Date: April 15 , 201 3 Memo 6

- 7 .1

To: Cuc Schroeder, EPA

From: John Rogers

Subject: Serial correlations for Steam Electric with and without adjustment for baseline values [ EP

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- 10

- 023 WA 2

- 6 (Steam Electric) ]

Revisions : This memo reflects wording changes and clarifications describing the correlation estimates originally delivered to EPA in Memo 6

- 4 dated September 27, 2012. The most recent change revised the tables to sho w only the correlations used in the develop ment the final effluent limitations.

1. Introduction

Under EPA Contract EP

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- 10

- 023 WA 2

- 6, EPA has asked Westat to calculate the 1

- day lag serial correlations to be used when calculating limits for steam electric plants and to calculate the limits for various technology options defined by EPA. This memo des cribes the process for calculating the correlations and provides estimates of the serial correlation using 1) the data, adjusted by a set of baseline values specified by EPA that were used for calculating the proposed limits for the Steam Electric Effluent Guidelines rulemaking; and 2) all valid reported results

, i.e., the data collected by EPA or provided by the plants with possible non

- detects but without baseline substitution. The autocorrelations obtained under these two scenarios were used to calculat e two sets of limits (with and without adjustment for baseline values).

EPA used the baseline values in the statistical analysis to account for the possibility that certain detected results may be at a lower concentration than can be reliably achieved by well

- operated laboratories. For calculating the limits from the data, both detected and non

- detected values that are below the baseline are set equal to the baseline and treated as not detected. In this memo, the term “non

- detect” or “not detected” refers to 1) values below the detection limit in the original reported results or 2) values below the detection limit in the original reported results or below the selected baseline , depending on the context

. Table 1 shows the baseline values established by EP A for the calculation of the serial correlations and limits.- 2

-

Table 1 Baseline values by pollutant Pollutant Baseline value Unit Arsenic 2 ug/L Selenium 5 ug/L Mercury 0.5 ng/L Nitrate Nitrite as N 0.05 mg/L Total Dissolved Solids 10 mg/L

For some pollutants and treatment technology options for FGD and gasification wastewater, the autocorrelations could not be calculated due to a limited amount of data. As a result, serial correlations could be calculated for only arsenic, selenium, and mercury for the biological treatment technology option for FGD wastewater. F or nitrate

- nitrite as N for this technology option, since there were only 8 observations available, autocorrelation could not be calcula ted

. For the biological technology option for FGD wastewater, EPA is proposing limits for arsenic, mercury, nitrate

- nitrite as N, and selenium . For arsenic and mercury, EPA transferred the limits from the chemical precipitation treatment system to the bio logical treatment for FGD wastewater. As a result, this memo only presents the serial correlation estimates for selenium for the biological treatment option. Of the baseline values considered by EPA, only those for arsenic and selenium affect the calculat ion of serial correlations and limits. The baseline for mercury and other pollutants are less than the smallest observed data value. This memo presents the serial correlations for selenium adjusted for the baseline values shown in Table 1 and using all val id reported results (without baseline adjustment).

2. Data Processing

The data processing and analysis was performed using SAS 9.2. Westat received the data from EPA in the SE_Sampling_ Data_10_1_12 .xls Excel file. The data were imported into SAS and proces sed by: 1) dropping values to be excluded from the calculations; 2) adjusting the concentrations for the baseline values; 3) creating some categorical variables to facilitate the data processing, 4) averaging duplicate samples collected on the same day by EPA, 5) averaging the results of two samples from the same day, one collected for EPA and one collected by the plant; and 6) calculating the log

- transformed concentration.

3. Calculation of 1

- day

- lag serial correlation

For a series of daily effluent measurements with all detected values, the calculation of the 1

- day lag serial correlation assumes the log

- transformed data can be described by an autoregressive model with one correlation term for 1

- day lag, an AR(1) model. The model assumes the log

- transformed data have a normal distribution. The AR(1) model has been used by EPA in the past to describe the serial correlation in effluent series. A preliminary analysis suggests that the AR(1) model is also a reasonable choice for the steam electric data.

In general, the serial correlation term can be estimated using time series ARIMA models. However, for the steam electric data, the estimation is complicated by small sample sizes, gaps in- 3

-

the series, and non

- detected values. With small sample sizes (particularly less than 30) the algorithms can have difficulty converging, the estimates will be particularly imprecise, and the estimates will

, on average , be closer to zero than the true correlation. The presence of non

- detects in the series is particularly problematic. In many analyses, the non

- detects are replaced by a substitute value, such as the detection limit (or baseline) or the detection limit over two. The presence of non

- detects and the choice of the substitute value can bias the correlation estimates. EPA generally substitutes the detection limit (or baseline) for the non

- detected values. If there are several different detection limits, the estimated serial correlation may be more closely related to the serial correlatio n of the detection limits than to the correlation of the concentrations being measured.

