MEMORANDUM 

To: Timothy Leighton, EPA 

From: Jonathan Cohen, ICF 

Date: 15 March, 2011 

Re: Contract No.: EP-W-06-091 WA 4-02: 

AEATF Wipe Study Using Trigger Spray and Wipe Statistical Review for HSRB 

Introduction and Summary 

In January 2011, AEATF submitted the final report for their study “A Study for Measurement of Potential Dermal and 

Inhalation Exposure during Application of a Liquid Antimicrobial Pesticide Product Using Trigger Spray 

and Wipe or Ready-to-Use Wipes for Cleaning Indoor Surfaces.” ICF were asked to analyze the data for Trigger Spray 

and Wipe from this study to investigate the relationship between dermal and inhalation exposures and the pesticide 

product usage. Note that much of the SAS code used for these analyses and some of the following description was 

adapted from Sarkar’s SAS code (which, in turn, was based on code provided by the AHETF) and his June 2010 

Statistical Review “Review of Statistical Analyses in Agricultural Handler Exposure Task Force (AHETF) 

Monographs.” 

The report describes the experimental study methodology and the measurements in detail. Briefly, three test sites in 

Fresno, CA, were selected to represent: cluster 1, an office space; cluster 2, a retail/shopping center, and; cluster 3, a 

meeting space. For each test site, six volunteer subjects were selected and randomly assigned different wiping time 

durations from 30 minutes to 210 minutes. Each subject was given inner and outer dosimeters to wear and was also 

given a personal air-sampling pump to monitor their inhalation exposure. Each subject then used a trigger spray bottle 

and rags to wipe horizontal and vertical surfaces for their assigned wiping time duration. In cluster 1, the DDAC 

pesticide was diluted as one part concentrated test substance and 64 parts tap water according to the product directions. 

In clusters 2 and 3, the directions were misinterpreted and the DDAC pesticide was diluted as one part concentrated test 

substance and 63 parts tap water. Subjects were allowed to take breaks on a chair away from the wiping site, as needed. 

The breaks were not included in the wiping time duration but the air-sampling pump remained on throughout the 

breaks. Although the study design was not a probability survey, some elements of the design used randomization. 

These statistical analyses treat the data as either a simple random sample or a stratified random sample, where the 

clusters are the strata. 

The exposure measurements in the report were corrected for the average percentage recovery of field fortification 

samples and for the removal efficiency of hand wash samples. These analyses used the corrected measurements. The 

data in the report were entered and compiled into an Excel spreadsheet. This included the units conversion of the 

amounts of active ingredient from mg into pounds and of the mass from μg to mg. The report data for inhalation 

exposure were unchanged other than the units conversions. The dermal exposure data were used to develop exposure 

measurements for three dermal exposure routes, as follows: 

• Long Dermal. This case represents the dermal exposure for a subject wearing long pants and long-sleeved 

shirts, without gloves. The exposure is the sum of the mass from hand wash, face and neck, and the six inner 

dosimeters (lower arm, upper arm, lower leg, upper leg, front torso, rear torso). 
Timothy Leighton, EPA 

15 March, 2011 

Page 2 

• Short Dermal. This case represents the dermal exposure for a subject wearing short pants and short-sleeved 

shirts, without gloves. The exposure is the sum of the mass from hand wash, face and neck, the outer dosimeters 

for the lower arm and lower leg, and the six inner dosimeters (lower arm, upper arm, lower leg, upper leg, front 

torso, rear torso). 

• Long Short Dermal. This case represents the dermal exposure for a subject wearing long pants and shortsleeved shirts, without gloves. The exposure is the sum of the mass from hand wash, face and neck, the outer 

dosimeter for the lower arm, and the six inner dosimeters (lower arm, upper arm, lower leg, upper leg, front 

torso, rear torso). 

The report only considered the dermal exposure for the “Long Dermal” case, and used the same definition. However 

for our analyses we chose a more consistent, and more conservative (i.e., more health protective) approach to deal with 

values reported as being below the level of quantitation (LOQ): 

Several of the measured values were below the level of quantitation (LOQ). The experimental protocol also required 

that measurements of the inner dosimeters were not taken if the outer dosimeter was below the LOQ. As a slightly 

more conservative (i.e., more health protective) approach than the method used in the report, we replaced any value that 

was either a non-detect or was not measured by one half the LOQ. If any inner or outer dosimeter value was below the 

LOQ, 3 μg, each such value was replaced by 1.5 μg = 0.0015 mg. For example, if all the inner dosimeters were below 

the LOQ, then the total would be replaced by 0.009 mg. If the face and neck measurement was below the LOQ, 50 ng, 

it was replaced by 25 ng = 0.000025 mg. All the hand wash measurements in the study were above the LOQ. 

In the body of this memorandum we present the analysis of the unit or normalized exposure defined as the dermal or 

inhalation exposure divided by the pounds of active ingredient handled. Estimates of the arithmetic and geometric 

means and standard deviation as well as the 95th percentile are computed using the empirical data as well as two 

statistical models: the lognormal simple random sampling model and the lognormal mixed model. The mixed model 

includes the possibility of clustering due to the effects of the location. For example, the results for different subjects at 

the office site might be correlated because of the physical similarities of the room, the physical similarities of the 

surfaces that were wiped, the building characteristics, the investigators at that location, or other factors that are due to 

the selected location. For each summary statistic we present confidence intervals. We also compute the fold relative 

accuracy of the summary statistics and compare with the study design benchmark of 3-fold accuracy. To evaluate the 

statistical models we present quantile-quantile plots to compare the fit of normal and lognormal distributions to the 

data. 

The statistical models for the normalized exposure assume that the mean value of the logarithm of the exposure is equal 

to an intercept plus the slope times the logarithm of the amount of active ingredient used, where the slope equals 1. To 

test this assumption, the regression model was fitted to the data either using simple random sampling or the mixed 

model and a 95% confidence interval for the slope was calculated. A statistical test was used to determine if the slope 

was 1 or 0, corresponding either to a valid normalized exposure model or to a case where the exposure is independent 

of the amount of active ingredient used. We applied this test to the three dermal exposures and to the inhalation 

exposure using the statistical mixed model. For dermal exposures it is reasonable to assume on physical grounds that 

the same patterns ought to apply to any type of dermal exposure, so that the slope should either be one for all types of 

dermal exposure or not one for all types of dermal exposure. To evaluate this issue we applied the same proportionality 

test to a hypothetical all dermal exposure case representing a janitor with no clothing, using a dermal exposure 

estimated as the sum of the exposures measured on the face and neck, hands, and all the inner and outer dosimeters. We 

also developed a statistical repeated measures model to analyze all three types of dermal exposure in a single statistical 
Timothy Leighton, EPA 

15 March, 2011 

Page 3 

model, accounting for within-worker correlations and within-location clustering. We also evaluated quadratic 

regression models. 

The results for long dermal and long short dermal exposure routes show that the estimated intra-cluster correlation 

(ICC) coefficient is zero, which implies that there are no clustering and location effects. However, the estimated intracluster correlation (ICC) coefficient for short dermal is 0.2 and for inhalation exposure is 0.3, which implies that there 

are some clustering and location differences, which one can hypothesize as being primarily due to differences in the 

wiping surfaces between different buildings. 

The mixed model results for the four dermal exposure routes all show a positive estimated slope but give inconsistent 

results for the tests of proportionality (slope equals one) and independence (slope equals zero). At the 5% level, the 

proportionality is rejected for long dermal and short dermal, but proportionality is not rejected for long short dermal 

and all dermal. At the 5% level, the independence is rejected for short dermal, long short and all dermal, but 

independence is not rejected for long dermal. Using the repeated measures model, proportionality and independence are 

both rejected. The mixed model results for inhalation exposure show a small slope of 0.2 and the proportionality is 

rejected. 

The appendix gives the detailed results of the corresponding analyses when the exposure is normalized by the wiping 

duration. The appendix also includes an analysis of inhalation exposure normalized by the product of the pounds of 

active ingredient handled and the wiping duration, and an analysis of inhalation exposure normalized by the surface 

area wiped. The results show similar patterns to the normalization by amount of active ingredient handled but the 

slopes are lower, showing less support for those alternative exposure models. For inhalation, the data suggest that the 

exposure is either independent of any of these normalizing variables or is a non-linear function of these normalizing 

variables. A summary comparing the normalizing approaches is presented at the end of this memorandum. The mixed 

model analyses show that in most cases (including the repeated measures model), the best of the normalizing options 

considered is the amount of active ingredient handled for dermal exposures, and is the surface area wiped for inhalation 

exposure. 

The inhalation exposure measure used for the main analyses presented here is the average air concentration (mg/m3

) 

over the entire period that the concentration was measured, which includes the breaks as well as the wiping durations. 

As an alternative approach we considered using the air DDAC mass (mg) which can be estimated as the average air 

concentration multiplied by the wiping duration (hours) and by an estimated 1 m3/hour of air breathed in by someone 

doing light activity. A similar statistical analysis found that the air DDAC mass is not proportional to amount of active 

ingredient × wiping duration, but there is not enough evidence at the 5% level to reject proportionality for the other 

three normalizing variables (amount of active ingredient, wiping duration, surface area wiped). As shown in the 

summary table 10 below, the best-fitting model for the normalized air DDAC mass normalizes for the amount of active 

ingredient used. The appendix includes the detailed results for this model. 

Analyses of exposure per pounds of active ingredient handled 

Table 1 summarizes the normalized exposure data (per lb active ingredient handled) with the summary statistics from 

the 18 measurements for each exposure route. 
Timothy Leighton, EPA 

15 March, 2011 

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Table 1. Summary statistics for normalized exposure. 

