MEMORANDUM
To:
Timothy Leighton, EPA; Timothy Dole, EPA; Bob Ross, Summitec Inc.; Andrew Yin, Summitec Inc.
From:
Jonathan Cohen, ICF International, Inc.
Date:
December 22, 2015
Re:
Contract No.: EP-W-11-014  TAF 4-08-5: 
CCA Wood Pressure Treatment Exposure Study Statistical Review

Contents
1.	Introduction and Summary	5
2.	Summary statistics of normalized exposure	8
Bootstrap simulation from lognormal mixed model (includes random site effect)	18
Bootstrap simulation from lognormal model (does not include site effect)	19
3.	Fold relative accuracy or K-factor	19
4.	Detailed summary statistics with parametric confidence intervals and fold relative accuracy	20
5.	Quantile plots	26
6.	Log-log-linearity analyses and estimated log-log slopes	47
7.	Threshold Analyses	63


Table 1. Summary statistics for hands only and total dermal iAs exposure using the empirical sampling model.	8
Table 2. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration iAs exposure using the empirical sampling model.	9
Table 3. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration Cr6 exposure using the empirical sampling model.	9
Table 4. Summary statistics for hands only and total dermal iAs normalized exposure per weighted percentage iAs using the empirical sampling model.	10
Table 5. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration iAs normalized exposure per weighted percentage iAs using the empirical sampling model.	10
Table 6. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration Cr6 normalized exposure per weighted percentage Cr6 using the empirical sampling model.	11
Table 7. Statistical tests comparing different jobs.	12
Table 8. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for iAs exposure.	14
Table 9. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for iAs exposure normalized by the weighted percentage iAs.	15
Table 10. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for Cr6 exposure.	16
Table 11. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for Cr6 exposure normalized by the weighted percentage Cr6.	17
Table 12. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the hands only iAs exposure (mg) for each job.	20
Table 13. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the total dermal iAs exposure (mg) for each job.	20
Table 14. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration iAs exposure (mg/m[3]) for each job.	21
Table 15. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose iAs exposure (mg) for each job.	21
Table 16. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration iAs exposure (mg/m[3]) for each job.	22
Table 17. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the hands only normalized iAs exposure (mg/%iAs) for each job.	22
Table 18. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the total dermal normalized iAs exposure (mg/%iAs) for each job.	22
Table 19. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration normalized iAs exposure ((mg/m[3])/%iAs) for each job.	23
Table 20. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose normalized iAs exposure (mg/%iAs) for each job.	23
Table 21. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average  concentration normalized iAs exposure ((mg/m[3])/%iAs) for each job.	23
Table 22. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration Cr6 exposure (mg/m[3]) for each job.	24
Table 23. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose Cr6 exposure (mg) for each job.	24
Table 24. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration Cr6 exposure (mg/m[3]) for each job.	24
Table 25. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration  normalized Cr6 exposure ((mg/m[3])/%Cr6) for each job.	25
Table 26. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose normalized Cr6 exposure (mg/%Cr6) for each job.	25
Table 27. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration  normalized Cr6 exposure ((mg/m[3])/%Cr6) for each job.	25
Table 28. SAS Methods for Computing the Fixed Effects Denominator Degrees of Freedom in PROC MIXED.	48
Table 29. 95 percent confidence intervals for the slope of log exposure versus log weighted percentage of active ingredient handled for iAs data.	49
Table 30. 95 percent confidence intervals for the slope of log exposure versus log weighted percentage of active ingredient handled for Cr6 data.	50
Table 31. Threshold values for the weighted percentage of active ingredient handled for iAs data.	64
Table 32. Threshold values for the weighted percentage of active ingredient handled for Cr6 data.	65

Figure 1. Quantile plot of iAs dermal exposure data with a normal distribution.	27
Figure 2. Quantile plot of iAs dermal exposure data with a lognormal distribution.	27
Figure 3. Quantile plot of iAs inhalation time-weighted average concentration exposure data with a normal distribution.	28
Figure 4. Quantile plot of iAs inhalation time-weighted average concentration exposure data with a lognormal distribution.	28
Figure 5. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data with a normal distribution.	29
Figure 6. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data with a lognormal distribution.	30
Figure 7. Quantile plot of iAs dermal exposure data normalized by the weighted percentage of iAs with a normal distribution.	31
Figure 8. Quantile plot of iAs dermal exposure data normalized by the weighted percentage of iAs with a lognormal distribution.	31
Figure 9. Quantile plot of iAs inhalation time-weighted average concentration exposure data normalized by the weighted percentage of iAs with a normal distribution.	32
Figure 10. Quantile plot of iAs inhalation time-weighted average concentration exposure data normalized by the weighted percentage of iAs with a lognormal distribution.	33
Figure 11. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data normalized by the weighted percentage of Cr6 with a normal distribution.	33
Figure 12. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data normalized by the weighted percentage of Cr6 with a lognormal distribution.	34
Figure 13. Q-Q plot of studentized residuals using a normal mixed model. Dermal iAs exposure for TOs.	35
Figure 14. Q-Q plot of studentized residuals using a lognormal mixed model. Dermal iAs exposure for TOs.	36
Figure 15. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs.	37
Figure 16. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs.	38
Figure 17. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs.	39
Figure 18. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs.	40
Figure 19. Q-Q plot of studentized residuals using a normal mixed model. Dermal iAs exposure for TOs normalized by the weighted percentage iAs.	41
Figure 20. Q-Q plot of studentized residuals using a lognormal mixed model. Dermal iAs exposure for TOs normalized by the weighted percentage iAs.	42
Figure 21. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs normalized by the weighted percentage of iAs.	43
Figure 22. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs normalized by the weighted percentage of iAs.	44
Figure 23. Q-Q plot of studentized residuals using a normal mixed model.  Inhalation time-weighted average concentration Cr6 exposure for TOs normalized by the weighted percentage of Cr6.	45
Figure 24. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs normalized by the weighted percentage of Cr6.	46
Figure 25. Mixed model regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job TO.	51
Figure 26. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job LO.	52
Figure 27. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job TO.	53
Figure 28. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job WH.	54
Figure 29. Mixed model regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job TO.	55
Figure 30. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job LO.	56
Figure 31. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job TO.	57
Figure 32. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job WH.	58
Figure 33. Mixed model regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job TO.	59
Figure 34. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job LO.	60
Figure 35. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job TO.	61
Figure 36. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job WH.	62
Figure 37. Lognormal mixed model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	65
Figure 38. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job LO.	66
Figure 39. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	67
Figure 40. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job WH.	68
Figure 41. Lognormal mixed model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	69
Figure 42. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job LO.	70
Figure 43. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	71
Figure 44. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job WH.	72
Figure 45. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	73
Figure 46. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job LO.	74
Figure 47. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	75
Figure 48. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job WH.	76
Figure 49. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	77
Figure 50. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job LO.	78
Figure 51. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	79
Figure 52. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job WH.	80
Figure 53. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	81
Figure 54. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job LO.	82
Figure 55. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job TO.	83
Figure 56. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job WH.	84
Figure 57. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	85
Figure 58. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.	86
Figure 59. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	87
Figure 60. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.	88
Figure 61. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	89
Figure 62. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.	90
Figure 63. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	91
Figure 64. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.	92
Figure 65. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	93
Figure 66. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.	94
Figure 67. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.	95
Figure 68. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.	96

