Date: June 1, 2015 Memo 6-9.3

To: Ronald Jordan, USEPA

From: Luke Smith, John Rogers

Subject: Average Percent Removals of Selenium Species for Chemical Precipitation followed 

by Biological Treatment at Allen and Belews Creek

This memorandum presents the results of the average percent removal calculations for various 

forms of selenium for the flue gas desulfurization (FGD) wastewater treatment system at Duke 

Energy’s Allen Station and Belews Creek Station. The average percent removals for each selenium 

species are calculated for each plant for the following treatment stages: (i) chemical precipitation 

treatment (the percent removal calculated based on the chemical precipitation effluent and the 

FGD purge); (ii) biological treatment (the percent removal calculated based on the bioreactor 

effluent and the chemical precipitation effluent (or bioreactor influent)); and (iii) the combination 

of chemical precipitation followed by the biological treatment (the overall percent removal 

calculated based on the bioreactor effluent and the FGD purge). The selenium species are 

selenate, selenite, methylseleninic acid (MeSe(IV)), selenocyanate (SeCN), and selenomethionine 

(SeMe).

The percent removals presented in this memorandum are based on the self-monitoring data from 

Allen and Belews Creek, which were provided to the EPA by Duke Energy. 

The sections below provide a brief description of the treatment system at Allen and Belews Creek, 

summary statistics of the data at each plant, description of the percent removal calculations, and 

the percent removal results for each of the plants together with the overall removals across both 

plants. 

1. Description of the treatment system at Allen and Data at Allen and Belews Creek

Allen and Belews Creek power plants use chemical precipitation followed by anoxic/anaerobic 

biological treatment to treat FGD wastewater. In the chemical precipitation stage of the 

treatment system, chemicals are added to the wastewater in a series of reaction tanks to convert 

soluble metals to insoluble metal hydroxide compounds, which precipitate from solution and are 

removed along with other suspended solids in a clarifier. Ferric chloride (or ferrous chloride) is 

also added to increase the removal of certain metals through iron coprecipitation, and to act as a 

coagulant to enhance solids removal in the clarifier. 
-2-

The biological stage of the treatment system is a fixed-film bioreactor that consists of an activated 

carbon bed that is inoculated with microorganisms that reduce selenium and other metals, as well 

as the nitrate-nitrite form of nitrogen. The top part of the bioreactor, where the plant introduces 

the wastewater exiting the chemical precipitation stage, is aerobic with a positive oxidationreduction potential (ORP) which allows nitrification to occur. As the wastewater moves down 

through the bioreactor, it enters an anoxic zone (negative ORP) where denitrification and 

chemical reduction of selenium occurs. The microorganisms reduce the selenium species to 

elemental selenium, which adheres to the cell walls of the microorganisms. Because the activated 

carbon substrate retains the microorganisms in the bioreactor, the selenium is essentially fixed to 

the activated carbon bed until it is removed from the system. For more details about the 

treatment system at these two plants, see the Technical Development Document for the steam 

electric effluent guidelines and standards (ELGs). 

2. Summary Statistics for the Self-Monitoring Data at Allen and Belews Creek

The summary statistics, percent removal estimates, and plots presented in this memorandum are 

based on data submitted by Allen and Belews Creek1. The Allen data spans dates from September 

9, 2009 to October 22, 2013. The Belews Creek data spans dates from August 4, 2008 to 

November 28, 2013. These timespans do not include the initial commissioning periods for FGD 

wastewater treatment systems at Allen and Belews Creek. 

Tables 1 and 2 below summarize the available selenium species data at Allen and Belews Creek. 

For example, there were a total of 168 sampling dates for one or more selenium species at Allen. 

Of the 168 sampling dates, there were 159 observations that were detected for the FGD Purge 

and 0 non-detected observations. For 9 of the 168 sampling dates, there were no samples

collected at the FGD Purge for selenate. 

 

 

1 Data were received from the plants in the following Excel spreadsheets: “Allen Biochem and PhyChem monitoring since inception.xls”, “BC 

WWTS Analytical Data since inception.xls”, and “Allen_3_28_14.xlsx, Belews_3_28_14.xlsx.” The database for this evaluation excludes analytical 

anomalies, samples collected during the initial commissioning period for the treatment system, and samples collected during the period when the 

Allen Station treatment system was in an upset condition due to a broken pipe. This is consistent with the approach and data exclusions used to 

develop the numeric effluent limitations for the ELGs. See “Effluent Limitations for FGD Wastewater, Gasification Wastewater, and Combustion 

Residual Leachate for the Final Effluent Limitations Guidelines and Standards for the Steam Electric Rulemaking.”
-3-

Table 1. Summary of selenium species data at Allen plant

Plant Species Summary 

Sampling Location 

FGD 

Purge 

Chemical Precipitation 

Effluent 

Bioreactor 

Effluent 

Allen

Selenate

Total Sampling Dates 168 168 168

N. Obs Detected 159 159 14

N. Obs Non-detected 0 1 144

N. Not Sampled 9 8 10

Selenite

Total Sampling Dates 168 168 168

N. Obs Detected 119 151 100

N. Obs Non-detected 40 9 58

N. Not Sampled 9 8 10

MeSe(IV)

