United States
Environmental Protection
Agency
Office of Water
(4303)
EPA-821-B-97-006
January 1998
Statistical Support
Document for Proposed
Effluent Limitations
Guidelines and Standards
for the Landfills
Point Source Category
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STATISTICAL SUPPORT DOCUMENT
FOR PROPOSED EFFLUENT LIMITATIONS GUIDELINES AND STANDARDS
FOR THE LANDFILLS POINT SOURCE CATEGORY
(EPA 821-B-97-006)
U.S. Environmental Protection Agency
Office of Water
Engineering Analysis Division (Mail Code 4303)
401M Street, SW
Washington, DC 20460
January, 1998
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ACKNOWLEDGEMENTS AND DISCLAIMER
This report has been reviewed and approved for publication by the Engineering Analysis Division, Office
of Science and Technology, U.S. Environmental Protection Agency. This report was prepared with the
support of Science Applications International Corporation under Contract 68-C4-0046, under the direction
and review of the Office of Science and Technology. Neither the United States government nor any of its
employees, contractors, subcontractors, or other employees makes any warranty, expressed or implied, or
assumes any legal liability or responsibility for any third party's use of, or the results of such use of, any
information, apparatus, product, or process discussed in this report, or represents that its use by such a third
party would not infringe on privately owned rights.
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TABLE OF CONTENTS
1. Overview of BPT Statistical Analyses
1
2. Description of Data Sources 2
3. Description of Data Conventions 2
3.1 Data Review 3
3.2 Data Types ...., 3
3.3 Data Aggregation 3
3.3.1 Data Aggregation Across Multiple Grabs and Field Duplicates 4
3.3.2 Aggregation of Multiple Grab Samples 4
3.3.3 Aggregation of Duplicates 5
4. Statistical Methodology
••- • 6
4.1 Overview of Methodology and Applicability to the Landfills BPT Data Base 6
4.1.1 Basic Overview of Delta-lognormal Distribution 6
4.1.2 Modifications to the Adapted Delta-Lognormal Model 9
4.1.3 Modification of the Discrete Spike 10
4.1.4 Modification of the lognorinal portion 11
4.2 Estimation Under the Modified Delta-Lognormal Model 12
4.2.1 Estimation of Long-Term Averages 13
4.2.2 Estimation of Variability Factors, Percentiles, and Limitations 14
4.2.3 Estimation of Facility-Specific 1-Day Variability Factors and 99th Percentiles
• • 14
4.2.4 Estimation of Facility-Specific 4-Day Variability Factors and 95th Percentiles of 4-
Day Averages 15
4.2.5 Estimation of Facility-Specific 20-Day Variability Factors and 95th Percentiles of
20-Day Averages 19
4.2.6 Estimation of 30-Day Variability Factors and 95th Percentiles of 30-Day
Averages 20
5. Estimation of Proposed Daily Maximum and Monthly Average Limitations 21
6. References 21
Appendix A. Proposed Limitations
Appendix B. Facilities, Sample Points, and Data Sources
Appendix C. Facility-Specific Statistics
Appendix D. Listing of Data Used to Develop Effluent Limitations
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1. Overview of BPT Statistical Analyses
The Development Document for Proposed Effluent Limitations Guidelines and Standards for the Landfills
Point Source Category (EPA 821-R-97-022) describes the evaluation and selection of (a) pollutants of
concern, (b) pollutants proposed for limitations, (c) treatment technology options, and (d) facilities and
sample points for which effluent sampling data was used to develop the proposed limitations. In general,
facilities are selected on the basis of the types of treatment systems in place and having treatable
concentrations of pollutants in the waste stream influent to the treatment system. Sample points (for
analyses leading to limitations, reported herein) are selected to obtain effluent from the treatment process
or combination of processes representing BPT treatment.
This document discusses in detail how effluent sampling data for selected pollutants and facilities were
used to develop the proposed BPT effluent limitations. The document provides an overview of the
statistical analyses; describes the sources of data; describes the modified delta-lognormal distribution
which was used to derive the proposed limitations; and evaluates an alternative statistical model which was
considered in developing the limitations.
EPA proposed effluent limitations for two subcategories: Non-Hazardous (Subtitle D), and Hazardous
(Subtitle C). For each subcategory, EPA evaluated a number of technology options, and proposed BPT
limitations guidelines for one selected technology option in each subcategory. Proposed BAT treatment
was based upon BPT treatment. The proposed BPT option for the Non-Hazardous subcategory was
equalization plus biological treatment, and for the Hazardous subcategory the proposed option was
equalization plus biological treatment plus metals precipitation. For the Non-Hazardous landfills
subcategory, EPA proposed limitations for the following pollutants: alpha-terpineol, ammonia, benzoic
acid, BOD5, p-cresol, phenol, toluene, TSS, and zinc. For the Hazardous landfills subcategory, EPA
proposed limitations for the following pollutants: alpha-terpineol, ammonia, aniline, arsenic, benzoic acid,
BOD5, chromium, naphthalene, p-cresol, phenol, pyridine, toluene, TSS, and zinc. Limits were also set for
pH, not reported here, based on engineering judgement and common practice.
Having selected BPT facilities, sample points, and pollutants, calculations were made as follows,
separately for each subcategory. Sampling data were aggregated, if necessary, by combining duplicates
and grab samples, to produce one concentration value per day. Then daily sampling data were used to
produce summary statistics and parameter estimates (long-term averages or LTAs, and variability factors or
VFs) for each combination of facility, sample point, and pollutant. These statistics and estimates are
referred to herein as "facility-specific statistics" or "facility-specific estimates". Then, for each pollutant,
LTAs and VFs were combined (aggregated) for all facilities (usually there is only one effluent sampling
point per facility at this stage of analysis). The median of the facility-specific LTAs was used as the
(aggregate) long-term average for that pollutant within a subcategory. The average of the reported facility-
specific VFs was used as the (aggregate) variability factor for that pollutant within a subcategory. Each
Limitation is the product of an aggregate long-term average and an aggregate variability factor. Details of
these statistical procedures and calculations are provided below.
Different variability factors must be estimated for different proposed frequencies of monitoring. Typical
monitoring frequencies are daily (weekdays; about twenty days per month) and weekly (about four days
per month). EPA establishes maximum limitations for monthly averages as well as for daily
measurements. Herein, monthly average limitations were proposed for only one frequency of monitoring,
either four or twenty days per month, for each pollutant.
