Annals of Occupational Hygiene Advance Access originally published online on May 2, 2006
Annals of Occupational Hygiene 2006 50(6):623-635; doi:10.1093/annhyg/mel021
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Attenuation in Risk Estimates in Logistic and Cox Proportional-Hazards Models due to Group-Based Exposure Assessment Strategy
Department of Public Health Sciences, The University of Alberta Canada
*Author to whom correspondence should be addressed. Tel: +1-780-492-3240; fax: +1-780-492-9677; e-mail: igor.Burstyn@ualberta.ca
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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In occupational epidemiology, it is often possible to obtain repeated measurements of exposure from a sample of subjects (workers) who belong to exposure groups associated with different levels of exposure. Average exposures from a sample of workers can be assigned to all members of that group including those who are not sampled, leading to a group-based exposure assessment. We discuss how this group-based exposure assessment leads to approximate Berkson error model when the number of subjects with exposure measurements in each group is large, and how the error variance approximates the between-worker variability. Under the normality assumption of exposures and with moderately large number of workers in each group, there is attenuation in the estimate of the association parameter, the magnitude of which depends on the sizes of the between-worker variability and the true association parameter. Approximate equations for attenuation have been derived in logistic and Cox proportional-hazards models. These equations show that the attenuation in Cox proportional-hazards models is generally more severe than in logistic regression. Furthermore, when the between-worker variability is large, our simulation study found that the approximation by equation is poor for the Cox proportional-hazards model. If the number of subjects is small, the approximation does not hold for either model.
Keywords: Berkson type error structure • between and within-worker variability • bias • ecological variable • epidemiology • homogenous error
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Appreciation for the effect of errors in exposure variables on the interpretation of epidemiological studies lies at the heart of modern epidemiology. In assessing occupational exposures, a group-based exposure assessment strategy is commonly used. In this strategy, subgroups of workers are constructed based on their job-titles, tasks or other features of the work environment; a sample of workers is measured within these groups on repeated days; and the group means are used as an estimate of exposure for all members of the corresponding groups. It is often necessary to employ a group-based strategy because study populations are typically large and it is impossible or not feasible to take measurements from individual workers whose health status is being studied. An early example of this strategy was a case–control study of brain cancer and leukemia in electric utility workers in which investigators collected repeated measurements on workers from groups expected to have different exposure levels (Kromhout et al., 1995).
It is recognized that non-differential errors in individual exposure variables lead to attenuation of exposure–response associations and that the use of grouped exposure data substantially eliminates the attenuation. Conventionally we deem it advantageous even though the group-based strategy results in larger standard errors compared with the individual-based strategy (Armstrong, 1990, 1998; Tielemans et al., 1998), but gains in accuracy (allowing small bias) often offset losses due to imprecision (Sexias and Shappard, 1996). This has been convincingly demonstrated where both exposure and outcome can be assumed to have normal distributions, and thus are amenable to ordinary least square regression analysis (Tielemans et al., 1998). Subsequently, it is often supposed that the association parameter estimates are unbiased in logistic and Cox proportional-hazards models when group-based exposure assessment is used.
However, it has been suggested that there may be attenuation in estimating association parameters with the group-based strategy in logistic and Cox proportional-hazards models even when the measurement error variance is homogeneous across groups. The same magnitude of attenuation for both models has been shown when the measurement error variance is heterogeneous (i.e. the variance of error depends on the observed value) (Deddens and Hornung, 1994).
Burr (1988), Reeves et al. (1998), and Heid et al. (2002) showed that probit-regression association parameter estimates, which approximate logistic-regression estimates, are asymptotically biased downward from the true association parameter when error structure is of the Berkson type. This indicates that even though the group means are known exactly, which is a true Berkson error, there is attenuation in the estimation of parameters for the logistic model. However, the attenuation is considered to be negligible if the error variance is small. It remains to be determined whether the group-based strategy gives attenuation in estimating association parameters in logistic and Cox proportional-hazards models with a homogenous error structure.