Statistic 

Normalized Longa

Dermal 

(mg/lb AI) 

Normalized Shortb

Dermal 

(mg/lb AI) 

Normalized Long Shortc

Dermal 

(mg/lb AI) 

Normalized 

Inhalation 

(mg/m3/lb AI) 

Arithmetic Mean 1065.8 1736.2 1544.7 10.2 

Arithmetic Standard 

Deviation 853.9 1083.4 1027.2 10.8 

Geometric Mean 859.2 1507.3 1325.1 6.7 

Geometric Standard 

Deviation 1.9 1.7 1.7 2.7 

Min 384.2 684.8 668.8 0.4 

 5% 384.2 684.8 668.8 0.4 

10% 455.3 785.3 672.2 3.0 

25% 551.6 1046.0 844.8 4.2 

50% 731.1 1276.1 1195.0 7.4 

75% 1139.9 2250.4 1927.0 12.8 

90% 2780.0 2988.6 2929.7 26.5 

95% 3478.6 5154.6 4841.8 46.3 

Max 3478.6 5154.6 4841.8 46.3 a

Long = Long pants and long sleeves 

b

Short = Short pants and short sleeves 

c

Long Short = Long pants and short sleeves

The summary analyses presented in Table 1 use the 18 measurements with a simple random sampling model that 

ignores the fact that the data were selected by choosing three “clusters” representing three different location types, and 

then measuring dermal and inhalation exposures on six subjects in each of the three clusters (a total of 18 different 

volunteer subjects). The six subjects in each cluster were randomly assigned six different amounts of wiping, as defined 

by the time to be spent wiping: 

• 30 to < 60 minutes 

• 60 to < 90 minutes 

• 90 to < 120 minutes 

• 120 to < 150 minutes 

• 150 to < 180 minutes 

• 180 to 210 minutes 

The statistical analyses use the following three alternative statistical models. Let X be the normalized exposure and X = 

exp(Y) so that Y = log (X), where log denotes the natural logarithm. LnGM is the log of the geometric mean. Let Z95 

be the 95th percentile of a standard normal distribution, approximately 1.645. 
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15 March, 2011 

Page 5 

• Empirical simple random sampling model. Code “s.” Assumes that the 18 values of X were randomly drawn 

from an unspecified distribution. Ignores clustering. Gives empirical estimates such as in Table 1 above. 

o Y = LnGM + Error. Error is independent and identically distributed with mean 0. 

o AMs = Arithmetic mean of X values 

o GMs = Geometric mean of X values = exp(LnGM) (= GMu) 

o GSDs = Geometric standard deviation of X values (= GSDu) 

o P95s = 95th percentile of X values 

• Lognormal simple random sampling model. Code “u.” Assumes that the 18 values of X were randomly drawn 

from a log-normal distribution. Ignores clustering. 

o Y = LnGM + Error. Error is normally distributed with mean 0, variance Vu, and standard deviation Su = 

√Vu. 

o AMu = Modeled arithmetic mean of X values = exp(LnGM) exp(1⁄2 Vu) 

o GMu = Modeled geometric mean of X values = exp(LnGM) 

o GSDu = Modeled geometric standard deviation of X values = exp(Su) 

o P95u = Modeled 95th percentile of X values = exp(LnGM) exp(Z95×Su) 

• Lognormal mixed model. Code “m.” Assumes that three cluster effects were randomly drawn from a normal 

distribution and that the 18 within-cluster error terms were independently randomly drawn from another normal 

distribution. The error term for each subject is the sum of the cluster effect for the subject’s cluster and the 

within-cluster error term. 

o Y = LnGM + Cluster + Error. Cluster is normally distributed with mean 0, variance Vc, and standard 

deviation Sc = √Vc. Error is normally distributed with mean 0, variance Vw, and standard deviation Sw 

= √Vw. Define V = Vc + Vw and S = √V. V is the variance of Y, and S is the standard deviation of Y. 

o ICC = Intra-cluster correlation coefficient = Vc/V. 

o AMm = Modeled arithmetic mean of X values = exp(LnGM) exp(1⁄2 V) 

o GMm = Modeled geometric mean of X values = exp(LnGM) 

o GSDm = Modeled geometric standard deviation of X values = exp(S) 

o P95m = Modeled 95th percentile of X values = exp(LnGM) exp(Z95×S) 

For the lognormal mixed model, the ICC value estimates the clustering effect and lies between 0 (no clustering) and 1 

(complete clustering and negligible within-cluster variation). If ICC = 0, then the lognormal mixed model is identical to 

the lognormal simple random sampling model and the parameters (AM, GM, GSD, and P95) are identical for those two 

models. 
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15 March, 2011 

Page 6 

Table 2 presents the arithmetic mean and 95th percentile estimates from the lognormal mixed model, together with 95% 

confidence intervals, for all the exposure routes. These are the values of AMm and P95m. The other summary statistics 

are presented in more detail below. 

Table 2. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized exposure. 

Exposure Route Clothing 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Dermal 

(mg/lb AI) 

Long pants and long sleeves 1047.6 (748.6, 1494.1) 2420.4 (1500.5, 3892.5) 

 Short pants and short sleeves 1739.9 (1224.5, 2499.9) 3638.0 (2266.6, 5835.9) 

 Long pants and short sleeves 1533.3 (1162.6, 2050.8) 3222.2 (2148.2, 4835.4) 

Inhalation (mg/m3/lb AI) 11.6 (4.8, 29.9) 37.5 (13.0, 109.6) 

For each exposure route, the above three statistical models were fitted to the observed data and the summary statistics 

listed above were calculated together with 95% confidence intervals. The 95% confidence intervals in Table 2 were 

computed using a parametric bootstrap. Confidence intervals computed using a non-parametric bootstrap are presented 

below. For these calculations, the parametric bootstrap simulations were all generated from the fitted lognormal mixed 

model, even for the empirical and simple random sample summary statistics, on the basis that the mixed model is the 

best choice for modeling the data, even if the summary statistics are developed from a simpler statistical model. 

The algorithm used was as follows: 

Step 1: 

Simulate 18 random variables Y, X from the estimated lognormal distribution superimposed upon the observed sampling structure ---; 

C = LnGM + RanNor(Seed)×Sc; 

Y = C + RanNor(Seed) ×Sw; 

X = exp(Y); 

where: 

LnGM = intercept of mixed effect log-log regression model 

Sc = square root of between cluster variance 

Sw = square root of within cluster variance under mixed-effect model. 

Step 2: 

For Y: 

Calculate GMs = exp(EAM) 

Calculate GSDs = exp(Su) 

Calculate AMu = GMs×exp(0.5×Su×Su) 

Calculate P95u = GMs× exp(Z95×S) 
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15 March, 2011 

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Fit mixed lognormal model to simulated Y values 

Under mixed-effects model: 

Calculate GMm= exp(intercept of mixed-effects model) 

Calculate GSDm = exp(square root (total variance V under mixed-effects model)) 

Calculate ICC = Vc /V 

Calculate AMm = exp(intercept + 0.5×V) 

Calculate P95m = exp(intercept + Z95×S) 

where: 

EAM = sample arithmetic average of Y 

Su = standard deviation of Y 

V = total variance under mixed-effects model 

S = square root of V 

Vc = between cluster variance. 

For X: 

Calculate arithmetic mean AMs

Calculate 95th percentile P95s

Step 3: Repeat Steps 1 and 2 10,000 times. 

Steps 1 to 3 result in 10,000 values each for GSDs, GSDm, ICC, GMs, GMm, AMs, AMm, AMu, P95s, P95m, and P95u. 95% confidence 

intervals can be defined for each parameter by the 2.5th and 97.5th percentiles (lower and upper, respectively) of the bootstrap distribution of 

that corresponding parameter. Note that by definition, GSDs = GSDu and GMs = GMu. Also note that GMs = GMm for this situation because 

the experiment is balanced with 3 clusters and 6 subjects in each cluster; this implies that the intercept is the same value for both the simple 

random sampling and mixed models. 

Fold relative accuracy (fRA) is a measure that can be used to determine how well a statistic can describe its population 

parameter. Let us assume θ is a parameter and T is the sample statistic of θ (i.e., an estimate of θ). If the 2.5th and 

97.5th percentiles of the sampling distribution of T can be denoted by T2.5 and T97.5, respectively, then the 95th percentile 

of sample fold relative accuracy can be theoretically calculated using the following formula provided in the AHETF 

Governing Document (AHETF, 2007; pg. 136): 

 fRA95 = Max (T97.5 / θ, θ / T2.5) 

The actual value of θ is unknown. Thus, fRA95 was calculated by substituting θ with T. If the fRA95 of a statistic were 

equal to 3, then it would be correct to say: “At least 95% of the time the sample statistic will be accurate to within 3-

fold of the population value”. According to the AHETF Governing Document, the statistical design of the exposure 

monitoring study should be adequate to produce a fRA95 less than or equal to 3. Thus the confidence intervals calculated 

in the above algorithm can be used to estimate the fold relative accuracy and compare the observed fRA with the study 

design benchmark of 3. If the observed fold relative accuracy is greater than 3, this means that the experiment did not 

meet the benchmark, which would be due to differences between the distributions of the CMA data used to design the 

study and the experimental data collected in the study. If the fold relative accuracy benchmark is not met, then it might 
Timothy Leighton, EPA 

15 March, 2011 

Page 8 

be desirable to collect more data for this scenario in order to meet the benchmark. Fold relative accuracy was not 

computed for the ICC since the estimated ICC is 0 in many cases. 

The HSRB reviewers of the statistical analyses of the Mop Study (“AEATF Mop Study Statistical Review for HSRB,” 

19 November 2010, original version dated 28 September, 2010) suggested that a non-parametric bootstrap approach 

should also be considered. The non-parametric bootstrap method should be more robust since it does not assume that 

the fitted parametric model is the correct one. For the non-parametric bootstrap, exactly the same approach was used 

except that Step 1 above is replaced by the following: 

Step 1: 

Simulate 18 random variables Y, X by resampling at random with replacement from the original data: 

For Cluster j, the original exposure data are X(1), X(2), ..., X(nj), where nj is the number of workers in cluster j (nj equals 6 for the 

wipe study data). 

Sample nj values at random with replacement from the exposure values X(1), X(2), ..., X(nj). This gives the 6 simulated random 

variables X from cluster j. 

Repeat for all three clusters. (j = 1, 2, and 3). 

Y = log(X). 

The Y, X values were independently resampled from the three clusters in order to preserve the covariance structure. 

Tables 3 to 6 present the estimates, parametric and non-parametric confidence intervals and fold relative accuracy 

values for all the summary statistics for the three dermal and one inhalation exposure routes, respectively. 