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2. Description of Data Sources
The data used to calculate the proposed limitations were derived from two sources: (1) EPA's sampling
effort ("SCC" data), and (2) self-monitoring data submitted by landfill facilities in response to a detailed
monitoring questionnaire ("DMQ").
"SCC" refers to EPA's Sample Control Center. The SCC data base contains the results of five-day
sampling episodes, conducted or supervised by EPA. "Episode" describes a sampling program conducted
on five consecutive days at a particular facility. These SCC sampling episodes were conducted at six
facilities between April 1995 and September 1995. For SCC data, five analytical data values were
available in most cases for each pollutant at each sampling point. Three of these facilities were selected to
represent BPT treatment, and were used as a source of data for developing limitations (Appendix B).
For all pollutants but "oil & grease", each daily sample was a 24-hour composite sample (six grab samples
were taken once every four hours over a 24-hour period, and were physically combined). Grab samples to
be analyzed for "oil & grease" were taken on the same schedule, but were not physically combined;
analytical data for these grab samples were arithmetically aggregated into a daily value as described later
(however, no limitations were proposed and thus no statistics are presented herein for "oil & grease"). At
one facility (identified as SCC number 4721), SCC samples were taken for only four consecutive days
because the landfill used a batch treatment system that made four batch treatment runs each week.
The second source of data was self-monitoring data supplied by twenty-six facilities in response to a
questionnaire ("DMQ", Detailed Monitoring Questionnaire) sent to selected facilities by EPA in 1995.
This data covered one or more years hi the range 1992-1994, depending on the facility and sampling point.
The sampling frequency varied from daily to monthly depending on the facility, sampling point, and
pollutant. The facilities to be surveyed were selected using information they supplied in the Detailed
Questionnaire, and were chosen to represent possible BPT and BAT treatment technologies. Eight of these
facilities were selected to represent BPT treatment, and were used as a source of data for developing
limitations (Appendix B).
In some cases, data from these two sources (SCC and DMQ) were available for the same facility. In such
cases, data from the two sources were analyzed separately when estimating averages and variability factors.
A listing of the data used to support BPT limitations development can be found in Appendix D. Extensive
documentation of the data quality reviews can be found in the record for the notice of proposed
rulemaking.
3. Description of Data Conventions
This section discusses the types of data in the landfills analytical data base and the hierarchy and
procedures for data aggregation.
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3.1 Data Review
The analytical sampling data in the analytical data bases were thoroughly reviewed by EPA. During this
review, the integrity of each sample was assessed to ensure that all specifications of the sampling protocol
were met. The reviewers determined that some samples should be excluded from the analyses. These
samples were flagged in the data base in a field labeled "SCC Qualifier." Samples with flags of
"EXCLUDE" or "DETECTED" (a value was detected but the concentration value was not recorded) were
excluded from analyses.
Also, during the data review, several samples were qualified as greater than, ">", the concentration value
reported in the data base. These samples were handled as right-censored samples in the analyses with a
lower bound equal to the reported concentration.
An engineering review of the data base also was conducted and a few additional data values were excluded
from the analyses for the reasons summarized in the record for the proposed rulemaking.
3.2 Data Types
The landfills analytical data base (from the SCC sampling effort and the questionnaire) contains the
following three different types of samples delineated by certain qualifiers in the data base:
• Noncensored (NC): a measured value, i.e., a sample measured above the minimum level of detection
for the specified analytical method
• Nondetect (ND): samples for which analytical measurement did not yield a concentration above the
sample-specific detection limit. These detection limits were either the Minimum Level (ML), or, for
metals, the Instrument Detection Limit (IDL). Statistical analyses used non-detect values as they were
reported in datasets.
• Right-censored (RC): these samples were qualified with a greater than (>) sign, signifying that the
reported value is considered a lower limit of the actual concentration.
The landfills effluent concentration data were characterized by a large number of measurements reported as
below the detection limit (ND). These detection limits were sample specific and, for many pollutants,
covered a wide range of values.
3.3 Data Aggregation
Data aggregation for the landfills analytical data was performed in two situations, (1) when there were
multiple grab samples for a given pollutant, sample date and sample point, and (2) when duplicate samples
(field duplicates) were collected.
In SCC sampling episodes, six grab samples were taken once every four hours over a 24-hour period.
These were physically combined into one daily, 24-hour composite sample in the case of all analytes
except oil & grease, for which the six grab samples were chemically analyzed separately. For oil & grease,
analytical data for the grab samples were arithmetically aggregated into a daily value as described below. '•
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Field duplicates are defined as samples collected at a particular sampling point at approximately the same
time on a given day, assigned different sample numbers, and flagged as duplicates for a single episode
number.
Data are aggregated in order to assign one concentration value to each sample point on a given sampling
date. The arithmetic average of measurements for four grab samples, taken six hours apart over a 24-hour
period, should approximate in value and precision a measurement made on a composite sample (a similar
set of four grab samples combined physically before analysis). When samples have been collected in both
these ways, a consistent and meaningful comparison of data requires arithmetic compositing (aggregation)
of grab samples over the 24-hour period so these can be compared with physically composited sample data.
3.3.1 Data Aggregation Across Multiple Grabs and Field Duplicates
If a given sample date and sample point ("sample set") had both multiple grabs and field duplicates, the
multiple grabs were aggregated first, then duplicates were aggregated. When all of the samples in a set
were noncensored, detected samples, the arithmetic average of the samples was used. When one or more
of the samples was nondetect or right-censored, the following methods were used to combine data. When
a noncensored (NC) sample and a nondetected (ND) sample were combined, the sample was labeled mid-
censored (MC), that is, a censored sample whose true value lies between two non-zero bounds (lower and
upper). For instance, the lower bound of the average is probably not zero (since one of the samples was
detected), but instead would equal the average of the NC and 0 (the lowest possible value of the
nondetect). Similarly, the upper bound would equal the average of the NC and the detection limit of the
ND (the highest possible value of the nondetect). Thus, the lower and upper bounds for this type of mid-
censored data point are
lower: NC/2
upper: (NC + ND)/2
where the value of ND is the detection limit for the nondetected sample, and the value of NC is the
observed concentration value. Tables 3-1 and 3-2 and the following two sections outline in greater detail
the methods for combining multiple grabs and field duplicate samples, respectively, for the statistical
analyses.
