Standard random effects models assume that the modelled random effects are uncorrelated. Probably, this assumption is often violated in occupational settings since exposures close in time tend to be more similar than exposures far apart in time [24,25]. In a study of upper arm postures in three occupations, Liv et al. [14] showed that the assumption of independent errors was less severely violated if a data sample was distributed among several short time periods during a shift than if the sample comprised fewer or only one longer time period. Due to autocorrelation, the observed variance of a mean exposure estimate was also larger than expected from theoretical formulae based on the posture variance components. Similar effects of autocorrelation can also be expected for sample estimates of variance components based on standard random effects models [14]. To our knowledge, this issue has not been investigated in the occupational exposure literature, and so the accuracy and precision of variance components derived by standard procedures under different data collection scenarios are only superficially understood.

The recordings from the different working days were of different duration, ranging from 240 minutes to 721. Most working days were close to 480 minutes. In a block bootstrap procedure, blocks of 30 minutes of simulated data were added to all days shorter than 480 minutes by resampling from the available data of the same day, until the day contained at least 480 minutes of data. All working days longer than 480 minutes were truncated at 480 minutes. This padding and truncating procedure has been described in detail and validated in a previous paper [14]. The described resulting data set was balanced and consisted of 480 minutes of upper arm elevation recordings from each of four days in each of 23 subjects. From this data set, we calculated minute-by-minute values of three exposure variables: average elevationpercentage time above 90° (percentage of time spent with an arm elevation larger than 90°), and percentage time below 15° (percentage of time spent with an arm elevation less than 15°). Prior to assessing average angles, each Abduflex interval recording was replaced by the central angle of that interval, except for angles above 90° that were assigned the value 105°. Figure ​Figure11 shows an example of the three posture variables during a working day of a car mechanic.

In Table ​Table3,3, accuracy and precision of the estimated variance components for percentage time above 90° are presented for the subset of all investigated sampling strategies for which n_s=10 or 20, n_d=2 or 4, t_tot=120 or 480 minutes and t_b=1, 15 or 60 minutes. A table containing results from all investigated sampling strategies and exposure variables is provided as an Additional file 1: Liv_complete_results.pdf.

For strategies with a block size of 1 minute, bias was absent or very small. In general, when block size increased, a stronger negative bias appeared for the within-day variance, and a stronger positive bias for the between-days variance (Table ​(Table3;3; illustrated in Figure ​Figure2).2). This effect was more pronounced with a smaller total sample size per subject (Figure ​(Figure2).2). The accuracy of the between-subjects variance was not affected by the sampling strategy to the same extent as the variance between and within days.

The present study showed that sampling data in large time blocks may lead to inaccuracy and imprecision in estimates of variance components. This was particularly prominent for strategies where small fractions of working days were sampled. Variance component estimates were particularly biased for strategies with small sample sizes and large block sizes. For percentage time above 90°, negative biases of up to 26% of the size of the “true” within-day variance component and positive biases of up to 1110% of the size of the “true” between-days variance component were observed. However, in many cases the total error in the variance component estimates was dominated by imprecision rather than inaccuracy. For the within-day variance component, which was estimated at 164.7 in the original data set, the 90% prediction interval ranged from 137 to 191 for the sampling strategy giving the best precision (n_s=20, t_tot=480, n_d=4, t_b=1). For the within-day variance component, the median width of the 90% prediction interval across all investigated sampling strategies was 89 while the median bias of the variance component was 1.4. Thus, our results suggested that imprecision will often be a more serious problem than inaccuracy for studies of the sizes simulated here. The results further indicated that the sample sizes investigated by us might not be sufficient to retrieve variance components with a satisfying precision. In occupational epidemiology, variance components are required for designing efficient exposure measurement strategies, and when combined with information on costs associated with data collection, they give a basis for deciding on efficient budget allocation [31-34]. Variance components can guide the selection of targets for interventions to reduce suspected hazardous exposures [2]; they are used in assessments of clinical reliability [35], and they are necessary inputs in conventional power analysis of, for instance, studies addressing exposure differences between groups or effects of an intervention [4,6]. The present paper clearly illustrates that the results of these applications of estimated variance components can be very uncertain, in particular if the estimates have been based on short and continuous exposure samples. This caveat is rarely addressed in the literature. Estimated variances are known to follow a positively skewed distribution; this was apparent even in the present study. Hence, a sample estimate of a variance is more likely too small compared to the true value than too large. Using an estimated variance in a power analysis of a planned intervention study will therefore more often lead to too “optimistic” (small) predictions of the necessary study size than to too “pessimistic”. The error may be considerable, as illustrated by the wide prediction intervals on variances in the present study. In order to account for variance estimation uncertainty in power analyses, some authors have suggested to use the 80^th percentile of the expected distribution of variance estimates as an input rather than the actual variance estimate [36]. A particular challenge appears if variance components per se are the exposure variables of interest, for instance in studies of exposure variation [6,7]. Variances are not normally distributed, and a conventional power analysis, which requires data to have this property, is not applicable. Developing power analysis procedures for studies addressing variance components is an interesting issue for further research.