 Introduction and Summary
The CCA Wood Pressure Treatment Exposure Study was completed in May 2015. The study report (MRID 49640101) was titled "Assessment of Potential Dermal and Inhalation Exposure to Inorganic Arsenic and Hexavalent Chromium Associated with Pressure-Treatment of Wood with Chromated Copper Arsenate (CCA) Wood Preservatives." ICF International, Inc. (ICF) was asked by EPA through Summitec Inc. to analyze the data from this study to investigate the relationship between dermal and inhalation exposures and the amount of inorganic arsenic ("iAs") and hexavalent chromium ("Cr6") used. The wood preservative Chromated Copper Arsenate (CCA) was applied in an aqueous solution containing approximately 3% CCA to wood products in an observational study of 19 volunteer workers. 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 study report describes the observational study methodology and the measurements in detail. Briefly, the study was carried out at six US pressure-treatment facilities identified by the letters A, B, C, D, E and F. 19 workers applied the treatment solution to wood poles, posts, plywood, and/or cut dimensional lumber products as part of their normal work activities. Each worker was monitored during their work shift, the monitoring event (ME), which ranged from about 7.2 to 9.6 hours. The ME sampling time excludes the time spent on meal, smoking and rest breaks. The ME sampling times differed for the dermal exposure measurements of iAs and the inhalation exposure measurements of iAs and Cr6. 
The 19 test subjects represented three weakly delineated job classes, determined by the study investigators. Treating System Operators' ("TOs") duties are to operate the treating system. Loader Operators' ("LOs") duties are to operate forklifts or other self-propelled loading equipment at least partially isolated from the treating area. Wood Handlers' ("WHs") duties are to directly handle the wood and treating equipment on the drip pad before and after treatment. As noted in the study report, treaters in any of these traditional job categories are likely to perform certain tasks associated with others. Over the course of this study, 9 TOs, 5 LOs, and 5 WHs were monitored, each for one worker shift.
Each subject was given a fresh cotton inner whole-body dosimeter to wear under a fresh cotton long-sleeved shirt, fresh cotton long pants, and a fresh leather work belt. An outer whole-body dosimeter was not worn. Each subject also wore a fresh pair of polypropylene work socks under their regular work shoes, but exposures to feet were not measured. Gloves were not worn. Each subject was also given a personal air-sampling pump worn on their belt, connected to an air filter used to measure breathing zone arsenic concentrations. Each subject was also given a second personal air-sampling pump worn on the opposite side of their belt, connected to an air filter used to measure breathing zone hexavalent chromium concentrations. The whole-body dosimeters were cut into sections for different body parts. The air sampling cassettes, face and neck pads, hand wash samples, and the sectioned whole body dosimeters were collected and were later analyzed by the laboratory to measure the mass of iAs for dermal and inhalation exposure and the mass of Cr6 for inhalation exposure only. Measurements were made for each worker on each day.
The statistical analyses presented in this memorandum are for Hands Only, Total Dermal, and the Inhalation Concentration, Dose, and 8-Hour Time-Weighted Average Concentration:
Hands (mg). iAs mass (mg) from the hand washes. Measured for iAs only.
Total Dermal (mg). Total iAs mass (mg) from the inner whole body dosimeters, face and neck pads, and hand washes. Measured for iAs only.
Inhalation Concentration (mg/m[3]). iAs or Cr6 concentration in air measured on the air filter. Calculated as air filter mass (mg) divided by m[3] of pumped air (sample length in minutes x average flow rate in m[3] per minute).
Inhalation Dose (mg). iAs or Cr6 concentration in air measured on the air filter (mg/m[3]) x breathing rate for moderate activity (1.5 m[3] per hour) x sample length (hours).
Inhalation 8-Hour Time-Weighted Average (TWA) Concentration (mg/m[3]).  For MEs with a sample length of at most 8 hours: Inhalation Concentration (mg/m[3]) x Sample length (minutes) / 480 minutes. For MEs with a sample length of more than 8 hours: Inhalation Concentration (mg/m[3]).
The dermal and inhalation exposure mass were adjusted for field recovery. The Cr6 inhalation exposure mass on the air filter was adjusted for storage stability (87.9%).
For these analyses, two small corrections were made to the data due to errors found in the Excel spreadsheet supplied by the study director. For the Cr6 inhalation mass measured on MEs 2 and 3 from site A, the wrong field spike adjustment factors of 82.3% and 82.7% respectively were applied instead of the average site A adjustment factor 78.1%. This was corrected in the data analyses reported here. For the Cr6 inhalation sampling duration for ME 2 from site D, the Cr6 air pump was not working correctly for the first 97 minutes of the total sampling time of 474 minutes and was replaced. (See footnote 2 of study Table XV). Fortunately, no CCA was being used at that time. The study report said that this issue was addressed by subtracting 97 minutes from the total air sampling time for that case (only for Cr6 on D2) but this was not properly implemented in the Excel spreadsheet, since nothing was subtracted. This was also corrected in the data analyses reported here.
In this memorandum we present the analysis of the unit or normalized exposure defined as the dermal or inhalation exposure divided either by one (1) or by the weighted percentage of active ingredient handled. The exposure divided by one is, of course, the same as the un-normalized exposure. The weighted percentage of active ingredient is the total volume of iAs or Cr6 used (summed over the wood charges) divided by the total volume of treatment solution used (summed over the wood charges), and multiplied by 100. The weighted percentage is a number between 0 and 100.
Estimates of the arithmetic and geometric means and standard deviations as well as the 95[th] percentile are computed using the empirical data for each job, a lognormal model for each job, and a lognormal mixed model for the job TO only. Because the exposure is expected to vary with the job, the main analyses reported here fit separate models for the three jobs. The empirical model calculates statistics for all the unit exposure measurements for a job assuming the data are statistically independent with a possibly different, but unspecified, distribution for each job. The lognormal model, denoted as model d in the SAS code and as model u in this memorandum, calculates statistics for all the unit exposure measurements for a job assuming the data are statistically independent with a possibly different lognormal distribution for each job; no site effect is assumed. The lognormal mixed model, denoted as model c in the SAS code and as model m in this memorandum, is fitted to the 9 exposure measurements for the job TO, and assumes that the logarithm of the exposure has a normal distribution with a constant mean, a random site effect, and a residual error. The random site effect is a random value associated with each of the 6 sites (A, B, C, D, E, and F) that represents the possibility of clustering or association between the measurements made at the same site, expressing the possibility that some facilities or times of the year tend to produce higher exposures. The lognormal mixed model could not be fitted to the measurements for jobs LO and WH because each site had no more than one LO and no more than one WH, so that variability between sites could not be distinguished from the between-worker variability that is the residual error.
Although not presented in this memorandum, the SAS program provided as an attachment calculates results for two additional models. The first additional model (denoted as model b in the SAS code) combines the data from the three jobs and fits a lognormal mixed model with a random site effect where the means and variances are the same for all three jobs. The second additional model (denoted as model a in the SAS code) combines the data from the three jobs and fits a lognormal mixed model where the mean (intercept) depends upon the job, but the variances of the random site effect and residual error are the same for all three jobs. The model denoted as model c in the SAS code is the lognormal mixed model m presented in this memorandum. The model denoted as model d in the SAS code is the lognormal model u presented in this memorandum.
For each summary statistic we present confidence intervals. We also compute the fold relative accuracy of the summary statistics and compare with the benchmark of 3-fold accuracy, which was not met for most of the models used. To evaluate the statistical models we present quantile-quantile plots of the data to evaluate the fit of lognormal distributions to the data. We also used quantile-quantile plots of the studentized residuals from the lognormal mixed models to evaluate the performance of those models.
The statistical models for the normalized exposure, normalized by the weighted percentage of active ingredient, assume that the mean value of the logarithm of the exposure is equal to an intercept plus the slope times the logarithm of the weighted fraction of active ingredient used, where the slope equals 1. To test this "log-log-linearity" assumption, a linear model with a slope term was fitted to the data 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 weighted percentage of active ingredient used.  We applied this test to the dermal and inhalation exposures using the lognormal mixed model m and the lognormal model u. Results for quadratic regression models are not reported here but were calculated in the provided SAS code.
The results for model m show that the estimated intra-cluster correlation (ICC) coefficient for the site effect for TOs was non-zero for iAs and Cr6 inhalation exposures, which implies that there are some site effects on inhalation exposure, and therefore, in particular, differences between exposures at different sites and/or times of the year, as might be expected. The ICC was zero for iAs dermal exposures. We do not have data that will allow us to determine if there are site effects for the other two jobs.
The linear model results for the dermal and inhalation exposure versus the weighted percentage of active ingredient show positive slopes in about 80% of the cases. The slopes are all less than one for the job TO but are either less than or greater than 1 for the other two jobs. The confidence intervals for the slopes are wide. The exposure models in all but one case do not reject log-log-linearity (slope equals one) and do not reject independence (slope equals zero) at the 5% significance level. These findings suggest that for this scenario, exposure estimates could be based only on the exposure route and job, and not on the weighted percentages of active ingredient handled. In other words the un-normalized exposure estimates given below (e.g., in  Table 8 and Table 10) could be used instead of the exposure estimates normalized by the weighted percentage of active ingredient (e.g., in Table 9 and Table 11). This conclusion assumes that all CCA wood treatment workers use similar amounts of active ingredient to the workers in this study and might change if more data becomes available. 
Summary statistics of normalized exposure
Table 1 to Table 6 summarize the normalized exposure data (per one or per weighted percentage of active ingredient handled) with the summary statistics from the 19 measurements for each dermal and inhalation exposure route, grouped by job, including the case where all three jobs are combined. These analyses assume that the exposure measurements for a given job come from some unspecified distribution. The normalized exposure "per one" is, of course, the same as the un-normalized exposure. 
Table 1. Summary statistics for hands only and total dermal iAs exposure using the empirical sampling model.
Statistic
Hands only (mg)
Total Dermal (mg)

                                      LO
                                      TO
                                      WH
                                      All
                                      LO
                                      TO
                                      WH
                                      All
Arithmetic Mean
0.4038
0.2636
0.4026
0.3371
0.9910
0.6466
1.8921
1.0650
Arithmetic Standard Deviation
0.4329
0.4410
0.2232
0.3799
0.9977
0.7436
1.0934
1.0059
Geometric Mean
0.1711
0.1019
0.3480
0.1613
0.5247
0.3932
1.6989
0.6235
Geometric Standard Deviation
5.7224
4.1711
1.8828
4.0297
4.2509
2.8390
1.6359
3.2603
Min
0.0229
0.0125
0.1411
0.0125
0.0737
0.1237
1.0336
0.0737
 5%
0.0229
0.0125
0.1411
0.0125
0.0737
0.1237
1.0336
0.0737
10%
0.0229
0.0125
0.1411
0.0229
0.0737
0.1237
1.0336
0.1237
25%
0.0311
0.0424
0.2708
0.0424
0.1961
0.1452
1.2862
0.1961
50%
0.3634
0.1031
0.3322
0.1686
0.7961
0.3474
1.6117
0.7961
75%
0.5282
0.1686
0.6111
0.5282
1.3735
0.7313
1.7454
1.6117
90%
1.0735
1.3851
0.6576
1.0735
2.5154
2.4466
3.7837
2.5154
95%
1.0735
1.3851
0.6576
1.3851
2.5154
2.4466
3.7837
3.7837
Max
1.0735
1.3851
0.6576
1.3851
2.5154
2.4466
3.7837
3.7837
Table 2. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration iAs exposure using the empirical sampling model.
Statistic
Inhalation Concentration  (mg/m[3])
Inhalation Dose  (mg)
Inhalation 8-hour Time Weighted Average  (mg/m[3])

LO
TO
WH
All
LO
TO
WH
All
LO
TO
WH
All
Arithmetic Mean
0.00071
0.00180
0.00348
0.00195
0.00717
0.01902
0.03805
0.02091
0.00060
0.00158
0.00317
0.00174
Arithmetic Standard Deviation
0.00050
0.00106
0.00509
0.00272
0.00496
0.01216
0.05778
0.03081
0.00041
0.00102
0.00482
0.00257
Geometric Mean
0.00049
0.00147
0.00160
0.00113
0.00486
0.01552
0.01590
0.01151
0.00041
0.00129
0.00132
0.00096
Geometric Standard Deviation
3.30208
2.07657
3.88385
3.03325
3.39360
2.04280
4.28709
3.14025
3.39360
2.06883
4.28709
3.14890
Min
0.00007
0.00033
0.00035
0.00007
0.00062
0.00419
0.00319
0.00062
0.00005
0.00033
0.00027
0.00005
 5%
0.00007
0.00033
0.00035
0.00007
0.00062
0.00419
0.00319
0.00062
0.00005
0.00033
0.00027
0.00005
10%
0.00007
0.00033
0.00035
0.00033
0.00062
0.00419
0.00319
0.00319
0.00005
0.00033
0.00027
0.00027
25%
0.00049
0.00114
0.00079
0.00058
0.00499
0.01237
0.00673
0.00605
0.00042
0.00103
0.00056
0.00050
50%
0.00058
0.00161
0.00123
0.00123
0.00605
0.01586
0.01182
0.01237
0.00050
0.00132
0.00099
0.00103
75%
0.00111
0.00236
0.00256
0.00236
0.01137
0.02460
0.02855
0.02460
0.00095
0.00205
0.00238
0.00205
90%
0.00129
0.00358
0.01247
0.00358
0.01280
0.03971
0.13994
0.03971
0.00107
0.00331
0.01166
0.00331
95%
0.00129
0.00358
0.01247
0.01247
0.01280
0.03971
0.13994
0.13994
0.00107
0.00331
0.01166
0.01166
Max
0.00129
0.00358
0.01247
0.01247
0.01280
0.03971
0.13994
0.13994
0.00107
0.00331
0.01166
0.01166

Table 3. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration Cr6 exposure using the empirical sampling model. 
Statistic
Inhalation Concentration  (mg/m[3])
Inhalation Dose  (mg)
Inhalation 8-hour Time Weighted Average  (mg/m[3])

LO
TO
WH
All
LO
TO
WH
All
LO
TO
WH
All
Arithmetic Mean
0.000264
0.000486
0.000760
0.000500
0.002670
0.004978
0.008218
0.005223
0.000223
0.000413
0.000685
0.000435
Arithmetic Standard Deviation
0.000157
0.000300
0.000870
0.000498
0.001610
0.003069
0.009932
0.005569
0.000134
0.000257
0.000828
0.000464
Geometric Mean
0.000223
0.000413
0.000433
0.000356
0.002232
0.004236
0.004289
0.003590
0.000186
0.000351
0.000357
0.000298
Geometric Standard Deviation
1.979800
1.813005
3.432442
2.282976
2.027565
1.812857
3.811521
2.379326
2.027565
1.822533
3.811521
2.381840
Min
0.000093
0.000215
0.000113
0.000093
0.000878
0.001944
0.001043
0.000878
0.000073
0.000162
0.000087
0.000073
5%
0.000093
0.000215
0.000113
0.000093
0.000878
0.001944
0.001043
0.000878
0.000073
0.000162
0.000087
0.000073
10%
0.000093
0.000215
0.000113
0.000113
0.000878
0.001944
0.001043
0.001043
0.000073
0.000162
0.000087
0.000087
25%
0.000131
0.000243
0.000144
0.000215
0.001320
0.002726
0.001232
0.001944
0.000110
0.000216
0.000103
0.000162
50%
0.000270
0.000312
0.000573
0.000312
0.002809
0.003630
0.005515
0.003538
0.000234
0.000303
0.000460
0.000295
75%
0.000357
0.000743
0.000730
0.000730
0.003538
0.008124
0.008144
0.008124
0.000295
0.000677
0.000679
0.000677
90%
0.000470
0.000993
0.002241
0.000993
0.004806
0.010327
0.025158
0.010327
0.000400
0.000861
0.002097
0.000861
95%
0.000470
0.000993
0.002241
0.002241
0.004806
0.010327
0.025158
0.025158
0.000400
0.000861
0.002097
0.002097
Max
0.000470
0.000993
0.002241
0.002241
0.004806
0.010327
0.025158
0.025158
0.000400
0.000861
0.002097
0.002097

