1 

MEMORANDUM 

TO: Docket ID No. EPA-HQ-OAR-2012-0522 

FROM: Tina Ndoh 

 Office of Air Quality Planning and Standards 

DATE: June 9, 2015 

SUBJECT: Approach for Applying the Upper Prediction Limit to Limited Datasets 

I. What is Our Approach for Applying the Upper Prediction Limit to Limited Datasets? 

The UPL approach addresses variability of emissions data from the best performing source or 

sources in setting MACT standards. The UPL also accounts for uncertainty associated with 

emission values in a dataset, which can be influenced by components such as the number of 

samples available for developing MACT standards and the number of samples that will be 

collected to assess compliance with the emission limit. The UPL approach has been used in 

many environmental science applications.1,2,3,4,5,6 As explained in more detail in the 

memorandum entitled “Use of the Upper Prediction Limit for Calculating MACT Floors”, which 

is located in the docket for this action, the EPA uses the UPL approach to reasonably estimate 

the emissions performance of the best performing source or sources to establish MACT floor 

standards. 

Sample size (the number of values in a particular dataset) is an important component of the UPL 

approach. First, the use of the UPL approach requires us to identify the distribution of the data, 

that is, whether a particular dataset is normally or non-normally distributed (see the 

memorandum entitled “Use of the Upper Prediction Limit for Calculating MACT Floors”, which 

is located in the docket for this action). To determine the distribution of the data, we use wellestablished tests (kurtosis and skewness tests), and these tests require at least three data points. In 

prior rulemakings, we used the kurtosis equation that is a built-in function in Excel software, but 

that equation requires at least 4 values in order to avoid dividing by zero, which precluded its use 

for datasets of 3. However, we recently further reviewed the application of the UPL where data 

                                                            

1

 Gibbons, R. D. (1987), Statistical Prediction Intervals for the Evaluation of Ground-Water Quality. Groundwater, 

25: 455–465 and Hart, Barbara F. and Janet Chaseling, Optimizing Landfill Ground Water Analytes—New South 

Wales, Australia, Groundwater Monitoring & Remediation, 2003, 23, 2. 

2

 Wan, Can; Xu, Zhao; Pinson, Pierre; Dong, Zhao Yang; Wong, Kit Po. Optimal Prediction Intervals of Wind 

Power Generation. 2014. IEEE Transactions on Power Systems, ISSN 0885-8950, 29(3): pp. 1166 – 1174. 

3

 Khosravi, Abbas; Mazloumi, Ehsan; Nahavandi, Saeid; Creighton, Doug; van Lint, J. W. C. Prediction Intervals to 

Account for Uncertainties in Travel Time Prediction. 2011. IEEE Transactions on Intelligent Transportation 

Systems, ISSN 1524-9050, 12(2):537 – 547. 

4

 Ashkan Zarnani; Petr Musilek; Jana Heckenbergerova. 2014. Clustering numerical weather forecasts to obtain 

statistical prediction intervals. Meteorological Applications, ISSN 1350-4827. 21(3): 605. 

5

 Rayer, Stefan; Smith, Stanley K; Tayman, Jeff. 2009. Empirical Prediction Intervals for County Population 

Forecasts. Population Research and Policy Review, 28(6): 773 – 793. 6

 Nicholas A Som; Nicolas P Zegre; Lisa M Ganio; Arne E Skaugset. 2012. Corrected prediction intervals for 

change detection in paired watershed studies. Hydrological Sciences Journal, ISSN 0262-6667, 57(1): 134 – 143
2 

are limited, and concluded that it is appropriate to use another kurtosis estimator7

 that provides a 

meaningful result with just 3 values. The use of this estimator is important, as many, if not most, 

of our new source emissions limits are based on 3 samples collected during an emissions test. 

Once we determine the data distribution, the appropriate equation to be used in the UPL 

approach is selected based on the data distribution. We recognize that the use of the UPL 

approach for limited datasets introduces some amount of uncertainty in the calculation of MACT 

standards, and, therefore, we are taking additional steps, discussed in this document, to ensure 

that the level of the MACT standards is reasonable. We also note that after MACT standards are 

promulgated, we are required to review those standards periodically, and for such reviews, we 

typically have significant additional HAP emissions data from the intervening years of 

compliance with which to further assess the actual performance of the various emission sources. 

