Assume Zero Background Response").
For s = 1,2,...,S,
Model 1
Pr(F >s\C,T) =H[as+p1*f1(C)+ p2*f2(T)\ Eq.1a
Model 2
Pr(Y>s\C,T)=H[as+pls*f1(0+ P2s * f2(T)\ Eq.1b
The left-hand side is read as follows: the probability that a response of severity level s or greater
occurs, given that concentration or dose is C and time is T (time refers to exposure duration). No
expression for s = 0 is included because this is the minimal category, and Y is always greater
than or equal to 0 (i.e., Pr(Y> 0|C,T) = 1).
The right-hand side is described as follows:
• H is a probability function taking values between 0 and 1, for which the user has three
choices: (1) logistic, (2) normal, and (3) Gumbel (described further in Appendix B).
• The parameter as is the intercept for severity level s, s = 2,...,S (to be called the intercept or
severity parameters). The severity parameters are ordered as a±> a2> ¦¦¦ > as. This
constraint is a consequence of the requirement that the probability of exceeding a lower
score is larger than the probability of exceeding a higher score for any fixed levels of C and T.
• In Model 1, the parameter ft determines the dependence of the response on concentration or
soe(C), whereas ft determines the dependence on time (T). In Model 2, the parameters are
also indexed by s because they may change values with severity level s.
• Current choices for/i and f2 are "untransformed" and "base-10 logarithm." Other
transformations of C and T may be obtained by transforming the input data.
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TECHNICAL BACKGROUND: MODELS
AND EXTRA RISK
• Parameters are as, pls (to be called the coefficient of concentration or dose), /?2s (to be called
the coefficient of time or duration), for severity levels s running from 1 to S. All parameters
may be stratified on variables, as discussed under "Stratifying" in Section 6.6.
The normal and logistic distributions are symmetric, each having median equal to zero. The
Gumbel distribution is skewed, with a lower tail similar to that of the logistic distribution and a
lighter upper tail. Figure 27 displays these three probability distributions. To compare the shapes
effectively, the distributions have been rescaled to have medians = 0 and equal 25th percentiles
(labeled as "EC25" on the horizontal axis). The scaled logistic and normal distributions are very
close for much of the range and differ substantively only in the extreme tails. The Gumbel
distribution is skewed, with a heavier tail on the left and a lighter tail on the right.
For each of the three choices of the probability function /-/there is an inverse function of H, called
the link function, that transforms it to a simple linear function in dose and time. CatReg requests
the name of the link function instead of the name of the probability function. The corresponding
link functions (in parentheses) are: logistic (logit), normal (probit), and Gumbel (cloglog). There is
further discussion of linking in Section 6.8 and Appendix B; the latter includes an example of how
link functions may be derived from a basic assumption that the ordinal severity score corresponds
to exceeding an underlying toxic response threshold.
Figure 27.—Normal, logistic, and Gumbel probability functions shifted and scaled to have equal medians
and 25th percentiles.
_Q
.a
o
CO
o
<£>
o
o
S|C = C,T = t)-Pr(Y>s\C=0,T=t)
l-Pr(Y>s\C=0,T=t)
Eq. 2
Forq between 1 and 100, inclusive, ERCq, at time T = t, is the concentration c for which equation
(4-2) equals q/100. For example, ERC10 at T = 2 (exposure time of 2 hours) for severity level 1 is
the value of c that satisfies
Pr(V > 1|C = c,T = 2)-Pr(y>i|c=o,r=2)
i-Pr(y>i|c=o,r=2)
Eq. 3
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AND EXTRA RISK
In words, ERC10 at T = 2 for severity level s is the exposure concentration or dose at which the
probability is 0.10 of an adverse effect of level s or higher due to exposure of two hours, i.e.,
given the adverse effect would not have occurred from other causes ("background causes")
during that time.
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