Table 3. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized long dermal exposure (mg/lb AI). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.9 1.5 2.3 1.2 1.5 2.1 1.3 

GSDm 1.9 1.5 2.3 1.2 1.5 2.2 1.3 

ICC 0.0 0.0 0.4 0.0 0.6 

GMs 859.2 626.2 1185.7 1.4 669.4 1129.4 1.3 

GMm 859.2 626.2 1185.7 1.4 669.4 1129.4 1.3 

AMs 1065.8 736.7 1468.3 1.4 737.4 1441.4 1.4 

AMu 1045.6 745.7 1483.8 1.4 733.3 1452.5 1.4 

AMm 1047.6 748.6 1494.1 1.4 740.2 1512.2 1.4 

P95s 3478.6 1484.4 5660.2 2.3 1185.7 3478.6 2.9 

P95u 2408.6 1483.8 3811.5 1.6 1297.7 3673.4 1.9 

P95m 2420.4 1500.5 3892.5 1.6 1342.7 4030.2 1.8 
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Table 4. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized short dermal exposure (mg/lb 

AI). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.7 1.4 2.0 1.2 1.4 1.9 1.2 

GSDm 1.7 1.4 2.1 1.2 1.4 2.0 1.2 

ICC 0.2 0.0 0.6 0.0 0.7 

GMs 1507.3 1076.5 2132.2 1.4 1233.9 1868.7 1.2 

GMm 1507.3 1076.5 2132.2 1.4 1233.9 1868.7 1.2 

AMs 1736.2 1209.0 2447.8 1.4 1337.2 2201.4 1.3 

AMu 1727.8 1216.4 2465.3 1.4 1334.3 2216.3 1.3 

AMm 1739.9 1224.5 2499.9 1.4 1342.0 2277.7 1.3 

P95s 5154.6 2251.5 7514.0 2.3 2516.6 5154.6 2.0 

P95u 3560.0 2245.3 5575.9 1.6 2295.0 5013.8 1.6 

P95m 3638.0 2266.6 5835.9 1.6 2327.2 5497.9 1.6 

Table 5. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized long short dermal exposure 

(mg/lb AI). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.7 1.4 2.1 1.2 1.4 1.9 1.2 

GSDm 1.7 1.4 2.1 1.2 1.4 2.0 1.2 

ICC 0.0 0.0 0.4 0.0 0.6 

GMs 1325.1 1016.8 1735.7 1.3 1063.3 1667.7 1.3 

GMm 1325.1 1016.8 1735.7 1.3 1063.3 1667.7 1.3 

AMs 1544.7 1150.0 2021.2 1.3 1158.1 1994.4 1.3 

AMu 1531.9 1160.2 2038.8 1.3 1154.1 1997.7 1.3 

AMm 1533.3 1162.6 2050.8 1.3 1157.9 2047.7 1.3 
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 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

P95s 4841.8 2138.5 6653.3 2.3 2071.1 4841.8 2.3 

P95u 3213.0 2138.3 4753.6 1.5 2009.8 4612.2 1.6 

P95m 3222.2 2148.2 4835.4 1.5 2034.8 4972.7 1.6 

Table 6. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized inhalation exposure (mg/m3/lb 

AI). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 2.7 1.9 4.1 1.5 1.7 3.9 1.6 

GSDm 2.9 1.9 4.7 1.6 1.8 4.5 1.6 

ICC 0.3 0.0 0.7 0.0 0.7 

GMs 6.7 3.1 14.4 2.2 4.5 9.5 1.5 

GMm 6.7 3.1 14.4 2.2 4.5 9.5 1.5 

AMs 10.2 4.5 25.2 2.5 6.5 14.5 1.6 

AMu 11.0 4.7 26.6 2.4 7.0 17.0 1.6 

AMm 11.6 4.8 29.9 2.6 7.2 19.8 1.7 

P95s 46.3 12.8 153.4 3.6 13.0 46.3 3.6 

P95u 34.8 12.6 92.8 2.8 16.2 61.1 2.1 

P95m 37.5 13.0 109.6 2.9 16.8 74.5 2.2 

Tables 3 and 5 show that the ICC estimated value is zero for the long dermal and long short dermal exposure routes 

showing that the estimated mixed and simple random sampling models are the same for those cases and that there is no 

variation between the different locations. Tables 4 and 6 show that the ICC estimated value is non-zero for the short 

dermal and inhalation exposure routes showing that the estimated mixed and simple random sampling models are not 

the same for those cases and that there is some variation between the different locations. All of the fold relative 

accuracy values met the study design benchmark of 3 except for the empirical 95th percentile for the normalized 

inhalation exposure. The parametric bootstrap confidence intervals were similar to the non-parametric bootstrap 

confidence intervals for the dermal exposure routes, but were generally wider than the non-parametric bootstrap 

confidence intervals for the inhalation exposure. 

Quantile-quantile plots of the normalized exposure values were used to evaluate whether the data were lognormally 

distributed, as implied by the assumed statistical models. In each case the quantile-quantile plot compared the observed 

quantiles of the 18 measured values with the corresponding quantiles of a normal or lognormal distribution. A perfect 

fit would imply that the plotted values lie in a straight line. The quantile plots are presented in Figures 1 to 8. They 
Timothy Leighton, EPA 

15 March, 2011 

Page 11 

clearly show that the lognormal distribution is a better fit than a normal distribution, and that the lognormal distribution 

fits reasonably well for all of the exposure routes. 

Quantile plot normalized long dermal exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

500

1000

1500

2000

2500

3000

3500

Normalized Long Dermal Exposure

Normal Quantiles

Figure 1 
Timothy Leighton, EPA 

15 March, 2011 

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Quantile plot normalized long dermal exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

5.5

6.0

6.5

7.0

7.5

8.0

8.5

log (Normalized Long Dermal Exposure)

Normal Quantiles

Figure 2
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15 March, 2011 

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Quantile plot normalized short dermal exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

1000

2000

3000

4000

5000

6000

Normalized Short Dermal Exposure

Normal Quantiles

Figure 3
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15 March, 2011 

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Quantile plot normalized short dermal exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

6.5

7.0

7.5

8.0

8.5

9.0

log (Normalized Short Dermal Exposure)

Normal Quantiles

Figure 4
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15 March, 2011 

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Quantile plot normalized long short dermal exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

1000

2000

3000

4000

5000

Normalized Long Short Dermal Exposure

Normal Quantiles

Figure 5
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15 March, 2011 

Page 16 

Quantile plot normalized long short dermal exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

6.5

7.0

7.5

8.0

8.5

lnnrmdermls

Normal Quantiles

Figure 6
Timothy Leighton, EPA 

15 March, 2011 

Page 17 

Quantile plot normalized inhalation exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

10

20

30

40

50

Normalized Inhalation Exposure)

Normal Quantiles

Figure 7
Timothy Leighton, EPA 

15 March, 2011 

Page 18 

Quantile plot normalized inhalation exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

-1

0

1

2

3

4

log (Normalized Inhalation Exposure)

Normal Quantiles

Figure 8 

 
Timothy Leighton, EPA 

15 March, 2011 

Page 19 

These statistical models for the normalized exposure assume that the exposure is proportional to the normalizing 

variable pounds of active ingredient handled. More precisely, the assumed statistical models are of the form 

 Log (Exposure) = Intercept + Slope × Log (Pounds of Active Ingredient) + Error Terms 

where Slope = 1. Possible alternative models include the same formulation with a Slope of zero, implying that the 

exposure does not depend upon the amount of active ingredient handled, even though the wiping times, and hence the 

amount of active ingredient handled, varied between the subjects as part of the study design. Other possible models 

include the same model with a slope not equal to zero or one, the quadratic models discussed below, or models with 

more complicated relationships between the exposure and the experimental conditions. To evaluate whether the slope is 

zero, one, or other possible values, we fitted the above statistical model and computed confidence intervals for the 

slope. In the appendix we investigate normalizing by the wiping duration or, for inhalation exposure, by the product of 

the pounds of active ingredient and wiping duration and by the number of wipes used. 

To analyze the proportionality, we also considered an additional hypothetical clothing scenario with no clothing at all. 

The dermal exposure for the No clothing scenario was calculated by summing all the inner and outer dermal exposure 

measurements: 

All Dermal. This case represents the dermal exposure for a subject wearing no clothes. The exposure is the sum of the 

mass from hand wash, face and neck, and the six inner and six outer dosimeters (lower arm, upper arm, lower leg, 

upper leg, front torso, rear torso). 

We can use these statistical models to calculate confidence intervals for the slope. The calculation of the confidence 

intervals depends upon the value of the denominator degrees of freedom for the mixed models used. A review of the 

alternative methods for calculating the denominator degrees of freedom for fixed effects in a mixed model using the 

SAS MIXED procedure is given in an article by Schaalje et al1

. Based on that article, the following Table 7 summarizes 

the five available methods: 

Table 7. SAS Methods for Computing the Fixed Effects Denominator Degrees of Freedom in PROC MIXED. 

DDF Method SAS Abbreviation Comments 

Residual residual Uses residual degrees of freedom. 

Ignores covariance structure as 

defined by the RANDOM and 

REPEATED statements. This method 

is not recommended. 

Containment contain Default method when RANDOM 

statements are present. Accounts for 

the minimum contribution of the 

random effects that syntactically 

                                                           

1 Schaalje, G. B., J. B. McBride, G. W. Fellingham. “Approximations to Distributions of Test Statistics in Complex Mixed Linear Models 

Using SAS® Proc MIXED” Proceedings of the Twenty Sixth Annual SAS Users Group International Conference. April 2001. Long Beach, 

CA. ISBN 1-58025-864-6. SAS Institute, Cary, NC 27513. 
Timothy Leighton, EPA 

15 March, 2011 

Page 20 

DDF Method SAS Abbreviation Comments 

contain the fixed effects of interest. 

Between-Within bw Default method when REPEATED 

statements are present and RANDOM 

statements are not present. Only exact 

when the data are balanced and the 

design is a repeated measures design 

with compound symmetry, and where 

the levels of the within-subjects 

effects are not replicated within any 

of the subjects. Otherwise the method 

is at best approximate and can be 

unpredictable. 

Satterthwaite / Fai-Cornelius satterth Designed to approximate the 

denominator degrees of freedom for 

split-plot designs with complicated 

covariance structures and/or 

unbalanced data sets. 

Kenwood-Rogers kr Designed to approximate the 

denominator degrees of freedom for 

designs with complicated covariance 

structures and/or unbalanced data 

sets. Results from simulations suggest 

better performance than the 

Satterthwaite method. If a covariance 

parameter has zero variance then this 

method ignores that covariance. 

