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D-101
-------
-------
Appendix E - Modified Delta-Lognormal Distribution
Appendix E
MODIFIED DELTA-LOGNORMAL DISTRIBUTION
-------
-------
Appendix E - Modified Delta-Lognormal Distribution
Appendix E
MODIFIED DELTA-LOGNORMAL DISTRIBUTION
This appendix describes the modified delta-lognormal distribution and the
estimation of the episode-specific long-term averages and variability factors used to calculate the
proposed limitations and standards.1 This appendix provides the statistical methodology that was
used to obtain the results presented in Section 12. . .
E.I
Basic Overview of the Modified Delta-Lognormal Distribution
• • EPA selected the modified delta-lognormal distribution to model pollutant effluent
concentrations from the iron and steel industry in developing the long-term averages and
variability factors. A typical effluent data set from a sampling episode or self-monitoring episode
(see Section 12 for a discussion of the data associated with these episodes) consists of a mixture
of measured (detected) and non-detected values. The modified delta-lognormal distribution is
appropriate for such data sets because it models the data as a mixture of measurements that
follow a lognormal distribution and non-detect measurements that occur with a certain
probability. The model also allows for the possibility that non-detect measurements occur at
multiple sample-specific detection limits. Because the data appeared to fit the modified delta-
lognormal model reasonably well, EPA has determined that this model is appropriate for these
data.
The modified delta-lognormal distribution is a modification of the 'delta
distribution' originally developed by Aitchison and Brown.2 While this distribution was originally
developed to model economic data, other researchers have shown the application to
environmental data.3 The resulting mixed distributional model, which combines a continuous
density portion with a discrete-valued spike at zero, is also known as the delta-lognormal
distribution. The delta in the name refers to the proportion of the overall distribution contained in
the discrete distributional spike at zero; that is, the proportion of zero amounts. The remaining
non-zero, non-censored (NC) amounts are grouped together and fit to a lognormal distribution.
EPA modified this delta-lognormal distribution to incorporate multiple detection
limits. In the modification of the delta portion, the single spike located at zero is replaced by a
discrete distribution made up of multiple spikes. Each spike in this modification is associated with
a distinct sample-specific detection limit associated with non-detected (ND) measurements in the
'In the remainder of this appendix, references to 'limitations' includes 'standards.'
2Aitchison, J. and Brown, J.A.C. (1963) "the Lognormal Distribution. Cambridge University Press, pages 87-99.
3Owen, W.J. and T.A. DeRouen. 1980. "Estimation of the Mean for Lognormal Data Containing Zeroes and Left-
Censored Values, with Applications to the Measurement of Worker Exposure to Air Contaminants." Biometrics,
36:707-719.
E-l
-------
Appendix E - Modified Delta-Lognormal Distribution
database.4 A lognormal density is used to represent the set of measured values. This modification
of the delta-lognormal distribution is illustrated in Figure E-l.
The following two subsections describe the delta and lognormal portion's of the
modified delta-lognormal distribution in further detail.
E.2
Continuous and Discrete Portions of the Modified Delta-Lognormal
Distribution
The discrete portion of the modified delta-lognormal distribution model'p the non-
detected values corresponding to the k reported sample-specific detection limits. In the model, 8
represents the proportion of non-detected values in the dataset and is the sum of smaller fractions,
8;, each representing the proportion of non-detected values associated with each distinct detection
limit value. By letting D; equal the value of the 1th smallest distinct detection limit in the data set
and the random variable XD represents a randomly chosen non-detected ineasurement, jfhe
cumulative distribution function of the discrete portion of the modified delta-lognormal model can
be mathematically expressed as:
0
-------
Appendix E - Modified Delta-Lognormal Distribution
Figure E-l
Modified Delta-Lognormal Distribution
Censoring Type
NC
ND
E-3
-------
Appendix E - Modified Delta-Lognormul Distribution
(E-4)
where the random variable Xc represents a randomly chosen detected measurement,
-------
Appendix E - Modified Delta-Lognormal Distribution
The expected value of the random variable U can be derived as a weighted sum of
the expected values of the discrete and continuous portions of the distribution (equations 2 and 5,
respectively) as follows
(E-9)
In a similar manner, the expected value of the random variable squared can be
written as a weighted sum of the expected values of the squares of the discrete and continuous
portions of the distribution as follows
(E-10)
Although written in terms of U, the following relationship holds for all random variables, U, XD,
= Var(U)+[E(U)]
(E-ll)
So using equation 1 1 to solve for Var(U), and applying the relationships in equations 9 and 10,
the variance of U can be obtained as
E.4
Episode-specific Estimates Under the Modified Delta-Lognormal
Distribution '
In order to use the modified delta-lognormal model to calculate the proposed
limitations, the parameters of the distribution are estimated from the data. These estimates are
then used to calculate the proposed limitations.