Regardless of the distribution of the data, UPL equations have three well-defined components: an 

average, the t-score and a measure of variability that includes the actual variability of the data, 

the sample size, and the number of data points that are averaged together to determine 

compliance with a particular emission limit. A t-score is a value that estimates the uncertainty 

and variability for a certain confidence level associated with a specific number of data points. 

For a constant confidence level, t-scores decrease as the number of samples increase. This means 

that, if the mean and variance remain constant, UPL values for a particular confidence level 

decrease as the number of samples increase. Figure 1 and Table 1 show the t-scores for various 

sample sizes, and demonstrate that the t-score is highly variable at the smallest sample sizes and 

becomes relatively constant once the sample size is larger than a few data points. Consequently, 

we recognize that we need to take special care when smaller, limited sample sizes are used to 

establish emission limits, as the t-scores can have a disproportionate effect, overwhelming the 

other components of the emission limit calculation. In addition, we recognize that for a sample 

size of fewer than three data points, which has a very large t-score and precludes the appropriate 

selection of a distribution, we should not develop emission limits using the UPL. In other words, 

if fewer than 3 data points are available for use in determining an emission limit for a particular 

source, and no other data from sources in the subcategory are available, we would have to 

establish a different procedure for establishing the MACT floor that does not rely on the UPL. 

                                                            

7

 Doric D, Nikolic-Doric E, Jevremovic V and Malisic J. 2009. On measuring skewness and kurtosis. Qual Quant. 

43:481-493. DOI 10.1007/s11135-007-9128-9. 
3 

Figure 1. Sample Sizes Versus One-sided, 99 Percent Confidence Interval T-Scores for 

Normal Distributions 

Table 1. Degrees of Freedom, T-Score for a One-Tailed, 99 Percent Confidence Interval, 

and Percent Difference in T-Score for Each Sample Size 

Sample Size 

(n) 

Degrees of Freedom 

(n-1) 

T-score 

(99 percent) 

Relative Percent 

Differencea

2 1 31.82 

3 2 6.96 78 

4 3 4.54 35 

5 4 3.75 17 

6 5 3.36 10 

7 6 3.14 7 

8 7 3.00 5 

9 8 2.90 3 a The relative percent difference between the t-score and the t-score shown on the preceding row.

Number of Samples

T-score

T-score vs Number of Samples

2 3 4 5 6 7 8 9

0

5

10

15

20

25

30

35

78% reduction from 2 to 3 samples

35% reduction from 3 to 4 samples

17% reduction from 4 to 5 samples

10% reduction from 5 to 6 samples

7% reduction from 6 to 7 samples

5% reduction from 7 to 8 samples
4 

In the case of a dataset with a normal distribution, the equation to calculate the UPL is as 

follows: 

ܷܲܮ ൌ ݔ̅ ൅ݐሾሺ௡ିଵሻ,ሺଵି஑ሻሿ

ඨݏଶ ൬

1

݊

൅

1

݉൰

Where ݔ = average or mean of test run data; 

ݐሾሺ௡ିଵሻ,ሺଵି஑ሻሿ = t score, the one-tailed t value of the Student’s t distribution for a specific 

degree of freedom (n-1) and confidence level (1- α); 

α = level of significance expressed as a decimal (e.g., 1% significance = 0.01), note that 

confidence level = 100 െ ሺߙ ൈ 100ሻ; 

ݏଶ = variance of the dataset (test run); 

݊ = number of values (test runs) in the dataset; and 

݉ = number of values used to calculate the test average (generally 3). 

As shown in the equation, the sample size (n) directly affects the UPL calculations in two ways. 

First, as mentioned earlier, the t-score is selected based on the sample size. Second, the sample 

size is a separate variable in the equation, and as with the t-score, as the sample size gets larger 

(all other variables being held constant), the UPL gets smaller (i.e., approaches the mean value). 

The sample size also influences both the mean (ݔ̅ (and variance (s2

) in that larger sample sizes 

provide better estimates of the population mean and variance. The confidence level also 

influences the t-score, and is represented in the UPL equation by the subscript “1- α.” As noted 

in the memorandum entitled “Use of the Upper Prediction Limit for Calculating MACT Floors”, 

which is located in the docket for this action, we typically choose the 99 percent UPL, which is 

another way of saying that α equals 0.01. 