To interpret this table for this study, note that the RANDOM statement was used to define the cluster effect. If the ICC 

equals zero, then there is no clustering and the cluster variance equals zero. The REPEATED statement was used to 

define the repeated measures model. A balanced data set is one where each treatment combination is applied to the 

same number of subjects. For this study, this implies that there are the same number of workers in every cluster, and 

each worker has the same number of measured exposure values. 

The study data were balanced since there were 6 workers in each cluster, each with the same number of exposure 

measurements. Based on this summary, the recommended methods are the containment method for the mixed models 

when the ICC parameter is zero, and the Kenwood-Rogers method for the mixed models where the ICC parameter is 

non-zero or for the repeated measures model (detailed below). (For other applications, the Kenwood-Rogers method 

would be preferred in general if the data are sufficiently unbalanced, but it is not easy to provide rules as to how this 

should be defined.) The confidence intervals for the regression coefficients presented in this memorandum follow these 

recommendations. In particular, since the ICC parameters in the mixed regression models for the logarithm of the long 
Timothy Leighton, EPA 

15 March, 2011 

Page 21 

dermal, short dermal, and long short dermal exposure against the logarithm of the pounds of active ingredient handled 

were all zero, the containment method was used in those cases. However, since the ICC parameter in the mixed 

regression models for the logarithm of the all dermal and inhalation exposure against the logarithm of the pounds of 

active ingredient handled was non-zero, the Kenwood-Rogers method was used for those mixed models. (As noted 

above the ICC for the normalized short dermal exposure was estimated to be 0.2, a small but nonzero value; this is not 

inconsistent with the regression analysis because the regression model allows the slope to be arbitrary but the model for 

the normalized short dermal exposure assumes the slope is equal to 1.) Note that this issue does not impact the 

calculated confidence intervals for the summary statistics in Tables 2 to 6, since they used a bootstrap method. 

Table 8 shows the 95% confidence intervals for the slope calculated from the above model, either assuming the 

lognormal simple random sampling model for the errors or the lognormal mixed model for the errors. Also shown is the 

width of the confidence interval for the slope. A confidence interval that includes one but not zero supports the 

assumptions of the normalized exposure models. A confidence interval that includes zero but not one suggests that the 

exposure does not depend on the amount of active ingredient handled. A confidence interval that includes both zero and 

one suggests that either the basic statistical model is incorrect or there are not enough data to statistically infer whether 

the slope is zero or one. The Repeated Measures statistical model (bottom row) is described and discussed below. 

Table 8. 95 percent confidence intervals for the slope of log exposure versus log pounds active ingredient handled. 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Dermal (mg) Long pants and long 

sleeves Mixed 0.37 -0.02 0.76 0.78 

 Simple 

Linear 0.37 -0.01 0.76 0.77 

 Short pants and short 

sleeves Mixed 0.57 0.21 0.93 0.73 

 Simple 

Linear 0.57 0.21 0.93 0.72 

 Long pants and short 

sleeves Mixed 0.58 0.20 0.97 0.77 

 Simple 

Linear 0.58 0/21 0.96 0.76 

 None Mixed 0.74 0.35 1.12 0.77 

 Simple 

Linear 0.65 0.27 1.02 0.74 

Inhalation 

(mg/m3

) Mixed 0.25 -0.46 0.96 1.42 

 Simple 

Linear 0.15 -0.53 0.83 1.36 

Dermal (mg) Any Repeated 

Measures 0.49 0.03 0.96 0.93 
Timothy Leighton, EPA 

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For the long dermal exposure route, the confidence interval for the slope includes 0 and excludes 1, and the estimated 

slope is approximately 0.4. Thus the assumption of proportionality was rejected at the 5% level but the assumption of 

independence was not rejected. For the short dermal exposure route, the confidence interval for the slope excludes 0 

and excludes 1, and the estimated slope is approximately 0.6. Thus the assumption of proportionality was rejected at 

the 5% level as was the assumption of independence. For the long short and all dermal exposure routes, the confidence 

interval for the slope includes 1 and excludes 0, and the estimated slope is approximately 0.6. Thus the assumption of 

proportionality was rejected at the 5% level and the assumption of independence was not rejected. For the inhalation 

exposure, there is sufficient evidence to reject the hypothesis of a slope of 1 at the 5% significance level. 

Based on the available CMA data, the experiment was designed to be able to detect whether the slope was 0 or 1 using 

a test at the 5% significance level with a power of 80%. On that basis, the experiment was designed to make the 

expected confidence interval width equal to 1.4: 

Given: 

Power = 1 – P(type II error) = 1-β = 0.80; 

Significance level = P(type I error) = α = 0.05; 

(Eq. 1) Predicted 95% Confidence Interval (two sided) for the regression slope = Sample estimate of slope ± 1.96× Standard Error 

Effect Size = (Z1-α/2 + Z1-β) × Standard Error 

Here, ZP is the pth percentile of a standard normal distribution: 

Z1-α/2 = Z0.975 = 1.96; 

Z1-β = Z0.80= 0.84 

Z1-α/2 + Z1-β = 2.8 

(Eq. 2) Standard Error = Effect Size ÷ 2.8 

Substituting Standard Error into (Eq. 1): 

Predicted 95% Confidence Interval (two sided) for the regression slope = Sample estimate ± 1.96× (Effect Size ÷ 2.8) = Sample estimate ± 

0.7×Effect Size 

Here, Effect Size = 1 – 0 = 1 (i.e., slope = 1 under H1 vs. slope = 0 under H0) 

So, expected width of confidence interval for slope parameter = 2×0.7 = 1.4

The results in Table 8 show that the actual confidence interval widths were less than 1.4 for the dermal exposure routes. 

However, the actual confidence interval widths were slightly greater than 1.4 for the inhalation exposure route because 

the standard errors of the slopes in the actual data were greater than the values estimated using the CMA data prior to 

the experiment. 

For the simple linear regression analyses based on the simple random sampling model, the relationship between the 

exposure and the amount of active ingredient is displayed in the following regression plots in Figures 9 to 13. 
Timothy Leighton, EPA 

15 March, 2011 

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lnderml = 2.4916 +0.373 lnnrm

N 

18 

Rsq 

0.2077

AdjRsq

0.1582

RMSE 

0.4896

Simple Linear Regression of Ln Long Dermal Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Long Dermal Exposure(mg)

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 9
Timothy Leighton, EPA 

15 March, 2011 

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lnderms = 4.401 +0.5711 lnnrm

N 

18 

Rsq 

0.4158

AdjRsq

0.3793

RMSE 

0.4549

Simple Linear Regression of Ln Short Dermal Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Short Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 10
Timothy Leighton, EPA 

15 March, 2011 

Page 25 

lndermls = 4.3634 +0.5845 lnnrm

N 

18 

Rsq 

0.4015

AdjRsq

0.3641

RMSE 

0.4797

Simple Linear Regression of Ln Long Short Dermal Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Long Short Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 11
Timothy Leighton, EPA 

15 March, 2011 

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lnderma = 5.2054 +0.6467 lnnrm

N 

18 

Rsq 

0.4586

AdjRsq

0.4247

RMSE 

0.4723

Simple Linear Regression of Ln All Dermal Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log All Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 12 
Timothy Leighton, EPA 

15 March, 2011 

Page 27 

lninh = -3.8988 +0.1477 lnnrm

N 

18 

Rsq 

0.0131

AdjRsq

-.0486

RMSE 

0.8615

Simple Linear Regression of Ln Inhalation Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Inhalation Exposure(mg/m3)

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 13 
Timothy Leighton, EPA 

15 March, 2011 

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These simple linear regression results are compared with the results for the lognormal mixed models in Figures 14 to 

18. Note that in the first three dermal exposure Figures 14 to 16, the regression lines are the same for the mixed and 

simple linear regression models because the estimated ICC parameter was zero. 

Figure 14 
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15 March, 2011 

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Figure 15 
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Figure 16 
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Figure 17 
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Figure 18 
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Finally, in Figures 19 and 20 we present regression plots showing the different regression lines for the three clusters 

based on the mixed models for all dermal and inhalation exposure. (For the long, short, and long short dermal exposure 

routes, the models predict the same values for all three clusters, i.e., they predict the values shown in Figures 14 to 16). 

The mixed model predicts the highest all dermal and inhalation exposures in Cluster 1 (office), and the lowest dermal 

all dermal and inhalation exposures in Cluster 2 (retail / shopping center), which is consistent with the data. These 

differences are likely attributable to the different wiping surfaces in the three buildings. 

Figure 19 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 20 

The above analyses of the proportionality for dermal exposure consistently show similar positive slopes, as one might 

expect on physical grounds. While all of the estimated slopes for dermal exposure are positive and approximately 0.5, 

the confidence intervals were inconsistent about whether they included zero or included one. To investigate this issue 

further, the following more complicated statistical model was fitted to the data of all three dermal exposures (excluding 

the unrealistic no clothing case) for all 18 subjects. The model was of the form: 

Log (dermal exposure) = LnGM (clothing type) + Slope × Log (Pounds of Active Ingredient) + Cluster + Error 

In this model, the intercept depends upon the clothing type, so there are three intercepts. The slope is the same for all 

three clothing types, The Cluster term accounts for possible clustering effects due to the location. Finally, to account 

for the expected correlations between different dermal exposure measurements on the same worker, the three error 

terms (one per clothing type) for each worker are assumed to be correlated (with an unspecified covariance matrix), but 

errors for different workers are assumed to be independent. Thus in SAS terminology, the Cluster effect is a RANDOM 

effect and the Error is a REPEATED effect where the subject is the worker. We will call this model the “Repeated 

Measures” model. The confidence interval for the slope using this statistical model is shown in the bottom row of Table 

8. Since the confidence interval does not include one and does not include zero, the proportionality for dermal exposure 

has not been shown using this statistical model. 
Timothy Leighton, EPA 

15 March, 2011 

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Quadratic models 

The proportionality test was based on a linear model for log exposure versus log pounds active ingredient handled. The 

HSRB suggested that a quadratic model should also be considered. 

There are two quadratic models that could be considered. Since the original linear model is of the form 

 Log (Exposure) = Intercept + Slope × Log (Pounds of Active Ingredient) + Error Terms, 

the main quadratic model is of the form 

Log (Exposure) = 

Intercept + Slope × Log (Pounds of Active Ingredient) + 

Quad × {Log (Pounds of Active Ingredient)}2

 + Error Terms. 