3.3.2 Aggregation of Multiple Grab Samples
If a sample set had both multiple grabs and field duplicates, the multiple grabs were aggregated first.
Within the SCC database, three pollutants, Oil and Grease (as HEM), Total Petroleum hydrocarbon (as
SGT-HEM), and Total Recoverable Oil and Grease, reported concentrations for multiple grab samples
taken during one-day sampling periods.
Multiple grab samples were aggregated within each sampling day/sample point combination. The
aggregation of the multiple grab samples was performed as identified in Table 3-1.
3.33 Aggregation of Duplicates
Another type of data aggregation for the Landfill SCC data was performed due to the identification of
duplicates in the data base. The duplicates are assumed to be field duplicates, which are defined as one or
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more samples collected for a particular sampling point at approximately the same time, assigned different
sample numbers, and flagged as duplicates for a single episode number/sampling point. The aggregation
method described in Table 3-2 was used:
Table 3-1.
Method.for Averaging Lab Duplicate Samples
If observations are:
AllNDs
AHNCs
AllRCs
NDandNC
ND and RC
NCandRC: NC>RC
NC and RC: NCXRC
Label of
"average"
ND
NC
RC
MC
RC
NC
RC
Value of "average" is:
mm(ND,,ND2)
(NC,+NC2)/2 :
max(RC,5RC2)
lower bound: NC/2
upper bound: (ND+NC)/2
(CND/2)+RC)/2
NC
(NC+RC)/2
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Table 3-2.
Method for Averaging Field Duplicate Samples
If observations are:
A11NC
A11ND
A11RC
All MC values
NCandND
NCandRC
NDandRC
NCandMC
MCandND
MC and RC where
MC,, > RC
MC and RC where
MC,,
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reported detection limit. In this sense, nondetect measurements represent in statistical terms what are
known as censored samples.
In general, censored samples are measurements bounded either by an upper or lower numerical limit for
which the exact value is not known. Nondetects qualify in this framework as left-censored samples,
which have an upper bound at the detection limit and a lower bound at zero. To model nondetects as left-
censored samples under a strictly lognormal density model, it is necessary to assume that the exact (but
unknown) values of these measurements follow the same lognormal distributional pattern as the rest of the
detected measurements and that they are positively-valued (i.e., greater than zero).
For all these reasons, two reasonably simple modifications to the lognormal density model have been used
by EPA for several years. The first modification is known as the classical delta-lognormal model, first
used in economic analysis to model income and revenue patterns (see Aitchison and Brown, 1969). In this
adaptation of the simple lognormal density, the model is expanded to include zero amounts. To do this, all
positive (dollar) amounts are grouped together and fit to a lognormal density. Then all zero amounts are
segregated into another group of measurements representing a discrete distributional "spike" at zero. The
resulting mixed distribution, combining a continuous density portion with a discrete-valued spike, is
known as the delta-lognormal distribution. The delta in the name refers to the percentage of the overall
distribution contained in the spike at zero, that is, the percentage of zero amounts.
Kahn and Rubin (1989) further adapted the classical delta-lognormal model ("adapted model") to account
for nondetect measurements in the same fashion that zero measurements were handled in the original delta-
lognormal. Instead of zero amounts and non-zero, positive amounts, the data consisted of nondetects and
detects. Rather than assuming that nondetects represented a spike of zero concentrations, these samples
were allowed to have a single positive value, usually equal to the minimum level of the analytical method.
Since each nondetect was assigned the same positive value, the distributional spike in this adapted model •
was located, not at zero, but at the minimum level. This adapted model was used in developing limitations
for the OCPSF and pesticides manufacturing rulemaking.
In the adapted delta-lognormal model, the delta again referred to those measurements contained in the
discrete spike, this time representing the proportion of nondetect values observed within the data set. By
using this approach, computation of estimates for the population mean and variance could be done easily
by hand, and nondetects were not assumed to follow the same distributional pattern as the detected
measurements. The adapted delta-lognormal model can be expressed mathematically as
Pr(Uzu) =
(1 -5) d> [(log(u) - (j)/oj
5 t (1 -5) D
(1.1)
where d represents the true proportion of nondetects (or the probability that any randomly drawn
measurement will be a nondetect), D equals the minimum level value of the discrete spike assigned to all
nondetects, <&(•) represents the standard normal cumulative distribution function, and jj. and a are the
parameters of the lognormal density portion of the model.This model assumes that all nondetected values
have a single detection limit D.
It is also possible to represent the adapted delta-lognormal model in another mathematical form, one in
which it is particularly easy to derive formulas for the expected value (i.e., long-term average [LTA]) and
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variance of the model. In this case, a random variable distributed according to the adapted delta-lognormal
distribution can be represented as the stochastic combination of three other independent random variables.
The first of these variables is an indicator variable, Iu, equal to 1 when the measurement u is a nondetect
and equal to 0 when u is a detected value. The second variable, XD, represents the value of a nondetect
measurement (discrete). In the adapted delta-lognormal, this variable is always a constant equal to the
concentration value assigned to each nondetect (i.e., equal to D in the adapted delta-lognormal model). In
general, however, XD need not be a constant, as will be seen below in the modified delta-lognormal model.
The final random variable, Xc, represents the value of a detected measurement, and is .distributed
according to a lognormal distribution (continuous) with parameters u and a.
Using this formulation, a random variable from the adapted delta-lognormal model can be written as
= /„ XD + d ~IU}XC
(1-2)
and the expected value of U is then derived by substituting the expected value of each quantity in the right-
hand side of the equation. Because the variables Iu, XD, and Xc are mutually independent, this leads to the
expression
E(U) = 5E(XD)-K1-5)E(XC) = 6D + (1 -5)exp(n + 0.5o2) (1.3)
where again 8 is the probability that any random measurement will be nondetect and the exponentiated
expression is the familiar mean of a lognormal distribution. In a similar fashion, the variance of the
adapted delta-lognormal model can be established by squaring the expression for U above, taking
expectations, and subtracting the square of E(U) to get
Var(U) = E(UZ) -
(1-5)l/ar(Xc) + 5(1 -5)[E(XD)-E(XC)]2. (1.4)
Since, in the adapted delta-lognormal formulation, XD is a constant, this expression can be reduced to the
following:
Var(U) = (1 -5)exp(2M+a2)[exp(o2)-(1 -5)] + 5(1 -5)D[D -2exp(y +0.5O2)]. (1.5)
In order to estimate the adapted delta-lognormal mean and variance from a set of observed sample
measurements, it is necessary to derive sample estimates for the parameters 8, u, and a. 8 is typically
estimated by the observed proportion of nondetects hi the data set. u and o are estimated using the logged
values of the detected samples where u is estimated using the arithmetic mean of the logged detected
measurements and a is estimated using the standard deviation of these same logged values; nondetects are
not included in the calculations. Once the parameter estimates are obtained, they are used in the formulas
above to derive the estimated adapted delta-lognormal mean and variance.