Note that the UPL equation for a lognormal distribution uses a z-score, instead of a t-score.8

 As 

with the normal UPL equation, the lognormal UPL equation is affected by sample size and the 

variance of the data, and the basic concepts outlined in this discussion apply to datasets 

regardless of the distribution. 

In summary, sample size is important for UPL calculations, as it is a key component of the tscore value and the calculation of the mean and variance of the sample. For the reasons presented 

above, for datasets below a certain size, further evaluation is warranted to ensure that the results 

are a valid representation of the performance of the emission sources whose data are included in 

the dataset. The remainder of this document provides our assessment of what constitutes a 

limited dataset and what additional analyses are appropriate for ensuring that the UPL 

                                                            

8

 Bhaumik, D.K. and Gibbons, R.D. “An Upper Prediction Limit for the Arithmetic Mean of a Lognormal Random 

Variable”, Technometrics, 46, 2004, 239-248. 
5 

calculations are valid representations of the performance of units as reflected in the available 

data. 

II. What is a limited dataset? 

While there are different options for choosing the number of samples that represents a limited 

dataset (or small sample size), we can use distinctions based on our observations in Figure 1 and 

Table 1 as an aid in defining what constitutes a limited dataset. As discussed earlier, the t-score 

is an influential parameter that can have a particularly large influence on an emission limit 

developed from a limited dataset. As such, the t-score is a key parameter for identifying what 

constitutes a limited dataset for purposes of MACT analyses. As shown in Figure 1 and Table 1, 

above, at the 99 percent confidence level there is a clear distinction between t-score values for 

sample sizes of 3 and fewer when compared to t-score values for sample sizes of more than 6. As 

Figure 1 and Table 1 show, the t-score changes drastically from sample size equal to 2 to sample 

size equal to 3. The changes in the t-score are considerably less dramatic as the sample size 

approaches 9 data points and larger. 

As noted earlier, we will not use the UPL approach for datasets with fewer than 3 values because 

of the high degree of uncertainty that results from the size of the dataset and the related t-score. 

In fact, the t-score (see Figure 1 and table 1) for a dataset of 3 is 78 percent lower than a t-score 

for a dataset of 2, meaning that given an equal variance, the component of the UPL that includes 

the t-score and variance would be more than 78 percent less when evaluating a 3-point dataset 

compared to a 2-point dataset. The table and figure also show that the relative change of the tscore for each additional data point decreases as the sample size gets larger. As sample size 

increases, uncertainty decreases, but there is always some amount of uncertainty. The discussion 

that follows presents our examination of the t-score at various sample sizes. We rely on this 

examination to help identify a sample size below which we would further evaluate the data and 

the application of the UPL to ensure that the amount of uncertainty is within a reasonable range. 

Based on Table 1, the lower bound of the sample size where such further evaluation is needed is 

3 values, and a reasonable value (where the size of the t-statistic has much less influence on the 

eventual emission limit) lies between 4 and 9 values. 

An additional factor to consider is that, in the context of MACT analyses, emission tests 

typically include 3 test runs (independent data points), and, therefore, our MACT floor dataset 

size typically is a multiple of 3. For this reason, we first considered sample sizes that are 

multiples of 3 when identifying the size below which further scrutiny is needed. As can be 

calculated from the t-scores shown in Table 1, the relative difference between the t-score for 

6 data points and the t-score for 9 data points is about 14 percent ((3.36-2.90)/3.36). The relative 

difference between the t-scores for 7 and 9 data points is about 8 percent ((3.14-2.90)/3.14). 