Total Sampling Dates 168 168 168

N. Obs Detected 54 100 4

N. Obs Non-detected 105 60 154

N. Not Sampled 9 8 10

SeCN

Total Sampling Dates 168 168 168

N. Obs Detected 34 8 2

N. Obs Non-detected 125 152 156

N. Not Sampled 9 8 10

SeMe

Total Sampling Dates 168 168 168

N. Obs Detected 13 24 4

N. Obs Non-detected 146 136 154

N. Not Sampled 9 8 10

Table 2. Summary of selenium species data at Belews Creek plant

Plant Species Summary 

Sampling Location 

FGD Purge Chemical 

Precipitation Effluent 

Bioreactor 

Effluent 

Belews

Creek

Selenate

Total Sampling Dates 203 203 203

N. Obs Detected 196 194 65

N. Obs Non-detected 0 1 118

N. Not Sampled 7 8 20

Selenite

Total Sampling Dates 203 203 203

N. Obs Detected 196 194 70

N. Obs Non-detected 0 1 113

N. Not Sampled 7 8 20

MeSe(IV)

Total Sampling Dates 203 203 203

N. Obs Detected 58 123 36

N. Obs Non-detected 87 21 96

N. Not Sampled 58 59 71

SeCN

Total Sampling Dates 203 203 203

N. Obs Detected 24 8 38

N. Obs Non-detected 121 136 94

N. Not Sampled 58 59 71

SeMe

Total Sampling Dates 203 203 203

N. Obs Detected 2 3 37

N. Obs Non-detected 143 141 95

N. Not Sampled 58 59 71
-4-

Tables 3 and 4 below provide summary statistics for concentrations for each selenium species at 

Allen and Belews Creek, respectively. The summary statistics were based on the detected and 

non-detected observations combined. Note that for non-detected observations, the concentration 

is set at the reporting limit.

Table 3. Summary statistics for selenium concentration (ug/L) for Allen

Plant Species Sampling Location N1 Summary Statistics (μg/L) 

Min Median Mean Max Std 

Allen

Selenate

FGD Purge 159 5.9 181 572.3 4800 807.5

Chemical Precipitation Effluent 160 0.5 197.7 458.7 3200 591.8

Bioreactor Effluent 158 0.1 0.5 4.2 530.9 42.2

Selenite

FGD Purge 159 0.4 33.1 202.4 4170 513.7

Chemical Precipitation Effluent 160 0.3 8.6 36.8 1440 140.4

Bioreactor Effluent 158 0.2 1 1.7 19.2 2.2

MeSe(IV)

FGD Purge 159 0.6 3 5.4 74.4 8.9

Chemical Precipitation Effluent 160 0.3 1 3.8 57.7 7.4

Bioreactor Effluent 158 0.2 0.6 0.6 2.3 0.3

SeCN

FGD Purge 159 0.3 2.5 11.1 217.5 31.3

Chemical Precipitation Effluent 160 0.1 0.5 1.4 119.6 9.4

Bioreactor Effluent 158 0.1 0.4 0.6 1.8 0.4

SeMe

FGD Purge 159 0.6 2.3 2.5 9.2 1.3

Chemical Precipitation Effluent 160 0.2 0.6 0.7 3.4 0.5

Bioreactor Effluent 158 0.2 0.5 0.6 2.3 0.3

N1: includes detected and non-detected observations. For the non-detected observations, the concentration is set at the reporting limits.

Table 4. Summary statistics for selenium concentration (ug/L) for Belews Creek

Plant Species Sampling Location N1 Summary Statistics (μg/L) 

Min Median Mean Max Std 

Belews

Creek

Selenate

FGD Purge 196 23.3 71.2 215 2470 431.5

Chemical Precipitation Effluent 195 1.5 62.8 181.5 2060 323.4

Bioreactor Effluent 183 0.1 0.6 1.5 31.1 3.2

Selenite

FGD Purge 196 9.2 88 141.5 1450 185.5

Chemical Precipitation Effluent 195 0.4 22.7 38.8 538.2 61.5

Bioreactor Effluent 183 0.2 0.8 1.2 13.5 1.5

MeSe(IV)

FGD Purge 145 0.6 3.3 3.7 13.8 2.1

Chemical Precipitation Effluent 144 0.2 2 2.5 9.3 1.7

Bioreactor Effluent 132 0.1 0.6 1 5.4 1.1

SeCN

FGD Purge 145 0.3 2.4 3.8 41.6 5.2

Chemical Precipitation Effluent 144 0.1 0.5 0.8 13.5 1.3

Bioreactor Effluent 132 0.1 0.6 1.1 6 1

SeMe

FGD Purge 145 0.5 2.4 2.9 16.8 2.2

Chemical Precipitation Effluent 144 0.1 0.6 0.7 2.4 0.5

Bioreactor Effluent 132 0.1 0.6 1 5.4 1.2

N1: includes detected and non-detected observations. For the non-detected observations, the concentration is set at the reporting limits.
-5-

3. Description of the Average Percent Removals Calculations 

The average percent removal is calculated for the following treatment stages:

i. Chemical precipitation treatment (the percent removal calculated based on the chemical 

precipitation effluent and the FGD purge); 

ii. Biological treatment (the percent removal calculated based on the bioreactor effluent and 

the chemical precipitation effluent (bioreactor influent)); 

iii. The combination of chemical precipitation followed by the biological treatment (the 

overall percent removal for the treatment system calculated based on the bioreactor 

effluent and the FGD purge). 

