MEMORANDUM 

To: Timothy Leighton, EPA 

From: Jonathan Cohen, ICF 

Date: 20 December, 2011 

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

AEATF Aerosol Study Statistical Review for HSRB 

Introduction and Summary 

In November 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 a Pressurized 

Aerosol Can for Indoor Surface Disinfecting” ICF were asked to analyze the data 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, a hotel/motel with a full kitchen; cluster 2, a hotel/motel without a 

food preparation area, and; cluster 3, a hotel/motel with a kitchenette. For each test site, six volunteer subjects were 

selected and randomly assigned different numbers of aerosol canisters containing C14 ADBAC to be sprayed from 1-

1.5 canisters to 3.5-4 canisters. Each subject was given inner and outer dosimeters to wear and was also given a 

personal air-sampling pump attached to an OVS air sampling tube and a second personal air-sampling pump attached to 

a three stage RespiconTM Particle Sampler to monitor their inhalation exposure. The Particle Sampler has three filters of 

sizes 100 μm, 10 μm, and 2.5 μm that sample particles of sizes 10 to 100 μm, 2.5 to 10 μm, and less than 2.5 μm, 

respectively. Subjects were also fitted with a silicone half-mask respirator which they could volunteer to not use or use 

during some or all of the spraying time. Each subject then used their assigned number of aerosol canisters to spray 

horizontal and vertical surfaces. The percent active ingredient in the first batch of canisters used for this study was 

0.126% C14 ADBAC and the percent active ingredient in the second batch of canisters was 0.104% C14 ADBAC.. 

Subjects were allowed to take breaks on a chair covered with plastic sheeting in a clean room, as needed. The breaks 

were not included in the spraying time duration but the air-sampling pumps 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 (90.3%) and face and neck (89.4%) wipe samples. The face and 

neck wipe samples were also corrected using a half-mask respirator and safety glasses correction factor of 1.43 to 

account for the area of the face that was covered by the use of personal protection equipment (PPE). 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. 
Timothy Leighton, EPA 

20 December, 2011 

Page 2 

The report data for inhalation exposure were unchanged other than the units conversions. The dermal exposure data 

were used to develop exposure measurements for four 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). 

• 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). 

• Hands Only. This case represents the dermal exposure to the hands only and is the mass from hand wash. This 

case is used for comparison purposes. 

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, although this did 

not in fact apply for any of the outer dosimeter measurements made in this study. 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, face and neck, and outer dosimeter measurements in the study were above the 

LOQ. 14 of the 108 inner dosimeter measurements were below the LOQ. 

For one subject (AE18, cluster 1), inner dosimeter values were much higher than all other subjects and were higher 

than the corresponding outer dosimeter pieces for areas of expected low exposure such as the torso. The study report 

suggests that this was due to significant C14 ADBAC dermal residues on the skin prior to reporting for the study, 

perhaps due to bathing using a product containing C14 ADBAC. No adjustments were made in these analyses for these 

potential outliers. 

Inhalation exposure was measured using the air sampling OVS tubes and using the three filters in the Particle Sampler. 

The inhalation exposure concentration (mg/m3

) was calculated by dividing the corrected residue mass by the volume of 

air drawn. For the Particle Sampler, the report tabulated the total C14 ADBAC concentration based on summing the 

corrected residues on the three filters, and the percentages of the total corrected residue mass captured on each filter. 

The following exposure concentrations are analyzed in this memorandum: 

• Inhalation Concentration (mg/m3). Concentration measured by the OVS tube.
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20 December, 2011 

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• 100μm Concentration (mg/m3

). Total concentration measured by the Particle Sampler. This was calculated by 

summing the corrected residues from the three filters at Stage 1 (size 2.5μm), Stage 2 (size 10μm), and Stage 3 

(size 100μm), and then dividing this total by the amount of air drawn. This measures the concentration of all 

particles less than 100 μm, and thus estimates the concentration of inhalable particles.

• 10μm Concentration (mg/m3

). Concentration of particles less than 10μm measured by the Particle Sampler. 

This value is the sum of the corrected residues from the two filters at Stage 1 (size 2.5μm) and Stage 2 (size 

10μm) divided by the amount of air drawn. This value was obtained from the study report tables by multiplying 

the reported 100μm concentration by the sum of the reported percentages for the Stage 1 and Stage 2 filters. 

• 2.5μm Concentration (mg/m3). Concentration of particles less than 2.5μm measured by the Particle Sampler. 

This value is the corrected residue from the filter at Stage 1 (size 2.5μm) divided by the amount of air drawn. 

This value was obtained from the study report tables by multiplying the reported 100μm concentration by the 

reported percentage for the Stage 1 filter. 

Note that the inhalation concentration and 100μm concentration were measured on different instruments connected to 

different air pumps. There were a few cases where the 100μm concentration exceeded the inhalation concentration for 

the same worker despite the fact that the RespiconTM Particle Sampler has an inlet that restricts particles to be less than 

100 μm in size. 

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 hotel/motel without a food preparation area might be correlated because of the physical similarities of the rooms, 

the physical similarities of the surfaces that were sprayed, the building characteristics, the investigators at that location, 

or other factors that are due to the selected location. The clustering includes both the random variation between 

locations as well as the fixed effect of the type of food preparation area (none, full kitchen, or kitchenette). Alternative 

statistical modeling approaches could have been used to separate out these random and fixed effects but this was not 

done due to the small sample size, the need to develop exposure factors applicable to all potential uses that do not 

depend upon the building type, and for consistency between these analyses and the previous analyses of the mop and 

wipe studies. 

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, which was met for all 3 clothing 

configurations (Long Dermal, Short Dermal, Long Short Dermal) and for the total and filtered particle inhalation 

exposure (Inhalation, 100μm Inhalation, 10μm Inhalation, and 2.5μm Inhalation). 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 total inhalation 
Timothy Leighton, EPA 

20 December, 2011 

Page 4 

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 the three main types of dermal exposure (excluding 

hands only) in a single statistical model, accounting for within-worker correlations and within-location clustering. We 

also evaluated quadratic regression models. 

The results for all the dermal and inhalation exposure routes show that the estimated intra-cluster correlation (ICC) 

coefficient is zero, which implies that there are no clustering and location effects, and therefore, in particular, no 

differences between exposures when spraying different types of food preparation areas. 

The mixed model results for the four dermal exposure routes all show a positive estimated slope of at least 0.7. These 

dermal exposure models consistently do not reject proportionality (slope equals one) at the 5% significance level. 

These dermal exposure models also consistently reject independence (slope equals zero) at the 5% significance level, 

with the sole exception of the mixed model for hands only exposure which had a slightly negative lower confidence 

bound. For inhalation, the estimated slope is 0.4, but proportionality is rejected and independence is not rejected. 

The appendix gives the detailed results of the corresponding analyses for dermal and inhalation exposure when the 

exposure is normalized by the spraying duration. The results show similar patterns to the normalization by amount of 

active ingredient handled but the slopes are lower, showing less support for that alternative exposure models. For 

inhalation, the data suggest that the exposure is either independent of spraying duration or is a non-linear function of 

spraying duration. A summary comparing the normalizing approaches is presented at the end of this memorandum. The 

mixed model analyses show that in most cases, the better of the two normalizing options considered is the amount of 

active ingredient handled. 

The inhalation exposure measures used for the main analyses presented here are the average air concentrations (mg/m3

) 

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

As an alternative approach we considered using the air C14 ADBAC mass (mg) which can be estimated as the average 

air concentration multiplied by the spraying duration (hours) and by an estimated 1 m3

/hour of air breathed in by 

someone doing light activity. These values were computed for the total inhalation mass as well as the inhalation mass 

for the three particle sizes. A similar statistical analysis found that the slope is significantly greater than 1. The 

appendix includes the detailed results for this model. 