Table 4. Summary statistics for hands only and total dermal iAs normalized exposure per weighted percentage iAs using the empirical sampling model.
Statistic
Hands only (mg/%iAs)
Total Dermal (mg/%iAs)

LO
TO
WH
All
LO
TO
WH
All
Arithmetic Mean
0.6475
0.4420
0.7459
0.5761
1.5973
0.9822
2.9849
1.6711
Arithmetic Standard Deviation
0.7449
0.9504
0.4597
0.7682
1.6741
1.6105
0.4858
1.5957
Geometric Mean
0.3083
0.1333
0.6042
0.2474
0.9457
0.5144
2.9499
0.9561
Geometric Standard Deviation
4.6170
4.4644
2.1847
4.2254
3.5106
2.8750
1.1929
3.2618
Min
0.0489
0.0105
0.2264
0.0105
0.1577
0.1214
2.2104
0.1214
 5%
0.0489
0.0105
0.2264
0.0105
0.1577
0.1214
2.2104
0.1214
10%
0.0489
0.0105
0.2264
0.0489
0.1577
0.1214
2.2104
0.1577
25%
0.0823
0.0806
0.3018
0.0823
0.5191
0.3383
2.8271
0.3502
50%
0.5324
0.1040
0.8792
0.2264
1.3844
0.4584
3.1643
0.8412
75%
0.6908
0.1410
1.0720
0.8792
1.5134
0.7370
3.3180
3.1643
90%
1.8831
2.9621
1.2501
1.8831
4.4121
5.2323
3.4046
4.4121
95%
1.8831
2.9621
1.2501
2.9621
4.4121
5.2323
3.4046
5.2323
Max
1.8831
2.9621
1.2501
2.9621
4.4121
5.2323
3.4046
5.2323

Table 5. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration iAs normalized exposure per weighted percentage iAs using the empirical sampling model.
Statistic
Inhalation Concentration  ((mg/m3)/%iAs)
Inhalation Dose  (mg/%iAs)
Inhalation 8-hour Time Weighted Average  ((mg/m3)/%iAs)

LO
TO
WH
All
LO
TO
WH
All
LO
TO
WH
All
Arithmetic Mean
0.00144
0.00246
0.00413
0.00263
0.01449
0.02511
0.04425
0.02735
0.00121
0.00209
0.00369
0.00228
Arithmetic Standard Deviation
0.00134
0.00161
0.00387
0.00243
0.01328
0.01526
0.04477
0.02675
0.00111
0.00128
0.00373
0.00223
Geometric Mean
0.00088
0.00193
0.00279
0.00173
0.00876
0.02030
0.02761
0.01764
0.00073
0.00168
0.00230
0.00147
Geometric Standard Deviation
3.52511
2.20079
2.83256
2.81096
3.60428
2.08892
3.13780
2.85248
3.60428
2.10907
3.13780
2.85994
Min
0.00014
0.00058
0.00074
0.00014
0.00132
0.00735
0.00683
0.00132
0.00011
0.00058
0.00057
0.00011
 5%
0.00014
0.00058
0.00074
0.00014
0.00132
0.00735
0.00683
0.00132
0.00011
0.00058
0.00057
0.00011
10%
0.00014
0.00058
0.00074
0.00050
0.00132
0.00735
0.00683
0.00503
0.00011
0.00058
0.00057
0.00042
25%
0.00050
0.00106
0.00138
0.00085
0.00503
0.01241
0.01181
0.00770
0.00042
0.00103
0.00098
0.00064
50%
0.00102
0.00255
0.00325
0.00211
0.01061
0.03022
0.03130
0.02161
0.00088
0.00252
0.00261
0.00180
75%
0.00211
0.00424
0.00487
0.00424
0.02161
0.03391
0.05427
0.03391
0.00180
0.00283
0.00452
0.00283
90%
0.00342
0.00450
0.01043
0.00487
0.03389
0.04675
0.11703
0.05427
0.00282
0.00390
0.00975
0.00452
95%
0.00342
0.00450
0.01043
0.01043
0.03389
0.04675
0.11703
0.11703
0.00282
0.00390
0.00975
0.00975
Max
0.00342
0.00450
0.01043
0.01043
0.03389
0.04675
0.11703
0.11703
0.00282
0.00390
0.00975
0.00975

Table 6. Summary statistics for inhalation concentration, dose, and 8-hour time-weighted average concentration Cr6 normalized exposure per weighted percentage Cr6 using the empirical sampling model.
Statistic
Inhalation Concentration  ((mg/m[3])/%Cr6)
Inhalation Dose  (mg/%Cr6)
Inhalation 8-hour Time Weighted Average  ((mg/m[3])/%Cr6)

LO
TO
WH
All
LO
TO
WH
All
LO
TO
WH
All
Arithmetic Mean
0.000465
0.000582
0.000938
0.000645
0.004686
0.005929
0.009881
0.006642
0.000391
0.000492
0.000823
0.000553
Arithmetic Standard Deviation
0.000343
0.000452
0.000687
0.000506
0.003466
0.004679
0.007671
0.005451
0.000289
0.000391
0.000639
0.000455
Geometric Mean
0.000350
0.000476
0.000654
0.000477
0.003503
0.004883
0.006484
0.004821
0.000292
0.000404
0.000540
0.000401
Geometric Standard Deviation
2.451465
1.892866
2.888825
2.259051
2.484504
1.848430
3.192942
2.314325
2.484504
1.853049
3.192942
2.316114
Min
0.000118
0.000216
0.000201
0.000118
0.001191
0.002337
0.001806
0.001191
0.000099
0.000195
0.000150
0.000099
 5%
0.000118
0.000216
0.000201
0.000118
0.001191
0.002337
0.001806
0.001191
0.000099
0.000195
0.000150
0.000099
10%
0.000118
0.000216
0.000201
0.000166
0.001191
0.002337
0.001806
0.001570
0.000099
0.000195
0.000150
0.000131
25%
0.000166
0.000315
0.000211
0.000216
0.001570
0.003475
0.001864
0.002337
0.000131
0.000290
0.000155
0.000195
50%
0.000396
0.000434
0.001242
0.000434
0.004118
0.003997
0.013061
0.004118
0.000343
0.000315
0.001088
0.000343
75%
0.000799
0.000645
0.001357
0.000846
0.008174
0.006173
0.013851
0.008379
0.000681
0.000514
0.001154
0.000698
90%
0.000846
0.001689
0.001677
0.001677
0.008379
0.017565
0.018825
0.017565
0.000698
0.001464
0.001569
0.001464
95%
0.000846
0.001689
0.001677
0.001689
0.008379
0.017565
0.018825
0.018825
0.000698
0.001464
0.001569
0.001569
Max
0.000846
0.001689
0.001677
0.001689
0.008379
0.017565
0.018825
0.018825
0.000698
0.001464
0.001569
0.001569

The summary statistics in Table 1 to Table 6 show that the geometric means for the exposures and the exposures per weighted percentage of active ingredient were consistently highest for WHs. A similar, but less consistent pattern was found in the arithmetic means.
To evaluate the exposure differences between the jobs, an analysis of variance was used to test for statistically significant differences. The following statistical mixed model, fitted separately for each exposure route and normalizing variable was used for this test:
      Log(normalized exposure) = job + site + error
For this model, the (fixed effect) job variable is one of the three jobs. Thus the geometric mean depends upon the job. The site term is a random effect that represents the correlation between multiple measurements from the same site; each site has its own random exposure effect that is added to the overall job mean (after taking logarithms). The site effects are drawn from a normal distribution with mean zero. Each of the measurements also has a random error term drawn from another normal distribution with mean zero. This mixed model corresponds to model a in the SAS code and is not the same as the lognormal model fitted separately to each job (model d in the SAS code) used elsewhere. 
This model was fitted to the data and then used to test for any difference between the three jobs using a pair of statistical contrasts to test if the three geometric means are equal. 
The results are shown in Table 7. P-values at or below 0.05 indicate statistically significant differences at the 5 percent significance level. All of the P-values for hands only exposure and for inhalation concentration, dose and TWA were not statistically significant, indicating that the geometric mean normalized exposures for the three jobs are not different. The differences were statistically significant at the 10% level but not the 5% level for total dermal iAs exposure normalized by one. The differences were statistically significant at the 5% level for total dermal iAs exposure normalized by the weighted percentage of iAs. Although the job effects were not statistically significant in most cases, to make these analyses consistent with the previous statistical analyses of wood treated with DDA, separate statistical mixed models were fitted to each job in order to calculate job-specific unit exposure values. Note that the Brown-Forsyth test described in the study report gives a robust comparison of the variances for the three jobs but does not test whether the job means or geometric means are the same.
Table 7. Statistical tests comparing different jobs.
Active Ingredient
Exposure Route
Normalizing Variable
P-Value
iAs
Hands
One
0.3125
iAs
Dermal
One
0.0835
iAs
Inhalation Concentration
One
0.1596
iAs
Inhalation Dose
One
0.1576
iAs
Inhalation Time Weighted Average
One
0.1615
Cr6
Inhalation Concentration
One
0.3925
Cr6
Inhalation Dose
One
0.4240
Cr6
Inhalation Time Weighted Average
One
0.4286
iAs
Hands
Weighted Percentage iAs
0.1731
iAs
Dermal
Weighted Percentage iAs
0.0270
iAs
Inhalation Concentration
Weighted Percentage iAs
0.1777
iAs
Inhalation Dose
Weighted Percentage iAs
0.1711
iAs
Inhalation Time Weighted Average
Weighted Percentage iAs
0.1747
Cr6
Inhalation Concentration
Weighted Percentage Cr6
0.4293
Cr6
Inhalation Dose
Weighted Percentage Cr6
0.4577
Cr6
Inhalation Time Weighted Average
Weighted Percentage Cr6
0.4598