The serial correlation is usually obtained by finding the parameter estimate that maximizes the likelihood of the data. The standard ARIMA models assume a likelihood f unction that is correct for serially correlated data with only detected values. Alternatively, the data can be analyzed using survival analysis using the SAS LIFEREG procedure. When using survival analysis, different likelihood functions are used for the d etects and the non

- detects where the likelihood function for the non

- detects depends on the detection limit

. Survival analysis estimates the parameters of the hypothetical distribution that would have been observed if the field samples had been analyzed

us ing a more precise analytical method such that no observations were below the detection limit. Although survival analysis uses a likelihood function that is correct for data with non

- detects , it does not account for or estimate the serial correlation. West at modified an algorithm developed in other work which combines the likelihood function for serially correlated data with the likelihood function for non

- detects . Based on simulations, the Westat procedure provides better estimates of the serial correlatio n for the concentration series than when using the ARIMA model with substitute values for the non

- detects. The Westat procedure was implemented using a Bayesian approach as described below.

Note that the correlation estimated by the Westat algorithm is th e serial correlation for the series that we would have observed if the samples had been analyzed using a laboratory technique with low detection limit such that all concentrations were detected (call this the underlying series)

. The adjustment for non

- dete cts is achieved by using a different likelihood function for the detected and the non

- detected values. Call th e resulting estimated correlation value for the underlying series Rho. However, this is not the serial correlation that is needed for the calculat ing the effluent limits. The limit calculations use the serial correlation of the series with values below the baseline replaced by the baseline and non

- detects above the baseline replaced by the detection limit (call this the baseline adjusted data). Call the serial correlation for the baseline adjusted data the baseline adjusted Rho. To estimate the baseline adjusted Rho we can either: 1) estimate Rho using the Westat macro applied to all valid reported results and adjust the estimate to account for the p resence of non

- detects (as described below in Section 6); or 2) calculate the baseline adjusted Rho from the baseline adjusted data. Although the second approach is relatively straight forward, the process of setting values below the baseline equal to the baseline reduces the information in the data and as a result the correlation estimates using the second approach are less precise than from the first approach (this was verified using simulations). In addition, the estimates can be affected by possible ser ial correlation in the detection limits. The first approach avoids these problems. Although the correlations calculated both ways are similar, the final correlations are based on estimates of Rho which are subsequently adjusted for the presence of non

- dete cts, as described below.- 4

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4. The Bayesian approach for estimating Rho

The more common and traditional statistical analysis procedures, called frequentist procedures, assume the parameters are fixed and the data are subject to random error. The resulting parameter provides an estimate of the unknown parameter and the 95% conf idence interval provides a range which is expected to cover the true parameter value 95% of the time. In contrast, Bayesian procedures assume the data are fixed and the parameter estimates have a distribution representing the possible values of the paramet er given the data and any prior information about the parameter (the Bayesian model we used assumes essentially no prior information, using a very diffuse prior distribution). For the non

- detects, the Bayesian model uses a substitute value to aid in estima ting the serial correlation but not for the likelihood for the non

- detects. Calculation of the parameter distributions for the Bayesian model involves integrating the likelihood function, which can be very difficult. As a result, the distributions are gene rally approximated using simulation. The mean and standard deviation of the simulated values are often used to summarize our knowledge of the parameter value. In practice, the frequentist and Bayesian approaches provide similar results. For this problem, t he Bayesian approach was easier to adapt than a frequentist approach for obtaining estimates of the serial correlation when some values are below the detection limit. Westat created a macro to estimate the serial correlation in data with non

- detects and wi th gaps in the data which used Bayesian methods implemented using the SAS MCMC procedure. To minimize the estimation error, the correlations are based on 200,000 simulated values.

5. Simulations to evaluate t he Bayesian approach for estimating Rho

The va lidity of this approach was assessed by simulating data with different characteristics and comparing two analysis approaches: 1) Analyze the data using the SAS ARIMA procedure with a substitute value for the non

detects (or baseline); or 2) Analyze the data usi ng the Westat Bayesian approach to estimate Rho and adjusting Rho for the presence of values below the detection limit.

The simulated series of daily data used the following parameters:

**•** Set the substitute value to the median of the expected distribution o f the non

- detects;

**•** 0%, 50%, or 80% of the daily values were randomly missing

**•** 25, 50, or 200 non

- missing observations;

**•** Rho = 0, .3, .6, or .9; and

**•** On average, 0%, 25%, 50%, or 75% of the non

- missing observations were below the detection limit.

The results are based on multiple simulated datasets for each combination of parameters above (10 datasets when simulating 200 non

- missing observations, 40 when simulating 50 non

- missing observations, and 80 simulated datasets when simulating 25 non

- missin g observations). These two approaches were compared using the bias in the estimated Rho (estimated Rho minus the true Rho used to generate the data) and the bias in the estimated standard error. If the standard error estimate from the ARIMA or Bayesian pro cedure is a good estimate of the true uncertainty in the estimate, the ratio of the pooled standard error across simulated datasets over the standard- 5

-

deviation of the estimates from the simulated datasets would be 1.0. The log of this ratio was used

as an estimate of the bias in the estimated standard error.