Analyses of exposure per pounds of active ingredient handled 

Tables 1 and 2 summarize the normalized exposure data (per lb active ingredient handled) with the summary statistics 

from the 18 measurements for each dermal and inhalation exposure route, respectively. 
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20 December, 2011 

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Table 1. Summary statistics for normalized dermal exposure.* 

Statistic 

Normalized Longa

Dermal 

(mg/lb AI) 

Normalized Shortb

Dermal 

(mg/lb AI) 

Normalized Long 

Shortc Dermal 

(mg/lb AI) 

Normalized Hands 

Only 

(mg/lb AI) 

Arithmetic Mean 251.5 653.3 365.7 145.3 

Arithmetic Standard 

Deviation 

191.4 257.1 200.4 117.9 

Geometric Mean 208.2 602.3 324.0 120.6 

Geometric Standard 

Deviation 

1.8 1.5 1.6 1.8 

Min 88.3 248.1 142.0 54.1 

 5% 88.3 248.1 142.0 54.1 

10% 97.5 325.2 149.5 54.4 

25% 140.2 471.2 248.7 81.8 

50% 193.2 648.3 325.8 121.7 

75% 253.3 837.4 430.0 154.6 

90% 606.0 992.3 688.2 258.6 

95% 850.2 1179.2 953.4 571.2 

Max 850.2 1179.2 953.4 571.2 

*Using Empirical simple random sampling model a

Long = Long pants and long sleeves 

b

Short = Short pants and short sleeves 

c

Long Short = Long pants and short sleeves 
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Table 2. Summary statistics for normalized inhalation exposure.* 

Statistic 

Normalized 

Inhalation 

(mg/m3

/lb AI) 

Normalized 100μm 

Inhalation 

(mg/m3/lb AI) 

Normalized 10μm 

Inhalation 

(mg/m3/lb AI) 

Normalized 2.5μm 

Inhalation 

(mg/m3/lb AI) 

Arithmetic Mean 59.5 47.6 25.3 11.3 

Arithmetic Standard 

Deviation 

28.1 25.0 16.1 8.5 

Geometric Mean 53.3 41.9 21.3 9.1 

Geometric Standard 

Deviation 

1.6 1.7 1.8 1.9 

Min 21.4 14.9 7.5 3.0 

 5% 21.4 14.9 7.5 3.0 

10% 23.4 20.6 9.7 4.2 

25% 40.8 29.2 14.4 5.2 

50% 54.3 41.8 17.9 8.5 

75% 82.5 54.4 30.2 14.2 

90% 114.0 91.2 57.4 25.8 

95% 114.2 92.1 61.0 36.8 

Max 114.2 92.1 61.0 36.8 

*Using Empirical simple random sampling model

The summary analyses presented in Tables 1 and 2 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 aerosol to be 

sprayed, as defined by the number of canisters: 

• 1 to 1.5 canisters 

• 1.5 to 2 canisters 

• 2 to 2.5 canisters 

• 2.5 to 3 canisters 

• 3 to 3.5 canisters 
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20 December, 2011 

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• 3.5 to 4 canisters 

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. 

• 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 Tables 1 and 2 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) 
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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. 

Table 3 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 3. 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 248.1 (185.1, 338.6) 551.5 (358.5, 844.6) 

 Short pants and short sleeves 660.5 (536.5, 819.6) 1221.2 (894.0, 1669.0) 

 Long pants and short sleeves 366.7 (287.4, 472.4) 734.6 (511.6, 1050.9) 

 Hands only 142.4 (106.7, 191.4) 311.2 (204.7, 471.2) 

Inhalation (mg/m3/lb AI) 60.3 (47.2, 77.6) 120.7 (84.1, 172.7) 

100μm Inhalation 

(mg/m3/lb AI) 

 48.1 (37.2, 63.0) 99.5 (67.9, 145.3) 

10μm Inhalation 

(mg/m3/lb AI) 

 25.5 (18.9, 35.1) 57.4 (37.0, 88.7) 

2.5μm Inhalation 

(mg/m3/lb AI) 

 11.3 (8.1, 16.1) 26.8 (16.6, 43.2) 

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 3 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. For 

example, in Tables 1 and 2, the empirical arithmetic means are presented, which are the arithmetic means of the 18 

measurements. To estimate the uncertainty of those empirical arithmetic means, data are simulated from the lognormal 

mixed model to calculate the parametric bootstrap confidence intervals. The arithmetic means in Table 3 are estimated 

using the lognormal mixed model, which is also used to estimate the confidence intervals in Table 3. The unit exposure 

estimates (from the lognormal mixed model) displayed in Table 3 are recommended over the empirical arithmetic 

means and 95th percentiles displayed in Tables 1 and 2. 

The algorithm used was as follows: 

Step 1: 
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20 December, 2011 

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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) 

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 
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20 December, 2011 

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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 

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 

aerosol 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 4 to 11 present the estimates, parametric and non-parametric confidence intervals and fold relative accuracy 

values for all the summary statistics for the four dermal and four inhalation exposure routes, respectively. 
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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 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 1.8 1.5 2.2 1.2 1.4 2.1 1.3 

GSDm 1.8 1.5 2.2 1.2 1.4 2.2 1.3 

ICC 0.0 0.0 0.4 0.0 0.6 

GMs 208.2 158.3 274.7 1.3 163.6 270.9 1.3 

GMm 208.2 158.3 274.7 1.3 163.6 270.9 1.3 

AMs 251.5 182.6 332.9 1.4 179.4 342.5 1.4 

AMu 248.1 184.1 336.6 1.4 180.2 343.3 1.4 

AMm 248.1 185.1 338.6 1.4 181.6 349.4 1.4 

P95s 850.2 357.2 1206.3 2.4 301.1 850.2 2.8 

P95u 551.5 356.5 833.0 1.5 315.3 863.3 1.7 

P95m 551.5 358.5 844.6 1.5 323.0 891.3 1.7 

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 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 1.5 1.3 1.8 1.2 1.4 1.7 1.1 

GSDm 1.5 1.3 1.8 1.2 1.4 1.7 1.1 

ICC 0.0 0.0 0.4 0.0 0.7 

GMs 602.3 492.8 737.5 1.2 504.3 722.2 1.2 

GMm 602.3 492.8 737.5 1.2 504.3 722.2 1.2 

AMs 653.3 532.9 812.1 1.2 546.6 767.6 1.2 

AMu 660.5 536.0 816.9 1.2 549.9 776.5 1.2 

AMm 660.5 536.5 819.6 1.2 556.1 777.8 1.2 

P95s 1179.2 893.7 2156.7 1.8 843.7 1179.2 1.4 

P95u 1220.9 888.7 1651.1 1.4 955.9 1446.4 1.3 

P95m 1221.2 894.0 1669.0 1.4 982.1 1473.4 1.2 
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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 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 1.6 1.4 1.9 1.2 1.4 1.8 1.2 

GSDm 1.6 1.4 2.0 1.2 1.4 1.9 1.2 

ICC 0.0 0.0 0.4 0.0 0.6 

GMs 324.0 257.5 409.1 1.3 263.7 403.3 1.2 

GMm 324.0 257.5 409.1 1.3 263.7 403.3 1.2 

AMs 365.7 285.6 466.3 1.3 287.0 460.1 1.3 

AMu 366.7 287.2 471.1 1.3 288.2 464.8 1.3 

AMm 366.7 287.4 472.4 1.3 290.5 466.3 1.3 

P95s 953.4 509.9 1417.9 1.9 442.7 953.4 2.2 

P95u 734.6 509.1 1038.7 1.4 500.0 1002.4 1.5 

P95m 734.6 511.6 1050.9 1.4 514.1 1016.6 1.4 

Table 7. 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 hands only 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 1.8 1.5 2.2 1.2 1.4 2.2 1.3 

GSDm 1.8 1.5 2.2 1.2 1.4 2.3 1.3 

ICC 0.0 0.0 0.4 0.0 0.5 

GMs 120.6 92.4 158.0 1.3 95.4 156.2 1.3 

GMm 120.6 92.4 158.0 1.3 95.4 156.2 1.3 

AMs 145.3 105.9 189.3 1.4 104.1 202.8 1.4 

AMu 142.4 106.7 191.4 1.3 104.7 200.2 1.4 

AMm 142.4 107.2 192.4 1.4 104.9 206.0 1.4 

P95s 571.2 203.9 666.6 2.8 174.0 571.2 3.3 

P95u 311.2 203.6 464.9 1.5 181.4 511.6 1.7 

P95m 311.2 204.7 471.2 1.5 183.5 547.8 1.8 
Timothy Leighton, EPA 

20 December, 2011 

Page 13 

Table 8. 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 1.6 1.4 1.9 1.2 1.4 1.8 1.2 