To calculate effluent limitations, it is also necessary to estimate upper percentiles from the underlying data
model. Using the delta-lognormal formulation above in equation (1.1)', letting Ua represent the 100*ath
percentile of random variable U, and adopting the standard notation of zs for the s* percentile of the
standard normal distribution, an arbitrary delta-lognormal percentile can be expressed as the following:
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exp(M+ozctf1_B) If (1-5)cD((|0g(D)-M)/a) sa
exp(|j+a za.B/1.5) if 5+(i-5)
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censored samples as well as noncensored samples (i.e., those detected measurements associated with
"exact" or known concentration values).
The necessity of these modifications became apparent upon examination of time series and probability
plots for pollutants of concern in the landfills data base.
4.1.3 Modification of the Discrete Spike
To appropriately modify the adapted delta-lognormal model for the observed landfills data base, the first
modification was made to the discrete, single-valued spike representing nondetect measurements. Because
nondetect samples have varying detection limits, the spike of the delta-lognormal model has been replaced
by a discrete distribution made up of multiple spikes. Each spike in this modification is associated with a
distinct detection limit observed in the landfills data base. Thus, instead of assigning all nondetects to a
single, fixed value, as in the adapted model, nondetects can be associated with multiple values depending
on how the detection limits vary.
In particular, because the detection limit associated with a nondetect sample is considered to be an upper
bound on the true value, which could range conceivably from 0 up to the detection limit, the modified
delta-lognormal model used here assigns each nondetect sample to its reported detection limit.
Once each nondetect has been associated with its reported detection limit, the discrete "delta" portion of
the modified model is estimated in a way similar to the adapted delta-lognormal distribution; only now,
multiple spikes are constructed and linked to the distinct detection limits observed in the data set. In the
adapted model, the parameter 8 is estimated by computing the proportion of nondetects. In the modified
model, 8 again represents the proportion of nondetects, but is divided into the sum of smaller fractions, 6;,
each representing the proportion of nondetects associated with a particular and distract detection limit.
Thus, it can be written
• -•»
(1.7)
If D( equals half the value of the Ith smallest distinct detection limit in the data set, and let the random
variable X represent a randomly chosen nondetect sample, then the discrete distribution portion of the
modified delta-lognormal model can be mathematically expressed as
The mean and variance of this discrete distribution (unlike the adapted delta-lognormal) also can be
computed with the variance of the modified spike being non-zero, using the following formulas:
0
and
Var(XD) =
(1.9)
It is important to recognize that, while replacing the single discrete spike in the adapted delta-lognormal
distribution with a more general discrete distribution of multiple spikes increases the complexity of the
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model, the discrete portion with multiple spikes plays a role in limitations development identically parallel
to the single spike case and offers flexibility for handling multiple observed detection limits.
4.1.4 Modification of the lognormal portion
To accommodate detected observations that are censored in some fashion, the lognormal portion of the
adapted delta-lognormal model also has been modified. A lognormal density is still used to represent the
set of detected measurements, but the manner of estimating the distributional parameters, u and o, has
been changed to allow for censored observations. In general, the method typically used to estimate the
parameters of the underlying lognormal density is known as maximum likelihood estimation (MLB). The
MLE method is based on the assumption that a group of independent observations follows a particular
distributional pattern, in this case the lognormal, and computing for each observed value the probability of
its occurrence or "likelihood" under the assumed model. By multiplying the likelihoods of all the
observations together, one can compute the overall likelihood of that particular group of data values.
Since the goal is to find the specific parameters of the distributional model that "best fit" or are most
consistent with the observed data, and these parameters are unknown, the overall product of individual
likelihoods becomes a function of the unknown parameters. Each time a different set of possible
parameters is plugged into the likelihood function, a different numerical value for the overall likelihood
results. The method used to find the best set of parameters relies on maximizing tiiis overall likelihood or
probability of occurrence, as a means of finding the specific probability distribution most consistent with
the data. In other words, the best fitting distribution should be the one that has the best chance of
generating the specific data observed.
In the case of the lognormal distribution, when none of the observed data is censored in any way, the
general overall likelihood function (L.F.) can be written as
L.F.= TI -L-
x,a
= (2n)
exp -
1
202'
(1.10)
where ((>(•) represents the standard normal probability density, and the X; are the observed data values. To
maximize this function with respect to the unknown parameters u. arid a, it is necessary to differentiate the
right-hand side and equate the result to 0, first with respect to n and then with respect to a.
As noted previously, if none of the detected measurements are censored, as in the adapted model,
calculation of the u and a estimates can be performed by hand using the log-transformed detected values.
This is possible because, when each value X; is known, the first derivatives of the likelihood function can
be explicitly solved for the unknown parameters, u and a. In fact, the solutions are essentially just the
mean and standard deviation, respectively, of the logged data values.
However, when some of the data observations are censored measurements and their values are not known
explicitly, simple "closed-form" solutions to the derivatives of the likelihood function cannot be obtained.
The likelihood function in this case has a slightly different form, depending on the pattern of censoring in
the data, but is one that in general involves a mixture of standard normal densities and cumulative normal
distribution terms.
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Maximizing the overall likelihood in this case requires the use of iterative numerical "search" techniques,
available via computer programs using a nonlinear maximization algorithm. The algorithm tests a series of
plausible parameter choices in the likelihood function until it can determine the approximate choice that
maximizes the overall probability of occurrence. The parameters that best maximize the overall likelihood
are MLEs. Once the MLEs are determined, these parameter estimates can be used to compute estimates
for the mean, variance, and upper-percentiles of either the continuous lognormal portion or the modified
delta-lognormal model as a whole.