This article provides an approximate description of the behavior of association estimates with the group-based exposure assessment strategy in the logistic and Cox proportional-hazards models. First, we show the circumstances under which this strategy leads to the Berkson error structure and demonstrate that the exposure measurement error variance approximates the between-worker variability. Second, we confirm that there is attenuation in association estimates in logistic and Cox proportional-hazards models when the group-based exposure assessment is used. We then derive an approximate relationship for the attenuation between the association parameters, based on true exposures and observed exposures, as a function of the between-worker variance and the true association parameters in the two regression models. These results and their limitations are explored in simulations. The overall objective of the article is to better understand how group-based exposure assessment influences association estimates in a typical case–control or cohort study in occupational epidemiology.
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Methods and results |
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TOPABSTRACTINTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Exposure model
In occupational epidemiology, the log-transformed exposure level (Ygij) is commonly assumed to satisfy the model
 | (1) |
gi is the random effect for i-th worker in g-th group that is normally distributed with a mean zero and between-worker variance 
, and 
gij is the random error term for j-th (day) level on i-th worker in the g-th group that is normally distributed with a mean zero and within-worker variance 
. The error variance of the repeated observations provides the degree of measurement error in the classical error model. It is assumed that gi and gij are mutually independent, and by letting µgi = µg + gi, we have the model as
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Derivation of a Berkson error model
In the group-based exposure assessment strategy, an average exposure ( 
) for a group g is taken to apply to all workers in the group (e.g. with the same job title), where 
and 
, Kg = k is the number of workers, and Ngi = n is the number of repeated measurement of each worker in a randomly selected sample. For each worker, this group mean 
is an approximation of his/her true exposure (µgi).
The conditional expectation of true exposures given observed exposures is
 | (2) |
. The derivation is made under the assumption of classical measurement error model (Appendix 1). If the number of workers in each group is sufficient for the true mean and the estimated group mean to be close in value, that is, 
, then we have
 | (3) |
and its mean is approximately the group mean. The situation is analogous to the Berkson error model, where the true exposure given the observed exposure has an expected value equal to the observed exposure.
The Berkson error model was originally proposed for experimental situations, in which the experimenter attempts to set a variable at a target quantity, but because of an imprecise control its true value may be randomly distributed around the target value, which is the observed value of the variable (Berkson, 1950). If the experiment is replicated many times with the same target value, the true value will be randomly distributed with an estimated mean value approaching the target value, and the errors are assumed to be independent of the target value. By showing an approximate property 
, we postulate a Berkson type error model with the assigned group mean () and true exposures (µgi), i.e.
 | (4) |
, egi) = 0 and cov(µgi,egi) = V(µgi|) 
0 (Appendix 2). We note that the condition of sufficiently large sample size in the classical model leads to the conditional expectation of the error given the group mean that is close to 0 in the Berkson error model. Also, the model (4) is not a truly Berkson error model since cov(,egi) = 0 does not imply that the observed value and the error are independent, which is required for truly Berkson error model. We re-postulate the model as Berkson type by selecting a moderately large sample of workers.
The error variance under the Berkson error model
We next demonstrate a relationship between the error variance and the between-worker variance under the Berkson error model, equation (4). The variance of the true exposure conditioned on the observed value is the variance of the error term egi, V(µgi|) = V(egi) (
), and the conditional variance is given by
 | (5) |
,µgi) = V(
) (Appendix 2). This equation implies that the error variance is approximately equal to the between-worker variance when the number of sampled workers (k) is sufficiently large.