GSDm 1.6 1.4 2.0 1.2 1.4 1.9 1.2 

ICC 0.0 0.0 0.4 0.0 0.5 

GMs 53.3 42.3 67.2 1.3 43.0 65.6 1.2 

GMm 53.3 42.3 67.2 1.3 43.0 65.6 1.2 

AMs 59.5 46.9 76.6 1.3 47.9 72.2 1.2 

AMu 60.3 47.2 77.4 1.3 48.1 73.6 1.3 

AMm 60.3 47.2 77.6 1.3 48.5 73.9 1.2 

P95s 114.2 83.8 233.0 2.0 83.6 114.2 1.4 

P95u 120.7 83.7 170.7 1.4 87.7 153.1 1.4 

P95m 120.7 84.1 172.7 1.4 89.1 157.2 1.4 

Table 9. 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 100μm 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 1.7 1.4 2.0 1.2 1.4 1.9 1.2 

GSDm 1.7 1.4 2.0 1.2 1.4 1.9 1.2 

ICC 0.0 0.0 0.4 0.0 0.5 

GMs 41.9 32.8 53.6 1.3 33.3 52.4 1.3 

GMm 41.9 32.8 53.6 1.3 33.3 52.4 1.3 

AMs 47.6 36.8 62.1 1.3 37.2 58.8 1.3 

AMu 48.1 37.1 62.8 1.3 37.3 60.0 1.3 

AMm 48.1 37.2 63.0 1.3 37.5 60.4 1.3 

P95s 92.1 67.6 199.4 2.2 83.5 92.1 1.1 

P95u 99.5 67.5 143.5 1.5 67.6 129.5 1.5 

P95m 99.5 67.9 145.3 1.5 68.4 132.6 1.5 
Timothy Leighton, EPA 

20 December, 2011 

Page 14 

Table 10. 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 10μm 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 1.8 1.5 2.2 1.2 1.5 2.1 1.2 

GSDm 1.8 1.5 2.3 1.2 1.5 2.1 1.2 

ICC 0.0 0.0 0.4 0.0 0.4 

GMs 21.3 16.1 28.2 1.3 16.4 28.0 1.3 

GMm 21.3 16.1 28.2 1.3 16.4 28.0 1.3 

AMs 25.3 18.7 34.5 1.4 18.8 33.1 1.3 

AMu 25.5 18.8 34.9 1.4 18.7 34.0 1.4 

AMm 25.5 18.9 35.1 1.4 18.8 34.1 1.4 

P95s 61.0 36.9 127.6 2.1 36.4 61.0 1.7 

P95u 57.4 36.8 87.5 1.6 36.0 80.9 1.6 

P95m 57.4 37.0 88.7 1.6 36.4 82.2 1.6 

Table 11. 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 2.5μm 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 1.9 1.6 2.4 1.2 1.6 2.2 1.2 

GSDm 1.9 1.6 2.4 1.3 1.6 2.3 1.2 

ICC 0.0 0.0 0.4 0.0 0.4 

GMs 9.1 6.7 12.3 1.4 6.8 12.4 1.4 

GMm 9.1 6.7 12.3 1.4 6.8 12.4 1.4 

AMs 11.3 7.9 15.7 1.4 7.8 15.4 1.4 

AMu 11.3 8.0 15.9 1.4 7.9 15.8 1.4 

AMm 11.3 8.1 16.1 1.4 7.9 15.9 1.4 

P95s 36.8 16.5 64.3 2.2 15.1 36.8 2.4 

P95u 26.8 16.5 42.5 1.6 15.7 40.8 1.7 

P95m 26.8 16.6 43.2 1.6 15.9 41.9 1.7 
Timothy Leighton, EPA 

20 December, 2011 

Page 15 

Tables 4 to 11 show that the ICC estimated value is zero for all the dermal and inhalation 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 and types of food preparation area. All of the fold relative accuracy values met the study 

design benchmark of 3 except for the empirical 95th percentile for the normalized hands only exposure using the nonparametric bootstrap confidence interval. The parametric bootstrap confidence intervals were similar to the nonparametric bootstrap confidence intervals for all the dermal and inhalation exposure routes. 

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-quantile plots are presented in Figures 1 to 16. 

They 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

200

400

600

800

1000

Normalized Long Dermal Exposure

Normal Quantiles

Figure 1 
Timothy Leighton, EPA 

20 December, 2011 

Page 16 

Quantile plot normalized long dermal exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

4.0

4.5

5.0

5.5

6.0

6.5

7.0

log (Normalized Long Dermal Exposure)

Normal Quantiles

Figure 2 
Timothy Leighton, EPA 

20 December, 2011 

Page 17 

Quantile plot normalized short dermal exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

200

400

600

800

1000

1200

Normalized Short Dermal Exposure

Normal Quantiles

Figure 3 
Timothy Leighton, EPA 

20 December, 2011 

Page 18 

Quantile plot normalized short dermal exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

5.50

5.75

6.00

6.25

6.50

6.75

7.00

7.25

log (Normalized Short Dermal Exposure)

Normal Quantiles

Figure 4 
Timothy Leighton, EPA 

20 December, 2011 

Page 19 

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

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

200

400

600

800

1000

Normalized Long Short Dermal Exposure

Normal Quantiles

Figure 5
Timothy Leighton, EPA 

20 December, 2011 

Page 20 

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

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

4.5

5.0

5.5

6.0

6.5

7.0

lnnrmdermls

Normal Quantiles

Figure 6 
Timothy Leighton, EPA 

20 December, 2011 

Page 21 

Quantile plot normalized hands only exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

100

200

300

400

500

600

Normalized Hands Only Exposure

Normal Quantiles

Figure 7 
Timothy Leighton, EPA 

20 December, 2011 

Page 22 

Quantile plot normalized hands only exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

3.5

4.0

4.5

5.0

5.5

6.0

6.5

log (Normalized Hands Only Exposure)

Normal Quantiles

Figure 8 
Timothy Leighton, EPA 

20 December, 2011 

Page 23 

 

Quantile plot normalized inhalation conc exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

20

40

60

80

100

120

Normalized Inhalation Conc Exposure)

Normal Quantiles

Figure 9 
Timothy Leighton, EPA 

20 December, 2011 

Page 24 

Quantile plot normalized inhalation conc exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

log (Normalized Inhalation Conc Exposure)

Normal Quantiles

Figure 10 
Timothy Leighton, EPA 

20 December, 2011 

Page 25 

Quantile plot normalized 100um conc exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

20

40

60

80

100

Normalized 100um Conc Exposure)

Normal Quantiles

Figure 11 
Timothy Leighton, EPA 

20 December, 2011 

Page 26 

Quantile plot normalized 100um conc exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

2.5

3.0

3.5

4.0

4.5

5.0

log (Normalized 100um Conc Exposure)

Normal Quantiles

Figure 12 
Timothy Leighton, EPA 

20 December, 2011 

Page 27 

Quantile plot normalized 10um conc exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

10

20

30

40

50

60

70

Normalized 10um Conc Exposure)

Normal Quantiles

Figure 13
Timothy Leighton, EPA 

20 December, 2011 

Page 28 

Quantile plot normalized 10um conc exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

2.0

2.5

3.0

3.5

4.0

4.5

log (Normalized 10um Conc Exposure)

Normal Quantiles

Figure 14 
Timothy Leighton, EPA 

20 December, 2011 

Page 29 

Quantile plot normalized 2.5um conc exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

5

10

15

20

25

30

35

40

Normalized 2.5um Conc Exposure)

Normal Quantiles

Figure 15
Timothy Leighton, EPA 

20 December, 2011 

Page 30 

Quantile plot normalized 2.5um conc exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

log (Normalized 100um Conc Exposure)

Normal Quantiles

Figure 16 
Timothy Leighton, EPA 

20 December, 2011 

Page 31 

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 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 spraying duration. 