For a given treatment stage, the percent removal is calculated by dividing the average of the 

effluent measurements by the average of the influent measurements. This method was selected 

because the calculation is transparent and facilitates interpretation, and because of its strong 

performance in simulations when compared to other possible approaches. In the Appendix we 

compare the selected method for calculating percent removal to two other methods that were 

considered. 

For those sampling dates where either the influent or the effluent was not sampled, both the 

influent and effluent measurements for that day were not included in the calculation. If the 

influent was not detected, then both measurements were removed. If the effluent was non-detect, 

the effluent result (i.e., value equal to the detection limit) was replaced by the detection limit 

divided by 2. This is consistent with the approach EPA used to estimate pollutant removals for 

the ELGs. This approach recognizes that pollutants present in the wastewater, but at a 

concentration below the level of detection/quantitation, are likely distributed below the 

detection/quantitation limit. For purposes of estimating pollutant removals, including percent 

removals, using “detection limit divided by 2” better reflects this distribution than simply using 

the detection limit itself for all non-detected observations.

After the average percent removal was obtained at each plant, the overall average percent removal 

across both plants was calculated by averaging the removals for both plants.

4. Summary of the Average Percent Removals for Allen and Belews Creek

Table 5 below provides the average percent removal for each treatment stage at each plant by 

selenium species. For selenate at Allen, the average percent removal for the chemical 

precipitation stage of the treatment system is 19.4%, based on a total of 159 observations that 

were available at both the FGD purge and chemical precipitation effluent sampling locations. The 

average percent removal for the biological treatment stage is 99.9%, based on a total of 157

observations that were available at both the chemical precipitation effluent (bioreactor influent)

and bioreactor effluent sampling locations. The average percent removal of selenate for the entire 

treatment system at Allen (i.e., chemical precipitation followed by the biological treatment) is 

99.3%. Further results for other selenium species and Belews Creek are in Table 5. 
-6-

Table 5. Percent removal by plant and sampling location

Plant Species 

Percent Removal (N1) 

Chemical Precipitation 

Treatment2 Biological Treatment3 Overall Removal4 

Allen

Selenate 19.4% (N = 159) 99.9% (N = 157) 99.3% (N = 157)

Selenite 82.9% (N = 119) 96.4% (N = 149) 99.5% (N = 119)

MeSe(IV) 18.7% (N = 54) 94.9% (N = 99) 97.4% (N = 54)

SeCN 90.8% (N = 34) 97.7% (N = 8) 99.4% (N = 34)

SeMe 74.7% (N = 13) 74.4% (N = 23) 93.1% (N = 13)

Belews

Creek

Selenate 16.7% (N = 192) 99.4% (N = 180) 99.5% (N = 180)

Selenite 72.8% (N = 192) 97.6% (N = 180) 99.4% (N = 180)

MeSe(IV) 32.5% (N = 57) 78.5% (N = 109) 92.1% (N = 55)

SeCN 88.5% (N = 22) 71.9% (N = 7) 95.1% (N = 22)

SeMe 87.9% (N = 2) 80.7% (N = 3) 95.5% (N = 2) 1: Number of observations available to calculate percent removal based on the criteria specified in Section 3.

2: The percent removal calculated based on the chemical precipitation effluent and the FGD purge 3: The percent removal calculated based on the bioreactor effluent and the chemical precipitation effluent

4: The percent removal calculated based on the bioreactor effluent and the FGD purge.

Table 6 below provides the average percent removal across both Allen and Belews Creek by 

treatment stage and selenium species. For example, the chemical precipitation treatment stage 

removed 18.0% of selenate, which is based on averaging the percent removals of 19.4% at Allen 

and 16.7% at Belews Creek. 

Table 6. Average percent removal by selenium species and treatment stage across Allen and 

Belews Creek

Selenium Species Average Percent Removal for Allen and Belews Creek 

Chemical Precipitation Effluent Biological Treatment Overall Removal 

Selenate 18.0 % 99.7 % 99.4 %

Selenite 77.8 % 97.0 % 99.5 %

MeSe(IV) 25.6 % 86.7 % 94.8 %

SeCN 89.7 % 84.8 % 97.2 %

SeMe 81.3 % 77.6 % 94.3 %

Tables 5 and 6 demonstrate that all forms of selenium were effectively removed across the 

treatment systems at Allen and Belews Creek, as measured by comparing data for the FGD Purge 

and bioreactor effluent. Percent removals for the five species of selenium ranged from 93.1% to 

99.5% at Allen, and from 92.1% to 99.5% at Belews Creek (see Table 5). When averaging the 

removals for the two plants, the percent removals for the five species of selenium ranged from 

94.3% to 99.5% (see Table 6).