Given that the test methods that are used to generate the data typically are accurate to within 

10 to 20 percent9

, this difference in t-score (between datasets with 7 and 9 runs) is less than the 

uncertainty in the test methods and would not have a large influence on the magnitude of an 

emission limit. Conversely, comparing the t-score for 3 runs to that for 7 runs, the relative 

difference in the t-score is 55 percent ((6.96-3.14)/6.96), indicating that there is a possibility, 

                                                            

9

 (ReMAP): PHASE 1, Precision of Manual Stack Emission Measurements; American Society of Mechanical 

Engineers, Research Committee on Industrial and Municipal Waste, February 2001. 
6 

especially if the variance is high, that the t-score for 3 test runs may introduce a large amount of 

uncertainty in an emission limit, and, therefore, additional analysis of a dataset of this size is 

warranted. We also conducted the same type of evaluation on datasets between 4 and 6 points, 

and while the change in t-score for datasets of these sizes could be considered to be sufficiently 

small, to be conservative, we conclude that a limited dataset, for purposes of MACT analyses, is 

a dataset that includes at least 3 test runs and fewer than 7 test runs. Because of the uncertainty 

associated with datasets at or below this size, further evaluation is warranted, and the types of 

additional analyses that we may choose to conduct are explained below. 

III. What approach will the EPA follow for MACT floors based on limited datasets? 

When a MACT floor for either existing or new sources is based on fewer than 7 data points, we 

will further evaluate each individual dataset in order to ensure that the uncertainty associated 

with a limited dataset does not cause the calculated emission limit to be so high that it does not 

reflect the average performance of the units upon which the limit is based after accounting for 

variability in the emissions of those units. The evaluation will include one or more of the 

following, depending on the specific dataset: confirming that the data distribution was selected 

correctly; after confirming the data distribution, ensuring that we use the most appropriate UPL 

equation10; and, as necessary, comparing UPL equation components for the individual unit upon 

which a new source floor is based with those of the units in the existing source floor to determine 

if our identification of the best unit is reasonable. Each of the additional evaluations are 

discussed below. 

Confirming the data distribution is important because UPL equations and the emission limits 

derived from those UPL equations depend on the distribution. In prior rulemakings, we had not 

identified a way of ascertaining the data distribution for units with three samples, so we either 

assumed a normal or lognormal distribution. This particularly affected new source emissions 

limits, many of which were based on three samples. However, we have since identified and 

adopted an established technique that compares the ratio of skewness to the standard error of the 

skewness for both the raw and log transformed data. The lower of the two ratios identifies the 

data distribution that best represents the sample set. This ability to more precisely ascertain the 

data distribution enables us to select the best option for the UPL equation given the small sample 

size. 

After confirming that we selected the best distribution based on the available data and ensuring 

that we used the correct equation to calculate a MACT floor value, we then examine key 

variables that factor into the calculated floor values. The variables that are factored into the 

emission limit include the following: average, t-score, confidence level, variance, number of test 

runs (i.e., sample size), and number of runs in each test (i.e., the number of data points that are 

averaged together to determine compliance with a particular emission limit). The considerations 

related to each variable are discussed below. 

                                                            

10 For datasets with lognormal distributions, we recently determined that the most appropriate UPL equation, which 

is especially important for the smallest datasets that we use, is based on an approach described in Bhaumik, 2004, 

op. cit.
7 

The average, which typically is calculated as the mean value of a dataset, does not require 

additional consideration. Once it is calculated, it is used to rank the performance of units, and 

also is the key value upon which each emission limit is based. As such, average values have 

already been compared within a particular dataset, and additional assessment is not necessary. 

However, in the overall assessment of variables, when multiple best performing units have 

emission averages that are similar, we may look to other variables like the variance to help to 

inform our decision as to which unit is the single best performer. 

As discussed earlier, the t-score is dependent on the sample size, and, therefore, we discuss these 

two variables together. As shown earlier in Figure 1, as the sample size decreases, the t-statistic 

increases. This is important because the t-statistic is multiplied by the variance and a factor that 

involves the sample size and the number of test runs used for compliance (i.e., the “m” term in 

the UPL equation). For the smallest datasets (3 data points) that we consider under the UPL 

approach, a large variance coupled with a t-score that is significantly greater than the t-score for 

a dataset of 7 or more can create a situation where the emission limit includes a large amount of 

uncertainty. In such cases, we use our technical expertise to assess what level of emissions could 

realistically be expected from the type of unit, controls, and other relevant factors, and would 

ensure that the emission limit is reasonable. 