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 12 

summarizes the five available methods: 

Table 12. 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 

20 December, 2011 

Page 32 

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 

20 December, 2011 

Page 33 

dermal, short dermal, long short dermal, all hands, and inhalation 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 exposure against the logarithm of the 

pounds of active ingredient handled was non-zero, although the value was only 0.002, the Kenwood-Rogers method 

was used for that mixed model. Note that this issue does not impact the calculated confidence intervals for the summary 

statistics in Tables 4 to 11, since they were based on a bootstrap method. 

Table 13 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. 

Regression models for the filter measurements by size fraction are not presented in this memorandum. 

Table 13. 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 0.81 0.08 1.54 1.47 

 Simple 

Linear 0.81 0.08 1.53 1.45 

 Short pants and short 

sleeves Mixed 1.01 0.47 1.55 1.07 

 Simple 

Linear 1.01 0.48 1.54 1.06 

 Long pants and short 

sleeves Mixed 0.87 0.25 1.48 1.24 

 Simple 

Linear 0.87 0.26 1.48 1.22 

 Hands Only Mixed 0.70 -0.01 1.40 1.40 

 Simple 

Linear 0.70 0.00 1.39 1.39 

 None Mixed 1.02 0.55 1.49 0.94 

 Simple 

Linear 1.02 0.56 1.48 0.92 

Inhalation 

(mg/m3

) Mixed 0.43 -0.11 0.97 1.08 

 Simple 

Linear 0.43 -0.11 0.96 1.07 
Timothy Leighton, EPA 

20 December, 2011 

Page 34 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Dermal (mg) Any Repeated 

Measures 1.28 0.78 1.77 0.99 

For all of the dermal exposure cases except for the mixed model for hands only, the confidence interval for the slope 

includes 1 but not 0. Thus, for these cases, the assumption of proportionality was not rejected and the assumption of 

independence was rejected. For the mixed model for hands only, the confidence interval for the slope includes both 1 

and 0 (although the lower bound is nearly zero, so that the assumptions of proportionality and independence were both 

not rejected. For the inhalation exposure, there is sufficient evidence to reject the hypothesis of a slope of 1 at the 5% 

significance level. 

A few of the slope values in Table 13 are slightly above 1. Since the confidence intervals include 1, these cases provide 

good support for the proportionality assumption. As discussed below, when the slope is greater than one, the predicted 

means from the normalized exposure model will underestimate the predicted means from the mixed model with an 

estimated slope (of the log exposure against the log amount of active ingredient) when the amount of active ingredient 

is high. However, this underestimation will be negligible when the slope is only slightly above 1, except if the amount 

of active ingredient handled is extremely high. 

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) 
Timothy Leighton, EPA 

20 December, 2011 

Page 35 

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

The results in Table 13 show that the actual confidence interval widths were less than 1.4 for all the exposure cases 

except for the long pants and long sleeves scenario where the width was slightly higher, 1.5. 

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 17 to 22. 

 

lnderml = 4.2195 +0.8093 lnnrm

N 

18 

Rsq 

0.2592

AdjRsq

0.2129

RMSE 

0.6047

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)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 17 
Timothy Leighton, EPA 

20 December, 2011 

Page 36 

lnderms = 6.4691 +1.0117 lnnrm

N 

18 

Rsq 

0.5049

AdjRsq

0.4740

RMSE 

0.4428

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)

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 18 
Timothy Leighton, EPA 

20 December, 2011 

Page 37 

lndermls = 4.995 +0.866 lnnrm

N 

18 

Rsq 

0.3608

AdjRsq

0.3208

RMSE 

0.5095

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)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 19 
Timothy Leighton, EPA 

20 December, 2011 

Page 38 

lnderma = 6.9721 +1.0178 lnnrm

N 

18 

Rsq 

0.5805

AdjRsq

0.5543

RMSE 

0.3824

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)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 20 
Timothy Leighton, EPA 

20 December, 2011 

Page 39 

lndermh = 3.0157 +0.6971 lnnrm

N 

18 

Rsq 

0.2208

AdjRsq

0.1721

RMSE 

0.5788

Simple Linear Regression of Ln Hands Only Exposure on Ln Pounds Active Ingredient Handled

Normalized by Pounds Active Ingredient Handled

log Hands Only Exposure (mg)

-2.25

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 21 
Timothy Leighton, EPA 

20 December, 2011 

Page 40 

lninh1 = 0.6138 +0.4269 lnnrm

N 

18 

Rsq 

0.1519

AdjRsq

0.0989

RMSE 

0.4459

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

Normalized by Pounds Active Ingredient Handled

log Inhalation Exposure (mg/m3)

-3.25

-3.00

-2.75

-2.50

-2.25

-2.00

-1.75

-1.50

-1.25

-1.00

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 22 
Timothy Leighton, EPA 

20 December, 2011 

Page 41 

These simple linear regression results are compared with the results for the lognormal mixed models in Figures 23 to 

28. In all these cases, the regression lines are the same for the mixed and simple linear regression models because the 

estimated ICC parameter was zero. Although these regression lines in these figures and the same as in Figures 17 to 22, 

they are included for consistency with the previous reviews.

Figure 23 
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20 December, 2011 

Page 42 

Figure 24 
Timothy Leighton, EPA 

20 December, 2011 

Page 43 

Figure 25
Timothy Leighton, EPA 

20 December, 2011 

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Figure 26 
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Figure 27 
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Figure 28 
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Finally, in Figure 29 we present a regression plot showing the slightly different regression lines for the three clusters 

based on the mixed model for all dermal exposure. For the other dermal clothing configurations and the inhalation 

exposure, the mixed models predict the same values for all three clusters. 

Figure 29 

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

expect on physical grounds. All of the estimated slopes for dermal exposure are positive and approximately 0.8, and all 

the confidence intervals contained 1. To investigate this issue further, the following more complicated statistical model 

was fitted to the data of all three dermal exposures (excluding the unrealistic hands only and 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 
Timothy Leighton, EPA 

20 December, 2011 

Page 48 

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 

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

been shown using this statistical model. 

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. 
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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. 

Table 14 presents the fitted quadratic models from the study for the mixed models of five exposure measurements 

(Long Dermal, Short Dermal, Long Short Dermal, Hands Only, 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 14. 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 -9.04 2.00 -137.34 119.27 1.86 0.00 256.61 

Long 

Dermal

Slope -3.67 13.00 -25.37 18.04 1.86 0.00 43.41 

Long 

Dermal

Quad -0.38 13.00 -2.20 1.45 1.86 0.00 3.64 

Short 

Dermal 

Intercept -4.86 2.00 -98.59 88.86 1.57 0.00 187.45 

Short 

Dermal

Slope -2.81 13.00 -18.67 13.04 1.57 0.00 31.71 

Short 

Dermal

Quad -0.32 13.00 -1.65 1.01 1.57 0.00 2.66 

Long 

Short 

Dermal 

Intercept -7.96 2.00 -115.82 99.90 1.68 0.00 215.72 

Long 

Short 

Dermal

Slope -3.51 13.00 -21.75 14.74 1.68 0.00 36.49 
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Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Long 

Short 

Dermal

Quad -0.37 13.00 -1.90 1.16 1.68 0.00 3.06 

Hands 

Only 

Intercept 20.71 2.00 -101.33 142.75 1.80 0.00 244.08 

Hands 

Only 

Slope 6.67 13.00 -13.98 27.31 1.80 0.00 41.29 

Hands 

Only 

Quad 0.50 13.00 -1.23 2.23 1.80 0.00 3.46 

Inhalation Intercept -38.66 13.19 -81.35 4.03 1.51 0.00 85.39 

Inhalation Slope -12.83 13.18 -27.21 1.55 1.51 0.00 28.76 

Inhalation Quad -1.11 13.17 -2.32 0.09 1.51 0.00 2.41 

Dermal 

Repeated 

Measures 

Slope -1.29 12.63 -15.98 13.41 NA NA 29.39 

Dermal 

Repeated 

Measures 

Quad -0.21 12.62 -1.45 1.02 NA NA 2.46 

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. 

