Ann. occup. Hyg., Vol. 47, No. 2, pp. 101-110, 2003
© 2003 British Occupational Hygiene Society
Published by Oxford University Press
Comparison of Measurement Strategies for Prospective Occupational Epidemiology
1 University Institute of Occupational Medicine, Lyon; 2 Department of Epidemiology, INRS, Nancy; 3 Rhoditech, Lyon, France
Received 10 August 2001; in final form 18 November 2002
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ABSTRACT |
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 TOP ABSTRACT 1. INTRODUCTION2. METHODS3. RESULTS4. DISCUSSIONREFERENCES |
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In the context of a prospective assessment of exposure for epidemiology, our objective is to obtain an optimal group-based design of allocation of a fixed total number of measurements. Such a design has been described by Ashford [Ashford JR. (1958) The design of a long-term sampling programme to measure the hazard associated with an industrial environment. J R Statist Soc A; 121: 331–47]. As this strategy is not operational, we developed three series of strategies: the first based on simplifications of Ashford’s strategy; the second based on a pilot study; and the third on an iterative assessment of the group specific standard deviation of exposure. These strategies are compared by simulating a day-to-day individual exposure in several industrial sites and the resulting health effect. Our criteria for comparing strategies are the mean squared error of the estimated exposure in each group weighted by the number of subjects and the mean squared error of the estimated linear regression coefficient in the dose–response relationship. Strategies relying on an iterative approach have been found to perform best whatever the circumstances, nearly as well as Ashford’s optimal strategy.
Keywords: occupational health; exposure assessment; prospective epidemiology; cross-sectional study; simulations
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1. INTRODUCTION |
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TOPABSTRACT1. INTRODUCTION2. METHODS3. RESULTS4. DISCUSSIONREFERENCES |
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A dose–response association is a main argument for a causal link between occupational hazards and diseases (Rothman and Greenland, 1998, pp. 24–8). However, the measurements of the effective individual dose are almost never available, and in the past, qualitative and semi-quantitative exposure estimates were often used as proxies of these doses (Stewart and Herrick, 1991). Recently, occupational epidemiology has relied more and more on quantitative exposure estimates (Blair and Stewart, 1992; Kauppinen, 1994; Stewart et al., 1996), based of individual exposure measurements. A series of recent papers discusses strategies on how to plan these individual exposure measurements with the aim of estimating the effect of exposure on health. These papers concentrate mostly on differences between individual-based and group-based strategies. In one of the last papers, Tielemans et al. (1998) provide ‘equations describing the influence of several factors on attenuation and on the standard error of an estimated linear regression coefficient relating a continuous exposure variable (measured with error) and a continuous health outcome via a simple linear regression model’. Our approach is very close to the latter insofar as a continuous health outcome is related to a single continuous exposure variable through a simple linear regression. However, a first difference from the approach of Tielemans et al. (1998) is that we assume a cumulative effect of exposure on health. The biomedical sciences paradigm considers the mass of a certain agent to be proportional to a toxic effect, and depending on the averaging time it becomes cumulative exposure. This cumulative exposure is best computed as the arithmetical mean exposure multiplied by the duration of exposure. If the variability in exposure is best described by a lognormal distribution, this arithmetical mean is different from the geometric mean and is difficult to compute. The equations developed by Tielemans et al. (1998) using arithmetical means of normal (Gaussian) distribution do not therefore apply. A second difference is that we do not consider individual-based strategies. The aim of this work, originating at the Rhône-Poulenc Group, was to obtain a strategy for measurements for future epidemiological studies. The numbers of measurements were restricted by costs, and strategies based on individual measurements were not considered feasible. We therefore consider only group-based strategies. The design of the epidemiological studies we consider is a prospective follow-up of a cohort of workers, exposed to a given substance, who will do a number of different jobs in the course of their employment history. During this follow up some new jobs will appear and some will disappear altogether, but these evolutions are impossible to predict ex ante. Within this framework the only choice left is an annual allocation of the measurements to the different tasks or exposure groups (EGs) occurring in these jobs. Ashford (1958) proposed a solution, in part theoretical, to this problem. By minimization ex post of the sum of the variances of the total cumulative exposure over each subject, he proposed a formula for allocation of the measurements to the EGs depending on a number of unknown parameters, among which was the standard deviation of the exposure in each EG.