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 for a given job. Let Z95 be the 95[th] percentile of a standard normal distribution, approximately 1.645. All three models are fitted separately to the X and Y data from each job, so that all the statistics depend upon the job. 
Empirical simple random sampling model. Code "s." This model was separately fitted to all three jobs. Assumes that all the values of X were randomly drawn from an unspecified distribution. Ignores within-site correlations. Gives empirical estimates such as in Tables 1 to 6 above.
Y = LnGM + Error. Error is independent and identically distributed with mean 0 and some variance 
AMs = Arithmetic mean of X values
GMs = Geometric mean of X values = exp(LnGM) 
GSDs = Geometric standard deviation of X values 
P95s = 95[th] percentile of X values 
Lognormal mixed model. Code "m." This model was only fitted to the job TO because the site effect could not be estimated for the other two jobs. (For the jobs LO and WH, only one worker was monitored at each site, so that site differences cannot be distinguished from worker differences.) Assumes that the site random effects were independently randomly drawn from a normal distribution and that the random error terms were independently drawn from another normal distribution. The error term for each site and exposure measurement is the sum of the site effect for that site and the within-site random error term. 
Y = LnGM + Site + Error. Site is normally distributed with mean 0, variance Vs, and standard deviation Ss = √Vs. Error is normally distributed with mean 0, variance Vr, and standard deviation Sr = √Vr. Define V = Vs + Vr and S = √V. V is the variance of Y, and S is the standard deviation of Y. All the parameters depend upon the job.
ICC = Intra-site correlation coefficient = Vs/V. 
AMm = Modeled arithmetic mean of X values = exp(LnGM) exp((1/2) V). 
GMm = Modeled geometric mean of X values = exp(LnGM). 
GSDm = Modeled geometric standard deviation of X values = exp(S).
P95m = Modeled 95[th] percentile of X values = exp(LnGM) exp(Z95xS). 
Lognormal model. Code "u." This model was separately fitted to all three jobs. Assumes that all the values of X were independently randomly drawn from a normal distribution. Ignores within-site correlations. This model is a special case of the lognormal mixed model where the ICC is assumed to equal zero. 
Y = LnGM + Error. Error is normally distributed with mean 0, variance V, and standard deviation S = √V. V is the variance of Y, and S is the standard deviation of Y. All the parameters depend upon the job.
AMu = Modeled arithmetic mean of X values = exp(LnGM) exp((1/2) V). 
GMu = Modeled geometric mean of X values = exp(LnGM). This is the same as GMs.
GSDu = Modeled geometric standard deviation of X values = exp(S). This is the same as GSDs.
P95u = Modeled 95[th] percentile of X values = exp(LnGM) exp(Z95xS).
For the lognormal mixed model, the ICC value estimates the clustering effect of multiple measurements from the same site and lies between 0 (no clustering) and 1 (complete clustering and negligible within-site variation).  An ICC of 0 is when repeated measurements from the same site are uncorrelated. An ICC of 1 is when all the exposure measurements from the same site are identical.
Table 8 (iAs exposure), Table 9 (iAs exposure normalized by weighted percentage iAs), Table 10 (Cr6 exposure), and Table 11 (Cr6 exposure normalized by weighted percentage Cr6) present the arithmetic mean and 95[th] percentile estimates from the lognormal mixed model for job TO (code m) and from the lognormal model for all three jobs (code u) together with 95% confidence intervals and K-factors, for each job and exposure route. These are the values of AMm, P95m, AMu, and P95u. The exposure estimates from the lognormal mixed model and lognormal model displayed in Table 8 to Table 11 are recommended over the empirical arithmetic means and 95[th] percentiles displayed in Table 1 to Table 6. The K-factor is either equal to upper bound of confidence interval / estimate, or estimate / lower bound of confidence interval, whichever is greater. The other summary statistics are presented in more detail below.
As shown in Table 8 to Table 11, the lognormal mixed model and lognormal model estimates of arithmetic mean and 95[th] percentile mostly fail to meet the benchmark of 3-fold accuracy. For iAs exposure, the highest K-factor was 21.00 for the hands only arithmetic mean for LOs. For iAs exposure normalized by the weighted percentage iAs, the highest K-factor was 11.32, also for the hands only arithmetic mean for LOs. For Cr6 exposure, the highest K-factor was 7.09 for the inhalation dose and inhalation time-weighted average arithmetic mean for WHs. For Cr6 exposure normalized by the weighted percentage Cr6, the highest K-factor was 5.36 for the inhalation dose and inhalation time-weighted average 95[th] percentile for WHs.
Table 8 to Table 11 also show that the arithmetic mean and 95[th] percentile estimates for the job TO using the lognormal mixed model (code m) are either the same as the estimates for the job TO using the lognormal model (code u) or are very similar. The confidence intervals for the mixed model are consistently wider and have higher K-factors. The estimates will be the same in the case where the estimated ICC for the mixed model is zero, indicating no site effect, but in that case the lognormal mixed model confidence intervals are wider than the lognormal model confidence intervals because of the extra uncertainty in the ICC. Furthermore, the lognormal mixed model is only available for the job TO. For these reasons the separate lognormal model for each job is recommended.
Table 8. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for iAs exposure.
Exposure Route
Model Code
Job
Arithmetic Mean
(95% confidence interval)
Arithmetic Mean
K-Factor
95[th] Percentile
(95% confidence interval)
95[th] Percentile
K-Factor
Hands (mg)
m
TO
0.28252 (0.07835, 1.49375)
5.29
1.06746 (0.23338, 5.13463)
4.81

u
LO
0.78323 (0.08581, 16.44835)
21.00
3.01467 (0.25472, 34.40772)
11.84

u
TO
0.28252 (0.08228, 1.26965)
4.49
1.06746 (0.24641, 4.51730)
4.33

u
WH
0.42510 (0.23322, 0.81029) 
1.91
0.98530 (0.39452, 2.38118)
2.50
Dermal (mg)
m
TO
0.67777 (0.29213, 1.84169)
2.72
2.18798 (0.72051, 6.89334)
3.15

u
LO
1.49505 (0.27375, 13.68369)
9.15
5.67112 (0.73005, 42.74627)
7.77

u
TO
0.67777 (0.30296, 1.67638)
2.47
2.18798 (0.74966, 6.27749)
2.92

u
WH
1.91762 (1.22244, 3.07732)
1.60
3.81733 (1.87312, 7.58302)
2.04
Inhalation Conc (mg/m[3])
m
TO
0.00184 (0.00096, 0.00377)
2.05
0.00485 (0.00195, 0.01203)
2.48

u
LO
0.00099 (0.00027, 0.00504)
5.08
0.00346 (0.00064, 0.01835)
5.43

u
TO
0.00193 (0.00114, 0.00338)
1.75
0.00491 (0.00232, 0.01026)
2.12

u
WH
0.00403 (0.00085, 0.02993)
7.43
0.01495 (0.00210, 0.09915)
7.12
Inhalation Dose (mg)
m
TO
0.01854 (0.01006, 0.03588)
1.94
0.04711 (0.02011, 0.11176)
2.37

u
LO
0.01025 (0.000264, 0.05518)
5.38
0.03627 (0.00642, 0.19962)
5.65

u
TO
0.02002 (0.01203, 0.03456)
1.73
0.05024 (0.02413, 0.10337) 
2.08

u
WH
0.04586 (0.00835, 0.44595)
9.72
0.17425 (0.02122, 1.32652)
8.21
Inhalation Time- Weighted Average Conc (mg/m[3])
m
TO
0.00155 (0.00083, 0.00306)
1.98
0.00399 (0.00167, 0.00966)
2.42

u
LO
0.00085 (0.00022, 0.00460)
5.38
0.00302 (0.00054, 0.01664)
5.65

u
TO
0.00167 (0.00100, 0.00292)
1.75
0.00425 (0.00201, 0.00885)
2.11

u
WH
0.00382 (0.00070, 0.03716)
9.72
0.01452 (0.00177, 0.11054)
8.21

Table 9. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for iAs exposure normalized by the weighted percentage iAs.
Exposure Route
Model Code
Job
Arithmetic Mean
(95% confidence interval)
Arithmetic Mean
K-Factor
95th Percentile
(95% confidence interval)
95th Percentile
K-Factor
Hands (mg/%iAs)
m
TO
0.40817 (0.10397, 2.47410)
6.06
1.56149 (0.31757, 8.09378)
5.18

u
LO
0.99344 (0.15840, 11.24103)
11.32
3.81740 (0.43716, 32.29003)
8.73

u
TO
0.40817 (0.10943, 2.07216)
5.08
1.56149 (0.33616, 7.07741)
4.65

u
WH
0.81998 (0.37902, 1.93579)
2.36
2.18495 (0.70554, 6.49708)
3.10
Dermal (mg/%iAs)
m
TO
0.89842 (0.38187, 2.49160)
2.77
2.92215 (0.94943, 9.33509)
3.19

u
LO
2.08065 (0.50917, 12.09059)
5.81
7.46155 (1.25964, 43.05855)
5.92

u
TO
0.89842 (0.39626, 2.26231)
2.52
2.92215 (0.98832, 8.49147)
2.96

u
WH
2.99610 (2.57081, 3.49360)
1.17
3.94255 (3.05491, 5.04167)
1.29
Inhalation Conc ((mg/m[3])/%iAs)
m
TO
0.00302 (0.00141, 0.00710)
2.35
0.00863 (0.00299, 0.02450)
2.88

u
LO
0.00194 (0.00047, 0.01136)
5.87
0.00695 (0.00117, 0.04035)
5.96

u
TO
0.00263 (0.00149, 0.00487)
1.85
0.00706 (0.00314, 0.01566)
2.25

u
WH
0.00479 (0.00159, 0.01763)
3.68
0.01544 (0.00343, 0.06596)
4.51
Inhalation Dose (mg/%iAs)
m
TO
0.02902 (0.01484, 0.06069)
2.09
0.07757 (0.03068, 0.19680)
2.54

u
LO
0.01993 (0.00469, 0.12270)
6.16
0.07218 (0.01174, 0.43212)
6.15

u
TO
0.02662 (0.01571, 0.04694)
1.76
0.06817 (0.03200, 0.14348)
2.13

u
WH
0.05308 (0.01535, 0.23894)
4.50
0.18109 (0.03464, 0.89210)
5.23
Inhalation Time- Weighted Average Conc ((mg/m[3])/%iAs)
m
TO
0.00243 (0.00122, 0.00517)
2.13
0.00656 (0.00254, 0.01686)
2.58

u
LO
0.00166 (0.00039, 0.01023)
6.16
0.00601 (0.00098, 0.03601)
6.15

u
TO
0.00222 (0.00130, 0.00395)
1.78
0.00574 (0.00267, 0.01219)
2.15

u
WH
0.00442 (0.00128, 0.01991)
4.50
0.01509 (0.00289, 0.07434)
5.23

Table 10. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for Cr6 exposure.
Exposure Route
Model Code
Job
Arithmetic Mean
(95% confidence interval)
Arithmetic Mean
K-Factor
95[th] Percentile
(95% confidence interval)
95[th] Percentile
K-Factor
Inhalation Conc (mg/m[3])
m
TO
0.000483 (0.000304, 0.000787)
1.63
0.001083 (0.000567, 0.002111)
1.95

u
LO
0.000282 (0.000145, 0.000574)
2.04
0.000686 (0.000261,  0.001779)
2.63

u
TO
0.000493 (0.000325, 0.000764)
1.55
0.001099 (0.000597, 0.002004)
1.84

u
WH
0.000926 (0.000236, 0.005086)
5.49
0.003291 (0.000553, 0.018372)
5.95
Inhalation Dose (mg)
m
TO
0.004834 (0.002935, 0.008280)
1.71
0.011036 (0.005438, 0.022725)
2.06

u
LO
0.002865 (0.001436, 0.006021)
2.10
0.007139 (0.002623, 0.019146)
2.72

u
TO
0.005055 (0.003337, 0.007836)
1.55
0.011269 (0.006119, 0.020552)
1.84

u
WH
0.010498 (0.002291, 0.074460)
7.09
0.038739 (0.005593, 0.250309)
6.93
Inhalation Time-Weighted Average Conc (mg/m[3])
m
TO
0.000401 (0.000242, 0.000691)
1.72
0.000921 (0.000450, 0.001917)
2.08