The Allen and Belews Creek treatment systems demonstrated similar efficacy, both in terms of 

overall treatment system performance and in regards to the performance of the chemical 

precipitation and biological treatment stages. Table 5 shows that, for both plants, selenite, SeCN, 

and SeMe were all removed more effectively by chemical precipitation than was the case for 

selenate and MeSe(IV). Table 5 also shows that the percent removals were lower in the biological 

treatment stage for SeMe at Allen, and for MeSe(IV), SeCN and SeMe at Belews Creek; however, 

these species were all still substantially removed by the biological treatment stage. Table 5 shows 

that the combination of chemical precipitation and biological treatment is highly effective at 

removing each of the selenium species. That finding is further illustrated by Table 6, where 
-7-

averaging the percent removals observed for Allen and Belews Creek shows that the combination 

of chemical precipitation and biological treatment results in every species of selenium being 

removed by more than 94%. 

As mentioned above, the Appendix presents two other approaches for calculating percent 

removal. In addition, the Appendix presents results of simulations to evaluate how the various 

approaches may be affected by the non-detected observations. The information in the Appendix 

shows that the approach used to calculate percent removal (as described above) is the better 

approach because the calculation is transparent and facilitates interpretation, and because of its 

strong performance in simulations when compared to other possible approaches. Furthermore, 

the information in the Appendix shows that, regardless of the approach used to calculate percent 

removals, the percent removals calculated by the three approaches evaluated are generally 

consistent. The results presented in this memorandum show that the combination of chemical 

precipitation and biological treatment effectively removes all five species of selenium.
-8-

Appendix

This Appendix discusses three methods for calculating percent removal, discusses the handling of 

non-detects, and compares results using the three different methods for calculating mean percent 

removal for selenium species, with a focus on selenate and selenite. Based on this evaluation, 

Method 2 (described below) was selected as the preferred approach because it demonstrates 

relatively small bias across a range of conditions. Additionally, the calculation using this method is 

transparent and facilitates interpretation. Using this approach, the average influent concentration 

and the average effluent concentration are calculated, and then these values are used to calculate 

the overall percent removal.

“Percent removal” refers to the fraction of the analyte in the influent that is removed by the 

processing system, generally between 0 and 1, which may be presented as a percentage by 

multiplying by 100.2 Percent removal is the fraction of the influent concentration that is removed 

by the treatment system based on comparing the long-term arithmetic average influent 

concentration to the long-term arithmetic average effluent concentration. 

The data used for the analysis are influent and effluent measurements from two facilities (Allen 

and Belews Creek). Percent removal is calculated for 1) chemical precipitation; 2) biological 

treatment; and 3) the combination of chemical precipitation and biological treatment. This 

analysis excludes sample days for which the influent measurement was below the detection limit 

and days for which either the influent or effluent measurement was missing. 

The three approaches for calculating the percent removal are:

 Method 1: Calculating the percent removal for each sampling day, and then averaging 

the daily percent removal across sampling days to estimate the overall percent 

removal.

 Method 2: Calculating the average influent concentration and the average effluent 

concentration, and then using these values to calculate the overall percent removal.

 Method 3: Calculating the mean and standard deviation of the log-transformed 

influent and effluent concentrations, from which the overall percent removal is 

estimated.

Although Method 1 appears to be a reasonable approach, its estimates of percent removal are 

biased and tend to be lower than the true percent removal. This bias persists even as the sample 

size increases. In contrast, in the absence of non-detects, the bias of Method 2 goes to zero as the 

sample size increases. Method 2 does not make assumptions about the distribution of the data; 

however, the method relies on large sample approximations and may be slightly biased with small 

samples and highly skewed data. Method 3 assumes that influent and effluent concentrations are 

log normally distributed, i.e., on the log scale are normally distributed. The distributions of 

influent and effluent of the selenium species often have characteristics of a lognormal distribution. 

However, Method 3 may perform poorly if the data do not behave like a lognormal distribution. 

 

2 In this Appendix, with the exception of Tables A.1 through A.4, the percent removal is presented as a fraction between 0 and 1. For Appendix 

Tables A.1 through A.4 and the main body of this memo, percent removal is presented as a percentage.
-9-

In our simulation study where the data are log normally distributed, the point estimates for 

Methods 2 and 3 are similar. 

The handling of the non-detects affects the estimates of percent removal, regardless of which 

method is used. This Appendix evaluates replacing the non-detects by the detection limit and half 

the detection limit. In the equations below Ii stands for the influent concentration on day i and Ei

stands for the effluent concentration on day i. Ei is equal to the detection limit or the detection 

limit divided by 2 (depending on the assumptions). Pairs of influent and effluent concentrations 

from the same day are excluded from the calculations if the influent concentration is missing or 

not detected and if the effluent concentration is missing. Table A1 summarizes the data used to 

calculate the removal ratios.