The effect of group-based exposure assessment on logistic and Cox regression: approximate equations and simulations
Attenuation in logistic model
Logistic regression takes the following form,
 | (6) |
(w) = 1/[1 + exp(–w)]. The logistic regression model and the probit regression models agree closely over a range of values of the predictor variable (µgi) with an adjusting value c:
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(t) is the cumulative density function of the standard normal distribution (McCullagh and Nelder, 1983). When we have a Berkson error structure with normality in the exposure variable [i.e. the conditional (exposure) variable µgi| is normally distributed with mean and variance 
], using the probit regression model, it can be shown that
 | (7) |
and 
denote the intercept and the association parameters, respectively, based on the group exposure values. These are a function of error variance in exposure variable, 
, i.e.
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It follows that when the measurement error variance is constant, even in the presence of a Berkson error structure, the logistic regression association parameter is biased downward from the true value ß1. With the group-based strategy, the approximate relationship, between the association parameters based on the true values and the observed values, is given by
 | (8) |
) approximates the between-worker variance (
), the association parameter ( 
) is also biased downward.
Attenuation in Cox proportional-hazards model
In general, the hazard function depends on both time and a set of covariates. The proportional-hazards model separates these components by specifying that the hazard rate for survival time of a subject is
 | (9) |
, is not necessarily the same as the association parameter (ß1) in the logistic model (Green and Symons, 1983). When the baseline hazard is a constant, 
, then
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with the mean (the property of the Berkson error structure) and variance , the expected Cox proportional hazard function is given (Deddens and Hornung, 1994) as
 | (10) |
However, we need to be careful with this approach because two approximations are used: one linking the logistic and probit models, which requires that the probability of diseases should lie between 0.1 and 0.9, and the other linking the logistic and Cox proportional-hazards models, which requires a condition that neither the cumulative incident over the follow-up period nor the combined effects of the covariate can be too large: 
and ß0 = ln(T). Therefore, the association parameters in logistic model will be approximately equal to that of Cox proportional-hazards model when the baseline hazard is constant with low risk factor and when the follow-up period is not too long, so that the probability of disease is neither rare nor too common.
Taylor series expansion is a power expansion up to order n of a function at a point and is used to approximate a function by a series. The parameters of each survival function are a function of the error variance, i.e. 
= function () and () equals; function (). If there is no measurement error in exposure, the error variance is 0, and the parameter estimated with the observed values is equal to the true parameter. We expand both parameters at the point, 
, in the first degree of the Taylor series and then let the coefficients of the series in the models be equal to find a relationship between parameters given the true exposures and the observed values with measurement error.
That is, by using (i) differentiation with respect to the observed exposure when it is zero and (ii) the Taylor series expansion of both survival functions [based on the equations (7) and (10)], around the error variance of zero, a relationship between the two parameters in a certain range of is given by
 | (11) |
indicates the association parameter based on the observed values in the Cox proportional-hazards model, which depends on measurement error variance, i.e. the between-worker variance with the group-based strategy (Appendix 4). Therefore, the estimate of the association parameter in the Cox proportional-hazards model for a given observed group mean exposure ( ) is approximately equivalent to the estimate with the true exposures divided by the attenuation factor of 
. It can be expected that this approximation will be poor when the between-worker variance is large and disease is either very rare or common.
Simulations
Simulations were performed to examine attenuation in association parameter estimates in logistic and Cox proportional-hazards models with group-based exposure assessment. We considered an occupational cohort study with time-invariant exposure that has five exposure groups, each group containing 1000 subjects and 100 exposed time periods (e.g. days, months) for each subject. We further assumed that disease risk depended only on exposure intensity, not on the duration of exposure. Lastly, we assumed that measurements of exposure for a sample of k workers in each group are obtainable (k = 10, 50, 100) and that measurements from n different time periods (e.g. days, n = 2) will be available for each sampled worker. We need to have repeated measurements in each sampled worker to construct a measurement error model of exposure assessment, since it is assumed that the true exposures of each worker are randomly varied among workers sampled with the between-worker variance. This is typical for an occupational exposure database that one might be able to obtain or construct for a given study.