While the MLEs are unbiased estimates of the distributional parameters p. and a, OMLE is a biased estimate
for a sample. As such, a,^ is adjusted to an unbiased sample estimate by multiplying o2MLE by (n/n-1),
°2AMLE- *n particular, once MLEs for u and a are determined using a lognormal density likelihood, the
mean and variance for the continuous portion of the modified delta-lognormal have exactly the same form
as the usual mean and variance of a lognormal distribution estimated with all noncensored data. The only
difference is that the MLEs for u and OAMLE are used hi place of the mean and standard deviation of the
logged data values. Expressions for the mean and variance of the lognormal portion of the model then take
the following form:
E(XC) =
V(XC) =
2n-l
and
- 1)exp(2MML£ + -^j-02AMLE).
(1.11)
4.2 Estimation Under the Modified Delta-Lognormal Model
Once the two basic modifications to the adapted delta-lognormal distribution are made, it is possible to fit a
wide variety of observed effluent data sets to the modified model. Multiple detection limits for nondetects
can be handled, as can detected samples with censored measurements. The same basic framework can be
used even if there are no nondetect values or censored data. Thus, the modified delta-lognormal model
offers a large degree of flexibility in modeling effluent data.
Combining the discrete portion of the model with the continuous portion, the cumulative probability
distribution of the modified delta-lognormal model can be expressed as follows, where Dn denotes half of
the largest distinct detection limit observed among the nondetects, and where the first summation is taken
over all those values D; that are less than u:
Y, 5,i-(1-5)[(log(ii)-M)/o)] if u
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where this time XD represents a random nondetect from the discrete portion of the model, Xc represents a
random detected measurement from the continuous lognormal portion (possibly a censored measurement),
and Iu is an indicator variable signaling whether any particular random measurement is detected or not.
Then the expected value and variance of U have forms somewhat similar to the standard delta-lognormal
model, namely
E(U\ = £5,0, + (1 -5)exp((j+0.5o2)
(1.14)
Var(U) =
5(1 -5)
-5)exp(2|j+o2)(exp(o2)-1)
-exp(|j+0.5o2)
(1.15)
where the D, equal half the individual detection limits for the nondetects, the 8; are the corresponding
proportions of not detected values with detection limit 2D;, and 8 = S8;.
4.2.1 Estimation of Long-Term Averages
For the purposes of estimating these long-term averages (equal to the expected value in the equation
(1.14)),it was necessary to divide the landfills data sets into three groups based on their size (number of
samples) and the type of samples in the subset. Thus, the computations differed for each of the following
groups:
Group 1: Less than 2 detected samples (NC,MC), or less than 4 total samples
Group 2: Two or more detected samples (NC,MC), but less than 2 noncensored samples (NC)
Group 3: Two or more noncensored samples (NC), and 4 or more total samples.
For Group 1, the long-term averages were calculated as the arithmetic average of the samples, since the
sample sizes for either the discrete portion or the continuous lognormal portion of the data were too small
to.allow distributional assumptions to be made. For the estimation of the long-term averages for Group 1,
nondetects were set to the detection limits, NC values remained the same, and MC values were set to the '
midpoint between the upper and lower bounds.
For Group 2, the long-term averages were calculated using the formula for E(U) in equation (1.14).
However, since the number of noncensored (NC) data values is one or none, the MLE method was
replaced by simply estimating the u and a parameters with the mean and variance of the logged NC values,
RC values, and MC midpoint values. The MLE method was not used because the information going into
the likelihood function for censored data is specified as a range, and if these ranges are not augmented by
two or more exact (noncensored) data values, then the algorithm may generate unreliable or nonconvergent
estimates.
13
-------
For Group 3, the long-term averages were calculated using the procedures outlined in the preceding section
using equation (1.14) and the MLEs for \i and o.
4.2.2 Estimation of Variability Factors, Percentiles, and Limitations
After determining estimated long-term average values for each pollutant at a particular sample point and
facility, EPA developed 1-day variability factors (VF1) for each pollutant and either 4-day or 20-day
monthly average variability factors (VF4 and VF20), dependent on the proposed frequency of monitoring.
Similar to the calculations for the long-term averages, the data were divided into the same three
computation groups based on the number and type of samples in each data subset as follows:
Group 1: Less than 2 detected (NC,MC) samples, or less than 4 total samples. Upper percentiles
and variability factors could not be computed using the modified delta-lognormal
methodology.
Group 2: Two or more detected samples (NC,MC), but less than 2 noncensored samples (NC). The
estimates of the parameters for the modified delta-lognormal distribution were calculated
empirically in the log-domain using the detection limit for nondetects and the average of
the upper and lower bounds for mid-censored values. Upper percentiles and variability
factors were calculated using these estimated parameters.
Group 3: Two or more noncensored samples (NC), and 4 or more total samples. The estimates of
the parameters for the modified delta-lognormal distribution of the data were calculated
using maximum likelihood estimation in the log-domain incorporating all types of
censoring. Upper percentiles and variability factors were calculated using these estimated
parameters.
Several data subsets belong in Group 1, and therefore have no estimate for the 99th percentile and
variability factor.
4.23 Estimation of Facility-Specific 1-Day Variability Factors and 99th Percentiles
The 1-day variability factors are a function of the long-term average, E(U), and the 99th percentile. An
iterative approach was used in finding the 99th percentile of each data subset using the modified delta-
lognormal methodology by first defining D0=0, S0=0, and Dk+1 = °° as boundary conditions, where D; equals
the i* smallest detection limit, and 8; is the associated proportion of nondetects at the i* detection limit. A
cumulative distribution function, p, for each data subset was computed as a step function ranging from 0 to
1. The general form, for a given value c, is:
;=0
Dm <, c < D
m+1'
(1.16)
14
-------
where <2> is the standard normal cumulative distribution function. The following steps were completed to
compute the estimated 99th percentile of each data subset:
1. k values of p at c=Dm, m=l,...k were computed and labeled pm.
2. The smallest value of m, such that pm > 0.99, was determined and labeled as Pj. If no such m
existed, steps 3 and 4 were skipped and step 5 was computed instead.