Threshold Analyses 

As described above, the following two mixed models were fitted to the dermal and inhalation exposure data. 

Linear mixed model: 

 Log (Exposure) = Intercept1 + Slope × Log (Pounds of Active Ingredient) + Cluster + Error 

Normalized exposure mixed model: 

 Log (Exposure) = Intercept2 + Log (Pounds of Active Ingredient) + Cluster + Error, 
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which is mathematically the same as 

 Log (Exposure/Pounds of Active Ingredient) = Intercept2 + Cluster + Error 

The intercepts for these two models are denoted as Intercept1 and Intercept2 since their estimated values will in general 

be different. 

Cluster is a normally distributed random effect variable with independent, identically distributed values for each 

cluster, and Error is a normally distributed error variable with independent, identically distributed values for each total 

exposure measurement. If the linear mixed model has a slope of 1, then the model is mathematically the same as the 

normalized exposure mixed model. 

These two statistical models can be compared by calculating the threshold value of the pounds of active ingredient at 

which both models predict the same mean exposure. The threshold values are computed as follows. 

Suppose first that the linear mixed model for log Exposure is correct. Then the predicted mean exposure for a given 

amount of active ingredient is given by the equation 

Predicted mean exposure from linear mixed model = 

exp(Intercept1) × (Pounds of Active Ingredient)Slope × exp( 1⁄2 V1) (1) 

where V1 is the total variance for the linear mixed model, calculated as the sum of the cluster variance and the error 

variance. The predicted mean exposure is the expected value of the exposure for a given amount of active ingredient 

and thus estimates the average (arithmetic mean) exposure for a large number of scenarios that are all using the same 

amount of active ingredient. 

Suppose instead that the normalized exposure mixed model for log Exposure is correct. Then the predicted mean 

exposure is given by the equation 

Predicted mean exposure from normalized exposure mixed model = 

exp(Intercept2) × (Pounds of Active Ingredient) × exp( 1⁄2 V2) = AMm × Pounds of Active Ingredient (2) 

where V2 is the total variance for the normalized exposure mixed model, calculated as the sum of the cluster variance 

and the error variance. AMm is the estimated average normalized exposure, 

 AMm = exp(Intercept2 + 1⁄2 V2) 

We now have two estimates of the predicted mean exposure for a given amount of active ingredient, equations (1) and 

(2). The graphs in Figures 30 to 34 below compare the predicted means for each clothing configuration (Long Dermal, 

Short Dermal, Long Short Dermal, Hands Only) and for total inhalation exposure. The two estimates (1) and (2) are 

equal if the pounds of active ingredient equals the Threshold value: 

 Threshold = {AMm / exp(Intercept1 + 1⁄2 V1)}1/(Slope – 1)

The Threshold values are tabulated in Table 15 below. 
Timothy Leighton, EPA 

20 December, 2011 

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Suppose that the estimated slope is less than 1 (which is true for all the cases studied here except for the Short Dermal 

exposure which had an estimated slope slightly greater than 1). The predicted mean exposure from the normalized 

exposure mixed model will be greater than the predicted mean exposure from the linear mixed model for amounts of 

active ingredient above the threshold (right hand side of the graph). The predicted mean exposure from the normalized 

exposure mixed model will be less than the predicted mean exposure from the linear mixed model for amounts of 

active ingredient below the threshold (left hand side of the graph). What this means is that if we assume proportionality 

and use the normalized exposure mixed model, then we will tend to over-predict the exposure unless the amount of 

active ingredient is below the threshold. (If the amount of active ingredient is below the threshold, then it will be low 

enough that the exposure will not usually be of concern). 

For the Short Dermal exposure, the estimated slope was greater than one. In this exceptional case the inequalities are 

reversed: The predicted mean exposure from the normalized exposure mixed model will be greater than the predicted 

mean exposure from the linear mixed model for amounts of active ingredient below the threshold (left hand side of the 

graph). The predicted mean exposure from the normalized exposure mixed model will be less than the predicted mean 

exposure from the linear mixed model for amounts of active ingredient above the threshold (right hand side of the 

graph). Although this case shows a tendency for exposure estimates from the normalized exposure mixed model to 

under-predict exposure, it is clearly seen from the graph that the two curves are numerically extremely close so that the 

under-prediction is very small. 

Table 15. Threshold values for the amount of active ingredient. 

Exposure 

Route Clothing Model Slope 

Threshold Level (lb active 

ingredient) 

Dermal (mg) Long pants 

and long 

sleeves 

Mixed 0.81 0.00295 

 Short pants 

and short 

sleeves 

Mixed 1.01 0.00174* 

 Long pants 

and short 

sleeves 

Mixed 0.87 0.00297 

 Hands only Mixed 0.70 0.00285 

Inhalation 

(mg/m3) 

 Mixed 0.43 0.00272 

*For this case, slope > 1 and so the normalized exposure mixed model under-predicts exposure for pounds of active 

ingredient above the threshold. 
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20 December, 2011 

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Figure 30 
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Figure 31 
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Figure 32 
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Figure 33 
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Figure 34 

Alternative normalizing and exposure variables 

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

the spraying duration. All the dermal exposure models are consistent with proportionality for the spraying duration. The 

estimated slopes for spraying duration are consistently lower than the slopes for the amount of active ingredient 

handled. The confidence intervals for the slopes are generally a little narrower than for the amount of active ingredient 

handled. The inhalation exposure models are inconsistent with proportionality for spraying duration. 

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 spraying 

duration (hours) and by an estimated 1 m3

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

the mass of C14 ADBAC 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, 

calculated for total inhalation and for the three particle size limits. 
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20 December, 2011 

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

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

locations including the type of food preparation area. 

Finally, to compare the two different normalizing variables, we present Table 16 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 16. The values in Table 16 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 exposure routes except for the hands only model with an unknown slope. 

Table 16. 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 

spraying 

duration 

Long Dermal Mixed Any 30.9 31.8 

 Mixed 1 31.2 33.8 

Short Dermal Mixed Any 19.6 25.0 

 Mixed 1 19.6 26.6 

Long Short 

Dermal Mixed Any 24.7 27.2 

 Mixed 1 24.9 29.8 
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20 December, 2011 

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Exposure 

Route 

Model Slope Normalized by 

pounds of active 

ingredient 

Normalized by 

spraying 

duration 

Hands Only Mixed Any 29.3 28.4 

 1 30.2 30.7 

Dermal Repeated 

Measures 

Any -34.2 -26.0 

 1 -33.0 -25.6 

Inhalation 

Concentration 

(mg/m3

) 

Mixed Any 19.9 21.7 

 Mixed 1 24.9 32.0 
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20 December, 2011 

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APPENDIX 

Analyses of exposure per spraying duration (hours) 

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

Table 1b. Summary statistics for normalized dermal exposure. 

Statistic 

Normalized Longa

Dermal 

(mg/hour) 

Normalized Shortb

Dermal 

(mg/hour) 

Normalized Long 

Shortc

 Dermal 

(mg/hour) 

Normalized Hands 

Only 

(mg/hour) 

Arithmetic Mean 1.73 4.57 2.54 0.98 

Arithmetic Standard 

Deviation 1.31 1.99 1.42 0.75 

Geometric Mean 1.41 4.07 2.19 0.82 

Geometric Standard 

Deviation 1.89 1.69 1.77 1.79 

Min 0.44 1.25 0.71 0.27 

 5% 0.44 1.25 0.71 0.27 

10% 0.63 1.90 0.98 0.44 

25% 0.96 2.85 1.43 0.50 

50% 1.40 5.04 2.35 0.77 

75% 1.95 6.02 3.24 1.16 

90% 3.85 7.00 4.38 1.68 

95% 5.87 8.15 6.59 3.63 

Max 5.87 8.15 6.59 3.63 a

Long = Long pants and long sleeves 

b

Short = Short pants and short sleeves 

c

Long Short = Long pants and short sleeves 
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20 December, 2011 

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Table 2b. Summary statistics for normalized inhalation exposure. 