The log of exposures was assumed to be normally distributed with means 1.1, 2.1, ... , 5.1 for the first group to fifthe group, respectively, with the between-worker standard deviation (
B) taking on values of 0, 0.1, 0.5, 1 and 1.5, and the within-worker standard deviation (W) taking on values of 0, 0.5, 1.5 and 3. We chose the variance components on the basis of the paper by Kromhout et al. (1993), which summarized the between- and within-worker variance components generally observed for airborne contaminants in workplaces.
The true association parameter was set to 0.2, 0.4 and 0.6 for both logistic and Cox proportional-hazards regression models, and –4 was used as the intercept parameter in the logistic model, and we set the constant baseline hazard at 0.01 in the Cox proportional-hazards model in order to make the survival functions in the logistic and Cox proportional-hazards models approximately equivalent.
For each case, 1000 simulations were performed. The estimated mean exposure for each group, formulated in accordance with equation (1), was assigned to all workers in a given group. A Bernoulli random variable was used for the status of disease in the logistic regression model with the true exposure covariates (µgi). On the basis of the specific probability, each subject was assigned either 1 or 0 disease status using RANBIN function in SAS software (version 8), and in the case of Cox proportional-hazards model, the survival times were generated using an exponential random variable with true hazard 0.01 exp(µgi); we assumed that there was no censoring. The association parameters were estimated using LOGISTIC (for logistic model, 
) and PHREG (for Cox proportional-hazards model, 
) procedures of SAS software (version 8). Attenuation (for graphical presentation of the results) was estimated as the ratio of the association parameter estimate obtained with the group mean to the true parameter value, i.e. 
for logistic model and 
for Cox proportional-hazards model. The ratios of the estimates based on group means with true exposures (
and 
) were calculated (not shown in the article): they showed very similar behavior to the ratio with the true parameter.
The true logistic disease model is given by P(Zgi = 1 | µgi] = (ß0 + ß1µgi) while the fitted disease model given the observed value, , is 
. The true Cox proportional-hazard model given the true exposure is h(t | µgi) = exp( µgi) and the fitted proportional-hazard model given the observed value is 
]. In Tables 1 and 2 we report association parameter estimates, absolute attenuation and mean square error (MSE) of the simulation based on the group exposure mean () in the logistic and Cox proportional-hazards models with ß1 = = 0.6, varying within-worker variability (W = 0, 0.5, 1.5 and 3) and the sampled number of workers (k = 10 and 100). The attenuation in association parameter estimates in the logistic model, when using the group exposure mean for each worker in the group, does not appear to be sensitive to the magnitude of the within-worker variability when the number of measured workers is sufficiently large. The attenuation is severe only when the between-worker variability is large and the number of measured workers is small. For example, if 100 workers are sampled per group, the true association parameter is 0.6 (ß1 = = 0.6) and the within-worker standard deviation is 0.5, the attenuation is 4% if the between-worker standard deviation (B) is equal to 1 and 17% if it is twice as large (B = 2). However, the change in attenuation is much larger, from 6 to 23%, if only 10 workers are sampled per exposure group. Also, if the number of sampled workers per group is small, the attenuation depends on both between- and within-worker variability (i.e. measurement error assumes a structure that is more of classical than Berkson type). For the Cox proportional-hazards model, the tables show similar pattern of results as in the logistic model, but the attenuation is more severe. For example, when the within-worker standard deviation is 0.5 and the between-worker standard deviation is 1, the attenuation in the Cox model (19%) is 
5 times greater than that in the logistic model (4%). MSEs of the association parameter estimates tend to increase with the between-worker variability. Similar patterns were observed for smaller true association parameters (0.4 and 0.2).