u
LO
0.000239 (0.000120, 0.000502)
2.10
0.000595 (0.000219, 0.001596)
2.72

u
TO
0.000420 (0.000276, 0.000654)
1.56
0.000942 (0.000508, 0.001727)
1.85

u
WH
0.000875 (0.000191, 0.006205)
7.09
0.003228 (0.000466, 0.020859)
6.93

Table 11. Arithmetic mean and 95[th] percentile estimates from lognormal mixed models and lognormal models for Cr6 exposure normalized by the weighted percentage Cr6.
Exposure Route
Model Code
Job
Arithmetic Mean
(95% confidence interval)
Arithmetic Mean
K-Factor
95[th] Percentile
(95% confidence interval)
95[th] Percentile
K-Factor
Inhalation Conc ((mg/m[3])/%Cr6)
m
TO
0.000601 (0.000366, 0.001025)
1.70
0.001414 (0.000706, 0.002898)
2.05

u
LO
0.000523 (0.000209, 0.001479)
2.83
0.001529 (0.000429, 0.005347)
3.56

u
TO
0.000584 (0.000372, 0.000941)
1.61
0.001360 (0.000706, 0.002591)
1.93

u
WH
0.001149 (0.000373, 0.004379)
3.81
0.003746 (0.000808, 0.016447)
4.64
Inhalation Dose (mg/%Cr6)
m
TO
0.006432 (0.003760, 0.011477)
1.78
0.015235 (0.007132, 0.032991)
2.17

u
LO
0.005300 (0.002082, 0.015336)
2.89
0.015650 (0.004311, 0.055743)
3.63

u
TO
0.005897 (0.003835, 0.009302)
1.58
0.013413 (0.007139, 0.024948)
1.88

u
WH
0.012721 (0.003588, 0.059264)
4.66
0.043769 (0.008163, 0.220927)
5.36
Inhalation Time- Weighted Average Conc ((mg/m[3])/%Cr6)
m
TO
0.000533 (0.000311, 0.000955)
1.79
0.001267 (0.000590, 0.002756)
2.18

u
LO
0.000442 (0.000174, 0.001278)
2.89
0.001304 (0.000359, 0.004645)
3.63

u
TO
0.000489 (0.000317, 0.000774)
1.58
0.001116 (0.000592, 0.002080)
1.88

u
WH
0.001060 (0.000299, 0.004939)
4.66
0.003647 (0.000680, 0.018411)
5.36

For each exposure route, the above two 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 8 to Table 11 were computed using a parametric bootstrap, separately calculated for each job. For the estimates based on the lognormal mixed model m, the parametric bootstrap simulations were all generated from the fitted lognormal mixed model. For the estimates based on the lognormal model u, the parametric bootstrap simulations were all generated from the fitted lognormal model. The parametric bootstrap simulations generated from the fitted lognormal model were also used to calculate the confidence intervals presented in the next section for the summary statistics from the empirical model. 
The bootstrap simulation algorithms were as follows:
Bootstrap simulation from lognormal mixed model (includes random site effect)
This simulation is used to estimate confidence intervals for the lognormal mixed model parameters only.
Step 1:
For each measurement, simulate a pair of random variables Y, X from the estimated lognormal distribution superimposed upon the observed sampling structure:
T = RanNor (Seed) xSs (6 values, one for each site)
Y = T + LnGM + RanNor(Seed)xSr
X = exp(Y)
where:
LnGM = intercept of mixed effect log-log regression model
Ss = square root of between site variance
Sr = square root of within site variance under mixed-effect model
Step 2:
For Y:
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 = Vs /V
Calculate AMm = exp(intercept + 0.5xV)
Calculate P95m = exp(intercept + Z95xS)
where:
V = total variance under mixed-effects model
S = square root of V 
Vs = between site variance.
Step 3:  Repeat Steps 1 and 2 10,000 times.
Steps 1 to 3 result in 10,000 values each for GSDm, ICC, GMm, AMm, and P95m.  95% confidence intervals can be defined for each parameter by the 2.5[th] and 97.5[th] percentiles (lower and upper, respectively) of the bootstrap distribution of that corresponding parameter.
Bootstrap simulation from lognormal model (does not include site effect)
This simulation is used to estimate confidence intervals for the lognormal and empirical model parameters. 
Step 1:
For each measurement, simulate a pair of random variables Y, X from the estimated lognormal distribution superimposed upon the observed sampling structure:
Y = LnGM + RanNor(Seed)xSr
X = exp(Y)
where:
LnGM = intercept of log-log regression model
Sr = square root of error variance
Step 2:
For Y:
Calculate geometric mean = GMs = GMu = exp(EAM)
Calculate geometric standard deviation = GSDs = GSDu
Fit lognormal model to simulated Y values
Under lognormal model:
Calculate AMu = exp(intercept + 0.5xV)
Calculate P95u = exp(intercept + Z95xS)
where:
EAM = sample arithmetic mean of Y 
V = error variance under lognormal model
S = square root of V 
For X:
Calculate arithmetic mean AMs
Calculate 95[th] percentile P95s
Step 3:  Repeat Steps 1 and 2 10,000 times.
Steps 1 to 3 result in 10,000 values each for GSDu, GMu, AMs, AMu, P95s, and P95u.   95% confidence intervals can be defined for each parameter by the 2.5[th] and 97.5[th] percentiles (lower and upper, respectively) of the bootstrap distribution of that corresponding parameter. 
Fold relative accuracy or K-factor
Fold relative accuracy (fRA) , also referred to as the K-factor, 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 2.5[th] and 97.5[th] percentiles of the sampling distribution of T can be denoted by T2.5 and T97.5, respectively, then the 95[th] percentile of sample fold relative accuracy can be theoretically calculated using the following formula provided in the AHETF Governing Document (AHETF, 2007, pg. 136 and AHETF, 2011, pg. 120):
 	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 benchmark of 3. If the observed fold relative accuracy is greater than 3, this means that the experiment did not meet the benchmark. 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.
Detailed summary statistics with parametric confidence intervals and fold relative accuracy
Table 12 to Table 27 present the estimates, parametric confidence intervals and fold relative accuracy values for selected summary statistics for the five exposure routes for iAs and the three exposure routes for Cr6, normalized by "one" and by the weighted fraction of active ingredient, respectively. Using the lognormal model (u) and bootstrap simulations from that same model, we present results for all the summary statistics for the lognormal model (u) and the empirical model (s). These results are calculated using the SAS code for model d. At the bottom of each table we also present confidence intervals for the ICC for job TO using the lognormal mixed model (m) and bootstrap simulations from that model; these results are calculated using the SAS code for model c. Results for the estimated arithmetic mean (AMm) and 95[th] percentile (P95m) based on the lognormal mixed model are presented in Table 8 to Table 11 above. Results for other summary statistics calculated using the lognormal mixed model are available using the SAS code for model c, but are not presented in this memorandum.  
Table 12. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the hands only iAs exposure (mg) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
5.72236
1.84236
18.40809
3.22
4.17112
2.12406
8.24445
1.98
1.88284
1.24609
2.88263
1.53
GMu
0.17106
0.03735
0.80498
4.71
0.10189
0.04031
0.26141
2.57
0.34797
0.20258
0.59903
1.72
AMs
0.40383
0.07204
3.83579
9.50
0.26360
0.07355
0.84568
3.58
0.40256
0.22751
0.74699
1.86
AMu
0.78323
0.08581
16.44835
21.00
0.28252
0.08228
1.26965
4.49
0.42510
0.23322
0.81029
1.91
P95s
1.07355
0.15968
16.39758
15.27
1.38507
0.18981
5.20047
7.30
0.65761
0.33616
1.80150
2.74
P95u
3.01467
0.25472
34.40772
11.84
1.06746
0.24641
4.51730
4.33
0.98530
0.39452
2.38118
2.50
ICC




0.00000
0.00000
0.91730






Table 13. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the total dermal iAs exposure (mg) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
4.25092
1.66018
11.20596
2.64
2.83896
1.73392
4.67038
1.65
1.63591
1.18665
2.27845
1.39
GMu
0.52469
0.14849
1.89641
3.61
0.39324
0.19971
0.78274
1.99
1.69885
1.11534
2.59209
1.53
AMs
0.99099
0.23813
5.68288
5.73
0.64655
0.28549
1.45859
2.26
1.89214
1.19956
2.93561
1.58
AMu
1.49505
0.27375
13.68369
9.15
0.67777
0.30296
1.67638
2.47
1.91762
1.22244
3.07732
1.60
P95s
2.51540
0.49558
23.11377
9.19
2.44659
0.61952
6.95781
3.95
3.78368
1.65385
6.10382
2.29
P95u
5.67112
0.73005
42.74627
7.77
2.18798
0.74966
6.27749
2.92
3.81733
1.87312
7.58302
2.04
ICC




0.00000
0.00000
0.91730






Table 14. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration iAs exposure (mg/m[3]) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
3.30208
1.51960
7.34984
2.23
2.07657
1.47025
2.94270
1.42
3.88385
1.60279
9.68030
2.49
GMu
0.00049
0.00017
0.00140
2.89
0.00147
0.00092
0.00239
1.62
0.00160
0.00050
0.00514
3.21
AMs
0.00071
0.00024
0.00300
4.23
0.00180
0.00111
0.00322
1.79
0.00348
0.00076
0.01423
4.56
AMu
0.00099
0.00027
0.00504
5.08
0.00193
0.00114
0.00338
1.75
0.00403
0.00085
0.02993
7.43
P95s
0.00129
0.00046
0.01105
8.54
0.00358
0.00203
0.01103
3.08
0.01247
0.00149
0.05451
8.37
P95u
0.00346
0.00064
0.01835
5.43
0.00491
0.00232
0.01026
2.12
0.01495
0.00210
0.09915
7.12
ICC




0.59751
0.00000
0.97238






Table 15. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose iAs exposure (mg) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accur-acy
GSDu
3.39360
1.53422
7.69314
2.27
2.04280
1.45759
2.87228
1.41
4.28709
1.65880
11.42000
2.66
GMu
0.00486
0.00167
0.01438
2.96
0.01552
0.00976
0.02486
1.60
0.01590
0.00458
0.05547
3.49
AMs
0.00717
0.00240
0.03192
4.45
0.01902
0.01176
0.03297
1.73
0.03805
0.00731
0.17537
5.20
AMu
0.01025
0.00264
0.05518
5.38
0.02002
0.01203
0.03456
1.73
0.04586
0.00835
0.44595
9.72
P95s
0.01280
0.00463
0.11878
9.28
0.03971
0.02118
0.11091
2.79
0.13994
0.01469
0.69825
9.53
P95u
0.03627
0.00642
0.19962
5.65
0.05024
0.02413
0.10337
2.08
0.17425
0.02122
1.32652
8.21
ICC




0.56158
0.00000
0.96950






Table 16. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration iAs exposure (mg/m[3]) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accur-acy
GSDu
3.39360
1.53422
7.69314
2.27
2.06883
1.46736
2.92650
1.41
4.28709
1.65880
11.42000
2.66
GMu
0.00041
0.00014
0.00120
2.96
0.00129
0.00080
0.00208
1.62
0.00132
0.00038
0.00462
3.49
AMs
0.00060
0.00020
0.00266
4.45
0.00158
0.00097
0.00278
1.76
0.00317
0.00061
0.01461
5.20
AMu
0.00085
0.00022
0.00460
5.38
0.00167
0.00100
0.00292
1.75
0.00382
0.00070
0.03716
9.72
P95s
0.00107
0.00039
0.00990
9.28
0.00331
0.00176
0.00951
2.87
0.01166
0.00122
0.05819
9.53
P95u
0.00302
0.00054
0.01664
5.65
0.00425
0.00201
0.00885
2.11
0.01452
0.00177
0.11054
8.21
ICC