Statistic 

Normalized Inhalation

(mg/m3

/hour) 

Arithmetic Mean 0.42 

Arithmetic Standard Deviation 0.24 

Geometric Mean 0.36 

Geometric Standard Deviation 1.84 

Min 0.12 

 5% 0.12 

10% 0.14 

25% 0.26 

50% 0.39 

75% 0.53 

90% 0.88 

95% 0.93 

Max 0.93 

Table 3b. 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 1.72 (1.25, 2.42) 4.01 (2.53, 6.34) 

 Short pants and short sleeves 4.72 (3.23, 6.99) 9.96 (6.02, 16.42) 

 Long pants and short sleeves 2.58 (1.90, 3.57) 5.62 (3.62, 8.69) 

 Hands only 0.97 (0.72, 1.31) 2.13 (1.40, 3.25) 

Inhalation (mg/m3/hour) 0.44 (0.29, 0.67) 1.01 (0.58, 1.75) 

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 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 1.89 1.53 2.34 1.24 1.54 2.16 1.23 

GSDm 1.89 1.54 2.36 1.25 1.57 2.20 1.21 
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20 December, 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 

ICC 0.00 0.00 0.39 0.00 0.76 

GMs 1.41 1.05 1.90 1.35 1.10 1.88 1.33 

GMm 1.41 1.05 1.90 1.35 1.10 1.88 1.33 

AMs 1.73 1.24 2.37 1.40 1.23 2.36 1.40 

AMu 1.72 1.25 2.41 1.39 1.25 2.39 1.38 

AMm 1.72 1.25 2.42 1.40 1.28 2.40 1.39 

P95s 5.87 2.52 9.31 2.34 2.16 5.87 2.72 

P95u 4.01 2.51 6.25 1.60 2.42 6.04 1.66 

P95m 4.01 2.53 6.34 1.59 2.54 6.12 1.58 

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 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 1.69 1.41 2.05 1.21 1.48 1.88 1.14 

GSDm 1.72 1.42 2.15 1.25 1.48 2.01 1.16 

ICC 0.21 0.00 0.66 0.00 0.84 

GMs 4.07 2.84 5.89 1.45 3.35 5.01 1.23 

GMm 4.07 2.84 5.89 1.45 3.35 5.01 1.23 

AMs 4.57 3.19 6.77 1.48 3.78 5.42 1.21 

AMu 4.68 3.21 6.85 1.46 3.84 5.56 1.22 

AMm 4.72 3.23 6.99 1.48 3.94 5.57 1.20 

P95s 8.15 5.96 20.77 2.55 6.31 8.15 1.29 

P95u 9.69 5.96 15.55 1.63 7.57 11.57 1.28 

P95m 9.96 6.02 16.42 1.66 7.92 12.28 1.26 
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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 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 1.77 1.47 2.13 1.21 1.52 1.95 1.16 

GSDm 1.77 1.47 2.16 1.22 1.54 2.01 1.15 

ICC 0.04 0.00 0.46 0.00 0.80 

GMs 2.19 1.64 2.95 1.35 1.76 2.79 1.28 

GMm 2.19 1.64 2.95 1.35 1.76 2.79 1.28 

AMs 2.54 1.87 3.51 1.38 1.98 3.21 1.28 

AMu 2.58 1.89 3.55 1.38 2.00 3.28 1.29 

AMm 2.58 1.90 3.57 1.38 2.05 3.28 1.27 

P95s 6.59 3.59 12.20 1.85 3.27 6.59 2.01 

P95u 5.59 3.59 8.52 1.56 3.93 7.37 1.42 

P95m 5.62 3.62 8.69 1.55 4.12 7.46 1.37 

Table 7b. 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 hands only 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 1.48 2.18 1.22 1.43 2.17 1.25 

GSDm 1.79 1.49 2.20 1.23 1.44 2.27 1.26 

ICC 0.00 0.00 0.39 0.00 0.59 

GMs 0.82 0.62 1.07 1.32 0.64 1.06 1.30 

GMm 0.82 0.62 1.07 1.32 0.64 1.06 1.30 

AMs 0.98 0.72 1.29 1.37 0.71 1.34 1.38 

AMu 0.97 0.72 1.31 1.35 0.71 1.34 1.38 

AMm 0.97 0.72 1.31 1.36 0.72 1.37 1.41 

P95s 3.63 1.39 4.62 2.61 1.19 3.63 3.05 

P95u 2.13 1.39 3.21 1.54 1.29 3.40 1.66 

P95m 2.13 1.40 3.25 1.53 1.31 3.64 1.70 
Timothy Leighton, EPA 

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Table 8b. 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 1.84 1.50 2.28 1.24 1.54 2.10 1.20 

GSDm 1.87 1.50 2.38 1.28 1.55 2.23 1.21 

ICC 0.17 0.00 0.62 0.00 0.66 

GMs 0.36 0.24 0.54 1.50 0.28 0.46 1.27 

GMm 0.36 0.24 0.54 1.50 0.28 0.46 1.27 

AMs 0.42 0.28 0.65 1.53 0.33 0.53 1.28 

AMu 0.43 0.29 0.66 1.52 0.34 0.55 1.29 

AMm 0.44 0.29 0.67 1.54 0.34 0.56 1.28 

P95s 0.93 0.58 2.35 2.52 0.63 0.93 1.47 

P95u 0.98 0.57 1.66 1.71 0.68 1.30 1.44 

P95m 1.01 0.58 1.75 1.74 0.71 1.39 1.42 
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20 December, 2011 

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Quantile plot normalized long dermal exposure data with a normal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

0

1

2

3

4

5

6

Normalized Long Dermal Exposure

Normal Quantiles

Figure 1b 
Timothy Leighton, EPA 

15 March, 2011 

Page 66 

Quantile plot normalized long dermal exposure data with a lognormal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

-1.0

-0.5

0

0.5

1.0

1.5

2.0

log (Normalized Long Dermal Exposure)

Normal Quantiles

Figure 2b
Timothy Leighton, EPA 

15 March, 2011 

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Quantile plot normalized short dermal exposure data with a normal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

1

2

3

4

5

6

7

8

9

Normalized Short Dermal Exposure

Normal Quantiles

Figure 3b
Timothy Leighton, EPA 

15 March, 2011 

Page 68 

Quantile plot normalized short dermal exposure data with a lognormal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

0

0.5

1.0

1.5

2.0

2.5

log (Normalized Short Dermal Exposure)

Normal Quantiles

Figure 4b
Timothy Leighton, EPA 

15 March, 2011 

Page 69 

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

Normalized by Hours Spraying Duration

-2 -1 0 1 2

0

1

2

3

4

5

6

7

Normalized Long Short Dermal Exposure

Normal Quantiles

Figure 5b
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15 March, 2011 

Page 70 

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

Normalized by Hours Spraying Duration

-2 -1 0 1 2

-0.5

0

0.5

1.0

1.5

2.0

lnnrmdermls

Normal Quantiles

Figure 6b
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15 March, 2011 

Page 71 

Quantile plot normalized hands only exposure data with a normal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Normalized Hands Only Exposure

Normal Quantiles

Figure 7b
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15 March, 2011 

Page 72 

Quantile plot normalized hands only exposure data with a lognormal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

log (Normalized Hands Only Exposure)

Normal Quantiles

Figure 8b 
Timothy Leighton, EPA 

15 March, 2011 

Page 73 

 

Quantile plot normalized inhalation conc exposure data with a normal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

0

0.2

0.4

0.6

0.8

1.0

Normalized Inhalation Conc Exposure)

Normal Quantiles

Figure 9b
Timothy Leighton, EPA 

15 March, 2011 

Page 74 

Quantile plot normalized inhalation conc exposure data with a lognormal distribution

Normalized by Hours Spraying Duration

-2 -1 0 1 2

-2.5

-2.0

-1.5

-1.0

-0.5

0

log (Normalized Inhalation Conc Exposure)

Normal Quantiles

Figure 10b 
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15 March, 2011 

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Table 13b. 95 percent confidence intervals for the slope of log exposure versus log spraying duration. 