$ U
The parameters l and ° are estimated from the data using the following formulas:
(E-13)
n
where nd is the number of non-detected measurements, dpj = -\ to nd, are the detection limits for
the non-detected measurements, n is the number of measurements (both detected and non-
detected) and I(...) is an indicator function equal to one if the phrase within the parentheses is
E-5
-------
Appendix E - Modified Delta-Lognortnal Distribution
true and zero otherwise. The "hat" over the parameters indicates that they are estimated from the
data.
ii
The expected value and the variance of the lognormal portion of the modified
delta-lognormal distribution can be calculated from the data as:
•k
7=1
k
' X '
/^
7=1
(E-14)
(E-15)
The parameters of the continuous portion of the modified delta-lognormal
distribution, " and ^ , are estimated by
(E-16)
I
where xt is the i* detected measurement value and nc is the number of detected measurements.
Note that n = nd + nc. i
The expected value and the variance of the lognormal portion of the modified
delta-lognormal distribution can be calculated from the data as: [
(E-17)
(E-18)
Finally, the expected value and variance of the modified delta-lognormal
distribution can be estimated using tihie following formulas:
(E-19)
E-6
-------
Appendix E -Modified Delta-Lognormal Distribution
Var(U] =
(E-20)
Equations 17 through 20 are particularly important in the estimation of episode-
specific long-term averages and variability factors as described in the following sections. These
sections are preceded by a section that identifies the episode data set requirements.
E.4.1
Episode Data Set Requirements
Estimates of the necessary parameters for the lognormal portion of the distribution
can be calculated with as few as two distinct detected values in a data set. (In order to calculate
the variance of the modified delta-lognormal distribution, two distinct detected values are the
minimum number that can be used and still obtain an estimate of the variance for the distribution.)
If an episode data set for a pollutant contained three or more observations with
two or more distinct detected concentration values, then EPA used the modified delta-lognormal
distribution to calculate long-term averages and variability factors. If the episode data set for a
pollutant did not meet these requirements, EPA used an arithmetic average to calculate the
episode-specific long-term average and excluded the dataset from the variability factor
calculations (because the variability could not be calculated).
In statistical terms, each measurement was assumed to be independently and
identically distributed from the .other measurements of that pollutant in the episode data set.
The next two sections apply the modified delta-lognormal distribution to the data
for estimating episode-specific long-term averages and variability factors for the iron and steel
industry. , . .
E.4.2
Estimation of Episode-specific Long-Term Averages
If an episode dataset for a pollutant mets the requirements described in the last
section, then EPA calculated the long-term average using equation 19. Otherwise, EPA
calculated the long-term average as the arithmetic average5 of the daily values where the sample-
specific detection limit was used for each non-detected measurement.
E.4.3
Estimation of Episode-Specific Variability Factors
For each episode, EPA estimated the daily variability factors by fitting a modified
delta-lognormal distribution to the daily measurements for each pollutant. In contrast, EPA
estimated monthly variability factors by fitting a modified delta-lognormal distribution to the
5EPA also used the arithmetic average of daily values in costing the technology options. See Section 12.7.1.
E-7
-------
Appendix E - Modified Delta-Lo'gnormhl Distribution
monthly averages for the pollutant at the episode. EPA developed these averages using the same
number of measurements as the assumed monitoring frequency for the pollutant. EPA is
assuming that all pollutants will be monitored weekly (approximately four times a month).6
E.4.3.1
Estimation of Episode-specific Daily Variability Factors
The episode-specific daily variability factor is a function of the expected value, and
the 99th percentile of the modified delta-lognoimal distribution fit to the daily concentration
values of the pollutant in the wastewater from the episode. The expected.value, was estimated
using equation 19 (the expected value is the same as the episode-specific long-term ayerage).