In this paper we propose a series of strategies consisting in allocation designs of a fixed total number of measurements. All these strategies are derived in various ways from Ashford’s design. These designs are then compared on the basis of exposure data and work histories simulated from real situations. These situations are descriptions of actual industrial populations, in terms of jobs and tasks within these jobs. A multidisciplinary task-group was set up, comprising occupational physicians, industrial hygienists, medical epidemiologists and a statistical epidemiologist, in order to describe all aspects of these. The group was especially concerned about the actual feasibility of the approach. It described seven industrial sites representing a whole range of plant typologies within the chemical industry. The sites are quite different in term of job turnover, number of tasks per job, duration of exposure, level and variability of exposure.
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2. METHODS |
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TOPABSTRACT1. INTRODUCTION2. METHODS3. RESULTS4. DISCUSSIONREFERENCES |
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2.1 Designs for allocating measurements to EGs
2.1.1 Ashford’s design
In Ashford’s computation (Ashford, 1958), the individual cumulative exposure for subject i is estimated by
(1)
where the expression tij is the time spent by subject j in the EGi and 
is the arithmetic mean of all measurements in EGi. The quantity, which is minimized, is the sum of the variances of the individual cumulative exposure estimates:
(2)
where 
i is the unknown (arithmetical) standard deviation of the exposure in EGi and ni the number of measurements in EGi.
The total number of measurements ni from the beginning of exposure to the study date in an EGi, corresponding to the minimum of (2) is proportional to
This quantity contains tij, the time spent by subject j in the EGi. If the population is followed up prospectively, which is the case we consider, these tij depend on the unknown future individual EGs. As it is impossible to predict the future, we use an approximation of this design, which we shall call Ashford’s restricted design. This approximation is obtained by the assumption that in a given short period, every subject has the same known duration of exposure di in each group and his/her group does not change. Thus the sum of subjects is
where Ni is the number of subjects in EGi. Finally, with these approximations the optimal allocation of the total number of measurements ni allocated to EGi is proportional to idi
Ni.
(3)
Ashford’s restricted design leads us to ignore to some extent the person-time spent in the different EGs, which was inherent in Ashford’s primary design. This last pattern was a strength insofar as it allowed the strategy to focus on lifetime (cumulative) exposure, but also its main weakness as this strategy relied on the future time spent in the different EGs.
Ashford’s restricted design is our reference strategy insofar as it is the best short-term strategy ignoring the future changes of EGs. However, even with this approximation, the is stay unknown.
A further complication is that the statistical distribution of exposure measurement data is usually considered to be a lognormal distribution within each EG. In this case, the variance of the exposure Xi in EGi depends both on the geometric mean gmi and the geometric standard deviation gsdi through the following formula:
(4)
2.1.2 Prospective strategies
In the context of a prospective strategy, the measurement allocation is performed each year in some EGs, assuming that the exposure situation is stable over 1 yr. Each year some tasks characterized as EGs may change, giving rise to new EGs or disappear altogether. The following subsections present different ways of estimating the is, corresponding to different strategies.
For a given strategy and a given year, the fixed number of available measurements is allocated according to equation (3). We thus obtain a non-integer number of measurements for each EG. We first allocate the integer part of this number and then the sum of the remainders to the highest remainders.
From equation (3), we see that the number of measurements allocated to an EG is proportional to the square-root of the number of subjects in this EG, proportional to the annual duration in this EG and by equation (4) it is proportional to the geometric mean and to a complicated function of the geometric standard deviation. The proposed strategies will be based on estimates of these parameters.
