Annals of Occupational Hygiene Advance Access originally published online on February 8, 2006
Annals of Occupational Hygiene 2006 50(4):371-377; doi:10.1093/annhyg/mei078
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© 2006 British Occupational Hygiene Society Published by Oxford University Press
Original Article |
Optimizing Occupational Exposure Measurement Strategies When Estimating the Log-Scale Arithmetic Mean Value—An Example from the Reinforced Plastics Industry
1 Occupational Medicine, Department of Public Health and Clinical Medicine, Umeå University, 901 87 Umeå, Sweden; 2 Department of Mathematical Statistics, Umeå University, 901 87 Umeå, Sweden
* Author to whom correspondence should be addressed. E-mail: erik.lampa{at}envmed.umu.se
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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When assessing occupational exposures, repeated measurements are in most cases required. Repeated measurements are more resource intensive than a single measurement, so careful planning of the measurement strategy is necessary to assure that resources are spent wisely. The optimal strategy depends on the objectives of the measurements. Here, two different models of random effects analysis of variance (ANOVA) are proposed for the optimization of measurement strategies by the minimization of the variance of the estimated log-transformed arithmetic mean value of a worker group, i.e. the strategies are optimized for precise estimation of that value. The first model is a one-way random effects ANOVA model. For that model it is shown that the best precision in the estimated mean value is always obtained by including as many workers as possible in the sample while restricting the number of replicates to two or at most three regardless of the size of the variance components. The second model introduces the ‘shared temporal variation’ which accounts for those random temporal fluctuations of the exposure that the workers have in common. It is shown for that model that the optimal sample allocation depends on the relative sizes of the between-worker component and the shared temporal component, so that if the between-worker component is larger than the shared temporal component more workers should be included in the sample and vice versa. The results are illustrated graphically with an example from the reinforced plastics industry. If there exists a shared temporal variation at a workplace, that variability needs to be accounted for in the sampling design and the more complex model is recommended.
Keywords: measurement strategy • variance components • exposure assessment
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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Measurement strategies for hazard control have to be efficient and effective (Kromhout, 2002
). Efficient in the sense that as much information about the exposure as possible is obtained by a minimal spending of resources, and effective in the sense that it should give valid information.
Kromhout et al. (1993) published the first comprehensive evaluation of temporal and personal variability of a large number of chemicals and workplaces and stated that exposures within the same industry tend to vary much, both between individuals and over time. Rappaport et al. (1995) published a theoretical framework for exposure assessment which accounted for these variances and introduced a Wald-type test (described in detail in Lyles et al., 1997) for testing whether the workers are likely to be overexposed (Tornero-Velez et al., 1997) in the long run or not.
An important aspect of the statistical methodology presented in Rappaport et al. (1995) is the need for repeated measurements (replicates) on the same worker over time. Of course, repeated measurements require more resources than single measurements, so how can a hygienist decide which the best combination of workers and replicates is? The answer to that question depends on what the objective of the measurements is. Rappaport et al. (1995) proposed an expression for determining the optimal sample size derived from the power of the Wald-type test. However, the Wald-type test is based on large-sample properties and may be unreliable for the small samples that are common today.
The aim of this study is to determine how to prioritize between spending resources on including many workers or many sampling days when striving to obtain the most precise estimate of the arithmetic mean (AM) value for a worker group. Elementary sampling theory states that the best estimate of a mean value is always obtained when one measurement is obtained from as many workers as possible (Cochran, 1977). That is however only valid when exposures are normally distributed, which is generally not the case as occupational exposures tend to be log-normally distributed. To explore this problem we investigate two models; the one-way random effects model previously used for assessing occupational exposures (see e.g. Kromhout et al., 1993 and Rappaport et al., 1995) and a two-way random effects model which introduces the concept of a temporal variability of the exposure common to all workers.
Here we derive the necessary equations and illustrate the application of the models by an example of exposure to styrene from the reinforced plastics industry.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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The one-way random effects model
Let {xij}, i = 1, ..., k workers, j = 1, ..., n replicates, be shift long measurements on the random continuous exposure variable X. By assuming
, X is then assumed to be log-normally distributed with mean 
and variance 
. The log-scale mean value, µy, can be estimated by 
which is an estimator for the log-scale geometric mean value (GM). The GM of {xij} is defined as 
. Taking logarithms yields 
, which is the estimator 
as stated above.