The 99th percentile of the modified delta-lognormal distribution fit to each data set
was estimated by using an iterative approach. First, the pollutant-specific detection limits were
ordered from smallest to largest. Next, the cumulative distribution function, p, for each detection
limit was computed. The general form, for a given value c, was:
P=
(E-21)
where <& is the standard normal cumulative distribution function. Next, the interval containing the
99* percentile was identified. Finally, the 99th percentile of the modified delta-lognormal
distribution was calculated. The following steps were completed to compute the estimated 99th
percentile of each data subset: [
Step 1 Using equation 21, k values of p at c=Dm, m=l,...,k were computed an|l labeled
Step 2
StepS
The smallest value of m (m=l,...,k), such that pm > 0.99, was determined and
labeled as PJ. If no such m existed, steps 3 and 4 were slapped and step 5 was
computed instead.
Computed p* = PJ -
8,.
"Compliance with the monthly average limitations will be required in the final rulemaking regardless of the number of
samples analyzed and averaged.
E-8
-------
Appendix E - Modified Delta-Lognormal Distribution
Step 4
Ifp*<0.99,then^)99 = Dj
else if p*_> 0.99, then
P99 = exp
f
" *• -at. — 1
LL H-cry?
V
. -i\
0.99 -£$
z=l
y\
1-5
-')
(E-22)
Step 5
where <&"' is the inverse normal distribution function.
If no such m exists such that pm > 0.99 (m=l,...,k), then
0.99-5
1-d
The episode-specific daily variability factor, VF1, was then calculated as:
(E-23)
VF1 =
P99
E(U)
(E-24)
E.4.3.2
Estimation of Episode-Specific Monthly Variability Factors
EPA estimated the monthly variability factors by fitting a modified delta-lognormal
distribution to the monthly averages. These equations use the same basic parameters, \i and a,
calculated for the daily variability factors. Episode-specific monthly variability factors were based
on 4-day monthly averages because the monitoring frequency was assumed to be weekly
(approximately four times a month).
In order to calculate the 4-day variability factors (VF4), the assumption was made
that the approximating distribution of jj , the sample mean for a random sample of four
independent concentrations, was also derived from the modified delta-lognormal distribution.7 To
obtain the expected value of the 4-day averages, equation 19 is modified for the mean of the
distribution of 4-day averages in equation 25:
E(U4)=84E(X4)D+(1-84)E(X4)(
(25)
7As described in section 12.4, when non-detected measurements are aggregated with non-censored measurements', EPA
determined that the result should be considered non-censored.
E-9
-------
'Appendix E - Modified Delta-Lognormdl Distribution
where * *'D denotes the mean of the discrete portion of the distribution of the average of four
independent concentrations, (i.e., when all observations are non-detected values) and
(X }
\ A'c denotes the mean of the continuous lognormal portion (i.e., when any observations are
detected).
First, it was assumed that the probability of detection (5) on each of theifour days
was independent of the measurements on the other three days (as explained in Section E.4.1, daily
measurements were also assumed to be independent) and therefore, 64 = S4. Because the
measurements are assumed to be independent, the following relationships hold:
(E-26)
Var
_Var(XD)
4
Substituting into equation 26 and solving for the expected value of the continuous
portion of the distribution gives: ;
(E-27)
1-8'
Using the relationship in equation 19 for the averages of 4 daily measurements and substituting
terms from equation 25 and solving for the variance of the continuous portion of jj gives:
1-5'
Using equations 17 and 18 and solving for the parameters of the lognormal distribution describing
the distribution of ;f gives:
E-10
(E-28)
-------
Appendix E - Modified Delta-Losnormal Distribi
Var(X4]
and
+ 1
(E-29)
In finding the estimated 95th percentile of the average of four observations/four
non-detects, not all at the same sample-specific detection limit, can gerierate an average that is not
necessarily equal to D,, D2,..., or Dk. Consequently, more than k discrete points exist in the '
distribution of the 4-day averages. .For example, the average of four non-detects at k=2 detection
limits, are at the following discrete points with the associated probabilities:
z
1
2
3
4
5
(3Dl+D2)/4
(2D1+2D2)/4
48,81
8*
When all four observations are non-detected values, and when k distinct non-
detected values exist, the multinomial distribution can be used to determine associated
probabilities. That is, .
Pr
u*=-£L
41
u^L.^l*-.!
(E-30)
where u; is the number of non-detected measurements in the data set with the D; detection limit.
The number of possible discrete points, k*, for k=l, 2,3,4, and 5 are as follows:
E-ll
-------
k
1
2
3
4
5
kl
1
5
15
35
70
I1
;.
i,
E
To find the estimated 95th percentile of the distribution of the average o|f four
observations, the same basic steps (described in Section 4.3.1) as for the 99th percentile of the
distribution of daily observations, were used with the following changes:
Step 1
Step 2
StepS
Step 4
StepS
Then, using
was calculated as:
Change P99 to P95, and 0.99 to 0.95. .