Table A1. Summary of how the percent removal calculations are performed for each date the

sample was taken at each plant

Influent Concentration Effluent Concentration Percent Removal Uses 

Detected Detected Reported influent and 

effluent concentrations

Detected Non-detected Reported influent 

concentration and the 

effluent detection limit 

divided by 2

Non-detected Detected Data from these days are 

excluded from the 

calculations

Non-detected Non-detected

Detected, Non-detected, not sampled Not sampled

Not sampled Detected, Non-detected

Methods 2 and 3 give similar results and have less bias than Method 1. Due to the ease of 

calculation and relatively small bias across a range of conditions, Method 2 was selected as the 

preferred approach for calculating percent removal.

The rest of this Appendix provides additional detail on the different methods, simulation results 

comparing the methods, and estimates of selenite and selenite percent removal using the three 

methods.

Methods for calculating percent removal

The analysis can estimate the ratio of the effluent to influent concentrations (call this the removal 

ratio):

R = E

I

,

from which the percent removal can be estimated:

P = 1 − R = I − E

I .

The removal ratio can be calculated in different ways.
-10-

Method 1

Using i to indicate different sampling days, calculating the arithmetic mean of the daily estimates 

of percent removal corresponds to:

R1 = ∑ Ri

N

i=1

N ; wheee Ri = EiIi

.

The percent removal is P1 = 1 − R1 . P1 is the estimate previously calculated by EPA.

This calculation weights each sampling day equally when averaging the daily removal ratios. Using 

Wi to indicate the weight, the equation above is equivalent to:

R1 =

∑ Ei

Ii

Wi

N

i=1

∑ Wi

N

i=1

; Wi = 1.

Thus days with small amounts of waste to process have the same weight (or influence on the 

calculated removal ratio) as days with large amounts of waste to process. Further, R1 converges to 

the mean of daily effluent divided by daily influent, which is not necessarily the same as R. 

Method 2 (the selected approach)

An alternative approach might weight days with higher influent concentration more than days 

with lower influent concentration. If the weight is proportional to the influent concentration (Ii), 

the calculations become:

R2 =

∑ Ei

Ii

Ii

N

i=1

∑ Ii

N

i=1

= ∑ Ei

N

i=1

∑ Ii

N

i=1

=

∑ Ei

N

i=1

N

∑ Ii

N

i=1

N

= E

I

,

where E is the sample mean of the effluent and I is the sample mean of the influent. 

Using this method, the percent removal is P2 = 1 − R2 . This is a simple method of moments 

estimator that is commonly used for estimating ratios of means. This value does converge to the 

true removal ratio as the sample size increases. The variance of R2 is approximately: 

VV (R2) = �

E

I

�

2 � VV (E)

E

2 + VV (I)

I

2 − 2 CCC(E,I)

E ∗ I �

N .
-11-

Method 3

The influent and effluent concentrations can often be reasonably described as having a lognormal 

distribution. The mean of a lognormal distribution can be calculated from the mean and standard 

deviation of the log-transformed concentrations. As a result, the ratio of the mean of the effluent 

concentrations to the mean of the influent concentrations can be calculated using the following 

formula:

R3 =

eee �M M�l (Ei)� +

VV �l (Ei)�

2 �

eee �M M�l (Ii)� +

VV �l (Ii)�

2 �

R3 is the ratio of the arithmetic mean of the effluent concentration to the arithmetic mean of the 

influent concentration where each mean is calculated from the mean and standard deviation of the 

log-transformed influent or effluent concentrations. The percent removal is P3 = 1 − R3. There 

is not a simple form for the standard error associated with R3. Bootstrap confidence intervals 

were calculated for the tables below.

Simulations comparing the various estimates

We performed simulations to evaluate the three approaches above for estimating percent removal. 

Separate simulations were run for combinations of settings for the true percent removal, the 

fraction of effluent measurements that were non-detects, and the substitute value used for the 

non-detects. The arithmetic mean influent concentration was either 2, 10, or 100 times the 

arithmetic mean concentration in the effluent (which was set to equal 1.0). These values 

correspond to removal percentages of 50%, 90%, and 99%. The fraction of effluent non-detects 

was set to either 0% or 50%. Non-detects were replaced by the detection limit divided by a scaling 

factor, either 1 or 2. For each set of simulation parameters we generated data for N = 100 days 

that was normally distributed (representing log transformed concentrations) with standard 

deviations σI = 0.5, 1, 2 and σE = 0.5, 1, 2 for influent and effluent respectively. Standard 

deviations of 0.5, 1, and 2 correspond to coefficients of variation (CVs) of the simulated 

concentrations of 0.53 (a slightly skewed distribution), 1.31 (clearly skewed), and 7.32 (highly 

skewed). The mean of influent and effluent concentrations was used to calculate the mean of the 

simulated normally distributed log-transformed concentrations. The detection limit was set to 

obtain the specified probability of effluent non-detects. That detection limit was used to 

determine if the simulated influent and effluent concentrations were non-detects. We also added a 

daily random effect to both log distributions that was normally distributed with mean 0 and 

standard deviation 0.3. For each combination of settings we performed 1000 replications. The 

different simulation settings are listed in Table A2. We compare the three estimators using mean 

absolute deviation (MAD), 95% confidence interval coverage probability, and bias. 