A remaining technical problem is that the allocation based on our estimated i is for the total number of measurements over the exposure period. But for any given year, the numbers of measurements allocated by a given strategy may be less than the number of measurements, which have already been obtained for some EGs (a minimum of one measurement is fixed for all EGs). In the other EGs the optimal number of remaining measurements is no longer proportional to idiNi. We use a simple constrained non-linear maximization algorithm to obtain the optimal allocation in this setting. For example, consider the case of 20 measurements to be allocated to three EGs: EG1, EG2 and EG3. Ten measurements have been already performed (respectively 3, 5 and 2). The allocation proportions are 10, 50 and 40% for a given strategy and so two measurements are to be made in EG1, 10 in EG2 and eight in EG3. But, as three measurements have already been made in EG1, no more measurements will be performed in this EG. So the allocation proportions remaining are 55% for EG2 and 45% for EG3 and respectively nine and eight measurements are to be made altogether (making 20 with the three in EG1). The 10 measurements to be done are allocated four to EG2 and six to EG3).
2.1.3 Strategies based on simplifications of Ashford’s restricted design
We obtain some strategies by applying equation (3) in which we compute the is according to simplification of equation (4).
· If the geometric standard deviations are unknown, we can only suppose that they are equal and the allocation in EGi becomes proportional to gmidiNi (called the gm.d.N hereafter in this paper). This presupposes, however, that the geometric means are all known.
· If, additionally, the geometric means are unknown, the only option is again to consider them equal, the allocation will then be proportional to diNi (d.N strategy).
· If, finally, we do not even know the number of subjects and the duration of each EG, these are also considered equal. The strategy consists then in allocating equal numbers of measurements to all EGs (hereafter called the balanced strategy).
2.1.4 Strategies based on a pilot study
In these strategies a pilot study is done, in which a fixed number of typical measurements is obtained from each EG. From these measurements we can compute different estimates of the is and, replacing these in equation (3), we obtain different strategies. In our simulations we use three measurements per EG, which is the minimum number from which we can compute a standard deviation.
· We can use the empirical standard deviation of the measurements in each EG of the pilot study for estimating the i in equation (3). This is the Emp strategy.
· If we assume, furthermore, a lognormal distribution for measurements in each EG we can use equation (4) with the geometric mean and geometric standard deviation of the measurements of pilot study for estimation of is (yielding the logN strategy).
Finally, in a logN+prior info strategy, we balance, as shown in equation (5) from Gelman et al. (1995), the empirical parameters from the pilot study and a prior knowledge on the size of the geometric standard deviations of exposure measurements and use a ilogN+prior info estimation of i.
(5)
where 
= 4 and 02 = [ln(1.8)]2. This prior knowledge states that the geometric standard deviations are thought to be between 1.4 and 5.2 with a 0.95 probability in all the EGs.
2.1.5 Iterative strategies
In the previous group of strategies, a fixed number of measurements in the pilot study are used for estimating the i in each EGi, which are used in allocating the total available number of measurements in EGs. An alternative is to update the variance estimators as information is collected each year in the main survey.
Estimations of i are made using the same three ways as in the previous global strategies and give rise to three further strategies: Iter.Emp, Iter.logN, Iter.logN+prior info.
Table 1 summarizes these different strategies.
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2.2 Simulations
The task-group described existing industrial sites using a series of job names, which appeared and disappeared from the beginning of the period of interest. All unexposed jobs were regrouped in a single job. The sites were further described in terms of the numbers of subjects hired each year and of the rate of subjects quitting employment either through retirement, which is then a function of the age of the subjects, or by dismissal or other reasons. Every job-period was then described in terms of the number of subjects holding it, its average age, its turnover-rate and, for all exposed jobs, of a list of tasks. Each of these tasks is considered as an EG. For each job-period, the number of subjects allocated to each task was also given. It was assumed that every subject carried out a single task each day, but that this task varied from day to day. All EGs were further characterized by a lognormal distribution with given geometric mean and geometric standard deviation. Tables 2 and 3 show the detailed description of the first site and summary descriptions for the six other sites.