In general, for both approaches: 1) The procedures performed well with little bias with 200 non

- missing observations and no or few non

- detects. 2) The estimated standard error was more likely to underestima te the true error with high correlations and, for the ARIMA model, small sample sizes. 3) The estimated Rho was more likely to underestimate the true Rho for smaller sample sizes, higher proportions of non

- detects, and higher correlations.

Across all simul ated conditions, the Bayesian approach provided a similar or less biased estimate of Rho and a more appropriate (or even conservative) estimate of the standard error. Across all simulations, the average bias provides a good comparison of the approaches. Th e average bias across all simulations was

- 0.036 for the Bayesian approach and

- 0.084 for the ARIMA approach. The average bias of the standard error was 0.027 for the Bayesian approach (i.e., standard errors were generally higher (conservative) by 2.7%) an d

- 0.184 for the ARIMA approach (standard errors were generally low by 18%).

The estimates from the Bayesian approach are essentially unbiased for data similar to the available data, i.e., 50 non

- missing values, of which half were below the detection limi t, spread across 100 days, with true Rho of 0.6. In contrast, using the ARIMA model, the estimates of Rho are low by 0.09 and the standard errors are low by roughly 28%.

The objective of the simulations was to verify that the Bayesian approach provides es timates of Rho that are comparable to or better than a simple ARIMA model for estimating Rho. Based on the simulations, we expect the Bayesian estimates to be more precise and less biased. Although the results depend somewhat on the substitute value used f or non

- detects, we expect the Bayesian approach to perform at least as well as the ARIMA approach. We also expect the Bayesian estimates to be less biased because the Bayesian procedure explicitly adjusts for the presence of

non

- detects. As a result, the B ayesian approach was used to estimate the serial correlations used when calculating the limits.

6. Calculation of the baseline adjusted Rho

The baseline adjusted Rho was calculated from the estimate of Rho and the proportion of non

detects (P) by: 1) Estimate Rho by applying the Bayesian algorithm to all valid reported results to estimate the serial correlation for the underlying series. 2) Calculating the proportion of non

- detects for a selected baseline (P) from the baseline adjusted data; 3) Simulating one millio n sequential daily observations with a standard normal distribution and serial correlation Rho; 4) Setting values below the P th percentile to the P th percentile of the distribution; and 5) Setting the baseline adjusted Rho equal to the 1

- day lag correlation of the resulting series

, calculated using the SAS ARIMA procedure

.- 6

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7. Correlations Used for the Limits

Separate correlations were calculated for each combination of treatment technology option, pollutant, plant, and baseline for FGD and gasification wastewate rs. However, for many of these combinations, there were very few observations (e.g., 8) or very few observations above the detection limit. Estimates based on such data would be very imprecise, or in some cases, reliable estimates could not be obtained due to convergence problems. As a result, Rho was estimated only for datasets with a reasonable number of detected values and total number of observations. For the estimates that were calculated, there were at least 20 detected values and at least 38 values

o verall. There was enough data to calculate the serial correlation for only biological treatment for FGD wastewater.

For biological treatment for FGD wastewater, Table 2 shows the estimated serial correlations and standard errors (in parentheses). In Table 2

, Rho is the estimated correlation for the underlying series (the series that we would have observed if the samples had been analyzed using a laboratory technique with a low detection limit such that all concentrations were detected)

.

Table 2 . Correlati on estimates for the biological treatment for FGD wastewater Analyte Plan t Name N N Detected Rho (±SE)

Selenium Allen 63 20 0.8552 (±0.0501) Belews Creek 77 36

- 0.1902 (±0.2946)

Table 3 shows the estimated serial correlation for the baseline adjusted data. In Table 3, Baseline is the assumed baseline value, N detected is the number of detect ed value s after adjusting for the baseline. Baseline Adjusted Rho is the estimated correlat ion for the baseline adjusted data. For a treatment option and pollutant, if Baseline Adjusted Rho is estimated for multiple plants, EPA uses the mean correlation across the plants on the assumption that the correlation can vary between plants and the mean is a better estimate of the correlation that might be found at plants covered by the regulations. The mean is shown in the last column ( Mean Baseline Adjusted Rho ) and is the correlation used for the limits. A baseline equal to zero is equivalent to not u sing a baseline.

Table 3

. Baseline adjusted c orrelation estimates for the biological treatment technology option for FGD wastewater

Pollutant Plant Name N Baseline N detected (A fter Baseline Adjustment ) Baseline Adjusted Rho Mean Baseline Adjusted Rho

Selenium

ug/L Allen 63 0 20 0.7835 0.329 Belews Creek 77 36

- 0.1254 Allen 63 5 4 0.6815 0.291 Belews Creek 77 26

- 0.0997