 
-12-

Table A2. Different simulation settings

Simulation setting ND % Scaling Factor Influent to effluent 

concentration ratio 

1 0% 2 100

2 50% 2 100

3 50% 1 100

4 0% 2 10

5 50% 2 10

6 50% 1 10

7 0% 2 2

8 50% 2 2

9 50% 1 2

Simulation results

Figure A1 shows the mean absolute deviation (MAD) of the different estimators under different 

simulation settings. Small values of MAD indicate that the estimated percent removal values are 

close to the true value. MAD may be high because the estimates are biased (higher or lower than 

the true percent removal on average) or because the estimates have more variability. MAD is 

highest for Method 1 in all except simulation settings 8 and 9. Method 3 usually outperforms 

method 2. Note that each panel in the figures below uses a different scale on the vertical axis. 

When the coefficient of variation of the effluent concentrations is relatively low (lower row of the 

plot), the MAD is much smaller than when the coefficient of variation is much larger. The MAD 

using F=1 is similar to or slightly higher than when using F=2, suggesting a preference for using 

the detection limit over 2 as the substitute value for non-detects as opposed to using the detection 

limit.

In general, the estimates of percent removal using Method 1 are below those for Method 2 and 3. 

In the simulation setting with 50% removal and 50% non-detects (simulation settings 8 and 9), a 

significant fraction of the influent concentrations may be non-detects. Influent-effluent pairs from 

the same day are excluded from the calculations if the influent measurement is not detected. The 

remaining observations used to calculate percent removal have higher average influent 

concentrations and thus appear to have high percent removals. As a result, for low percent 

removal and a high proportion of non-detects, estimates using Method2 and 3 tend to 

overestimate the percent removal. Estimates using Method 1 are also higher than otherwise, but 

are still lower than those using Method 2 and 3 and generally lower than the true percent removal. 

Thus, for simulation settings 8 and 9, Methods 2 and 3 tend to over- estimate the percent removal 

and Method 1 tends to under-estimate the percent removal, resulting in similar MAD for some 

combinations of influent and effluent CVs. 

As the simulation setting number increases the true percent removal decreases which is often 

associated with a corresponding increase in the MAD (i.e., percent removal estimates are more 

variable when the true percent removal is near 50% than when it is near 100%). In addition, the 

percent removal estimates for Method 1 are particularly sensitive to the fraction of non-detects, 

resulting in the “jagged” patterns for Method 1. As the simulation setting number increases, the 

drop in the MAD for Method 1 corresponds to a change from 0% to 50% non-detects in the 

effluent data.
-13-

Figure A1. Summary of mean absolute deviation (MAD). Simulation setting is on the x axis, and 

MAD is on the y axis. The simulation settings are described in Table A2

Mean Absolute Deviation

0.005 0.020 0.100

E = 

2

0.005 0.020 0.100 0.500

5e-03 5e-02 5e-01 5e+00

0.001 0.005 0.020 0.100

E = 1

0.002 0.010 0.050 0.500

5e-03 5e-02 5e-01 5e+00

5e-04 2e-03 1e-02 5e-02

I = 0.5

E = 0.5

1 2 3 4 5 6 7 8 9

Method 1

Method 2

Method 3

0.001 0.005 0.050 0.500

I = 1

1 2 3 4 5 6 7 8 9

5e-03 5e-02 5e-01 5e+00

I = 2

1 2 3 4 5 6 7 8 9
-14-

Figure A2 shows the bias of the different estimators under different simulation settings. The bias 

is the average difference between the estimated and true percent removal. Across all simulation 

settings Method 1 is biased low, giving estimates of percent removal that are, on average less or 

much less than the true percent removal. In contrast, Method 2 and 3 are relatively unbiased (have 

bias close to zero on the vertical axis). However, as noted in the discussion above, for simulation 

settings 8 and 9, Method 2 and 3 can be biased high, sometimes by a similar amount that Method 

1 is biased low. Note that each panel in Figure A2 uses a different scale on the vertical axis. When 

the effluent has a high coefficient of variation (top row) the Method 1 estimates of percent 

removal can be negative and quite variable. 
-15-

Figure A2. Summary of bias. Simulation setting is on the x axis, and bias is on the y axis. The 

simulation settings are described in Table A2

Bias

-0.1 0.0 0.1 0.2

E = 

2

-0.8 -0.6 -0.4 -0.2 0.0 0.2

-25 -20 -15 -10 -5 0

-0.20 -0.10 0.00 0.05

E = 1

-0.8 -0.6 -0.4 -0.2 0.0

-25 -20 -15 -10 -5 0

-0.15 -0.10 -0.05 0.00 0.05

I = 0.5

E = 0.5

1 2 3 4 5 6 7 8 9

Method 1

Method 2

Method 3

-0.8 -0.6 -0.4 -0.2 0.0 0.2

I = 1

1 2 3 4 5 6 7 8 9

-25 -20 -15 -10 -5 0

I = 2

1 2 3 4 5 6 7 8 9
-16-

Figure A3. Summary of coverage probabilities of the estimators. Simulation setting is on the x 

axis, and 95% coverage probabilities are on the y axis. The simulation settings are 

described in Table A2

95% Coverage Probability

Figure A4 shows boxplots for transformed estimated removal ratios from the simulations where 

the true percent removal is 0.90, and 50% of the effluent observations are non-detects replaced by 

the detection limit over two (simulation setting 5). The boxplots are provided to illustrate the 

typical distribution of the percent removal estimates. The transformation was chosen to make the 

distributions roughly symmetric. The dashed black line shows the true percent removal in the 

transformed scale and the red dashed line shows a percent removal of zero. The inner axis label 