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In the first site, there are five exposed jobs: salinization (in the BH3 workshop), distillation (BHC), hydrogenation (HK), reconditioning (BH) and foreman (AM). In each of the four first of these, two tasks are involved (for a total of six different tasks): moving operator (R.1–R.4) and process operator (Op.1 for the HK, BHC and BH3 and Op.2 for BH). Moving operators have to collect some product samples and they are more exposed than the process operators. The foremen have the most exposed jobs as they intervene during loss of process control (incidents).
From these descriptions of actual sites, we simulated cohorts of subjects, their daily exposure with the resulting health effect and finally the measurement strategies and the resulting individual exposure estimates.
2.2.1 Cohort simulation
For the simulations of a cohort in a site, we generate the age of subjects who hold various jobs (exposed or not) at the beginning of exposure. Each year, some subjects in each job are randomly switched according to the turnover rates. In the same spirit, all subjects who reach the age of retirement leave the plant along with others sampled from the study base to leave for other reasons. Others whose age and job are sampled according to the information obtained for each job-group replace leavers. We proceed so until a date fixed to be the end of the study. So we obtain for all subjects of a cohort, from the beginning of exposure to the end of the study, the jobs they hold each year.
2.2.2 Simulation of the daily individual exposure
The exposure of a subject on a given day depends on the task performed by the subject on this day and this task depends on the job he holds (see Table 2, for example). Therefore we first sample for every subject, for every day and for every replication of the simulation the task performed during this day, and then sample the daily exposure from the corresponding lognormal distribution.
2.2.3 The subsequent health effect
For each subject of the cohort, the cumulative exposure is obtained by summing all the daily exposures. The baseline health parameter for each individual before exposure is obtained by sampling from a normal distribution based on external knowledge on the health effect considered. This parameter is then assumed to vary linearly with the individual cumulative exposure (in the case of a nervous motor speed affected by exposure, it would decrease; if it were a reaction time, it would increase). In order to achieve comparability of the different sites, the slope was chosen for each site to achieve a 80% statistical power in a cross-sectional study among workers still employed at the end of the study.
2.2.4 Simulation of the allocation strategies
A strategy consists of the allocation of an annual number of measurements to be performed in each EG. A simulation of a strategy consists therefore in an annual random sample of measurements in the database of the daily exposure values, simulated as described in section 2.2.2. With this given number of measurements for each strategy, we calculate the arithmetic mean of all measurements in each EG, Xi, and thus obtain an estimation of the individual cumulative exposure as given in equation (1). For the iterative strategies, the parameter estimates of each EG are updated every year. An example will be presented in the Results section.
2.2.5 Replication of simulations
The population and the distribution of the daily individual exposures are simulated once. The resulting health effect is simulated 100 times (subscripted by h = 1 to H, where H = 100), and within each simulation of the health effect, the observed exposure levels are simulated 100 times (subscripted by k = 1 to K, where K = 100), yielding 10 000 simulations of the measurements.
2.3 Criteria
We use two criteria for comparing the strategies.
The MSE–EG criterion is the mean squared error of the estimated exposure in each group weighted by the number of subjects and the duration of exposure. It quantifies the error in the individual cumulative exposure due to replacing the actual exposure by that estimated using exposure measurements with different strategies.
If Mi is the true arithmetic mean of exposure in the EGi with Ni exposed persons-years with annual duration di and 
is the arithmetic mean of the measurements obtained for EGi, simulated health effect h and simulated exposure measurements k according to the strategy s, then
(6)
for a total number I of EGs. We note that this MSE–EG is independent of the health effect and thus of the regression coefficient of a linear dose–response relationship.