3. Computed p* = PJ - 8j.
4. If p*< 0.99, then P99 = Dj,
else if p*> 0.99, then
P99=exp
0.99-£ 8,
k J=0 ,
d -
5. If no such m exists, such that pm > 0.99 (m=l,...k), then
P99 =exp
h-1
0.99-6
(1-6)
(1.17)
(1.18)
The daily variability factor, VF1, was then calculated as:
P99
VF1 =
E(U)
(1.19)
4.2.4 Estimation of Facility-Specific 4-Day Variability Factors and 95th Percentiles of 4-Day
Averages
It was necessary to calculate a variability factor for monthly averages based on the distribution of 4-day
averages, because EPA is proposing that some pollutants will be monitored weekly (approximately 4 times
a month). In order to calculatejhe 4-day variability factor (VF4), the assumption was made that the
approximating distribution of U4, the sample mean for a random sample of 4 independent concentration
values, also is derived from this modified delta-lognormal distribution, with the same mean as the
distribution of the concentration values. The mean of this distribution of 4-day averages is (Kahn and
Rubin, 1989):
E(U4) = 54E(X4)
-54)£(X4)C
(1.20)
where (X4)D denotes the mean of the discrete portion of the distribution of the average of four independent
concentration values (i.e., when all observations are not detected), and (X4)c denotes the mean of the
continuous lognormal portion of the distribution. ,
15
-------
First, it is assumed that the probability of detection (8) on each of the four days is independent of that on
the other days, since these samples are not taken on consecutive days and are therefore not correlated such
that 64 = 64. Also, since ECX^ = E(XD), then
5-D
(1.21)
and since E(04) = E(U), then
M4 =
-54)
-0.50%.
(1.22)
The expression for a\ was derived from the following relationship:
Var(U4) = 54Va/t(X4)D) + (1 -54)Var((X4)c) + 54(1 -54)[E(X4)D-E(X4)C]2.
(1.23)
Since
—
Var((X4)D)=
VartX
^
), and 64=5
=4
then,
Var(Uj =
This further simplifies to
k k
Var(UJ = —^-
-54)
+ (1-54)exp(2|J4+o24)[exP(o24)-1]
452
I2
E-^ -exp(M4+0.5o24)
f=i 6 J
and, furthermore,
(1.24)
4)c) + 54(1 -64)[E(XD)-E(X4)cf. (1.25)
(1.26)
16
-------
exp(o24)-1 =
Then, from (1.21) above,
-52(1 -54) £ 5,£>.-5exP(M4 + 0.5o24)
(1-54)exp(2jJ4+024)
(1.27)
exp(M4+0.5G24) =
-54)
since E(t/J=E(l/)
4
(1'28)
and letting
H = E(l/)-58£5,D, then expdJ4+0.5o2 ) = —Q_
' (1 -54)
1=1
(1.29)
Furthermore,
O24=log
1 +
d-54)n2
(1-54)2
Since Var(U4) = Var(U)/4, and by rearranging terms,
024 = log
k k
52
(1.30)
(1.31)
Thus, estimates of u4 and a4 were derived by using estimates of 81,...6k (sample proportion of nondetects at
observed detection limits D,,...Dk), u (MLE of logged values), and a2 (MLE logvariance) in the equations
above.
In finding the estimated 95* percentile of the average of four observations, four nondetects, not all at the
same detection limit, can generate an average that is not necessarily equal to Dl5 D2,..., or Dk.
Consequently, more than k discrete points exist in the distribution of the 4-day averages. For example, the
17
-------
average of four nondetects at k=2 detection limits are at the following discrete points with the associated
probabilities:
D",
6*;
1
2
3
4
5
DI
(3D, +D2)/4
(2Dn+2D2)/4
(D1+3D2)/4
D,
V
45^5,
6512Q2:
45A8
s,4
In general, when all four observations are not detected, and when k detection limits exist, the multinomial
distribution can be used to determine associated probabilities, that is,
Pr
^4=-
4!
U1!u2!...i
(1.32)
The number of possible discrete points, k*, for k= 1,2,3,4, and 5, are given below:
k
1
2
3
4
5
k!
1
5
15
35
70
To find the estimated 95th percentile of the distribution of the average of four observations, the same basic
steps as used for the 99th percentile of the distribution of daily observations, were followed with the
following changes:
1. Change Pgg to P95, and 0.99 to 0.95.
2. Change Dm to Dro*, the weighted averages of the detection limits.
3. Change 8; to 8i*.
4. Change k to k*, the number of possible discrete points based on k. detection limits.
5. Change the estimates of 8, \L, and a to estimates of 64, u4, and o4, respectively.
18
-------
Then, the estimate of the 95th percentile 4-day mean variability factor is as follows:
P95
VF4 =
since E(U4) = E(U).
(1.33)
4.2.5 Estimation of Facility-Specific 20-Day Variability Factors and 95th Percentiles of 20-Day
Averages
Because some pollutants (BOD5 and TSS) were proposed to be monitored daily, the monthly average
limitations assume that monitoring will be conducted for 20 days per month. Concentration values
measured on consecutive days are likely to be positively correlated, which means that such concentration
values tend to be similar. Accurate estimation of the variability factors would require making some
allowance for autocorrelation. The effect of accounting for a positive autocorrelation is to increase the
variability factor applied in calculating a limitation. For an autocorrelation of 0.5, the variability factor
might be increased by 10% to 20% over its value when autocorrelation is zero (autocorrelation may range
in value from zero to one). The SCC data consisted of only four or five consecutive daily measurements;
the DMQ data consisted of data collected mostly at 1-week to 1-month intervals. Therefore, at this time
EPA does not have sufficient data to reliably estimate autocorrelation between concentration values
measured on consecutive days, and the variability factors do not make allowance for autocorrelation.
Thus, at this time it is assumed that the daily concentration values are effectively independent of one
another, and
E(U
-..
2o
20
(1.34)
where E(U) and V(U) are calculated as in equations (x) and (y). Finally, since U20 is approximately
normally distributed by the Central Limit Theorem, the estimate of the 95th percentile of a 20-day mean
and the corresponding 20-day average variability factor (VF20) are approximately
P9520 =
0-1(0.95)*(\?d/20))
(1.35)
and
VF20 =
P95
20
E(U20)
P95
20
(1.36)
where <&~l(Q.95) is the 95th quantile of the standard normal distribution.