Note that the quadratic term is the square of the logarithm of the pounds of active ingredient rather than the logarithm 

of the square; the latter approach produces an ill-defined model with two multiples of the logarithm of the pounds of 

active ingredient. 

Another approach might be to consider a quadratic model for exposure: 

Exposure = 

Intercept + Slope × (Pounds of Active Ingredient) + 

Quad × (Pounds of Active Ingredient)2 + Error Terms. 

We do not recommend this second approach for these data since the exposures are known to be non-negative and the 

quantile plots indicate that the exposure data are better modeled using a log-normal distribution than using a normal 

distribution. Furthermore, unless the intercept is zero, this model predicts a nonzero exposure when the pounds of 

active ingredient is zero, and so a more realistic (though possibly poorer-fitting) model of this form would have a zero 

intercept. For other exposure data a proportionality test could be carried out by fitting the zero intercept model 

Exposure = Slope × (Pounds of Active Ingredient) + Quad × (Pounds of Active Ingredient)2

 + Error Terms 

and testing if Quad equals zero. 

The parsimony principle suggests that the appropriate statistical procedure for this study is to first fit the quadratic 

regression model for the logarithm of the exposure 

Log (Exposure) = 

Intercept + Slope × Log (Pounds of Active Ingredient) + 

Quad × {Log (Pounds of Active Ingredient)}2

 + Error Terms. 

If the coefficient Quad is statistically significant at the 5% level, which is equivalent to requiring that the 95% 

confidence interval does not include zero, than the quadratic model is supported. Otherwise the linear model should be 

used. 
Timothy Leighton, EPA 

15 March, 2011 

Page 36 

We will first present the results of fitting these quadratic models to the study data. We will then consider whether these 

statistical models produce meaningful and useful physical models for antimicrobial exposure. 

Table 9 presents the fitted quadratic models from the study for the mixed models of the four exposure measurements 

(Long Dermal, Short Dermal, Long Short Dermal, Inhalation) and for the Repeated Measures model for Dermal 

exposures. For the Repeated Measures model, the model has different intercepts (but the same Slope and Quad 

coefficients) for the three different dermal exposures. In view of the earlier discussion about denominator degrees of 

freedom, the confidence intervals for cases where the ICC parameter is non-zero and for the Repeated Measures model 

are calculated using the Kenwood-Rogers method. The confidence intervals for other cases where the ICC parameter is 

zero are calculated using the containment method. 

Table 9. Quadratic mixed models with 95% confidence intervals for the log exposure versus log pounds active 

ingredient handled. 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Long 

Dermal 

Intercept 9.80 2.00 -32.28 51.87 1.64 0.00 84.16 

Long 

Dermal

Slope 2.55 13.00 -3.70 8.79 1.64 0.00 12.49 

Long 

Dermal

Quad 0.16 13.00 -0.30 0.62 1.64 0.00 0.92 

Short 

Dermal 

Intercept 19.19 14.22 0.05 38.32 1.55 0.07 38.27 

Short 

Dermal

Slope 4.94 14.09 -0.68 10.56 1.55 0.07 11.23 

Short 

Dermal

Quad 0.32 13.96 -0.09 0.73 1.55 0.07 0.82 

Long 

Short 

Dermal 

Intercept 19.96 2.00 -18.23 58.15 1.57 0.00 76.37 

Long 

Short 

Dermal

Slope 5.22 13.00 -0.44 10.89 1.57 0.00 11.34 

Long 

Short 

Quad 0.34 13.00 -0.07 0.76 1.57 0.00 0.83 
Timothy Leighton, EPA 

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Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Dermal

Inhalation Intercept 10.83 13.88 -25.31 46.98 2.44 0.19 72.29 

Inhalation Slope 4.42 13.79 -6.20 15.04 2.44 0.19 21.24 

Inhalation Quad 0.31 13.70 -0.47 1.08 2.44 0.19 1.55 

Dermal 

Repeated 

Measures 

Slope 4.29 13.30 -2.04 10.62 NA NA 12.64 

Dermal 

Repeated 

Measures 

Quad 0.27 13.16 -0.19 0.74 NA NA 0.93 

Since the 95% confidence intervals for Quad include zero in every case, the quadratic coefficient is not statistically 

significant and the quadratic models are not supported. 

Physical Implications

Let us now consider the physical implications of these quadratic models. Exponentiating both sides of the exposure 

regression equation (and assuming zero error) produces the model 

Exposure = exp[Intercept + Slope × Log (Pounds of Active Ingredient) + Quad × {Log (Pounds of Active 

Ingredient)}2

}] 

= A(Pounds of Active Ingredient)Slope × exp[Quad × {Log (Pounds of Active Ingredient)}2

}] 

where A = exp(Intercept). The final term does not simplify unless Quad equals zero. 

Under the quadratic model, the predicted exposure will not double if the amount of active ingredient is doubled, which 

would happen if the same amounts of liquid were sprayed onto the rags but the concentration in the bottle is doubled, or 

if the duration is doubled and the concentration in each spray bottle is the same. In the first case, the physical model 

might be explained by some non-linearity in atmospheric or physical processes. In the second case, a physical 

explanation might be something like the idea that if you wipe for twice as long, you get more tired which increases the 

probability of larger accidental exposures, or alternatively that if you wipe twice as long, you get more experienced 

which decreases the probability of exposures. For the DDAC experiment the concentration in each bottle was the same 

(other than the unplanned slight concentration differences of 1/65 versus 1/64 between cluster 1 and clusters 2 and 3), 

so we only have data for the second case and cannot estimate what would happen in the first case without making 

additional assumptions. According to this logic, if the quadratic model is true then the results of the DDAC experiment 

could only be used directly to estimate exposures when the liquid concentration is the same as used in the experiment, 
Timothy Leighton, EPA 

15 March, 2011 

Page 38 

and you would need to make additional assumptions in order to use the same model to predict exposures when DDAC 

is used at a different concentration or, more importantly, when another antimicrobial is used instead of DDAC. 

If Quad equals zero, then the model becomes the Linear model for log Exposure and the exposure equation becomes 

Exposure = A(Pounds of Active Ingredient)Slope

Unless Slope = 1, the predicted exposure will not double if the amount of active ingredient is doubled, which again 

implies non-linearity in the exposure. However if Quad equals 0 and Slope = 1, then the exposure is proportional to the 

amount of active ingredient and the model implies that normalized exposure is physically meaningful, and hence that 

the results of the DDAC experiment can be used directly to estimate exposures when DDAC is used at a different 

concentration or, more importantly, when another antimicrobial is used instead of DDAC. 

The primary goal of the analysis should be to test whether or not the exposures are proportional to the amount of active 

ingredient, rather than trying to find the best-fitting statistical model for the exposure as a function of the amount of 

active ingredient. The first step should be to use the Linear model for log Exposure for the proportionality test, which 

compares the proportionality model with the model Exposure = A(Pounds of Active Ingredient)Slope. If the 

proportionality is rejected, then the Quadratic model and other complicated model formulations could then be 

considered. However in this case it would be desirable to obtain exposure data using different liquid concentrations of 

DDAC and/or different antimicrobials to develop physically realistic models without making untested assumptions. 

Over-prediction from normalized exposure model. 

As an approximation, one can use the normalized exposure model to estimate exposures even if the proportionality 

assumption is rejected by the statistical test. Suppose that the linear model for log Exposure is correct. Then the 

exposure is given by the equation 

Exposure estimate from linear model = exp(Intercept) × (Pounds of Active Ingredient)Slope

If the normalized exposure model is used to estimate the expected exposure, then the estimated exposure is given by 

the pounds of active ingredient multiplied by the average normalized exposure, which is estimated by AMm, the 

arithmetic mean from the lognormal mixed model. Thus we have 

 Exposure estimate from normalized exposure model = AMm × Pounds of Active Ingredient 

Suppose that the estimated slope is less than 1. Then the exposure estimate from the normalized exposure model will 

over-predict the exposure from the linear model if, and only if, the pounds of active ingredient exceeds the following 

threshold: 

 Threshold = {AMm / exp(Intercept)}1/(Slope – 1)

These threshold values are given in Table 10. 
Timothy Leighton, EPA 

15 March, 2011 

Page 39 

Table 10. Minimum Pounds of Active Ingredient for Which Normalized Exposure Model Over-Predicts Dermal and 

Inhalation Exposure. 

Exposure 

Route Clothing Model Slope Threshold Level (lb AiaH) 

Dermal (mg) Long pants 

and long 

sleeves 

Mixed 0.37 0.00081 

 Short pants 

and short 

sleeves 

Mixed 0.57 0.00080 

 Long pants 

and short 

sleeves 

Mixed 0.58 0.00078 

Inhalation 

(mg/m3) 

 Mixed 0.25 0.00053 

Alternative normalizing variables 

The appendix gives the tables and graphs with the detailed results of the analysis when the exposure is normalized by 

the wiping duration or, for inhalation exposure only, by the product of pounds of active ingredient handled and wiping 

duration or by surface area wiped. The long dermal and repeated measures exposure models are inconsistent with 

proportionality for the wiping duration. The short dermal, long dermal, and all dermal exposure models are consistent 

with proportionality for the wiping duration, but the confidence intervals are much wider than for the amount of active 

ingredient. The inhalation exposure models are inconsistent with proportionality for product of pounds of active 

ingredient handled and wiping duration and are only just consistent with proportionality for the surface area wiped 

(upper bound of 95% confidence interval is 1.00). 

The appendix also gives the tables and graphs with the detailed results of the analysis when the inhalation exposure 

concentration is converted to an estimated mass, calculated as the average air concentration multiplied by the wiping 

duration (hours) and by an estimated 1 m3

/hour of air breathed in by someone doing light activity. This value estimates 

the mass of DDAC breathed in by each participant. However this is an approximation because actual breathing rates are 

not constant and will vary with the activity as well as the individual. Moreover, the air concentration was measured and 

averaged over the entire air pumping period, which includes the resting periods (breaks) as well as the wiping 

durations. The analysis shown in the appendix is for the mass normalized by the amount of active ingredient. 