Exposure Route Clothing Model Estimate Lower Upper 

Confidence Interval 

Width 

Dermal (mg) Long pants and long 

sleeves Mixed 0.61 -0.01 1.22 1.23 

 Simple 

Linear 0.61 -0.00 1.22 1.22 

 Short pants and short 

sleeves Mixed 0.73 0.19 1.28 1.10 

 Simple 

Linear 0.67 0.17 1.18 1.01 

 Long pants and short 

sleeves Mixed 0.61 0.07 1.14 1.07 

 Simple 

Linear 0.61 0.07 1.14 1.07 

 Hands Only Mixed 0.61 0.05 1.18 1.12 

 Simple 

Linear 0.61 0.06 1.17 1.11 

 None Mixed 0.79 0.33 1.25 0.92 

 Simple 

Linear 0.71 0.27 1.15 0.88 

Inhalation 

(mg/m3

) Mixed 0.23 -0.30 0.76 1.06 

 Simple 

Linear 0.22 -0.24 0.68 0.92 

Dermal (mg) Any Repeated 

Measures 0.91 0.36 1.45 1.09 
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lnderml = 0.0014 +0.608 lnnrm

N 

18 

Rsq 

0.2183

AdjRsq

0.1694

RMSE 

0.6212

Simple Linear Regression of Ln Long Dermal Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log Long Dermal Exposure (mg)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 17b
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15 March, 2011 

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lnderms = 1.121 +0.6737 lnnrm

N 

18 

Rsq 

0.3341

AdjRsq

0.2925

RMSE 

0.5135

Simple Linear Regression of Ln Short Dermal Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log Short Dermal Exposure (mg)

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 18b
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15 March, 2011 

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lndermls = 0.4423 +0.6062 lnnrm

N 

18 

Rsq 

0.2637

AdjRsq

0.2177

RMSE 

0.5468

Simple Linear Regression of Ln Long Short Dermal Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log Long Short Dermal Exposure (mg)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 19b
Timothy Leighton, EPA 

15 March, 2011 

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lnderma = 1.62 +0.7108 lnnrm

N 

18 

Rsq 

0.4224

AdjRsq

0.3863

RMSE 

0.4487

Simple Linear Regression of Ln All Dermal Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log All Dermal Exposure (mg)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 20b 
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15 March, 2011 

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lndermh = -0.5389 +0.6143 lnnrm

N 

18 

Rsq 

0.2558

AdjRsq

0.2093

RMSE 

0.5656

Simple Linear Regression of Ln Hands Only Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log Hands Only Exposure (mg)

-2.25

-2.00

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 21b 
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15 March, 2011 

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lninh1 = -1.6948 +0.2248 lnnrm

N 

18 

Rsq 

0.0629

AdjRsq

0.0043

RMSE 

0.4687

Simple Linear Regression of Ln Inhalation Conc Exposure on Ln Hours Spraying Duration

Normalized by Hours Spraying Duration

log Inhalation Exposure (mg/m3)

-3.25

-3.00

-2.75

-2.50

-2.25

-2.00

-1.75

-1.50

-1.25

-1.00

log Hours Spraying Duration

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Figure 22b 
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15 March, 2011 

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Figure 23b 
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Figure 24b 
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Figure 25b
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Figure 26b 
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Figure 27b 
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Figure 28b 
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Figure 29.1b 
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Figure 29.2b 
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Figure 29.3b 
Timothy Leighton, EPA 

15 March, 2011 

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Figure 29.4b 

Table 14b. Quadratic mixed models with 95% confidence intervals for the log exposure versus log spraying duration. 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Long 

Dermal 

Intercept -0.22 2.00 -1.68 1.24 1.85 0.00 2.93 

Long 

Dermal

Slope -0.23 13.00 -1.87 1.41 1.85 0.00 3.28 

Long 

Dermal

Quad -0.50 13.00 -1.41 0.41 1.85 0.00 1.82 

Short 

Dermal 

Intercept 0.99 11.16 0.32 1.66 1.67 0.11 1.34 
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 

Short 

Dermal

Slope 0.03 13.35 -1.30 1.35 1.67 0.11 2.65 

Short 

Dermal

Quad -0.42 13.04 -1.15 0.30 1.67 0.11 1.45 

Long 

Short 

Dermal 

Intercept 0.25 2.00 -1.04 1.54 1.72 0.00 2.57 

Long 

Short 

Dermal

Slope -0.14 13.00 -1.58 1.31 1.72 0.00 2.88 

Long 

Short 

Dermal

Quad -0.44 13.00 -1.24 0.36 1.72 0.00 1.60 

Hands 

Only 

Intercept -0.60 2.00 -1.99 0.79 1.79 0.00 2.78 

Hands 

Only 

Slope 0.39 13.00 -1.17 1.94 1.79 0.00 3.11 

Hands 

Only 

Quad -0.14 13.00 -1.00 0.73 1.79 0.00 1.73 

Inhalation Intercept -1.85 12.33 -2.45 -1.24 1.59 0.04 1.21 

Inhalation Slope -0.40 13.61 -1.65 0.85 1.59 0.04 2.51 

Inhalation Quad -0.38 13.21 -1.07 0.30 1.59 0.04 1.37 

Dermal 

Repeated 

Measures 

Slope 0.32 12.60 -1.03 1.68 NA NA 2.71 

Dermal 

Repeated 

Measures 

Quad -0.35 12.49 -1.09 0.40 NA NA 1.49 
Timothy Leighton, EPA 

15 March, 2011 

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Table 15b. Threshold values for the amount of active ingredient. 

Exposure 

Route Clothing Model Slope Threshold Level (hours) 

Dermal (mg) Long pants 

and long 

sleeves 

Mixed 0.61 0.40908 

 Short pants 

and short 

sleeves 

Mixed 0.73 0.40208 

 Long pants 

and short 

sleeves 

Mixed 0.61 0.40425 

 Hands only Mixed 0.61 0.40758 

Inhalation 

(mg/m3) 

 Mixed 0.23 0.37555 
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15 March, 2011 

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Figure 30b 
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Figure 31b 
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Figure 32b 
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Figure 33b 
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Figure 34b 
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15 March, 2011 

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Analyses of inhalation mass exposure per amount of active ingredient 

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

Inhalation mass exposure (mg) is estimated as air concentration (mg/m3) × breathing rate (1 m3/hour) × spraying 

duration (hours). 

Table 2c. Summary statistics for normalized inhalation mass exposure. 

Statistic 

Normalized 

Inhalation Mass 

(mg/lb AI) 

Normalized 100μm 

Inhalation Mass 

(mg/lb AI) 

Normalized 10μm 

Inhalation Mass 

(mg/lb AI) 

Normalized 2.5μm 

Inhalation Mass 

(mg/lb AI) 

Arithmetic Mean 24.6 19.2 10.0 4.4 

Arithmetic 

Standard Deviation 10.5 8.7 5.1 2.6 

Geometric Mean 22.3 17.6 8.9 3.8 

Geometric 

Standard Deviation 1.6 1.5 1.6 1.8 

Min 6.7 8.7 3.9 1.0 

 5% 6.7 8.7 3.9 1.0 

10% 13.3 9.5 4.8 1.9 

25% 15.2 14.2 5.9 2.3 

50% 23.9 16.0 8.5 3.9 

75% 35.7 25.0 13.6 5.6 

90% 39.1 32.0 20.7 8.8 

95% 42.5 41.1 20.9 11.1 

Max 42.5 41.1 20.9 11.1 
Timothy Leighton, EPA 

15 March, 2011 

Page 100 

Table 3c. Arithmetic mean and 95th percentile estimates from lognormal mixed model for normalized inhalation mass 

exposure. 