4.2.6 Estimation of 30-Day Variability Factors and 95th Percentiles of 30-Day Averages
No pollutants are proposed for monitoring on 30 days per month. However, for completeness, Appendix C
reports variability factors for 1-day, 4-day, 20-day, and 30-day averages. Therefore, the methodology for
computing variability factors for 3-day averages is described here. As explained above, EPA does not
have sufficient data with which to estimate and make allowance for autocorrelation, so the following
methodology assumes zero autocorrelation.
19
-------
Thus, at this time it is assumed that the consecutive measurement values are effectively independent of one
another, and
E(U30) = E(U) and
(1.37)
where E(U) and V(U) are calculated as in equations (1) and (2). Finally, since U30 is approximately
normally distributed by the Central Limit Theorem, the estimate of the 95th percentile of a 30-day mean
and the corresponding 3 0-day average variability factor (VF30) are approximately
P9530 = E(U30) + 0-1(0.95)*(V(i;30))2
(1.38)
and
P95
VF30 = — =
30
P95
(1.39)
where $"l(0.95) is the 95th quantile of the standard normal distribution.
5. Estimation of Proposed Daily Maximum and Monthly Average Limitations
Proposed limitations were derived by the following steps, for each subcategory and pollutant, after final
selection of a facility and sample point representing BPT treatment.
1. For a given pollutant, facility, and sampling point, sampling data (Appendix D) were aggregated
(Section 3.3), if necessary, by combining duplicates and grab samples, to produce one concentration value
per day (daily data). All digits reported for the data were retained.
2. Daily data were used to produce summary statistics and parameter estimates (long-term averages or
LTAs, and variability factors or VFs, Section 4.2) for each combination of facility, sample point, and
pollutant. These statistics and estimates are referred to herein as "facility-specific statistics" or "facility-
specific estimates" (Appendix C), and were rounded to three significant figures before the next step.
3. For each pollutant, LTAs and VFs were combined (aggregated) for all facilities (usually there is only
one effluent sampling point per facility at this stage of analysis). The median of the facility-specific LTAs
was used as the (aggregate) long-term average for a given pollutant within a subcategory. The average of
the reported facility-specific VFs was used as the (aggregate) variability factor for a given pollutant within
a subcategory. Aggregate LTAs and VFs were rounded to three significant figures.
4. Each Limitation is the product of an aggregate long-term average and an aggregate variability factor
(Appendix A). Limitations were rounded to two significant figures.
Monthly average limitations were proposed for only one frequency of monitoring for each pollutant, either
4 days or 20 days per month. Different VFs are calculated for different monitoring frequencies (Section
4.2).
20
-------
If all or nearly all of the data for a pollutant were nondetects, EPA used the minimum level of the
analytical method in place of the LTA, and transferred a variability factor (Appendix A) when one could
not be estimated (Section 4.2.2). Some of the transferred variability factors were those reported for similar
treatment systems at industrial facilities in the organic chemical industry (U.S. Environmental Protection
Agency, 1987). In other cases, the variability factor for phenol, estimated using landfill effluent data, was
transferred to other pollutants in Appendix A. Sources of transferred VFs, and instances of LTAs being set
equal to minimum levels, are identified in the footnotes to Tables in Appendix A.
6. References
Aitchison, J., and Brown, J.A.C., 1969, The Lognormal Distribution. Cambridge: Cambridge University
Press.
Kahn, H.D., and Rubin, M.B., 1989, Use of Statistical Methods in Industrial Water Pollution Control
Regulations inthe United States, Environmental Monitoring and Assessment, 12: 129-148
U. S. Environmental Protection Agency, 1987, Development Document for Effluent Limitations
Guidelines and Standards for the Organic Chemicals, Plastics, and Synthetic Fibers Point Source
Category. .EPA 440/1-87/009, Volume I.
21
-------
-------
APPENDIX A. PROPOSED LIMITATIONS
This Appendix lists the proposed limitations and the aggregate Long Term Averages (LTAs) and
variability factors (VFs) used to calculate each limitation.
Facilities representing BPT treatment were selected as described in the Development Document.
All digits reported for the daily data (Appendix D) were used during calculations of facility-
specific statistics, which were then reported with three significant figures (Appendix C). The
median of the reported facility-specific long-term averages was used as the (aggregate) long-term
average for that pollutant within a subcategory. The average of the reported facility-specific
variability factors was used as the (aggregate) variability factor for that pollutant within a
subcategory. These medians and averages were rounded to three significant digits. Each
limitation is the product of an aggregate long-term average and an aggregate variability factor,
and is rounded to two significant digits. Monthly average limitations were proposed for only one
frequency of monitoring, either 4 days or 20 days per month, for each pollutant. The notation
"n/a" ("not applicable") indicates that a monthly average limitation was not proposed for the
corresponding monitoring frequency indicated at the head of the column.
A-l
-------
TABLE A-l. PROPOSED LIMITATIONS FOR NON-HAZARDOUS SUBCATEGORY
Pollutant
Alpha Terpineol
Ammonia
Benzoic Acid
BOD5
P-Cresol
Phenol
Toluene
TSS
Zinc
CAS
number
98555
7664417
65850
C-002
106445
108952
108883
C-009
7440666
Long-
Term
Average
(mg/1)
0.0182**
1.43
0.0911**
24.1
0.0182**
0.0182**
0.0100*
20.1
0.0682
Variability Factors
Daily
3.26 A
4.09
2.49 B
6.55
2.49 B
2.49
7.95 c
4.41
2.97
Monthly,
4-day
average
1.60A
1.75
1.42B
n/a .
1.42B
1.42
2.57 c
n/a
1.60
Monthly,
20-day
average
n/a
n/a
n/a
1.67
n/a
n/a
n/a
1.33
n/a
Proposed Limitations
(mg/L)
Daily
Maximum
0.059
5.9
0.23
160
0.046
0.045
0.080
89
0.20
Maximum
Monthly
Average
0.029
2.5
0.13
40
0.026
0.026
0.026
27
0.11
* Set at the Minimum Level (ML) published for EPA Method 1624
** Set at the detection limit reported with these samples, and exceeds the ML
ATransferred - set at the value reported in OCPSF, subcategory I, for 2,4-dimethyl phenol
B Transferred - set at the value reported for phenol in this table
c Transferred - set at the value reported in OCPSF, subcategory I, for toluene
OCPSF: U. S. Environmental Protection Agency, 1987, Development Document for Effluent Limitations
Guidelines and Standards for the Organic Chemicals, Plastics, and Synthetic Fibers Point Source Category.