Using both normalizing variables (amount of active ingredient, wiping duration), the ICC value is either zero or is 

small (at most 0.2) for all three dermal routes, showing only a small amount of variability between the clusters, i.e., the 

locations. Using the four normalizing variables (amount of active ingredient, wiping duration, amount of active 

ingredient × wiping duration, number of wipes) the ICC value for normalized inhalation exposure (as a concentration) 

was non-zero and at most 0.3, also showing that the inhalation exposure does not vary very much between the clusters. 

For dermal exposure routes normalizing by wiping duration, the slope of 0 was rejected only for the all dermal 
Timothy Leighton, EPA 

15 March, 2011 

Page 40 

exposure, and the slope of one was rejected only for long dermal exposure and the repeated measures model. For 

inhalation exposure as a concentration, the confidence intervals for the slope included zero and excluded one when 

normalizing by amount of active ingredient, wiping duration, or amount of active ingredient × wiping duration, but 

included both zero and one when normalizing by the surface area wiped. The slope estimates for the inhalation 

exposure as a concentration were all small (below 0.3) and positive. For the inhalation exposure as a mass normalized 

by the amount of active ingredient, the ICC value was non-zero and the confidence interval for the slope included one 

and excluded zero. 

Finally, to compare the four different normalizing variables, we present Table 11 that gives the values of minus twice 

the log-likelihood “-2LL” for the alternative approaches and exposure routes. -2LL is a measure of how well a 

statistical model fits the data and this can be used to compare different models for the same data. (-2LL is a relative 

measure that can only be used to compare different models for the same data, such as the various models for the 18 

long dermal exposure values. -2LL cannot be used to compare different data such as the long dermal and short dermal 

exposure). Using this measure, the models with the lowest -2LL values are preferred, and the preferred models in each 

row are shown in bold in Table 11. The values in Table 11 are -2LL for each of the mixed and repeated measures 

models. The log-likelihood values were computed by fitting the models using the maximum likelihood method. The 

rows where the slope is 1 give the -2LL values for the models where the regression slope is set to be 1 so the exposure 

is normalized by dividing the exposure by the normalizing variable. The rows where the slope = “Any” give the -2LL 

values for the more general models where the regression slope is arbitrary so the exposure is normalized by dividing 

the exposure by some power of the normalizing variable, NRMp

. The mixed models using the pounds of active 

ingredient are the preferred models for all the dermal exposure routes except for the long dermal model with a known 

slope of one. For inhalation exposure, the preferred model for normalized exposure as a concentration uses the surface 

area wiped. The preferred model for normalized inhalation exposure as a mass uses the surface area wiped when the 

slope is allowed to vary and uses the pounds of active ingredient when the slope equals one. 

Table 11. Minus twice the log-likelihood for different mixed models (smaller is better). Preferred model is shown in 

bold. 

Exposure 

Route 

Model Slope Normalized by 

pounds of active 

ingredient 

Normalized by 

wiping duration 

Normalized by 

pounds of active 

ingredient and 

wiping duration 

Normalized by 

surface area 

wiped 

Long Dermal Mixed Any 23.3 26.0 

 Mixed 1 33.2 31.8 

Short Dermal Mixed Any 20.6 27.1 

 Mixed 1 26.5 30.4 

Long Short 

Dermal 

Mixed Any 22.5 29.1 

 Mixed 1 27.8 32.2 

Dermal Repeated 

Measures 

Any -2.6 1.1 

 1 4.5 6.7 
Timothy Leighton, EPA 

15 March, 2011 

Page 41 

Exposure 

Route 

Model Slope Normalized by 

pounds of active 

ingredient 

Normalized by 

wiping duration 

Normalized by 

pounds of active 

ingredient and 

wiping duration 

Normalized by 

surface area 

wiped 

Inhalation 

Concentration 

(mg/m3) 

Mixed Any 43.3 43.7 43.6 42.1 

 Mixed 1 49.1 48.5 60.0 47.3 

Inhalation 

Mass (mg) 

Mixed Any 41.8 43.7 42.3 41.7 

 Mixed 1 41.9 43.7 49.1 42.4 
Timothy Leighton, EPA 

15 March, 2011 

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APPENDIX 

Analyses of exposure per wiping duration (hours) 

Table and Figure Numbers are consistent with the main text (add “b”). 

Table 1b. Summary statistics for normalized exposure. 

Statistic 

Normalized Longa

Dermal 

(mg/hour) 

Normalized Shortb

Dermal 

(mg/hour) 

Normalized Long Shortc

Dermal 

(mg/hour) 

Normalized 

Inhalation 

(mg/m3

/hour) 

Arithmetic Mean 0.62 1.06 0.95 0.0056 

Arithmetic Standard 

Deviation 0.45 0.65 0.62 0.0050 

Geometric Mean 0.51 0.90 0.79 0.0040 

Geometric Standard 

Deviation 1.83 1.79 1.84 2.61 

Min 0.25 0.39 0.38 0.0002 

 5% 0.25 0.39 0.38 0.0002 

10% 0.27 0.49 0.39 0.0016 

25% 0.32 0.56 0.46 0.0027 

50% 0.40 0.82 0.69 0.0048 

75% 0.76 1.79 1.30 0.0079 

90% 1.66 2.07 1.97 0.0086 

95% 1.71 2.54 2.39 0.0228 

Max 1.71 2.54 2.39 0.0228 a

Long = Long pants and long sleeves 

b

Short = Short pants and short sleeves 

c

Long Short = Long pants and short sleeves

Table 2b. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized exposure. 

Exposure Route Clothing 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Dermal 

(mg/hour) 

Long pants and long sleeves 0.61 (0.45, 0.84) 1.38 (0.89, 2.12) 

 Short pants and short sleeves 1.06 (0.80, 1.43) 2.32 (1.53, 3.53) 

 Long pants and short sleeves 0.95 (0.70, 1.31) 2.15 (1.38, 3.33) 

Inhalation 

(mg/m3

/hour) 

 0.0064 (0.0034, 0.0128) 0.0198 (0.0087, 0.0443) 
Timothy Leighton, EPA 

15 March, 2011 

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Table 3b. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized long dermal exposure (mg/hour). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.83 1.50 2.23 1.2 1.50 1.99 1.2 

GSDm 1.83 1.50 2.26 1.2 1.54 2.05 1.2 

ICC 0.00 0.00 0.39 0.00 0.57 

GMs 0.51 0.39 0.68 1.3 0.39 0.68 1.3 

GMm 0.51 0.39 0.68 1.3 0.39 0.68 1.3 

AMs 0.62 0.45 0.83 1.4 0.44 0.82 1.4 

AMu 0.61 0.45 0.84 1.4 0.43 0.84 1.4 

AMm 0.61 0.45 0.84 1.4 0.44 0.85 1.4 

P95s 1.71 0.88 3.05 1.9 0.82 1.71 2.1 

P95u 1.38 0.88 2.09 1.6 0.78 1.98 1.8 

P95m 1.38 0.89 2.12 1.5 0.81 2.02 1.7 

Table 4b. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized short dermal exposure (mg/hour). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.79 1.47 2.17 1.2 1.52 1.96 1.2 

GSDm 1.79 1.48 2.19 1.2 1.52 2.00 1.2 

ICC 0.00 0.00 0.39 0.00 0.47 

GMs 0.90 0.69 1.18 1.3 0.70 1.16 1.3 

GMm 0.90 0.69 1.18 1.3 0.70 1.16 1.3 

AMs 1.06 0.79 1.41 1.3 0.78 1.36 1.4 

AMu 1.06 0.79 1.43 1.3 0.77 1.39 1.4 

AMm 1.06 0.80 1.43 1.4 0.78 1.41 1.4 

P95s 2.54 1.52 5.00 2.0 1.81 2.54 1.4 

P95u 2.32 1.52 3.48 1.5 1.45 3.19 1.6 

P95m 2.32 1.53 3.53 1.5 1.46 3.27 1.6 
Timothy Leighton, EPA 

15 March, 2011 

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Table 5b. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized long short dermal exposure 

(mg/hour). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 1.84 1.50 2.26 1.2 1.55 2.02 1.2 

GSDm 1.84 1.51 2.28 1.2 1.56 2.06 1.2 

ICC 0.00 0.00 0.39 0.00 0.43 

GMs 0.79 0.59 1.05 1.3 0.60 1.04 1.3 

GMm 0.79 0.59 1.05 1.3 0.60 1.04 1.3 

AMs 0.95 0.69 1.28 1.4 0.69 1.23 1.4 

AMu 0.95 0.70 1.30 1.4 0.68 1.26 1.4 

AMm 0.95 0.70 1.31 1.4 0.68 1.28 1.4 

P95s 2.39 1.37 4.80 2.0 1.67 2.39 1.4 

P95u 2.15 1.37 3.28 1.6 1.30 2.98 1.6 

P95m 2.15 1.38 3.33 1.6 1.31 3.06 1.6 

Table 6b. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized inhalation exposure 

(mg/m3/hour). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 2.61 1.89 3.62 1.4 1.64 3.78 1.6 

GSDm 2.65 1.90 3.80 1.4 1.65 4.16 1.6 

ICC 0.12 0.00 0.56 0.00 0.60 

GMs 0.0040 0.0022 0.0071 1.8 0.0026 0.0057 1.5 

GMm 0.0040 0.0022 0.0071 1.8 0.0026 0.0057 1.5 

AMs 0.0056 0.0032 0.0118 2.1 0.0038 0.0078 1.5 

AMu 0.0063 0.0033 0.0123 2.0 0.0043 0.0090 1.5 

AMm 0.0064 0.0034 0.0128 2.0 0.0043 0.0100 1.6 

P95s 0.0228 0.0086 0.0744 3.3 0.0083 0.0228 2.8 

P95u 0.0192 0.0086 0.0418 2.2 0.0095 0.0319 2.0 

P95m 0.0198 0.0087 0.0443 2.3 0.0097 0.0363 2.0 
Timothy Leighton, EPA 

15 March, 2011 

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Table 8b. 95 percent confidence intervals for slope of log exposure versus log hours of wiping duration. 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Dermal (mg) Long pants and long 

sleeves Mixed 0.32 -0.26 0.91 1.17 

 Simple 

Linear 0.32 -0.26 0.90 1.16 

 Short pants and short 

sleeves Mixed 0.49 -0.11 1.10 1.21 

 Simple 

Linear 0.49 -0.10 1.09 1.20 

 Long pants and short 

sleeves Mixed 0.48 -0.16 1.12 1.28 

 Simple 

Linear 0.48 -0.15 1.11 1.26 

 None Mixed 0.63 0.001 1.26 1.26 

 Simple 

Linear 0.63 0.008 1.25 1.24 

Inhalation 

(mg/m3/hour) Mixed 0.04 -0.89 0.98 1.87 

 Simple 

Linear 0.02 -0.94 0.97 1.90 

Dermal (mg) Any Repeated 

Measures 0.38 -0.21 0.97 1.18 

Table 9b. Quadratic mixed models with 95% confidence intervals for the log exposure versus log hours wiping 

duration. 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Long 

Dermal 

Intercept -0.24 2.00 -1.23 0.74 1.72 0.00 1.97 

Long 

Dermal

Slope 0.24 13.00 -1.79 2.28 1.72 0.00 4.08 

Long Quad 0.07 13.00 -1.72 1.86 1.72 0.00 3.57 
Timothy Leighton, EPA 

15 March, 2011 

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Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Dermal