(closest to the plot) shows the scale for the transformed percent removals. The outer axis label 

shows the corresponding values of percent removal. Note that, particularly for measurements with 

E = 

2

0

0.25

0.5

0.75

1

E = 1

0

0.25

0.5

0.75

1

I = 0.5

E = 0.5

1 2 3 4 5 6 7 8 9

0

0.25

0.5

0.75

1

Method 1

Method 2

Method 3

I = 1

1 2 3 4 5 6 7 8 9

I = 2

1 2 3 4 5 6 7 8 9
-17-

a high coefficient of variation, the estimated percent removal can be negative (below the red 

dashed line) and occasionally highly negative. 

Using the transformed scale for this simulation setting, the percent removal estimated using 

Method 1 is less or much less than using Method 2 or 3. Estimates from Method 2 are generally 

similar to or closer to the true percent removal than using Method 3.

Figure A4. Distribution of simulated percent removal (P) estimates using the three calculation 

methods (true percent removal = 0.90, 50% non-detects replaced by DL/2)

In Tables A3 to A6 below we have calculated confidence intervals for the estimated percent 

removal using data from the Allen and Belews Creek plants. Figure A3 provides additional 

information on the performance of those confidence intervals. For strong estimators, we expect 

the confidence interval to cover the true percent removal 95% of the time (see the horizontal line -2 0 2 4

-30

-9

-3

0

0.7

0.9

0.97

0.99

P (Transformed scTransformed P = -ln

E = 

2

-2 0 2 4

-30

-9

-3

0

0.7

0.9

0.97

0.99

P (Transformed scale)Transformed P = -ln(1-P)

E = 1

Method1 Method2 Method3

-2 0 2 4

-30

-9

-3

0

0.7

0.9

0.97

0.99

P (Transformed scale)Transformed P = -ln(1-P)

E = 0.5

E = 0.5

Method1 Method2 Method3

E = 1

Method1 Method2 Method3

E = 2
-18-

at 0.95). While Method 3 performs well across the board with respect to MAD, the coverage 

probabilities of Method 3 are uneven particularly when the coefficient of variation of the influent 

measurements is high (last column), as shown in Figure A3. The coverage probabilities are 

particularly sensitive to the fraction of non-detects and the substitute value used for the nondetects (i.e., F = 1 or 2), resulting in the “jagged” patterns in some panels of the plot. Simulation 

setting numbers 1, 4, and 7 have no non-detects, simulation setting numbers 2, 5, and 8 have 50% 

non-detects replaced by the detection limit over two, and simulation setting numbers 3, 6, and 9 

have 50% non-detects replaced by the detection limit. 

Method 2 and 3 have similar coverage probabilities with Method 2 better than Method 3 for some 

panels in the plot and the reverse in other panels. Excluding simulation settings 8 and 9, the 

minimum coverage for the Method 2 confidence intervals is higher than the minimum for Method 

3. Excluding simulation settings 8 and 9, the coverage for Method 1 is consistently less than for 

the other methods and Method 1 has instances where the coverage is essentially zero.

The simulations were constructed to compare three relatively easy methods for calculating percent 

removal. Of these, we recommend Method 2. Method 2 is simple to calculate, and is competitive 

with or better than Method 1 and 3 with respect to MAD, bias, and coverage probability across 

simulation settings. We believe the different patterns associated with a combination of low 

percent removal and a high proportion of non-detects and the sensitivity of some estimates to the 

presence and handling of non-detects might be addressed using a model based approach; 

however, the calculations would be much more complex.

Recommended Procedure for Calculating Percent Removal

Because the calculations for Method 2 are much easier than for Method 3 and the performance of 

Method 2 and 3 are otherwise fairly similar and usually much better than Method 1, we 

recommend using Method 2 for calculating percent removal. EPA has previously substituted the 

detection limit over two for non-detects when calculating percent removal. The simulation results 

provide no reason to modify that approach. Thus, we recommend the following analysis steps:

 If most or all influent and effluent measurements can be paired, such as influent and 

effluent measurements from the same day, exclude values that are unpaired. This 

minimizes the influence of day-to-day variation on the results.

 Exclude paired data for which the influent is less than the detection limit. Such 

samples provided little information on percent removal.

 Replace the non-detects by the detection limit divided by 2.

 Calculate the arithmetic average of the influent and effluent measurements, E and I.

 Calculate the percent removal:

P = 1 − E

I
-19-

 If desired, calculate the variance of the influent and effluent measurements and their 

covariance from which the variance of the estimate of percent removal is calculated 

as follows (N is the number of paired measurements):

VV (P) = �

E

I

�

2 � VV (E)

E

2 + VV (I)

I

2 − 2 CCC(E,I)

E ∗ I �

N .