The MSE–ß criterion is the mean squared error of the estimated linear regression coefficient in the dose–response relationship. It quantifies the error in the estimation of the exposure effect on health due to replacing the actual exposure by that estimated using exposure measurements with different strategies. In the case of a dose–response relationship of the form y = 
+ ßx, if ßhk is the estimated regression coefficient for simulated of health effect h and simulated exposure measurements k according to the strategy s and if ßh is the real coefficient, obtained with the exact cumulative exposure, then
(7)
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3. RESULTS |
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TOPABSTRACT1. INTRODUCTION2. METHODS3. RESULTS4. DISCUSSIONREFERENCES |
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3.1 An example of allocation strategies
We detail here the allocation of a total of 50 measurements in 5 yr (1994–98) to the seven EGs of site 1. In the first group of strategies, 10 measurements would be allocated each year. A balanced strategy would, for instance, allocate one measurement to each EG each year and share the remaining three among the EGs in order to get a balanced result of seven measurements per EG after 5 yr. In the second and the third group of strategies, a pilot study of three measurements per EG would be done in the first year, so that seven measurements remain for each of the four remaining years. In the second group of strategies the estimations of the
is of the three measurements in the year 1994 would fix the proportions allocated for the remaining years. In the third group of strategies (iterative strategies), the estimations of the is are updated every year. Table 4 summarizes the results of one simulation of the Iter.logN+prior info strategy over the 5 yr and the corresponding annual percentage of allocation by EGs. The results are contrasted with the theoretically optimal design we called Ashford’s restricted design, which is available as all is known. In this example 3 x 7 = 21 measurements are spent for the pilot study, so that seven measurements are left each year, assuming a total duration of 5 yr. Each year, these seven measurements are allocated based on the allocation proportions computed from the estimated standard deviations. For instance, the seven measurements in 1995 are allocated as follows. The non-integer numbers of measurements are (0.11, 1.29, 1.74, 1.69, 1.63, 0.29, 0.26). First the integer parts are attributed, leaving three measurements to share between the groups. The remainders are (0.11, 0.29, 0.74, 0.69, 0.63, 0.29, 0.26) so that they are attributed to the third, fourth and fifth groups. The actual total allocation proportion is therefore only comparable with the optimal one for the last 28 measurements. In this example the theoretically optimal numbers would be (0, 5, 7, 5, 10, 0, 1) which compares well with the numbers obtained (0, 5, 10, 7, 6, 0, 0). The main difference is for the third and the fifth groups in which the computed proportions are overestimated for the third and underestimated for the fifth. Note that the standard deviations given in Table 4 are not geometric standard deviations, so that the geometric standard deviations for high exposures are much higher than for low exposures, which explains that some differences between true and estimated variances were quite large. In this example, the pilot study induced a strategy, which was already very close to the optimal allocation, and the iterative allocation did not improve the results substantially.
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3.2 Comparison of the strategies based on the simulations
Table 5 shows how the strategies compare for the first site. The MSEs shown are not their actual value but the percentage of their value with respect to Ashford’s restricted design with 300 measurements. For all the sites, the total number of measurements considered are 50, 100, 150 and 300 allocated in a 5 yr period. The detailed results for the other sites could not be shown but the following general observations can be made.
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3.2.1 For the MSE–EG criterion
For each strategy, the MSE–EG is approximately proportional to the inverse of the total number of measurements. All strategies can thus be compared using a single number of measurements. We can base our quantitative comparisons on a single column. We chose to present the strategy for 300 measurements, as in this case, chance variations are lower than with lower numbers of measurements, and a direct comparison with the theoretically optimal strategy can be immediately read from the table. All general statements made in the following paragraphs could, however, have been made for any number of measurements.
The third group of strategies with iterative estimate of is is always the best. They are usually within 110% of Ashford’s restricted design. In site 1 (see Table 5), the MSE–EG is between 103 and 110% of the optimal, well below the other strategies (157–226%). Only the gm.d.N strategy (121%) was nearly as good, but it relies on the knowledge of the geometric means.
Within these iterative strategies the Iter.logN+prior info strategy is always the best although sometimes marginally (in site 1, 103% versus 110%). An exception is site 3 in which two EGs have very small geometric standard deviations, which do not agree with the prior information (geometric standard deviations between 1.4 and 5.2 with a 95% probability). Usually (though not in site 1) the Iter.emp strategy seems to perform better than the Iter.logN. This was expected as the slight bias of the latter estimator is known. In site 2, all EGs disappear after 2 yr of measurements, to be replaced by other EGs. In this case, the iterative strategies are only marginally better than the other strategies, as the measurements done in the first 2 yr carried no information as to an optimal allocation for the subsequent EGs.