Short 

Dermal 

Intercept 0.24 2.00 -0.77 1.25 1.75 0.00 2.02 

Short 

Dermal

Slope 0.05 13.00 -2.04 2.15 1.75 0.00 4.18 

Short 

Dermal

Quad 0.40 13.00 -1.43 2.24 1.75 0.00 3.66 

Long 

Short 

Dermal 

Intercept 0.12 2.00 -0.95 1.19 1.81 0.00 2.14 

Long 

Short 

Dermal

Slope 0.10 13.00 -2.11 2.31 1.81 0.00 4.43 

Long 

Short 

Dermal

Quad 0.35 13.00 -1.59 2.29 1.81 0.00 3.88 

Inhalation Intercept -4.94 8.03 -5.87 -4.02 2.47 0.11 1.86 

Inhalation Slope 0.23 13.14 -2.99 3.45 2.47 0.11 6.45 

Inhalation Quad -0.17 13.13 -3.00 2.65 2.47 0.11 5.64 

Dermal 

Repeated 

Measures 

Slope 0.13 15.01 -1.92 2.19 NA NA 4.11 

Dermal 

Repeated 

Measures 

Quad 0.23 15.01 -1.57 2.03 NA NA 3.60 
Timothy Leighton, EPA 

15 March, 2011 

Page 47 

Table 10b. Minimum Hours Wiping Duration for Which Normalized Exposure Model Over-Predicts Dermal and 

Inhalation Exposure. 

Exposure 

Route Clothing Model Slope Threshold Level (hours) 

Dermal (mg) Long pants 

and long 

sleeves 

Mixed 0.32 1.4 

 Short pants 

and short 

sleeves 

Mixed 0.49 1.3 

 Long pants 

and short 

sleeves 

Mixed 0.48 1.3 

Inhalation 

(mg/m3) 

 Mixed 0.04 1.1 
Timothy Leighton, EPA 

15 March, 2011 

Page 48 

Quantile plot normalized long dermal exposure data with a normal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

0.25

0.50

0.75

1.00

1.25

1.50

1.75

Normalized Long Dermal Exposure

Normal Quantiles

Figure 1b
Timothy Leighton, EPA 

15 March, 2011 

Page 49 

Quantile plot normalized long dermal exposure data with a lognormal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

-1.5

-1.0

-0.5

0

0.5

1.0

log (Normalized Long Dermal Exposure)

Normal Quantiles

Figure 2b
Timothy Leighton, EPA 

15 March, 2011 

Page 50 

Quantile plot normalized short dermal exposure data with a normal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized Short Dermal Exposure

Normal Quantiles

Figure 3b
Timothy Leighton, EPA 

15 March, 2011 

Page 51 

Quantile plot normalized short dermal exposure data with a lognormal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

-1.0

-0.5

0

0.5

1.0

log (Normalized Short Dermal Exposure)

Normal Quantiles

Figure 4b
Timothy Leighton, EPA 

15 March, 2011 

Page 52 

Quantile plot normalized long short dermal exposure data with a normal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

0

0.5

1.0

1.5

2.0

2.5

Normalized Long Short Dermal Exposure

Normal Quantiles

Figure 5b
Timothy Leighton, EPA 

15 March, 2011 

Page 53 

Quantile plot normalized long short dermal exposure data with a lognormal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

-1.0

-0.5

0

0.5

1.0

lnnrmdermls

Normal Quantiles

Figure 6b 
Timothy Leighton, EPA 

15 March, 2011 

Page 54 

Quantile plot normalized inhalation exposure data with a normal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

0

0.005

0.010

0.015

0.020

0.025

Normalized Inhalation Exposure)

Normal Quantiles

Figure 7b 
Timothy Leighton, EPA 

15 March, 2011 

Page 55 

Quantile plot normalized inhalation exposure data with a lognormal distribution

Normalized by Wiping Duration

-2 -1 0 1 2

-9

-8

-7

-6

-5

-4

-3

log (Normalized Inhalation Exposure)

Normal Quantiles

Figure 8b
Timothy Leighton, EPA 

15 March, 2011 

Page 56 

lnderml = -0.2469 +0.3213 lnnrm

N 

18 

Rsq 

0.0796

AdjRsq

0.0220

RMSE 

0.5278

Simple Linear Regression of Ln Long Dermal Exposure on Ln Wiping Duration

Normalized by Wiping Duration

log Long Dermal Exposure(mg)

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

log Wiping Duration

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 9b
Timothy Leighton, EPA 

15 March, 2011 

Page 57 

lnderms = 0.2066 +0.4948 lnnrm

N 

18 

Rsq 

0.1612

AdjRsq

0.1087

RMSE 

0.5452

Simple Linear Regression of Ln Short Dermal Exposure on Ln Wiping Duration

Normalized by Wiping Duration

log Short Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Wiping Duration

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 10b
Timothy Leighton, EPA 

15 March, 2011 

Page 58 

lndermls = 0.0882 +0.4782 lnnrm

N 

18 

Rsq 

0.1387

AdjRsq

0.0849

RMSE 

0.5754

Simple Linear Regression of Ln Long Short Dermal Exposure on Ln Wiping Duration

Normalized by Wiping Duration

log Long Short Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Wiping Duration

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 11b 
Timothy Leighton, EPA 

15 March, 2011 

Page 59 

lnderma = 0.4131 +0.6287 lnnrm

N 

18 

Rsq 

0.2237

AdjRsq

0.1752

RMSE 

0.5655

Simple Linear Regression of Ln All Dermal Exposure on Ln Wiping Duration

Normalized by Wiping Duration

log All Dermal Exposure(mg)

-0.5

0.0

0.5

1.0

1.5

2.0

log Wiping Duration

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 12b 
Timothy Leighton, EPA 

15 March, 2011 

Page 60 

lninh = -4.913 +0.0151 lnnrm

N 

18 

Rsq 

0.0001

AdjRsq

-.0624

RMSE 

0.8671

Simple Linear Regression of Ln Inhalation Exposure on Ln Wiping Duration

Normalized by Wiping Duration

log Inhalation Exposure(mg/m3)

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

log Wiping Duration

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 13b
Timothy Leighton, EPA 

15 March, 2011 

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Figure 14b 
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15 March, 2011 

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Figure 15b 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 16b 
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15 March, 2011 

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Figure 17b 
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Figure 18b 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 20b
Timothy Leighton, EPA 

15 March, 2011 

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Analyses of inhalation exposure per pounds of active ingredient handled and hours of wiping duration. 

Table and Figure Numbers are Consistent with main text (add “c”). 

Table 1c. Summary statistics for normalized exposure. 

Statistic 

Normalized Inhalation

(mg/m3/lb AI/hour) 

Arithmetic Mean 7.82 

Arithmetic Standard Deviation 12.84 

Geometric Mean 3.57 

Geometric Standard Deviation 3.75 

Min 0.12 

 5% 0.12 

10% 0.99 

25% 1.90 

50% 2.93 

75% 7.44 

90% 22.71 

95% 54.49 

Max 54.49 

Table 2c. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized exposure. 

Exposure Route Clothing 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Inhalation 

(mg/m3

/lb 

AI/hour) 

 8.94 (3.15, 28.92) 33.15 (10.06, 108.51) 

Table 6c. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized inhalation exposure 

(mg/m3

/lb AI/hour). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDs 3.75 2.40 5.98 1.6 2.06 5.89 1.8 
Timothy Leighton, EPA 

15 March, 2011 

Page 68 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold Relative 

Accuracy Lower Bound Upper Bound 

Fold Relative 

Accuracy 

GSDm 3.88 2.42 6.56 1.7 2.08 6.83 1.9 

ICC 0.16 0.00 0.61 0.00 0.61 

GMs 3.57 1.53 8.53 2.4 2.09 5.90 1.7 

GMm 3.57 1.53 8.53 2.4 2.09 5.90 1.7 

AMs 7.82 2.83 23.02 2.9 3.44 13.43 2.3 

AMu 8.56 3.08 25.75 3.0 3.89 18.10 2.2 

AMm 8.94 3.15 28.92 3.2 4.00 22.79 2.5 

P95s 54.49 9.89 207.51 5.5 8.41 54.49 6.5 

P95u 31.43 9.85 97.04 3.2 10.70 69.44 2.9 

P95m 33.15 10.06 108.51 3.3 10.96 85.96 3.0 

Table 8c. 95 percent confidence intervals for slope of log exposure versus log (pounds of active ingredient × hours of 

wiping duration). 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Inhalation 

(mg/m3/lb AI/ 

hour) 

Mixed 0.09 -0.33 0.51 0.83 

 Simple 

Linear 0.06 -0.35 0.47 0.82 

Table 9c. Quadratic mixed models with 95% confidence intervals for the log exposure log (pounds of active ingredient 

× hours of wiping duration). 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Inhalation Intercept 0.78 14.16 -15.11 16.68 2.44 0.14 31.79 

Inhalation Slope 1.77 14.04 -3.35 6.88 2.44 0.14 10.23 

Inhalation Quad 0.13 13.95 -0.27 0.54 2.44 0.14 0.81 
Timothy Leighton, EPA 

15 March, 2011 

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Table 10c. Minimum Pounds of Active Ingredient × Hours of Wiping Duration for Which Normalized Exposure Model 

Over-Predicts Inhalation Exposure. 