0.57600
0.00000
0.97074






Table 17. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the hands only normalized iAs exposure (mg/%iAs) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
4.61696
1.70891
12.86324
2.79
4.46441
2.20158
9.11492
2.04
2.18471
1.31221
3.69689
1.69
GMu
0.30832
0.08119
1.19916
3.89
0.13328
0.05045
0.35763
2.68
0.60421
0.30977
1.18176
1.96
AMs
0.64750
0.13661
4.07744
6.30
0.44201
0.09649
1.29206
4.58
0.74591
0.36446
1.66039
2.23
AMu
0.99344
0.15840
11.24103
11.32
0.40817
0.10943
2.07216
5.08
0.81998
0.37902
1.93579
2.36
P95s
1.88305
0.29026
16.85774
8.95
2.96211
0.25575
8.20254
11.58
1.25007
0.57900
4.60351
3.68
P95u
3.81740
0.43716
32.29003
8.73
1.56149
0.33616
7.07741
4.65
2.18495
0.70554
6.49708
3.10
ICC




0.00000
0.00000
0.91730






Table 18. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the total dermal normalized iAs exposure (mg/%iAs) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
3.51056
1.55254
8.14098
2.32
2.87503
1.74551
4.75828
1.66
1.19285
1.06323
1.34318
1.13
GMu
0.94572
0.31626
2.88404
3.05
0.51440
0.25911
1.03247
2.01
2.94987
2.53705
3.43199
1.16
AMs
1.59734
0.46259
6.68907
4.19
0.98215
0.37302
1.95444
2.63
2.98486
2.56464
3.48113
1.17
AMu
2.08065
0.50917
12.09059
5.81
0.89842
0.39626
2.26231
2.52
2.99610
2.57081
3.49360
1.17
P95s
4.41212
0.90001
25.25469
5.72
5.23230
0.81487
9.42346
6.42
3.40460
2.92163
4.66455
1.37
P95u
7.46155
1.25964
43.05855
5.92
2.92215
0.98832
8.49147
2.96
3.94255
3.05491
5.04167
1.29
ICC




0.00000
0.00000
0.91730






Table 19. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration normalized iAs exposure ((mg/m[3])/%iAs) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
3.52511
1.55479
8.19740
2.33
2.20079
1.51601
3.20638
1.46
2.83256
1.43620
5.70870
2.02
GMu
0.00088
0.00029
0.00268
3.06
0.00193
0.00116
0.00325
1.68
0.00279
0.00114
0.00681
2.44
AMs
0.00144
0.00043
0.00625
4.34
0.00246
0.00144
0.00460
1.87
0.00413
0.00150
0.01248
3.02
AMu
0.00194
0.00047
0.01136
5.87
0.00263
0.00149
0.00487
1.85
0.00479
0.00159
0.01763
3.68
P95s
0.00342
0.00083
0.02363
6.90
0.00450
0.00272
0.01693
3.77
0.01043
0.00263
0.04168
4.00
P95u
0.00695
0.00117
0.04035
5.96
0.00706
0.00314
0.01566
2.25
0.01544
0.00343
0.06596
4.51
ICC




0.67895
0.00000
0.97874






Table 20. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose normalized iAs exposure (mg/%iAs) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
3.60428
1.56693
8.50713
2.36
2.08892
1.47486
2.96859
1.42
3.13780
1.48823
6.77483
2.16
GMu
0.00876
0.00286
0.02735
3.12
0.02030
0.01258
0.03300
1.63
0.02761
0.01039
0.07367
2.67
AMs
0.01449
0.00424
0.06580
4.54
0.02511
0.01532
0.04462
1.78
0.04425
0.01424
0.15333
3.47
AMu
0.01993
0.00469
0.12270
6.16
0.02662
0.01571
0.04694
1.76
0.05308
0.01535
0.23894
4.50
P95s
0.03389
0.00833
0.25063
7.40
0.04675
0.02797
0.15429
3.30
0.11703
0.02594
0.53885
4.60
P95u
0.07218
0.01174
0.43212
6.15
0.06817
0.03200
0.14348
2.13
0.18109
0.03464
0.89210
5.23
ICC




0.59378
0.00000
0.97210






Table 21. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average  concentration normalized iAs exposure ((mg/m[3])/%iAs) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
3.60428
1.56693
8.50713
2.36
2.10907
1.48235
3.01098
1.43
3.13780
1.48823
6.77483
2.16
GMu
0.00073
0.00024
0.00228
3.12
0.00168
0.00104
0.00275
1.64
0.00230
0.00087
0.00614
2.67
AMs
0.00121
0.00035
0.00548
4.54
0.00209
0.00127
0.00374
1.79
0.00369
0.00119
0.01278
3.47
AMu
0.00166
0.00039
0.01023
6.16
0.00222
0.00130
0.00395
1.78
0.00442
0.00128
0.01991
4.50
P95s
0.00282
0.00069
0.02089
7.40
0.00390
0.00233
0.01312
3.37
0.00975
0.00216
0.04490
4.60
P95u
0.00601
0.00098
0.03601
6.15
0.00574
0.00267
0.01219
2.15
0.01509
0.00289
0.07434
5.23
ICC




0.60557
0.00000
0.97302






Table 22. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration Cr6 exposure (mg/m[3]) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
1.979800
1.270296
3.128252
1.58
1.813005
1.368670
2.408105
1.33
3.432442
1.535398
7.872499
2.29
GMu
0.000223
0.000123
0.000409
1.83
0.000413
0.000281
0.000612
1.48
0.000433
0.000151
0.001248
2.88
AMs
0.000264
0.000141
0.000519
1.97
0.000486
0.000321
0.000740
1.52
0.000760
0.000216
0.002924
3.85
AMu
0.000282
0.000145
0.000574
2.04
0.000493
0.000325
0.000764
1.55
0.000926
0.000236
0.005086
5.49
P95s
0.000470
0.000217
0.001331
2.83
0.000993
0.000535
0.002126
2.14
0.002241
0.000405
0.010667
5.54
P95u
0.000686
0.000261
0.001779
2.63
0.001099
0.000597
0.002004
1.84
0.003291
0.000553
0.018372
5.95
ICC




0.171853
0.000000
0.933545






Table 23. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose Cr6 exposure (mg) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
2.027565
1.280948
3.255295
1.61
1.812857
1.368611
2.407816
1.33
3.811521
1.592351
9.380589
2.46
GMu
0.002232
0.001205
0.004181
1.87
0.004236
0.002878
0.006271
1.48
0.004289
0.001366
0.013525
3.15
AMs
0.002670
0.001397
0.005396
2.02
0.004978
0.003292
0.007585
1.52
0.008218
0.002054
0.036475
4.44
AMu
0.002865
0.001436
0.006021
2.10
0.005055
0.003337
0.007836
1.55
0.010498
0.002291
0.074460
7.09
P95s
0.004806
0.002171
0.014179
2.95
0.010327
0.005488
0.021794
2.11
0.025158
0.003987
0.138767
6.31
P95u
0.007139
0.002623
0.019146
2.72
0.011269
0.006119
0.020552
1.84
0.038739
0.005593
0.250309
6.93
ICC




0.482072
0.000000
0.962605






Table 24. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration Cr6 exposure (mg/m[3]) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
2.027565
1.280948
3.255295
1.61
1.822533
1.372459
2.426821
1.33
3.811521
1.592351
9.380589
2.46
GMu
0.000186
0.000100
0.000348
1.87
0.000351
0.000238
0.000521
1.49
0.000357
0.000114
0.001127
3.15
AMs
0.000223
0.000116
0.000450
2.02
0.000413
0.000272
0.000633
1.53
0.000685
0.000171
0.003040
4.44
AMu
0.000239
0.000120
0.000502
2.10
0.000420
0.000276
0.000654
1.56
0.000875
0.000191
0.006205
7.09
P95s
0.000400
0.000181
0.001182
2.95
0.000861
0.000456
0.001832
2.13
0.002097
0.000332
0.011564
6.31
P95u
0.000595
0.000219
0.001596
2.72
0.000942
0.000508
0.001727
1.85
0.003228
0.000466
0.020859
6.93
ICC




0.493171
0.000000
0.963555






Table 25. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation concentration  normalized Cr6 exposure ((mg/m[3])/%Cr6) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
2.451465
1.369032
4.469591
1.82
1.892866
1.400146
2.566420
1.36
2.888825
1.446058
5.899680
2.04
GMu
0.000350
0.000160
0.000776
2.22
0.000476
0.000315
0.000725
1.52
0.000654
0.000264
0.001627
2.49
AMs
0.000465
0.000200
0.001191
2.56
0.000582
0.000366
0.000904
1.59
0.000938
0.000349
0.003051
3.25
AMu
0.000523
0.000209
0.001479
2.83
0.000584
0.000372
0.000941
1.61
0.001149
0.000373
0.004379
3.81
P95s
0.000846
0.000338
0.003653
4.32
0.001689
0.000629
0.002759
2.69
0.001677
0.000618
0.010303
6.14
P95u
0.001529
0.000429
0.005347
3.56
0.001360
0.000706
0.002591
1.93
0.003746
0.000808
0.016447
4.64
ICC




0.163762
0.000000
0.932472






Table 26. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation dose normalized Cr6 exposure (mg/%Cr6) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
2.484504
1.375467
4.570631
1.84
1.848430
1.382711
2.477929
1.34
3.192942
1.497268
6.975211
2.18
GMu
0.003503
0.001584
0.007859
2.24
0.004883
0.003277
0.007323
1.50
0.006484
0.002403
0.017565
2.71
AMs
0.004686
0.001989
0.012248
2.61
0.005929
0.003778
0.008978
1.57
0.009881
0.003321
0.037422
3.79
AMu
0.005300
0.002082
0.015336
2.89
0.005897
0.003835
0.009302
1.58
0.012721
0.003588
0.059264
4.66
P95s
0.008379
0.003379
0.037867
4.52
0.017565
0.006381
0.026506
2.75
0.018825
0.006086
0.132423
7.03
P95u
0.015650
0.004311
0.055743
3.63
0.013413
0.007139
0.024948
1.88
0.043769
0.008163
0.220927
5.36
ICC




0.526429
0.000000
0.966484






Table 27. Arithmetic mean, geometric mean, geometric standard deviation, 95[th] percentiles, and ICC (with 95% confidence intervals and fold relative accuracy), for different statistical models of the inhalation time-weighted average concentration  normalized Cr6 exposure ((mg/m[3])/%Cr6) for each job.
Param-eter
LO
TO
WH

Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
Estimate
Lower Bound
Upper Bound
Fold Relative Accuracy
GSDu
2.484504
1.375467
4.570631
1.84
1.853049
1.384533
2.487080
1.34
3.192942
1.497268
6.975211
2.18
GMu
0.000292
0.000132
0.000655
2.24
0.000404
0.000271
0.000608
1.50
0.000540
0.000200
0.001464
2.71
AMs
0.000391
0.000166
0.001021
2.61
0.000492
0.000313
0.000746
1.57
0.000823
0.000277
0.003119
3.79
AMu
0.000442
0.000174
0.001278
2.89
0.000489
0.000317
0.000774
1.58
0.001060
0.000299
0.004939
4.66
P95s
0.000698
0.000282
0.003156
4.52
0.001464
0.000529
0.002211
2.77
0.001569
0.000507
0.011035
7.03
P95u
0.001304
0.000359
0.004645
3.63
0.001116
0.000592
0.002080
1.88
0.003647
0.000680
0.018411
5.36
ICC