Within the global allocation strategies based on a pilot study, the ranks are the same as within iterative strategies. Depending on the site, MSE–EGs are between 120 and 250% of Ashford’s restricted design. In site 1, these strategies are of the same order of magnitude (157–226%) as the balanced design (172%) The high MSE–EG values are obtained for the two sites in which some geometric standard deviations are very large. In this case, a pilot study is not sufficient to obtain stable estimates for is, and iterative strategies considerably improve the MSE–EG.
As expected, the gm.d.N strategy usually performs better than the d.N strategy. An exception is site 7, which has highly contrasting geometric standard deviations, so that is are not proportional to the geometric means. The balanced strategy is better than the d.N strategy when most subjects have low exposure. The balanced strategy can be much worse if this is not the case. In five of our seven sites, the balanced strategy and the d.N strategy perform poorly (~200%).
The results for the MSE–ß are very similar to the results for MSE–EG with a notable exception for site 4 in which, due to the very high turnover between the jobs, the between-subject cumulative exposure variance is very small, so that this exposure is virtually proportional to the exposure duration. Even if the EGs are well characterized, this has no influence on the MSE–ß and therefore all the strategies are equivalent (MSE–ßs between 100 and 120%).
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4. DISCUSSION |
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TOPABSTRACT1. INTRODUCTION2. METHODS3. RESULTS4. DISCUSSIONREFERENCES |
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In this paper, we do not consider individual-based exposure assessment; neither do we consider strategies for grouping individuals into EGs (Werner and Attfield, 2000). Our problem is to allocate prospectively numbers of measurements to EGs that are supposed to be well identified by industrial hygienists. In this respect, they are comparable to the situation in coalmines considered by Ashford. A difference from his approach, however, is that these EGs are variable, appearing and disappearing. Similarly, job rotation may be important and no long-term measurement assessment is possible. In this context, the only possible strategy is to reassess the measurement allocation for every change of EG.
We work in the context of a prospective epidemiological study so that a minimum of 50 measurements on which we based our conclusions seems realistic. Our search for an optimal allocation strategy is aimed at maximizing the information obtained for a given measurement cost. An alternative way to incorporate information on exposure is, especially in the epidemiological context, to use some expert judgements on exposure in each EG as described, for instance, in Wild et al. (2002).
As mentioned in the Introduction, our approach is very similar to that of Tielemans et al. (1998), who take up the formulation of Kromhout et al. (1993) and Kromhout and Heederik (1995). As in these papers, we have a continuous single health outcome as obtained in a cross-sectional epidemiological study, which is a linear function of the exposure and which is allowed to differ from worker to worker even when they are members of the same EG. However, in these papers, the health outcome is regressed on the log-transformed exposure data. And as acknowledged by Tielemans et al. (1998), in most studies, estimates of exposure are based on arithmetic means. This means that if exposure is twice as high, the health effect will also be doubled. If, however, the health effect is linear with the log-transformed exposure, then if the exposure is twice as high, the health effect will be augmented by ß x ln(2), whatever the dose. This is in contradiction with the assumption of a cumulative effect of dose on health, which is the model we consider and which underlies Ashford’s approach. Our, in our view, more realistic model, becomes, however, mathematically intractable, and we had to rely on simulations.
We sample the daily exposure from a lognormal distribution whose parameters, geometric mean and geometric standard deviation (gsd(ww) in Table 2), are given for each task. We are well aware that this is an extreme simplification insofar as we ignore the between-worker variance and the possibility of isolated incidents. Ongoing work is focused on the effect of these exposure modifiers. However, we want to stress that Ashford’s approach does not presuppose homogeneous EGs. The only feature he uses for his EGs are their variances as shown in section 2.1.1. If we are able in other contexts to obtain reasonable estimates of these variances (e.g. making use of our knowledge of between-worker variance and/or incidents), his mathematical result concerning optimal allocation still applies. We can therefore reasonably assume that our approximations of Ashford’s design are reasonable.