Exposure 

Route Clothing Model Slope Threshold Level (lb AI hour) 

Inhalation 

(mg/m3) 

 Mixed 0.09 0.00076 

Quantile plot normalized inhalation exposure data with a normal distribution

Normalized by PoundsAI*Duration

-2 -1 0 1 2

0

10

20

30

40

50

60

Normalized Inhalation Exposure)

Normal Quantiles

Figure 7c
Timothy Leighton, EPA 

15 March, 2011 

Page 70 

Quantile plot normalized inhalation exposure data with a lognormal distribution

Normalized by PoundsAI*Duration

-2 -1 0 1 2

-3

-2

-1

0

1

2

3

4

log (Normalized Inhalation Exposure)

Normal Quantiles

Figure 8c 
Timothy Leighton, EPA 

15 March, 2011 

Page 71 

lninh = -4.5559 +0.0563 lnnrm

N 

18 

Rsq 

0.0053

AdjRsq

-.0569

RMSE 

0.8649

Simple Linear Regression of Ln Inhalation Exposure on Ln PoundsAI*Duration

Normalized by PoundsAI*Duration

log Inhalation Exposure(mg/m3)

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

log PoundsAI*Duration

-8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0

Figure 13c 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 18c 
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15 March, 2011 

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Figure 20c
Timothy Leighton, EPA 

15 March, 2011 

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Analyses of inhalation exposure per surface area wiped. 

Table and Figure Numbers are Consistent with main text (add “d”). 

Table 1d. Summary statistics for normalized exposure. 

Statistic 

Normalized Inhalation

(mg/m3/sq ft) 

Arithmetic Mean 0.0000063 

Arithmetic Standard Deviation 0.0000044 

Geometric Mean 0.0000048 

Geometric Standard Deviation 2.5242839 

Min 0.0000002 

 5% 0.0000002 

10% 0.0000024 

25% 0.0000036 

50% 0.0000053 

75% 0.0000069 

90% 0.0000155 

95% 0.0000180 

Max 0.0000180 

Table 2d. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized exposure. 

Exposure Route Clothing 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Inhalation 

(mg/m3/sq ft) 

 0.0000075 (0.0000041, 

0.0000142) 

0.0000225 (0.0000104, 

0.0000481) 

Table 6d. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized inhalation exposure 

(mg/m3

/sq ft). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold 

Relative 

Accuracy 

Lower 

Bound Upper Bound

Fold Relative 

Accuracy 

GSDs 2.52 1.86 3.44 1.4 1.50 3.91 1.7 
Timothy Leighton, EPA 

15 March, 2011 

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 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold 

Relative 

Accuracy 

Lower 

Bound Upper Bound

Fold Relative 

Accuracy 

GSDm 2.56 1.87 3.59 1.4 1.51 4.34 1.7 

ICC 0.09 0.00 0.53 0.00 0.54 

GMs 0.0000048 0.0000028 0.0000082 1.7 0.0000032 0.0000067 1.5 

GMm 0.0000048 0.0000028 0.0000082 1.7 0.0000032 0.0000067 1.5 

AMs 0.0000063 0.0000039 0.0000132 2.1 0.0000046 0.0000082 1.4 

AMu 0.0000074 0.0000041 0.0000138 1.9 0.0000051 0.0000104 1.4 

AMm 0.0000075 0.0000041 0.0000142 1.9 0.0000052 0.0000117 1.6 

P95s 0.0000180 0.0000103 0.0000802 4.5 0.0000085 0.0000180 2.1 

P95u 0.0000220 0.0000103 0.0000457 2.1 0.0000100 0.0000372 2.2 

P95m 0.0000225 0.0000104 0.0000481 2.2 0.0000101 0.0000434 2.2 

Table 8d. 95 percent confidence intervals for slope of log exposure versus log (surface area wiped). 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Inhalation 

(mg/m3/sq ft) Mixed 0.34 -0.32 1.00 1.32 

 Simple 

Linear 0.35 -0.25 0.95 1.21 

Table 9d. Quadratic mixed models with 95% confidence intervals for the log exposure versus log (surface area wiped). 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Inhalation Intercept -15.79 13.89 -61.42 29.85 2.38 0.12 91.27 

Inhalation Slope 2.70 13.79 -10.05 15.44 2.38 0.12 25.49 

Inhalation Quad -0.16 13.71 -1.05 0.72 2.38 0.12 1.77 
Timothy Leighton, EPA 

15 March, 2011 

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Table 10d. Minimum Square Feet Surface Area Wiped for Which Normalized Exposure Model Over-Predicts 

Inhalation Exposure. 

Exposure 

Route Clothing Model Slope Threshold Level (sq ft) 

Inhalation 

(mg/m3) 

 Mixed 0.34 792.1 

Quantile plot normalized inhalation exposure data with a normal distribution

Normalized by Surface Area Wiped

-2 -1 0 1 2

0

5.0E-06

1.0E-05

1.5E-05

2.0E-05

Normalized Inhalation Exposure)

Normal Quantiles

Figure 7d
Timothy Leighton, EPA 

15 March, 2011 

Page 77 

Quantile plot normalized inhalation exposure data with a lognormal distribution

Normalized by Surface Area Wiped

-2 -1 0 1 2

-16

-15

-14

-13

-12

-11

-10

log (Normalized Inhalation Exposure)

Normal Quantiles

Figure 8d
Timothy Leighton, EPA 

15 March, 2011 

Page 78 

lninh = -7.474 +0.35 lnnrm

N 

18 

Rsq 

0.0863

AdjRsq

0.0292

RMSE 

0.8289

Simple Linear Regression of Ln Inhalation Exposure on Ln Surface Area Wiped

Normalized by Surface Area Wiped

log Inhalation Exposure(mg/m3)

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

log Surface Area Wiped

5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25

Figure 13d 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 18d 
Timothy Leighton, EPA 

15 March, 2011 

Page 80 

Figure 20d 

Analyses of inhalation mass exposure per amount of active ingredient. 

Table and Figure Numbers are Consistent with main text (add “e”). 

Inhalation mass exposure (mg) is estimated as air concentration (mg/m3

) × breathing rate (1 m3/hour) × wiping duration 

(hours). 

Table 1e. Summary statistics for normalized exposure. 

Statistic 

Normalized Inhalation

(mg/lb AI) 

Arithmetic Mean 16.64 

Arithmetic Standard Deviation 12.14 

Geometric Mean 12.48 
Timothy Leighton, EPA 

15 March, 2011 

Page 81 

Statistic 

Normalized Inhalation

(mg/lb AI) 

Geometric Standard Deviation 2.31 

Min 1.52 

 5% 1.52 

10% 6.26 

25% 7.71 

50% 10.88 

75% 27.00 

90% 37.82 

95% 39.37 

Max 39.37 

Table 2e. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized exposure. 

Exposure Route Clothing 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Inhalation (mg/lb 

AI) 

 18.51 (8.53, 42.33) 53.77 (20.47, 143.56) 

Table 6e. Arithmetic mean, geometric mean, geometric standard deviation, and 95th percentiles (with 95% confidence 

intervals and fold relative accuracy), for different statistical models of the normalized inhalation exposure 

(mg/lb AI). 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold 

Relative 

Accuracy 

Lower 

Bound Upper Bound

Fold Relative 

Accuracy 

GSDs 2.31 1.70 3.37 1.5 1.78 2.99 1.3 

GSDm 2.43 1.70 3.87 1.6 1.83 3.35 1.4 

ICC 0.37 0.00 0.77 0.00 0.84 

GMs 12.48 6.25 24.88 2.0 9.21 16.67 1.4 

GMm 12.48 6.25 24.88 2.0 9.21 16.67 1.4 

AMs 16.64 8.11 36.85 2.2 12.64 20.78 1.3 

AMu 17.74 8.34 38.31 2.2 12.97 23.00 1.4 

AMm 18.51 8.53 42.33 2.3 13.25 25.09 1.4 

P95s 39.37 20.37 174.00 4.4 30.91 39.37 1.3 

P95u 49.59 19.90 120.76 2.5 31.05 71.54 1.6 
Timothy Leighton, EPA 

15 March, 2011 

Page 82 

 Parametric Bootstrap Non-parametric Bootstrap 

Parameter Estimate 

Lower 

Bound 

Upper 

Bound 

Fold 

Relative 

Accuracy 

Lower 

Bound Upper Bound

Fold Relative 

Accuracy 

P95m 53.77 20.47 143.56 2.7 32.21 84.53 1.7 

Table 8e. 95 percent confidence intervals for slope of log exposure versus log (amount of active ingredient). 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Inhalation 

(mg/lb AI) Mixed 0.94 0.30 1.59 1.28 

 Simple 

Linear 0.77 0.10 1.44 1.34 

Table 9e. Quadratic mixed models with 95% confidence intervals for the log exposure versus log (amount of active 

ingredient). 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Inhalation Intercept 9.50 13.57 -24.13 43.14 2.50 0.33 67.26 

Inhalation Slope 3.13 13.49 -6.76 13.02 2.50 0.33 19.77 

Inhalation Quad 0.16 13.44 -0.56 0.88 2.50 0.33 1.45 

Table 10e. Minimum Pounds of Active Ingredient for Which Normalized Exposure Model Over-Predicts Inhalation 

Mass Exposure. 

Exposure 

Route Clothing Model Slope Threshold Level (lb AiaH) 

Inhalation 

(mg) 

 Mixed 0.94 0.00000086 
Timothy Leighton, EPA 

15 March, 2011 

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Quantile plot normalized inhalation mass exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

10

20

30

40

Normalized Inhalation Mass Exposure)

Normal Quantiles

Figure 7e
Timothy Leighton, EPA 

15 March, 2011 

Page 84 

Quantile plot normalized inhalation mass exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

log (Normalized Inhalation Mass Exposure)

Normal Quantiles

Figure 8e
Timothy Leighton, EPA 

15 March, 2011 

Page 85 

lninh2 = 0.9659 +0.7709 lnnrm

N 

18 

Rsq 

0.2705

AdjRsq

0.2250

RMSE 

0.8507

Simple Linear Regression of Ln Inhalation Mass Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Inhalation Mass Exposure(mg)

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

log Pounds Active Ingredient Handled

-8.00 -7.75 -7.50 -7.25 -7.00 -6.75 -6.50 -6.25 -6.00 -5.75 -5.50 -5.25

Figure 13e 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 18e 
Timothy Leighton, EPA 

15 March, 2011 

Page 87 

Figure 20e