0.529497
0.000000
0.966790






The ICC estimated values in Table 12 to Table 27 are: zero for hands only and dermal iAs exposure and normalized exposure; approximately 0.2 for inhalation concentration Cr6 exposure and normalized exposure, and; range from 0.5 to 0.7 for inhalation dose and time-weighted average concentration iAs and Cr6 exposure and normalized exposure. As explained above, these ICC values are based on the lognormal mixed model m (code c in the SAS code) and apply only to the job TO. These results show that for the job TO, for inhalation exposure but not dermal exposure, there is some variation between sites and some correlation between measurements from the same sites. 
Most of the summary statistics in Table 12 to Table 27 did not meet the benchmark of 3 for the fold relative accuracy. The fold relative accuracy values were generally higher for iAs, with the highest values for hands only exposure by TOs of 21.0 for the lognormal model arithmetic mean, 15.27 for the empirical model 95[th] percentile, and 11.84 for the lognormal model 95[th] percentile. For the Cr6 inhalation measurements, the fold relative accuracy values mostly met the benchmark of 3 for LOs and TOs, but mostly exceeded the benchmark for WHs.
Quantile plots
Quantile-quantile plots of the exposure and normalized exposure values were used to evaluate whether the data were lognormally distributed, as implied by the assumed statistical lognormal mixed and lognormal models, instead of being normally distributed. For the lognormal mixed models, these quantile-quantile plots are not intended to evaluate the overall model performance, which is better assessed using quantile-quantile plots of the model residuals, presented later. However, for the lognormal models, where no site random effect is included, the following quantile-quantile plots of the exposure and normalized exposure values can also be used to assess the overall model performance because the quantile-quantile plots of the model residuals are identical apart from a rescaling of the y-axis. In each case the quantile-quantile plot compares the observed quantiles of the measured values with the corresponding quantiles of a normal or lognormal distribution. We present the plots for total dermal and the inhalation time-weighted average concentration for the job TO only. Quantile-quantile plots for the other jobs and for the other exposure routes are not shown here, to avoid a voluminous report, but can be made available. Since there were only 5 LOs and 5 WHs, quantile-quantile plots for those two jobs would not be very useful in determining the population distribution. There were 9 TOs. The arithmetic mean normalized exposure for the given job was subtracted from the normalized exposure when creating the normal distribution plots, and the corresponding arithmetic mean logarithm of the normalized exposure for the given job was subtracted from the logarithm of the normalized exposure when creating the lognormal distribution plots. A perfect fit would imply that the plotted values lie in a straight line. The quantile-quantile plots for the job TO are presented in Figure 1 to Figure 12.Figure 1. Quantile plot of iAs dermal exposure data with a normal distribution. They clearly show that the lognormal distribution is a better fit than a normal distribution, and that the lognormal distribution fits reasonably well.
 
                                       
Figure 1. Quantile plot of iAs dermal exposure data with a normal distribution.
                                       
Figure 2. Quantile plot of iAs dermal exposure data with a lognormal distribution.
                                       
                                       
Figure 3. Quantile plot of iAs inhalation time-weighted average concentration exposure data with a normal distribution.
                                       
Figure 4. Quantile plot of iAs inhalation time-weighted average concentration exposure data with a lognormal distribution.
                                       
                                       
Figure 5. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data with a normal distribution.


                                       
Figure 6. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data with a lognormal distribution.

                                       
Figure 7. Quantile plot of iAs dermal exposure data normalized by the weighted percentage of iAs with a normal distribution.
                                       
                                       
Figure 8. Quantile plot of iAs dermal exposure data normalized by the weighted percentage of iAs with a lognormal distribution.
                                       
                                       
Figure 9. Quantile plot of iAs inhalation time-weighted average concentration exposure data normalized by the weighted percentage of iAs with a normal distribution.



                                       
Figure 10. Quantile plot of iAs inhalation time-weighted average concentration exposure data normalized by the weighted percentage of iAs with a lognormal distribution.
                                       
                                       
Figure 11. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data normalized by the weighted percentage of Cr6 with a normal distribution.


                                       
Figure 12. Quantile plot of Cr6 inhalation time-weighted average concentration exposure data normalized by the weighted percentage of Cr6 with a lognormal distribution.

To evaluate the model performance of the statistical mixed models, a better approach is to use a quartile-quantile plot of the studentized residuals from the normal mixed model or the lognormal mixed model. The normal mixed model is defined exactly as the lognormal mixed model described above, except that the dependent variable is the normalized exposure and not its logarithm. The residuals for the normal mixed model are defined as the normalized exposure minus its predicted value for the job (based on the fixed effects only). The residuals for the lognormal mixed model are defined as the logarithm of the normalized exposure minus its predicted value for the job (based on the fixed effects only). The studentized residual is calculated as the residual divided by an estimate of its standard deviation, so that the studentized residuals each have a standard deviation of 1, approximately. Figure 13 to Figure 24 present the quantile-quantile plots of the studentized residuals for total dermal and the inhalation time-weighted average concentration for the job TO only. Quantile-quantile plots for the other exposure routes are not shown here, to avoid a voluminous report, but can be made available. These plots clearly show that the lognormal mixed model fits well. The normal mixed model fits about as well as the lognormal mixed model in the cases shown.

                                       
Figure 13. Q-Q plot of studentized residuals using a normal mixed model. Dermal iAs exposure for TOs.


                                       
Figure 14. Q-Q plot of studentized residuals using a lognormal mixed model. Dermal iAs exposure for TOs.

                                       
Figure 15. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs.


                                       
Figure 16. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs.


                                       
Figure 17. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs.

                                       
                                       
Figure 18. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs.

                                       
                                       
Figure 19. Q-Q plot of studentized residuals using a normal mixed model. Dermal iAs exposure for TOs normalized by the weighted percentage iAs.

                                       
                                       
Figure 20. Q-Q plot of studentized residuals using a lognormal mixed model. Dermal iAs exposure for TOs normalized by the weighted percentage iAs.


                                       
Figure 21. Q-Q plot of studentized residuals using a normal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs normalized by the weighted percentage of iAs.

                                       
                                       
Figure 22. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration iAs exposure for TOs normalized by the weighted percentage of iAs.

                                       
                                       
Figure 23. Q-Q plot of studentized residuals using a normal mixed model. 
Inhalation time-weighted average concentration Cr6 exposure for TOs normalized by the weighted percentage of Cr6.

                                       
                                       
Figure 24. Q-Q plot of studentized residuals using a lognormal mixed model. Inhalation time-weighted average concentration Cr6 exposure for TOs normalized by the weighted percentage of Cr6.

                                       
Log-log-linearity analyses and estimated log-log slopes
The following analyses evaluate whether the logarithm of exposure is linear in the logarithm of the weighted percentage of active ingredient handled (perc AI) with a slope of 1.  These analyses only apply to the exposure normalized by the weighted percentage of iAs or Cr6, since the slope cannot be distinguished from the intercept if we normalize by one. We now refer to these analyses as the log-log-linearity analyses. In the Governing Documents and in some of our statistical memoranda analyzing previous AEATF studies we have referred to these analyses as a "proportionality" analysis, but this has caused some confusion because the statistical models do not assume that the exposure is directly proportional to the perc AI but instead assume that the logarithm of the exposure is linear in the logarithm of perc AI with a slope of 1, which is a related finding but a very different model, as explained here. We have therefore changed the terminology from "proportionality" to "log-log-linearity."
For the lognormal mixed model (code m), the Site Error Terms are assumed to be normally distributed with a mean of zero and a variance of Varsite. For the lognormal model (code u), there is no random site effect, but we can put this model into the same mathematical framework by assuming that the site error terms are identically zero and Varsite = 0, which means that the Site Error Terms are assumed to be normally distributed with a mean of zero and a variance of Varsite = 0. The Random Error Terms are assumed to be normally distributed with a mean of zero and a variance of Varerror. These Error terms are assumed to be statistically independent of each other.  The error terms are also assumed to be independent of the weighted percentage of active ingredient, which is the explanatory variable in this regression model.  Proportionality (log-log-linearity) of exposure to the weighted percentage of active ingredient is statistically modeled by assuming a Slope equal to 1 in the more general log-log linear model:
	Log (Exposure) = Intercept + Slope x Log (perc AI) + Site Error + Random Error.
The values of Intercept, Slope, Varsite, and Varerror are parameters of the fitted model. For the lognormal mixed model, these parameters all vary with the job. For the lognormal model, the only job modeled is TO and Varsite is known to be zero. 
This general log-log-linear model is for the Exposure rather than the Normalized Exposure (Exposure / perc AI)
Using this general model, taking exponentials of both sides gives
	Exposure = e[Intercept] x (perc AI)[Slope] x e[Site] Error + Random Error, so that
 	E{Exposure | perc AI} = Expected Exposure Given the Weighted Percentage of Active Ingredient
 	= C x (perc AI)[Slope], where 
      C = Expected Value {e[Intercept] x e[Site] Error + Random Error} = e[Intercept] x e[(Var][site] +Varerror}/2
The value of E{Exposure | perc AI} is the arithmetic mean of the distribution of exposures for a future set of randomly selected workers that are all using exactly the same weighted percentage of active ingredient. The parameters Intercept, Varsite, and Varerror are unknown, but are estimated by fitting the general log-log-linear model to the data.
Therefore, the expected exposure given the weighted percentage of active ingredient will be proportional to the weighted percentage of active iugredient if and only if the Slope in the log-log linear model equals 1. Note that the proportionality constant is C, which is very different to the estimated value of Slope. 
The above calculation using the general model is for what we refer to as the "`linear model." If instead the slope is assumed to be 1, then we get the "normalized exposure model" and in that case the same calculations give
      E{Exposure | perc AI} = Expected Exposure Given the Weighted Percentage of Active Ingredient
 	= C* x (Weighted Percentage of Active Ingredient), where 
      C* = Expected Value {e[Intercept*] x e[Site] Error [+ Random Error]} = e[Intercept*] x e[(Var][site][* +Varerror*}/2]
These parameters are shown with asterisks to emphasize that they will in general be different from the ones for the model with a slope parameter not necessarily equal to 1. These two sets of equations are used later in this memorandum in the derivation of Threshold values.
Possible alternative models include the same general formulation with a Slope of zero, implying that the exposure does not depend upon the weighted percentage of active ingredient handled, even though the observed weighted percentage of active ingredient handled varied between the subjects. If the true slope for the weighted percentage of active ingredient equals zero, then the resulting models are exactly the same as the models described above where the exposure is normalized by one. 
Other possible models include the same model with a slope not equal to zero or one, the quadratic models analyzed in the SAS code (but not presented in this memorandum), or models with more complicated relationships between the exposure and the observed 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. 
We can use these statistical models to calculate confidence intervals for the slope. For the lognormal mixed model, 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 al. Based on that article, the following Table 28 summarizes the five available methods:
Table 28. 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 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 site effect. If the ICC equals zero, then there is no site clustering and the site variance equals zero. A balanced data set is one where each treatment combination is applied to the same number of subjects. For this study, this implies that for each job there are the same number of measurements at each site.
The study data were not balanced. Based on this summary, the recommended method is the Kenwood-Rogers method for the lognormal mixed models. The confidence intervals for the regression coefficients presented in this memorandum follow these recommendations. Note that this issue does not impact the calculated confidence intervals for the summary statistics in Table 8 to Table 27, since they were based on a bootstrap method.
Table 29 and Table 30 show the 95% confidence intervals for the slope calculated from the above lognormal mixed models and lognormal models for the exposure against the weighted percentage of active ingredient, for the iAs and Cr6 data, respectively. 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 weighted percentage 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. 
Table 29. 95 percent confidence intervals for the slope of log exposure versus log weighted percentage of active ingredient handled for iAs data.
Exposure Route
Model Code
Job
Estimate
Lower
Upper
Confidence Interval Width
Hands (mg)
m
TO
0.03
−2.74
2.80
5.54