Change Dm to Dm*, the weighted averages of the sample-specific detection limits.
Change 8; to 8;*.
Change k to k*, the number -of possible discrete points based on k detection limits.
Change the estimates of 8, ^ ,and G to estimates of 84, ™ and 4 respectively.
, the estimate of the episode-specific 4-day variability fadjor, VF4,
E.4.3.3
:E(U)
Evaluation of Episode-Specific Variability Factors
Estimates of the necessary parameters for the lognormal portion of the ^distribution
can be calculated with as few as two distinct measured values in a data set (in order to. calculate
the variance); however, these estimates can be unstable (as can estimates from larger data sets).
As stated in Section E.4.1, EPA used the modified delta-lognormal distribution to develop
episode-specific variability factors for data sets that had a three or more observations with two or
more distinct measured concentration values. \
To identify situations producing unexpected results, EPA reviewed all pf the
variability factors and compared daily to monthly variability factors. EPA used several criteria to
determine if the episode-specific daily and monthly variability factors should be included in
calculating the option variability factors. One criteria that EPA used was that the daily and
monthly variability factors should be greater than 1.0. A variability factor less than 1.0 would
result in a unexpected result where the estimated 99th percentile would be less than the long-term
average. This would be an indication that the estimate of ° (the log standard deviation) was
unstable. A second criteria was that the daily variability factor had to be greater than the monthly
variability factor. A third criteria was that not all of the sample-specific detection limits could
exceed the values of the non-censored values. All the episode-specific variability factbrs used for
the proposed limitations and standards met these criteria. j
E-12
-------
Appendix E - Modified Delta-Li
llDiitrihi
E.5
References
Aitchison,J. and J.A.C. Brown. 1963. The Lognormal Distribution. Cambridge University
Press, New York. '
Barakat,R. 1976. "Sums of Independent Lognormally Distributed Random Variables." Journal
of the Optical Society of America, 66:211-216.
Cohen, A. Clifford. 1976. Progressively Censored Sampling in the Three Parameter Log-Normal
Distribution. Technometrics, 18:99-103.
Crow,E.L,andK.Shimizu. 1988. Lognormal Distributions: Theory and Applications Marcel
Dekker, Inc., New York.
Kahn, H.D., and M.B. Rubin. 1989. "Use of Statistical Methods in Industrial Water Pollution
Control Regulations in the United States." Environmental Monitoring and Assessment
Vol. 12:129-148.
Owen, W.J. and T.A. DeRouen. 1980. Estimation of the Mean for Lognormal Data Containing
Zeroes and Left-Censored Values, with Applications to the Measurement of Worker Exposure to
Air Contaminants. Biometrics, 36:1'01'-719. .
U.S. Environmental Protection Agency. 2000. Development Document for Effluent Limitations
Guidelines and Standards for the Centralized Waste Treatment Point Sniifr.p Tat^gn^ Volume T
Volume II. EPA 440/1 -87/009. ~ •
E-13
-------
-------
Appendix F - Attachments for Section 12
Appendix F
ATTACHMENTS FOR SECTION 12
-------
-------
Appendix F - Attachments for Section 12
Appendix F
ATTACHMENTS FOR SECTION 12
Subcategory Abbreviations:
Abbreviation Subcategorv
COKE_BYPROD
FINISHING
INT_HOTFORM
INT_STEEL
IRON
NONINT_STEEL_HOTFORM
OTHER
Option Abbreviations:
Abbreviation
CARBON_BAT1
SPECIALTY_BAT1
DRI BPT
Cokemaking, By-Product Segment
Steel Finishing
Integrated and Stand-Alone Hot Forming
Integrated Steelmaking
Lronmaking
Non-Integrated Steelmaking and Hot Forming
Other Operations
Option
Carbon and Alloy Segment, Option BAT1
Stainless Steel Segment, Option BAT1
Direct Iron Reduction, Option BPT
Other Abbreviations:
Abbreviation
CAS_NO
Est. ,
LTA
ND
Obs
STD
V.F.
Definition
Chemical Abstract Service Number
Estimated
Long-Term Average
Non-Detect .
Number of Daily Values; OR Observed (e.g., Qbs Mean)
Standard Deviation
Variability Factor
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