The epidemiologic study, for which the data was originally sampled, attempted a random collection of subjects and working weeks [21,26,27]. The present data is therefore a likely representative sample of car mechanics’ exposure to elevated upper arms. The original study also included measurements on house painters and machinists [21,26,27]. The car mechanics, spending on average 4.7% time with the right arm elevated above 90°, worked more with elevated arms than machinists (1.6% time >90°), but less than house painters (8.8% time >90°). The size of exposure variability in the three groups differed in a similar fashion; the car mechanics showed more variability than the machinists did, but less than the house painters did. The autocorrelation function for percentage time above 90° at lag 1 was 0.31 for machinists and 0.46 for house painters, compared to an autocorrelation of 0.52 for the car mechanics [14]. This implies violation of the assumption of independence in the error term also for machinists and house painters. This leads us to believe that the principal effects of sampling strategy on variance component estimators shown in the present study are relevant also to data collections of other exposure variables and in other occupational groups. The magnitude of these effects, however, probably varies between variables and occupational groups, and our numerical results should therefore be applied outside the group of car mechanics only with great caution.

Variability of upper arm elevation has been reported in the literature for other occupations [6,22,37], but the posture variables were different from the ones used in the present study and accuracy and precision of the reported variance components were not explored. Consistent with our findings, estimates of variance components were shown to be associated with considerable imprecision in previous studies on muscle activity during assembly work [3,4] and on posture and electromyography data from short-cycle manual handling [6].

The present study determined exposure variability between and within subjects using a random effects model, which assumes that effects are uncorrelated. Results showed that this assumption was violated since the car mechanics exhibited considerable autocorrelation between measurements within a working day. As demonstrated by David [38], the ordinary sample variance estimator, , underestimates the population variance if observations are not independent and if the sample size is not large. An equivalent effect can be expected on the estimator for within-day variance. This is a likely explanation why within-day variance estimates were inaccurate for sampling strategies with larger block sizes, while they were not for strategies with block size 1, where observations will be (close to) unaffected by autocorrelation. Since variance components are partitions of the total (constant) variance present in the data, a negative bias in the within-day variance estimate propagates to the other variance components, in particular showing up as a positive bias in the between-days variance. When block size increases, the time span between the observations in the sample decreases. Hence, the sample will be more autocorrelated, which leads to a larger bias. We believe that increased autocorrelation explains the occasional larger bias of variance components estimated by strategies where a particular sampling time was distributed across four days rather than two. This will lead to a smaller sampling time per day and – if the block size is large – to a more dominant effect of autocorrelation.

Violations of other assumptions of the random effects model (equation 1) than independence also occurred. Visual inspections of plots of predicted values of the random effects and their residuals suggested that the assumption of constant variance across subjects and days was violated in some cases. The residuals also had positively skewed distributions. Although the model in equation (1) makes no assumptions of the distributional form of the random effects, the REML estimators that we used to estimate the variance components assume normal distributions. However, REML estimators are identical to ANOVA estimators when data is balanced [39], and as ANOVA estimators are not based on distributional assumptions we do not consider this to be a problem. We did not transform the exposure data because we could not identify a transformation that improved the fit of the random effects model to any noticeable extent.

More complex random effects models are available that can model time dependence in the error term by incorporating correlation structures according to different time series models, like autoregressive models (AR) or moving average models (MA) [40]. A successful fit of such a model might result in unbiased measures of variability between and within subjects. However, for data sets as large as that used in the present study (23 subjects*4days*480 minutes=44160 observations), computations with very large variance and covariance matrices will be involved. Thus, it may not be possible to fit these models. Moreover, identifying a reasonable model of the time dependence of our posture variables is not a trivial matter.

Since inaccuracy and imprecision of variance component estimators is an issue in limited exposure sampling, occupational research and practice addressing variance components for descriptive, epidemiologic or intervention purposes may face the need for studies of a considerable size – larger than commonly done – if this inaccuracy and imprecision is to be reduced to levels normally pursued in studies addressing mean exposures.

Our findings lead us to the following guidance for sampling strategies addressing variance components:

· A larger data sample will be required to reach a satisfying precision of variance components than when the purpose is to estimate a mean value with good precision.

· Distributing a within-day sample across the whole day and in small blocks will lead to variance component estimators with a better accuracy and precision than if the sample is collected in larger time blocks.

· Increasing the total within-day sample size increases the accuracy and precision of variance component estimators and reduces the adverse effects of sampling in larger blocks.