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) regressions with different values of between- (
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) and Cox (Figures 1–4 further illustrate the patterns seen in Tables 1 and 2 and compare them with theoretical predictions of equation (8) for the logistic model and equation (11) for the Cox proportional-hazards model. As expected, the trend of attenuation is close to the theoretically predicted one if between-worker variability is small and for larger values of between-worker variability if the number of measured workers is large. Figure 1 indicates how vulnerable our theoretical results are to small sample size (k = 10). It would appear that attenuation is related mostly to increase in
B, as with large sample sizes. However, when W > B even at small B we observe some attenuation that is greater than predicted by theory. For large sample size, the equation for the logistic model appears to be quite appropriate in the range of between-worker standard deviation <2, but the equation for attenuation in the Cox proportional-hazards model provides a good approximate solution in a narrower range: with between-worker standard deviation <1 (Fig. 2). Figures 3 and 4 show that the equation for logistic regression predicts the attenuation well when the number of measured workers is large (
50) regardless of the magnitude of within-worker variability (W = 0.5 and W = 1.5). The situation is more complex with the Cox proportional-hazards model. The theoretically predicted and observed attenuation in Cox model are in close agreement when between-worker variability is small and the number of measured workers is large (50) regardless of the magnitude of within-worker variability (as with the logistic regression). However, as the between-worker variability increases, predictions of attenuation become progressively worse regardless of the number of measured workers, with the theory overestimating the actual attenuation. If the number of sampled workers is small, the within-worker variability affects the attenuation.
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Figure 5 shows the attenuation of various association parameter values (0.2, 0.4 and 0.6) for constant values of the within-worker variability (
W = 0.5) and the number of sampled workers (k = 100). In each case, the pattern of attenuation is similar, but smaller association parameters give smaller attenuation. Similar patterns were observed for large values of within-worker variability.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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We have demonstrated that the group-based strategy leads to a Berkson type error structure when there are large numbers of subjects with exposure measurements in each group, and the error variance approximates the between-worker variance. It should be noted that a pure Berkson error structure is not achieved through grouping in observational studies.We developed approximate equations for attenuation in association parameters that results from the application of a group-based exposure assessment. The approximation for the equations holds quite well in a small range of between-worker (subject) variances for both logistic and Cox proportional-hazards models. The association parameter estimates in Cox proportional-hazards models are more severely attenuated than those in logistic models.
There is a belief (Armstrong, 1990) that there is no attenuation in the estimation of the association parameters in the logistic and Cox proportional-hazards models if the error variance is constant for all observations and exposure is assessed on a group level, since the group-based strategy leads to the Berkson error model. However, when a random sample from a population is used for exposures assessment, it produces a classical error model, not a truly Berkson error model. Classical error structures arise when we wish to ascertain the value of some unknown quantity. We measure repeatedly from a population and the observed value is considered to be measured with unbiased error. When errors are independent of the true quantity, we have a classical error model. Classical measurement errors produce attenuation in the association parameter estimates in the linear regression models (Berskon, 1950) and in logistic regression models when the measurement error variance is the same for all subgroups of subjects (Carroll et al., 1984; Reeves et al., 1998; Spiegelman and Valanis, 1998; Heid et al., 2002) and it appears possible to adjust for bias in Cox proportional-hazards models (Nakamura, 1992; Wang et al., 1997; Kong, 1999; Augustin, 2004; Li and Ryan, 2004).
A true Berkson error structure arises when a nominal measured amount is used instead of the unknown true exposure, so that the assigned value is fixed by design and the true value varies due to errors (Carroll et al., 1995). Therefore, an exposure model that reflects the group-based exposure assessment strategy cannot lead to a truly Berkson error model, but it gives a property of the Berkson error model (
) when there are enough workers with exposure measurements in each group. With this property, the situation is approximately analogous to the Berkson type error model.