Exposure Route 

Arithmetic Mean 

(95% confidence interval) 

95th percentile 

(95% confidence interval) 

Inhalation mass (mg/lb 

AI) 

25.1 (19.8, 32.0) 49.4 (34.8, 69.8) 

100 μm inhalation mass 

(mg/lb AI) 

19.4 (14.1, 26.9) 36.7 (24.0, 56.1) 

10 μm inhalation mass 

(mg/lb AI) 

10.1 (7.8, 13.3) 20.2 (13.8, 29.5) 

2.5 μm inhalation mass 

(mg/lb AI) 

4.5 (3.3, 6.2) 10.0 (6.5, 15.6) 

Table 8c. 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 mass 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 1.6 1.4 1.9 1.2 1.4 1.9 1.2 

GSDm 1.6 1.4 1.9 1.2 1.4 1.9 1.2 

ICC 0.0 0.0 0.4 0.0 0.5 

GMs 22.3 17.9 28.0 1.3 17.9 27.4 1.2 

GMm 22.3 17.9 28.0 1.3 17.9 27.4 1.2 

AMs 24.6 19.7 31.7 1.3 20.1 29.3 1.2 

AMu 25.1 19.8 32.0 1.3 20.3 29.8 1.2 

AMm 25.1 19.8 32.0 1.3 20.5 29.9 1.2 

P95s 42.5 34.7 93.4 2.2 36.2 42.5 1.2 

P95u 49.4 34.6 69.1 1.4 36.7 60.5 1.3 

P95m 49.4 34.8 69.8 1.4 37.0 62.2 1.3 
Timothy Leighton, EPA 

15 March, 2011 

Page 101 

Table 9c. 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 100μm inhalation mass 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 1.5 1.3 1.8 1.2 1.4 1.7 1.1 

GSDm 1.6 1.3 1.9 1.2 1.4 1.8 1.1 

ICC 0.2 0.0 0.7 0.0 0.7 

GMs 17.6 12.9 24.0 1.4 14.9 20.7 1.2 

GMm 17.6 12.9 24.0 1.4 14.9 20.7 1.2 

AMs 19.2 14.0 26.4 1.4 16.0 22.6 1.2 

AMu 19.3 14.1 26.6 1.4 16.0 22.7 1.2 

AMm 19.4 14.1 26.9 1.4 16.1 23.1 1.2 

P95s 41.1 23.7 67.0 1.7 26.2 41.1 1.6 

P95u 35.8 23.7 53.3 1.5 26.7 44.1 1.3 

P95m 36.7 24.0 56.1 1.5 27.2 47.5 1.4 

Table 10c. 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 10μm inhalation mass 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 1.6 1.4 1.9 1.2 1.4 1.8 1.2 

GSDm 1.6 1.4 2.0 1.2 1.4 1.9 1.1 

ICC 0.0 0.0 0.5 0.0 0.7 

GMs 8.9 6.9 11.5 1.3 7.3 11.0 1.2 

GMm 8.9 6.9 11.5 1.3 7.3 11.0 1.2 

AMs 10.0 7.7 13.1 1.3 8.0 12.3 1.2 

AMu 10.1 7.8 13.2 1.3 8.0 12.5 1.3 

AMm 10.1 7.8 13.3 1.3 8.1 12.6 1.3 

P95s 20.9 13.7 39.6 1.9 13.8 20.9 1.5 

P95u 20.1 13.7 28.9 1.5 14.3 26.2 1.4 

P95m 20.2 13.8 29.5 1.5 14.5 27.1 1.4 
Timothy Leighton, EPA 

15 March, 2011 

Page 102 

Table 11c. 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 2.5μm inhalation mass 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 1.8 1.5 2.2 1.2 1.5 2.1 1.2 

GSDm 1.8 1.5 2.2 1.2 1.5 2.2 1.2 

ICC 0.0 0.0 0.4 0.0 0.6 

GMs 3.8 2.9 5.1 1.3 3.0 4.9 1.3 

GMm 3.8 2.9 5.1 1.3 3.0 4.9 1.3 

AMs 4.4 3.3 6.1 1.4 3.4 5.6 1.3 

AMu 4.5 3.3 6.2 1.4 3.5 5.7 1.3 

AMm 4.5 3.3 6.2 1.4 3.5 5.8 1.3 

P95s 11.1 6.4 22.1 2.0 5.7 11.1 1.9 

P95u 10.0 6.4 15.3 1.6 6.8 13.7 1.5 

P95m 10.0 6.5 15.6 1.6 6.9 14.5 1.5 

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

50

Normalized Inhalation Mass Exposure)

Normal Quantiles

Figure 9c 
Timothy Leighton, EPA 

15 March, 2011 

Page 103 

Quantile plot normalized inhalation mass exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

1.5

2.0

2.5

3.0

3.5

4.0

log (Normalized Inhalation Mass Exposure)

Normal Quantiles

Figure 10c
Timothy Leighton, EPA 

15 March, 2011 

Page 104 

Quantile plot normalized 100um mass exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

5

10

15

20

25

30

35

40

45

Normalized 100um Mass Exposure)

Normal Quantiles

Figure 11c 
Timothy Leighton, EPA 

15 March, 2011 

Page 105 

Quantile plot normalized 100um mass exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

log (Normalized 100um Mass Exposure)

Normal Quantiles

Figure 12c 
Timothy Leighton, EPA 

15 March, 2011 

Page 106 

Quantile plot normalized 10um mass exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

Normalized 10um Mass Exposure)

Normal Quantiles

Figure 13c 
Timothy Leighton, EPA 

15 March, 2011 

Page 107 

Quantile plot normalized 10um mass exposure data with a lognormal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

log (Normalized 10um Mass Exposure)

Normal Quantiles

Figure 14c 
Timothy Leighton, EPA 

15 March, 2011 

Page 108 

Quantile plot normalized 2.5um mass exposure data with a normal distribution

Normalized by Pounds Active Ingredient Handled

-2 -1 0 1 2

0

2

4

6

8

10

12

Normalized 2.5um Mass Exposure)

Normal Quantiles

Figure 15c 
Timothy Leighton, EPA 

15 March, 2011 

Page 109 

Quantile plot normalized 2.5um 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

log (Normalized 100um Mass Exposure)

Normal Quantiles

Figure 16c 
Timothy Leighton, EPA 

15 March, 2011 

Page 110 

Table 13c. 95 percent confidence intervals for the slope of log inhalation exposure mass versus log pounds active 

ingredient handled. 

Exposure Route Model Estimate Lower Upper

Confidence Interval 

Width 

Inhalation mass 

(mg) Mixed 1.55 1.02 2.08 1.05 

 Simple 

Linear 1.55 1.03 2.07 1.04 

lninh5 = 6.3267 +1.5491 lnnrm

N 

18 

Rsq 

0.7136

AdjRsq

0.6957

RMSE 

0.4338

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)

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

log Pounds Active Ingredient Handled

-6.8 -6.6 -6.4 -6.2 -6.0 -5.8 -5.6 -5.4 -5.2

Figure 22c 
Timothy Leighton, EPA 

15 March, 2011 

Page 111 

Figure 28c 

Table 14c. Quadratic mixed models with 95% confidence intervals for the log inhalation mass exposure versus log 

pounds active ingredient handled. 

Exposure Parameter Estimate Degrees 

of 

Freedom 

Lower 

Bound 

Upper 

Bound 

GSD ICC Width of 

Confidence 

Interval 

Inhalation 

mass 

Intercept -18.51 2.00 -106.93 69.92 1.53 0.00 176.85 

Inhalation 

mass

Slope -6.83 13.00 -21.79 8.13 1.53 0.00 29.92 

Inhalation 

mass

Quad -0.70 13.00 -1.96 0.55 1.53 0.00 2.51 
Timothy Leighton, EPA 

15 March, 2011 

Page 112 

Table 15c. Threshold values for the amount of active ingredient. 

Exposure 

Route Model Slope 

Threshold Level (lb active 

ingredient) 

Inhalation 

mass (mg) 

Mixed 1.55 0.00295* 

*For this case, slope > 1 and so the normalized exposure mixed model under-predicts exposure for pounds of active 

ingredient above the threshold. 

Figure 34c