The variance is another key variable that we would evaluate. Generally speaking, if our 

evaluation showed a very small amount of variability (e.g., a small variance), the emission limit 

would not contain an unacceptable amount of uncertainty and variability because a small 

variance would result in an emission limit that would be reasonably close to the mean of the data. 

With larger datasets, it is more likely that the demonstrated variance is truly representative of the 

variation in the source’s emissions simply because there is more evidence of the source’s 

emissions over time. On the other hand, when a limited dataset includes a large amount of 

variability (e.g., a large variance), we would need to carefully evaluate the data. For instance, 

consider the case where the pool of best performing units all have similar averages and the unit 

identified as the single best performer (based on average emissions only) has a limited dataset 

and a variance that greatly exceeds the variance of the other similarly-controlled best-performing 

units. In such a case, the MACT floor analysis may yield an emission limit for that unit (i.e., the 

new source MACT floor) that is higher than the existing source MACT floor, which is an 

indicator that further analysis is warranted. Careful consideration of variance will allow us to 

better ascertain which unit should be considered the best performer. 

Another tool that could be used to adjust unreasonable emission limits is the selection of the 

confidence level. In certain instances, we may determine that a limited dataset includes such a 

high degree of variability that the 99 percent confidence level results in an emission limit that our 

experience suggests is higher than the average emissions limitation achieved by the best 

performing source or sources after accounting for variability. In such cases, we may choose to 

select a confidence level of 95 percent or 90 percent. As an example, consider a unit identified as 

the best performing new source with a specific process or control device that has been 

demonstrated to operate far more efficiently (thus having the potential to lower emissions) in 

similar units, processes, or control devices. Under these conditions, we may choose to 

acknowledge the better operation by lowering the confidence level (which lowers the emissions 

limit). 
8 

Establishing and using this approach on a case-by-case basis for limited datasets will ensure 

consistent application of emissions limit development procedures, which will mitigate the 

additional uncertainty that could otherwise result from calculating MACT floor standards with 

limited data.11

IV. How did we apply the approach for limited datasets to limited datasets in the phosphoric acid 

manufacturing source category? 

For the phosphoric acid manufacturing source category, we have a limited dataset for the 

pollutant total F from phosphate rock calciners. Therefore, we evaluated this specific dataset to 

determine whether it is appropriate to make any modifications to the approach used to calculate 

the MACT floor. 

For the phosphate rock calciner dataset, we performed the steps outlined above, including: 

ensuring that we selected the data distribution that best represents the dataset; ensuring that the 

correct equation for the distribution was then applied to the data; and comparing individual 

components of the limited dataset to determine if the total F standard based on the limited dataset 

reasonably represents the performance of the units included in the dataset. The results of this 

analysis are presented below. 

The MACT floor dataset for total F from new and existing phosphate rock calciners includes 3 

test runs from 1 phosphate rock calciner. After determining that the dataset is best represented by 

a normal distribution and ensuring that we used the correct equation for the distribution, we 

considered the selection of a lower confidence level for determining the emission limit by 

evaluating whether the calculated limit reasonably represents the performance of the unit upon 

which it is based. In this case, the calculated emission limit is about twice the short-term average 

emissions from the best performing source, indicating that the emission limit is not unreasonable 

compared to the actual performance of the unit upon which the limit is based and is within the 

range that we see when we evaluate larger datasets using our MACT floor calculation 

procedures. Therefore, we determined that no changes to our standard floor calculation 

procedure are warranted for this pollutant and subcategory, and we are setting the MACT floor at 

9.0E-04 lb/ton of rock feed for total F from new and existing phosphate rock calciners.

                                                            

11 We note that in the recent D.C. Circuit Court of Appeals decision in National Assn. of Clean Water Agencies v. 

EPA (NACWA), which involved challenges to EPA’s MACT standards for sewage sludge incinerators, the court 

identified two instances where we replaced a new source floor value with the existing source value because the new 

source value was less stringent than the existing source value (the “anomalous result”). See 734 F.3d 1115. The 

procedures outlined here would eliminate the anomalous result in both instances in the rule at issue in NACWA. The 

first instance in which the court in the NACWA decision identified the anomalous result is where we established an 

emission limit using the UPL on a dataset that included only 2 data points. The other instance in which the court 

identified the anomalous result is where we did not apply the most appropriate lognormal UPL equation.