We found that exposure error variances in group-based exposure assessments (with a sufficient number of measured workers per group) can be approximated by the between-worker variance. This allows us to bridge the gap between the literature on group-based exposure assessment and measurement error papers that have not traditionally reflected the measurement error model that has emerged in occupational epidemiology. Usually, the parameters are unidentifiable in measurement error models, so that the error variance must be known a priori, or estimated from a separate validation dataset. Our results allow estimation of error variances in the group-based exposure assessment without resorting to additional assumptions or conducting supplementary experiments. Furthermore, we have derived an approximate equation for attenuation in Cox proportional-hazards model with Berkson type error structure. The equation for the attenuation enabled us to examine the approximate behavior of the association parameter estimates in populations with different characteristics using a group-based exposure assessment strategy. This may enable us in the future to adjust for attenuation in association parameter estimates obtained in such studies. Finally, we showed, for the first time, that the logistic model and the Cox proportional-hazards model could not be used equivalently with measurement errors in exposure.
However, if the number of sampled workers is small, the attenuation becomes severe when both between- and within-worker variabilities get large: since the properties of the Berkson error structure cannot be satisfied, this leads to the classical error model in which the attenuation can be expected to depend on both between- and within-worker variabilities. Also, if the probability of disease is rare for all groups (p < 0.02), the approximation relating the logistic model to the probit model is poor.
We note that the probability of disease should satisfy the approximation conditions (McCullagh and Nelder, 1983). The approximation from logistic to probit functions holds for 0.1 < p < 0.9 and the quantity 
should fall in the range of (0, 1) for the Taylor approximation from logistic to Cox proportional-hazards models. In the simulations, the expected true probability of diseases [according to P(Zgi = 1|µg) = (ß0 + ß1µg)] among groups was between 0.03 and 0.28 for ß1 = 0.6, between 0.02 and 0.12 for ß1 = 0.4 and between 0.02 and 0.05 for ß1 = 0.2 (the attenuation is still close to the expected line, which is not shown in Fig. 5) with the same amount of true exposures.
We have demonstrated that the group-based strategy in logistic models works well, leading to little bias in estimating the association parameter of the model. The strategy is simple to implement with standard software using a sufficient statistic of the true common mean. For the Cox proportional-hazards models, the group-based strategy does not seem to address the problem effectively under the assumptions we considered, namely fixed baseline hazard, short length of following-up, time independent exposures and no censoring.
It is also noted that the group-based strategy does not lead to the common calibration method for individual-based main study/validation study design (Rosner et al., 1989; Spiegelman et al., 1997, 2001), because in our case only a sample of subjects has any measurements, unlike the classical calibration method in which all subjects have one measurement, but a subset also has repeated and/or gold standard measurements. The common calibration method considers neither sampling nor grouping.
Our results have the following important limitations. First, our results apply to the situations where logarithms of exposure are reasonable proxies for biologically relevant doses, i.e. with a log–log model, log(rate) = ß0 + ß1 x log(exposures). However, toxicological arguments indicate that log-linear models, log(rate) = ß0 + ß1 x (exposures), may be more plausible for many exposure-disease associations (Rappaport, 1991) and therefore they should lead to more realistic inferences (e.g. risk assessments) (Steenland and Deddens, 2004). Application of a group-based strategy when the true exposure-disease is log-linear is a subject of ongoing research. Second, the group means are considered fixed instead of random effects. Third, we consider constant within and between-worker variability. Fourth, although the condition of equal numbers of workers per group and measurements per worker are rather restrictive, the purpose of this study is to provide an evidence of impact of using the group-based strategy in logistic and Cox proportional-hazards models for further development of exposure assessment methodology. Fifth, we considered the case of homogenous variance components, but Burstyn et al. (2006) studied the heteroscedastic measurement error under group-level exposure assessment. The results indicate that small-to-negligible bias can be expected to result from heteroscedastic between-worker variances: Cox proportional-hazards models can produce attenuated risk estimates, while logistic regression may result in an overestimation of risk gradient. In the case of small number of workers observed in each group, we need more repeated measurements of each worker to lead to a Berkson error structure with the group-based assessment. If the number of workers in each group is different in such a case, the situation leads to the heteroscedastic variance components [suggested by equation (5)]. Sixth, the fundamental assumption about exposure assessment that we consider is that it is based on sufficient number of measurements from each group to satisfy the condition under which approximate Berkson type error emerges (3). We do not address the question of how many measurements (and of what type—more workers or repeats) are needed to meet this condition. Our simulations suggest that if we are able to monitor 50 workers on 2 days from each group, then the sample size is sufficient. A quantitative answer to this query awaits further research.