The one-way random effects analysis of variance (ANOVA) model can be written as
 | (1) |
ij is the residual term. It is assumed in the model that 
and that 
and that they are mutually independent. 
and 
are called the between- and within-worker variance components, respectively, and 
. We see that for Model 1 E(Yij) = µy, i.e. equation (1) implies a stationary behaviour of the mean value, which means that there should be no systematic changes in the exposure during the time period that the samples are collected. The variance components can be estimated by several methods (Searle et al., 1992) but our method applies to variance components estimated by the ANOVA estimators. Methods in statistical software, e.g. PROC MIXED in SAS, estimate the variance components by restricted maximum likelihood (REML) by default, but in the case of completely balanced designs, as we focus on here, the estimators are identical (Searle et al., 1992).
The AM value is estimated by 
with the total variance estimated as the sum of the estimated variance components, i.e. 
. For calculations to be relatively easy we will focus our interest to the log-transformed AM. Maximizing the precision is equivalent to minimizing the variance so our problem is to find a set of k and n which minimizes 
which can be written as,
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and the variance component estimators are statistically independent (Searle et al., 1992) so the last term in equation (2) is equal to zero. We then have according to Cochran (1977) and Searle and Fawcett (1970).
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Here K is the worker population total and N is the number of replicates theoretically available. We can think of N as a very large number and therefore approximate 
and 
. In the derivation, the worker effects are assumed to be a random sample from an infinite population and although the worker population, unless very large, is finite we approximate the first pair of parentheses in 
to unity. The variance of the variance estimator can be expanded to 
. Searle et al. (1992) give explicit expressions for the sampling variances of the variance components and we can write equation (2) as
 | (3) |
The two-way random effects model
For the one-way model E[Yij|j] = µy regardless of the sampling occasion. If we assume that the choice of sampling day might bias the mean estimate, i.e. E[Yij|j] = µy + 
j for sampling day j, we must account for that in the sampling design. Since the within-worker variability is the sum of all variability in the data not explained by the between-worker variance component we can partition the within-worker variance into a temporal variance assumed common for all workers and the residual variance, i.e. 
. Again assuming shift long measurements and no interaction between factors the resulting model can be written as,
 | (4) |
, where 
represents the shared temporal variability, and that 
. The derivation for the variance of the estimated log-scale mean value for the two-way model is in Appendix A.
A practical example
To illustrate the use of the two models styrene exposure data (described in Liljelind et al., 2001) from five industries were used. The sixth industry in the original data was not used due to the small number of workers included (K = 3). Data were analysed with the one-way and the two-way random effects model, respectively, and the variance components were estimated. Distinct variance components were assumed for each industry. The estimates were then used to graphically illustrate the effect of the sample allocation on the precision of the estimated mean value. The plots were constructed using MATLAB 6.5 (The MathWorks, Inc.). We have not taken into account that sampling costs are often different for the inclusion of more workers as compared to more replicates.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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Table 1 shows the results of the analyses. We see that in the first three industries, the between-worker variance is larger than the within-worker variance. For those three industries, the shared temporal variance
is only slightly larger than zero indicating that only a minor part of the variability is a day to day variation that the workers have in common. For the other two industries the situation is reversed. Here the within-worker component is larger than the between-worker component and the shared temporal component is a large part of the total variability. As expected, the shared temporal variance reduces the within-worker/residual variances. In industry number 4 we also note that the between-worker variance decreased and that there was a relatively large residual variance. Such effects are not uncommon when using REML as the estimation procedure. In all cases the distributional assumptions of the random effects in the models could not be rejected on the 5%-level using the Shapiro–Wilk test on the estimated random effects. For further graphical illustrations we chose industries 1 and 5 since they reflect two different conditions.
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In order to obtain the most precise estimate of the mean value in a group of workers, using the simpler model (Model 1), as many subjects as possible should be included in the sample, regardless of the sizes of the between-worker and the within-worker variances. This is illustrated in Figs 1 and 2 where the solid lines indicate equivariance lines. The numbers on the lines indicate the variance of the estimated log-transformed mean value. The lower the variance is, the more precise is the estimate of the mean. The broken lines indicate possible alternatives for distribution of 8, 16 and 32 measurements. We have here chosen to show continuous lines, though in practice only integers are possible on the Subjects as well as the Replicates axes. By following the broken lines it is obvious that the lowest variance, and thus the best precision in the estimate of the mean, is obtained when as many subjects as possible are studied.
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k
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However, when the workers' exposure on a particular calendar day to some extent covaries, e.g. when the workers are in the same hall with a single source of exposure in common or when exposure is influenced by day-to-day variability in production, we need to block for that effect in the sampling design and use the two-way model. In this case it is intuitively expected that if the shared temporal variance (i.e. the covarying part of the within-worker variance) is larger than the between-worker variance, then increasing the number of replicates contributes more to the precision than does increasing the number of subjects, as is the case in Fig. 3.