u
LO
3.21
−3.46
9.88
13.33

u
TO
0.03
−2.74
2.80
5.54

u
WH
−0.05
−2.71
2.61
5.32
Dermal (mg)
m
TO
0.44
−1.55
2.42
3.97

u
LO
2.50
−3.29
8.29
11.58

u
TO
0.44
−1.55
2.42
3.97

u
WH
1.05
0.32
1.79
1.47
Inhalation Conc (mg/m[3])
m
TO
0.11
−2.32
2.55
4.87

u
LO
−0.12
−6.21
5.97
12.19

u
TO
0.29
−1.10
1.68
2.78

u
WH
2.48
−0.94
5.91
6.86
Inhalation Dose (mg)
m
TO
0.25
−1.95
2.46
4.39

u
LO
−0.08
−6.32
6.15
12.47

u
TO
0.42
−0.91
1.76
2.66

u
WH
2.63
−1.14
6.40
7.54
Inhalation Time- Weighted Average Conc (mg/m[3])
m
TO
0.25
−2.00
2.51
4.51

u
LO
−0.08
−6.32
6.15
12.47

u
TO
0.43
−0.92
1.79
2.71

u
WH
2.63
−1.14
6.40
7.54

Table 30. 95 percent confidence intervals for the slope of log exposure versus log weighted percentage of active ingredient handled for Cr6 data.
Exposure Route
Model Code
Job
Estimate
Lower
Upper
Confidence Interval Width
Inhalation Conc (mg/m[3])
m
TO
0.33
−2.06
2.72
4.79

u
LO
−0.84
−4.02
2.34
6.36

u
TO
0.37
−0.78
1.51
2.29

u
WH
1.58
−2.84
5.98
8.82
Inhalation Dose (mg)
m
TO
0.38
−1.66
2.41
4.07

u
LO
−0.80
−4.15
2.54
6.69

u
TO
0.44
−0.68
1.56
2.24

u
WH
1.70
−3.10
6.49
9.59
Inhalation Time- Weighted Average Conc (mg/m[3])
m
TO
0.38
−1.67
2.43
4.11

u
LO
−0.80
−4.15
2.54
6.69

u
TO
0.45
−0.68
1.58
2.26

u
WH
1.70
−3.10
6.49
9.59

In all cases except for iAs, dermal, job WH, model u, the confidence interval for the slope includes both 0 and 1. For iAs, dermal, job WH, model u, the confidence interval for the slope includes 1 but not 0. Thus in all but one case the assumption of independence is not rejected and the assumption of log-log-linearity with slope 1 is also not rejected. The sign of the slope is positive in 25 of the 32 cases (78%). Models with a negative slope are unrealistic since they imply that the exposure decreases when the weighted percentage of active ingredient increases. However the wide confidence intervals show that the true slope has substantial uncertainty, due to the small numbers of workers measured. For the job TO, all the slope estimates lie between zero and one.
The regression results for the exposure routes dermal and inhalation time-weighted average are shown in Figure 25 to Figure 36. The plots show the data points together with the fitted regression line.
                                       
Figure 25. Mixed model regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job TO.


                                       
Figure 26. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job LO.

                                       
                                       
                                       
Figure 27. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job TO.

                                       
                                       
Figure 28. Simple linear regression plot for iAs log dermal exposure versus log weighted percentage iAs. Job WH.

                                       
                                       
Figure 29. Mixed model regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job TO.

                                       
                                       
Figure 30. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job LO.

                                       
                                       
Figure 31. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job TO.

                                       
                                       
Figure 32. Simple linear regression plot for iAs log inhalation time-weighted average exposure versus log weighted percentage iAs. Job WH.

                                       
                                       
Figure 33. Mixed model regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job TO.

                                       
                                       
Figure 34. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job LO.

                                       
                                       
Figure 35. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job TO.

                                       
                                       
Figure 36. Simple linear regression plot for Cr6 log inhalation time-weighted average exposure versus log weighted percentage Cr6. Job WH.

                                       
Threshold Analyses
As described above, alternative statistical models with an estimated slope or with a slope set to be one were fitted to the dermal and inhalation exposure data, either using a mixed lognormal model with a random site effect for job TO or using a lognormal model for each job without a random site effect. The model for exposure with an estimated slope is a linear model. The model for exposure with a slope of 1 is equivalent to a model for the normalized exposure, normalized by the weighted percentage of active ingredient. The alternative models produce different estimates for the arithmetic mean exposure for a given weighted fraction of active ingredient, perc AI. The equations are:
	E{Exposure | perc AI}linear model = C x (perc AI)[Slope]
 	E{Exposure | AI}normalized exposure model = C* x (perc AI)
Therefore, if we define the Threshold value as
      Threshold = {C*/C}[1/(][S][lope  -  1)],
then the estimated exposure values from the two models will be equal when perc AI equals Threshold. It also follows that
If Slope < 1
      E{Exposure | AI}linear model < E{Exposure | AI}normalized exposure model when perc AI > Threshold
      E{Exposure | AI}linear model > E{Exposure | AI}normalized exposure model when perc AI < Threshold
If Slope > 1
      E{Exposure | AI}linear model < E{Exposure | AI}normalized exposure model when perc AI < Threshold
      E{Exposure | AI}linear model > E{Exposure | AI}normalized exposure model when perc AI > Threshold
The value of Slope is less than 1 for all the cases for job TO, but slopes above and below 1 are found for the other two jobs. In all those cases where Slope < 1 we can use the normalized exposure model to provide conservative (in the sense of being health-protective) estimates of the arithmetic mean exposure for a given weighted percentage of active ingredient if the weighted percentage of active ingredient is large. The normalized exposure model is easier to apply than the linear model because it is based on the normalized, or unit exposures (exposure per weighted percentage of active ingredient). Applying the linear model requires that the exposures are evaluated as exposures per weighted percentage[s][lope].
We have two estimates of the predicted arithmetic mean exposure for a given weighted percentage of active ingredient. The graphs in Figure 37 to Figure 68 below compare the predicted arithmetic means for each active ingredient, job and exposure route. For the job TO, separate graphs are provided for the mixed linear model (with a site effect) and the simple linear regression model (without a site effect). For the jobs LO and WH, graphs are provided only for the simple linear regression model (without a site effect). Exposure is plotted against the weighted percentage of active ingredient. The letter codes denote the site. The green curve gives the predictions for the linear model (i.e. when the slope is allowed to vary). The purple line gives the predictions for the normalized exposure model (when the slope equals 1). The two estimates are equal if the weighted percentage of active ingredient equals the Threshold value:
	Threshold = {C*/C}[1/(][S][lope  -  1)]
The Threshold values are tabulated in Table 31 (iAs) and Table 32 (Cr6) below.
As proven above, in cases where the slope was less than one, the predicted arithmetic mean exposure from the normalized exposure model will be greater than the predicted arithmetic mean exposure from the linear model for weighted percentages of active ingredient above the threshold (right hand side of the graph). The predicted arithmetic mean exposure from the normalized exposure mixed model will be less than the predicted mean exposure from the linear model for weighted percentages of active ingredient below the threshold (left hand side of the graph). What this means is that if we assume log-log-linearity and use the normalized exposure model, then we will tend to over-predict the exposure unless the weighted percentage of active ingredient is below the threshold. (If the weighted percentage of active ingredient is below the threshold, then it will often be low enough that the exposure will not usually be of concern.) For cases where the slope is greater than one, the opposite inequalities apply so that the predicted arithmetic mean exposure from the normalized exposure model will be less than the predicted arithmetic mean exposure from the linear model for weighted percentages of active ingredient above the threshold (right hand side of the graph).
Table 31. Threshold values for the weighted percentage of active ingredient handled for iAs data.
Exposure Route
Model Code
Job
Slope
Threshold Weighted Percentage iAs (%)
Hands (mg)
m
TO
0.03
0.327295

u
LO
3.21
0.559334

u
TO
0.03
0.327295

u
WH
−0.05
0.455142
Dermal (mg)
m
TO
0.44
0.739090

u
LO
2.50
1.101892

u
TO
0.44
0.739090

u
WH
1.05
1.576418
Inhalation Conc (mg/m[3])
m
TO
0.11
0.001967

u
LO
−0.12
0.001235

u
TO
0.29
0.001967

u
WH
2.48
0.002950
Inhalation Dose (mg)
m
TO
0.25
0.019448

u
LO
−0.08
0.012972

u
TO
0.42
0.020296

u
WH
2.63
0.032877
Inhalation Time- Weighted Average Conc (mg/m[3])
m
TO
0.25
0.001631

u
LO
−0.08
0.001081

u
TO
0.43
0.001700

u
WH
2.63
0.002740





Table 32. Threshold values for the weighted percentage of active ingredient handled for Cr6 data.
Exposure Route
Model Code
Job
Estimate
Threshold Weighted Percentage Cr6 (%)
Inhalation Conc (mg/m[3])
m
TO
0.33
0.000488

u
LO
−0.84
0.000307

u
TO
0.37
0.000493

u
WH
1.58
0.000587
Inhalation Dose (mg)
m
TO
0.38
0.004947

u
LO
−0.80
0.003133

u
TO
0.44
0.005036

u
WH
1.70
0.006645
Inhalation Time- Weighted Average Conc (mg/m[3])
m
TO
0.38
0.000411

u
LO
−0.80
0.000261

u
TO
0.45
0.000419

u
WH
1.70
0.000554

                                       
Figure 37. Lognormal mixed model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 38. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 39. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 40. Lognormal model arithmetic mean exposure predictions for iAs hands only versus weighted percentage iAs. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 41. Lognormal mixed model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 42. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 43. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 44. Lognormal model arithmetic mean exposure predictions for iAs total dermal versus weighted percentage iAs. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 45. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 46. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 47. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 48. Lognormal model arithmetic mean exposure predictions for iAs inhalation concentration versus weighted percentage iAs. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 49. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 50. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 51. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 52. Lognormal model arithmetic mean exposure predictions for iAs inhalation dose versus weighted percentage iAs. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 53. Lognormal mixed model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 54. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 55. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 56. Lognormal model arithmetic mean exposure predictions for iAs inhalation time-weighted average versus weighted percentage iAs. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 57. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 58. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 59. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 60. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation concentration versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 61. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 62. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 63. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 64. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation dose versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.

                                       
                                       
Figure 65. Lognormal mixed model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 66. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job LO.

                                       
                                       
Figure 67. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job TO.

                                       
                                       
Figure 68. Lognormal model arithmetic mean exposure predictions for Cr6 inhalation time-weighted average versus weighted percentage Cr6. Letter codes show the data from that site. Job WH.
                                       
                                       