In summary, our results provide several new insights into the estimation of association parameters in the logistic and Cox proportional-hazards model when exposures are assessed with the group-based strategy. The understanding of the approximate behavior of the association parameter estimate in the models can help design efficient grouping strategy in practical situations. In general, it would seem that maximizing the number of workers with exposure measurements in each group is desirable. In collecting measurements for exposure assessment, the need to have a repeated measures design is apparent to enable the investigators to study measurement error structure.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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1. Proof of Properties under Classical Error Structure
The joint distribution of (µgi,
) is given by
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Where
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2. Proof of properties under Berkson error structure
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3. Attenuation for Logistic Model
, where 
c = 0.588 and 0.1 < p < 0.9. (t) is the cumulative density function for the standard Normal distribution.
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4. Attenuation for Cox Proportional-Hazard Model
The logistic regression function with true risk factor Z with coefficients ß0 and ß1 is given as
 | (A1) |
 | (A2) |
is that the coefficients of the risk factor Z in the logistic regression will be approximately equal to the coefficient for the Cox model. This result holds when exp(ß0 + ß1Z) is small, implying that sum of ß0 = ln(T) and the linear combination of risk factors is small (Cox, 1972).
When the covariates Z are not available, but X are observable with error in the standard Cox proportional-hazards model, it has been shown that there exists a solution of maximum partial Cox likelihood function and that the naive estimate of (based on X) is a consistent estimator of the solution, 
,and is a function of , which reflects the average magnitude of the measurement error (Nakamura, 1992; Li and Ryan, 2004; Heid et al., 2002). We consider a range of 
, where 
0 and [ß1 and 
] and [ and 
] are approximately equivalent. Then, an approach to find the relationship between estimates of based on the true exposure and those of * based on observed exposures is to use approximate equivalence of survival functions of the two models (logistic and Cox) given the observed exposures (X) when the disease risk is low in this range.
Suppose that true exposure conditioned on the observed exposure, Z|X, is normally distributed with mean E(Z|X) and variance 
, the expected probability of getting diseases given the observed value X in the logistic regression model (10) is approximately
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Where
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When the conditional probability function is concentrated (Prentice, 1982) the expected hazard in Cox regression model is
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The logistic survival function and Cox model survival function will be approximately equivalent when
 | (A3) |
.
In order to make a relationship of the slope parameters between two models when exposures are measured correctly, we need a condition that the baseline risk in the logistic is equivalent to the logarithm of the baseline risk times the duration of the follow-up: 
. Since we consider a small range of the error variance, where the parameter given the observed value is approximately equivalent to the parameter given the true value 
, the equivalence of two survival functions holds when
 | (A4) |
Then, the equivalence (A4) with Berkson error model, E(Z | X) = X and 
, can be rewritten as
 | (A5) |
where
By taking derivative of the equation (A5) with respect to X and at X = 0, we obtain
 | (A6) |
By the Taylor series expansion at on the right side of (A6), the approximation
 | (A7) |
Therefore, an approximate expression of the relationship between the parameters and in a range 
is
 | (A8) |
with large number of workers in each group, the equation (11) holds.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMethods and resultsDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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The authors wish to thank Drs Nicola Cherry, Ben Armstrong and James A. Deddens for their helpful comments. This work was funded by the grant from the Canadian Cancer Etiology Research Network (http://www.ccern.org/).
Received November 4, 2005; in final form March 5, 2006
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