 If desired, calculate an approximate 95% confidence interval for the percent removal:

95% C = P ± 1.96�VV (P).

Estimates of percent removal using the different methods

Tables A3 through A6 show estimated percent removal for the Allen and Belews Creek data for 

selenate and selenite using the three methods described above. In general, the estimates using 

Method 2 and 3 are more similar to each other than to the estimate using Method 1. The percent 

removals calculated using Method 2 and 3 generally are larger than from Method 1; however, 

there are some exceptions. Approximate confidence intervals are provided for the estimates. Note 

that when the removal percentages approach 100% any differences between the methods become 

difficult to see. The differences in the methods are apparent when the removal percentages are 

much less than 100%.

Table A3. Estimated percent removal for selenate at the Allen plant, with 95% confidence 

intervals using three different calculation methods

Substitution 

method for 

non-detect 

Method 

Treatment 

Chemical Precipitation Biological Both 

(Chem Precip + Bio) 

ND=DL

1 0.1% (-11.7%, 11.9%) 99.1% (98.8%, 99.4%) 98.4% (96.4%, 100.5%)

2 19.4% (9.2%, 29.6%) 99.8% (99.8%, 99.9%) 99.2% (98.1%, 100.4%)

3 2.6% (-25%, 19.9%) 99.9% (99.8%, 99.9%) 99.9% (99.8%, 99.9%)

ND=DL/2

1 0.1% (-11.7%, 11.9%) 99.5% (99.4%, 99.7%) 98.7% (96.6%, 100.7%)

2 19.4% (9.2%, 29.6%) 99.9% (99.8%, 99.9%) 99.3% (98.1%, 100.5%)

3 0.3% (-32%, 19.6%) 99.9% (99.9%, 99.9%) 99.9% (99.8%, 99.9%)

Table A4. Estimated percent removal for selenate at the Belews Creek plant, with 95% 

confidence intervals using three different calculation methods

Substitution 

method for 

non-detect 

Method 

Treatment 

Chemical Precipitation Biological Both 

(Chem Precip + Bio) 

ND=DL

1 -3.3% (-16.8%, 10.1%) 98.1% (97.6%, 98.7%) 98.6% (98.3%, 98.9%)

2 16.7% (2.9%, 30.4%) 99.2% (99.0%, 99.5%) 99.4% (99.1%, 99.6%)

3 4.6% (-10.4%, 18.4%) 99.2% (99.0%, 99.5%) 99.3% (99.0%, 99.5%)

ND=DL/2

1 -3.3% (-16.8%, 10.1%) 98.7% (98.2%, 99.2%) 99.1% (98.9%, 99.3%)

2 16.7% (2.9%, 30.4%) 99.4% (99.2%, 99.7%) 99.5% (99.3%, 99.7%)

3 3.5% (-12.5%, 17.6%) 99.4% (99.2%, 99.6%) 99.5% (99.3%, 99.6%)
-20-

Table A5. Estimated percent removal for selenite at the Allen plant, with 95% confidence 

intervals using three different calculation methods

Substitution 

method for 

non-detect 

Method 

Treatment 

Chemical Precipitation Biological Both 

(Chem Precip + Bio) 

ND=DL

1 61.7% (52.8%, 70.6%) 71.2% (62.1%, 80.2%) 94.0% (90.9%, 97.0%)

2 82.9% (74.9%, 90.9%) 96.1% (93.6%, 98.5%) 99.4% (99.2%, 99.7%)

3 87.8% (81.3%, 91.7%) 93.9% (90.8%, 96.2%) 99.4% (99.0%, 99.6%)

ND=DL/2

1 61.7% (52.8%, 70.7%) 72.6% (63.5%, 81.8%) 94.2% (91.1%, 97.2%)

2 82.9% (74.9%, 90.9%) 96.4% (94.1%, 98.7%) 99.5% (99.2%, 99.7%)

3 87.4% (80.7%, 91.5%) 94.2% (91.3%, 96.4%) 99.4% (99.1%, 99.7%)

Table A6. Estimated percent removal for selenite at the Belews Creek plant, with 95% 

confidence intervals using three different calculation methods

Substitution 

method for 

non-detect 

Method 

Treatment 

Chemical Precipitation Biological Both 

(Chem Precip + Bio) 

ND=DL

1 54.6% (46.0%, 63.3%) 92.2% (90.0%, 94.4%) 97.8% (97.3%, 98.4%)

2 72.8% (68.6%, 77%) 96.9% (96.0%, 97.9%) 99.2% (98.9%, 99.4%)

3 73.6% (69.1%, 77.4%) 96.9% (95.9%, 97.6%) 99.2% (98.9%, 99.3%)

ND=DL/2

1 54.6% (46.0%, 63.3%) 93.9% (92.0%, 95.8%) 98.3% (97.8%, 98.8%)

2 72.8% (68.6%, 77%) 97.6% (96.8%, 98.5%) 99.4% (99.2%, 99.5%)

3 73.4% (68.9%, 77.4%) 97.7% (96.9%, 98.3%) 99.4% (99.2%, 99.5%)