EPA 440/1-87/009, Volume I.
A-2
-------
TABLE A-2. PROPOSED LIMITATIONS FOR HAZARDOUS SUBCATEGORY
DIRECT DISCHARGE
Pollutant
Alpha Terpineol
Ammonia
Aniline
Arsenic
Benzene
Benzoic Acid
BOD5
Chromium
Naphthalene
P-Cresol
Phenol
Pyridine
Toluene
TSS
Zinc
CAS
number
98555
7664417
62533
7440382
71432
65850
C-002
7440473
91203
106445
108952
110861
108883
C-009
7440666
Long-
Term
Average
(mg/1)
0.0100*
1.43
0.0100*
0.331
0.0100*
0.0500*
24.1
0.244
0.0100*
0.0100*
0.0201
0.0100*
0.0100*
20.1
0.149
Variability Factors
Daily
4.20 A
4.09
2.37 B
3.03
13.5 c
2.37 B
6.55
3.54
5.89 c
2.37 B
2.37
7.18 c
7.95 c
4.41
2.47
Monthly,
4-day
average
1.86A
1.75
1.45B
1.56
3.64 c
1.45 B
n/a
1.62
2.16C
1.45B
1.45
2.45 c
2.57 c
n/a
1.38
Monthly,
20-day
average
n/a
n/a
n/a
n/a
n/a
n/a
1.67
n/a
n/a
n/a
n/a
n/a
n/a
1.33
n/a
Proposed Limitations
(mg/L)
Daily
Maximum
0.042
5.9
0.024
1.0
0.14
0.12
160
0.86
0.059
0.024
0.048
0.072
0.080
89
0.37
Maximum
Monthly
Average
0.019
2.5
0.015
0.52
0.036
0.073
40
0.40
0.022
0.015
0.029
0.025
0.026
27
0.21
* Set at the Minimum Level (ML) reported for EPA Method 1624 or 1625, as appropriate
A Transferred - set at the value reported in OCPSF, subcategory I, for 2,4-dimethyl phenol
B Transferred - set at the value reported for phenol in this table
c Transferred - set at the value reported in OCPSF, subcategory I, for this analyte
OCPSF: U. S. Environmental Protection Agency, 1987, Development Document for Effluent Limitations
Guidelines and Standards for the Organic Chemicals, Plastics, and Synthetic Fibers Point Source Category
EPA 440/1-87/009, Volume I. °
A-3
-------
TABLE A-3. PROPOSED LIMITATIONS FOR HAZARDOUS SUBCATEGORY,
INDIRECT DISCHARGE
Pollutant
Alpha Terpineol
Ammonia
Aniline
Benzoic Acid
P-Cresol
Toluene
CAS
number
98555
7664417
62533
65850
106445
108883
Long-
Term
Average
(mg/1)
0.0100*
1.43
0.0100*
0.0500*
0.0100*
0.0100*
Variability Factors
Daily
4.20 A
4.09
2.37 B
2.37 B
2.37 B
7.95 c
Monthly,
4-day
•average
1.86A
1.75
1.45 B
1.45B
1.45B
2.57 c
Monthly,
20-day
average
n/a
n/a
n/a
n/a
n/a
n/a
Proposed Limitations
(mg/L)
Daily
Maximum
0.042
5.9
0.024
0.12
0.024
0.080
Maximum
Monthly
Average
0.019
2.5
0.015
0.073
0.015
0.026
* Set at the Minimum Level (ML)
A Transferred - set at the value reported in OCPSF, subcategory I, for 2,4-dimethyl phenol
B Transferred - set at the value reported for phenol in table A-2 (Landfills, Hazardous, direct discharge)
c Transferred - set at the value reported hi OCPSF, subcategory I, for this toluene
OCPSF: U. S. Environmental Protection Agency, 1987, Development Document for Effluent Limitations
Guidelines and Standards for the Organic Chemicals, Plastics, and Synthetic Fibers Point Source Category.
EPA 440/1-87/009, Volume I.
A-4
-------
APPENDIX B
FACILITIES, SAMPLE POINTS, AND DATA SOURCES
Table B-1. Facilities, Sample Points (SP), and Data Sources (DMQ and SCC) Used to Develop
Long-Term Averages and Variance Factors Upon Which Limitations Are Based.
Facilities and Sample Points. Providing Data Used to Develop LTAs and YFs
for Conventional, >fan-conven£ional and. Priority Pollutants
in the Non-Hazardous Subcategory
Questionnaire
Facility ID
16118
16132
16058
16120
16253
16122
16041
Questionnaire
Effluent SP
SP002
SP 004
SP 001
SP 002
SP002
SP 003
SP004
DMQ
Facility ID
17013
17023
17004
17015
17027
17016
17002
DMQ
Effluent SP
SP002
SP 004
SP001
SP002
SP002
SP 003
SP004
SCC Episode
ID
4626
4721
SCC Effluent
SP
SP08
SP02
Facilities and Sample Points Used to Develop LTAs and VFs
for Conventional, Non-conventional, and Priority Pollutants
in the Hazardous Subcategory
Questionnaire
Facility ID
16041
16087
Questionnaire
Effluent SP
SP 004
SP003 .
DMQ
Facility ID
17002
17006
DMQ
Effluent SP
SP004
SP005
SCC Episode
ID
4721
4759
SCC Effluent
SP
SP02
SP03
B-1
-------
-------
APPENDIX C
FACILITY-SPECIFIC STATISTICS
This Appendix lists the facility-specific statistics which were used to calculate effluent
limitations (Appendix A). The sources of data, data types, and procedures for aggregation and
calculation are described in the main body of this document.
C-l
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-------
-------
APPENDIX D
LISTING OF DATA USED TO DEVELOP EFFLUENT LIMITATIONS
This Appendix lists the data used to calculate facility-specific long-term averages and variability
factors (Appendix C), which were used to calculate effluent limitations (Appendix A). The
sources of data, data types, and procedures for aggregation and calculation are described in the
main body of this document.
D-l
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