On the other hand, if there is substantial between-worker variance, the group-based estimates will not be valid for each worker. In this case, only individual sampling strategies can be expected to yield valid information (Kromhout et al., 1996; Seixas and Sheppard, 1996). Some preliminary simulations suggest that the group-based sampling strategies for measurements are still valid for gsd(bw) = 1.2 but seem to be more questionable for gsd(bw) = 1.4. However, even in this case the attenuation of exposure–response relationships when using the individual strategy may be higher than that of the group-based strategy (Van Tongeren et al., 1999). Individual strategies are probably only an option when quite large numbers of measurements can be obtained.
Apart from the optimality criteria we present, we studied two other criteria: the first is the statistical power of the estimation of the health effect; the second is the global variance of the cumulative exposure for which Ashford derived his design. We did not use the statistical power as it did not discriminate well between the strategies. As the restricted form of Ashford’s design ignores the individual durations of exposure, the criterion that is then optimal is in fact the criterion we called MSE–EG, which was therefore our natural choice. This MSE–EG is also more natural for industrial hygienists as it measures the quality of the quantitative exposure assessment at a given time and does not depend on the health status and therefore on the form of the dose–response relationship. It depends neither on the total duration of EGs nor on the turnover rates between EGs. The MSE-ß, on the other hand, depends both on the bias of the slope estimate, e.g. due to attenuation (Boleij et al., 1995; Preller et al., 1995), and on its intrinsic variance. Attenuation should not be a problem here, as we rely on group-based strategies which induce a Berkson-type error (Armstrong, 1990). This criterion depends critically, however, on the assumption of linearity of the dose–response relationship.
A series of recommendations as to how to design simulation studies for epidemiology has been proposed (Maldonado and Greenland, 1997). We checked the different points (to be realistic, to cover a broad range of scenarios, to check whether random error can be ignored, etc.) for the validity of our simulations. We can be reasonably sure that our results are not an artefact of insufficient simulations.
An alternative to simulations would have been to take very large exposure data sets and to compare our strategies by sampling the exposure measurements in these data sets. Such a data set, containing 37 000 measures for 1000 subjects and a lot of health effects measurements, was presented by Heederik and Attfield (2000), but with different objectives in mind. Such data sets are, however, rare but would be interesting as a possibility of validation. On the other hand, such a data set does not allow one to change the different parameters that may influence the results of strategies like the presence of incidents, the slope of the dose–response relationship and the gsd(bw).
Are our results reliable? As mentioned above, we did not use an actual data set for validation, and rely therefore on our representation of the industrial sites as a series of stable EGs. Furthermore the computed optimality is a mean optimality and cannot be verified on actual implementation of the designs in a site. However, within this framework, our qualitative results seem reliable: there are theoretical and practical justifications. The different allocation strategies make use of different amounts of information. The simplest (balanced) strategy makes use only of the number of EGs, and the most sophisticated one makes use at any moment of all the measurements that have already been obtained as well as a hygienic knowledge of the possible range of reasonable geometric standard deviations. We expected, from a theoretical point of view, that the strategies using more information should perform better, which was what we found in our simulations. On the other hand, we empirically examined these strategies in a number of rather different situations in which the expected a priori ranking was observed with slight differences we could explain from the specific features. However, while the advantage in using the information from previous measurements is sometimes small, it always improved the strategy in the mean. We are therefore, based on this theoretical argument and its practical consequence in our sites, rather confident in the generalizability of our results. We feel that these strategies can provide a practical guidance for anyone planning further research.
Acknowledgements—We acknowledge the helpful comments from two anonymous reviewers and the associate editor Dr Van Tongeren who helped to improve this paper significantly. The work of the first author was done in the framework of his Ph.D., work paid for by Rhône-Poulenc Group.
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FOOTNOTES |
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* Author to whom correspondence should be addressed. E-mail: pascalwd{at}inrs.fr
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