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In Fig. 4 the shared temporal variance is estimated as almost zero and then we see a similar pattern to that of Fig. 1. This is also easy to see from the model formulation in equation (4); if
the two-way model collapses into the one-way model.
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From the figures we see the impact of the between-worker and the shared temporal variances on the precision of the estimated mean value. The residual variance determines the spacing between the solid lines and the curvature of them. A large residual variance will result in equivariance lines farther apart and less curved than would a smaller residual variance.
A wide range of values for the variance components were applied, three-dimensional plots examined, and the results were in line with the expectations (data not shown).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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In this study we have described two models for optimization of a measurement strategy by minimizing the variance of the estimated AM value, based on the different variance components (the one-way and the two-way random effects models). In order to do this we have further developed the one-way model by separating the shared temporal variability from the variability within workers. Thus, we end up with three variances: the between-worker variance, the shared temporal variance and the residual variance. The main findings are that the optimal sample allocation depends on the relative sizes of the between-worker component and the shared temporal component.
How do these variances affect the optimization? In order to answer this, we let 
denote the quotient of the between-worker variance and the shared temporal variance. A 
> 1 thus means that measurements on many subjects should be obtained whereas a < 1 means that efforts should be focused on obtaining many replicates from each worker. A = 1 means that equal efforts should be put into obtaining samples from workers and replicates from them. If the residual variance is very large the optimum will be drawn towards a more equal weight for replicates and subjects. The proposed two-way random effects model and the concept of a shared temporal variation separated from the within-worker variance have, to the authors' knowledge, not previously been described in the literature.
This paper proposes a tool for determining the optimal sample size if the study objective is to determine the mean value as precisely as possible given certain restrictions and is not meant to be used in the analysis of determinants of exposure or for testing if workers are overexposed or not. The expressions for the variance of the mean value and the resulting graphs can be implemented in, for example, MATLAB and serve as a graphical tool for sample size determination.
We have assumed that the values of the estimated variance components reflect the true variances. A prerequisite for this is a large well-designed pilot study, which may not be available. We have chosen not to investigate the effect of poorly estimated variance components on the optimal sample allocation. The minimal sample size required to estimate all the variance components from the two-way model are two workers measured twice with both workers measured on the same day at least once but we do support Rappaport et al. (1995)'s recommendation that a minimal sample size for a pilot study should be at least five workers measured twice.
For the hygienist our results imply that even when the within-worker variance is large, there are certain situations when the optimal number of replicates for a precise estimation of the mean value may not be large. For example, if the shared temporal variance is zero or small compared to the other variance components, the two-way model collapses into the one-way model and then increasing the number of replicates does very little to contribute to the precision.
It should be mentioned that our estimates of the variance components are somewhat different from those in Liljelind et al. (2001), as we have not assumed that industries could be pooled due to similar variance component characteristics.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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The optimal numbers of workers and replicates for a sampling strategy is closely linked to the variance components. If no shared temporal variance exists, the best precision in the mean value is always obtained by including the maximum number of workers available in the sample. If there is a shared temporal variance, the optimal design is determined by the sizes of the variance components. If the between-worker component is larger than the shared temporal component, more workers should be included in the sample and vice versa. However, this is only valid when the costs for inclusion of more workers are the same as those for the inclusion of more replicates. Of course, more advanced cost models could be applied to the equations.
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APPENDIX A—THE DERIVATION FOR THE VARIANCE OF THE ESTIMATED LOG-SCALE MEAN VALUE FOR THE TWO-WAY MODEL |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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Let m be a vector of mean squares and let
2 be a vector of variance components of the same order as m. Suppose a matrix P is such that E(m) = P2. The ANOVA estimator of 2 is then 
. It follows that Var.
Searle et al. (1992) show that under the normality assumption, for a mean square Mi and a degree of freedom fi; 
for which an unbiased estimator is 
. If we define D = V(m) as a diagonal matrix with 
as the elements, then .
For the two-way model, we can write
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For unbiased estimation, D is replaced by
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For the two-way model 
and 
. An expression for the variance of the estimated log-scale mean value can then be obtained by noting that the covariance matrix 
has the form
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIX A--THE DERIVATION FOR...ACKNOWLEDGEMENTSREFERENCES |
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This study was funded by the Swedish Council for Working Life and Social Research and the Faculty of Medicine, Umeå University.
Received October 17, 2005; in final form December 8, 2005
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Rappaport SM, Lyles RH, Kupper LL. (1995) An exposure-assessment strategy accounting for within- and between-worker sources of variability. Ann Occup Hyg; 39: 469–95.
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