Journal of Exposure Science and Environmental Epidemiology (2010), 1–10 r 2010 Nature Publishing Group All rights reserved 1559-0631/10/$32.00
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Bias in the estimation of exposure effects with individual- or group-based exposure assessment HYANG-MI KIMa,e, DAVID RICHARDSONb, DANA LOOMISc, MARTIE VAN TONGERENd AND IGOR BURSTYNe a
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada Department of Epidemiology, University of North Carolina, Chapel Hill, USA c Department of Environmental and Occupational Health, University of Nevada, Nevada, USA d Institute of Occupational Medicine, Riccarton, UK e Department of Medicine, University of Alberta, Edmonton, Alberta, Canada b
In this paper, we develop models of bias in estimates of exposure–disease associations for epidemiological studies that use group- and individual-based exposure assessments. In a study that uses a group-based exposure assessment, individuals are grouped according to shared attributes, such as job title or work area, and assigned an exposure score, usually the mean of some concentration measurements made on samples drawn from the group. We considered bias in the estimation of exposure effects in the context of both linear and logistic regression disease models, and the classical measurement error in the exposure model. To understand group-based exposure assessment, we introduced a quasi-Berkson error structure that can be justified with a moderate number of exposure measurements from each group. In the quasi-Berkson error structure, the true value is equal to the observed one plus error, and the error is not independent of the observed value. The bias in estimates with individual-based assessment depends on all variance components in the exposure model and is smaller when the between-group and between-subject variances are large. In group-based exposure assessment, group means can be assumed to be either fixed or random effects. Regardless of this assumption, the behavior of estimates is similar: the estimates of regression coefficients were less attenuated with a large sample size used to estimate group means, when between-subject variability was small and the spread between group means was large. However, if groups are considered to be random effects, bias is present, even with large number of measurements from each group. This does not occur when group effects are treated as fixed. We illustrate these models in analyses of the associations between exposure to magnetic fields and cancer mortality among electric utility workers and respiratory symptoms due to carbon black. Journal of Exposure Science and Environmental Epidemiology advance online publication, 24 February 2010; doi:10.1038/jes.2009.74
Keywords: quasi-Berkson type error structure, non-differential measurement error, bias, mixed exposure model, homogenous error.
Introduction In epidemiological cohort studies of occupational and environmental exposures, individual exposure measurements are often not available for all members of the study, whereas health outcome measures are obtained for each individual. In such settings, a commonly employed approach is to derive exposure estimates through a group-based strategy (Loomis and Kromhout, 2004) (also know as semi-individual or semiecological study design). Individuals are grouped according to shared attributes, such as job title or work area, and assigned an exposure score, usually the mean of some concentration measurements made on samples drawn from the group.
1. Address all correspondence to: Dr. Hyang-Mi Kim, Department of Mathematics and Statistics, The University of Calgary, 2500 University Drive N.W, Calgary, AB, Canada, T2N 1N4. Tel: þ 403 220 5691. Fax: þ 403 282 5150. E-mail:
[email protected] Received 24 August 2009; accepted 30 November 2009
Interestingly, in some settings, the use of a group-based strategy for assigning exposure scores can result in a less biased estimate of an exposure–disease association than would be achieved through individual exposure measurements. It is well-known that non-differential measurement errors in individual exposure estimates may lead to bias in estimates of exposure–response associations; a group-based strategy can minimize this attenuation bias by creating an error structure that has some properties of a Berkson-type error. The Berkson error model was originally proposed for experimental situations, in which the investigator attempted to set the exposure at a target value, but because of imprecision of instrumentation, its true value was randomly distributed around the target (Berkson, 1950). If the experiment was replicated many times with the same target value, the true value would be randomly distributed with an estimated mean approaching the target value: the errors would be independent of the target value. Kim et al. (2006) showed that the group-based strategy leads to an approximate Berkson measurement error structure when data are
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available for a large number of subjects in each group. It is approximate in the sense that assigned group means may not be independent of error. To account for this complexity, we formally introduce a novel quasi-Berkson error model in this paper. When members of a cohort are grouped, mixed-effects models are often used to fit to exposure data, as these models allow an analyst to treat the group as either a fixed or a random effect. A question arises as to whether estimation using random grouping methods (RGE) produces exposure– response results that are consistent with those obtained using fixed group effect (FGE) modeling. In some settings, the rationale for treating exposure groups as fixed is clear. An occupational cohort study that makes use of a previously published job–exposure matrix implies an exposure assessment in which groups are fixed. If such an assumption is made, then conclusions can be drawn only about exposure– response association in the occupational groups that were investigated. This may well be desirable in narrowly targeted studies of uncommon exposures (that only occur in the studied workplaces) or in investigations undertaken by one company0 s health and safety department (wherein the goal is to simply understand health risk to employees of a given enterprise). In contrast, an occupational cohort in which the investigator wishes to draw conclusions about exposure– response not only among a fixed set of studied occupational groups but also in all possible occupational groups, it is more natural to assume that groups are created through a random draw of all possible groupings. This desire to generalize findings beyond, say, one occupation in a given factory to all similar jobs in a specific industry requires us to assume that the observed groups provide information about the characteristics of all possible exposure groups. This assumption enables an investigator to estimate the variation in exposure between groups (Goldstein, 2003). The following assumptions were made for the purposes of our study: a normal exposure distribution (a log transformation for log-normally distributed exposures was applied in the examples), known constant error variance components, no systematic error, non-differential error and no correlation among errors. We also focused on the scenario in which the disease under study was neither common nor extremely rare. Throughout this paper, we define exposure as intensity or concentration of substance, ignoring complications that arise from time-varying exposure patterns and accumulation of dose due to long-term exposure. Our first aim is to examine, from a theoretical perspective, the use of fixed and random group-based strategies for assigning exposure scores in an epidemiological cohort. Next, we illustrate how a researcher may obtain valid estimates of exposure–disease associations through linear or logistic regression methods, even when exposure measurements for all subjects are not available, as long as an adequate 2
sample of measured values for each group is drawn and the between-group variability is large. The impact of different grouping schemes on parameter estimation is illustrated in two examples: (a) occupational exposure to magnetic fields among workers with any cancer (Kromhout et al., 1995; Saviz and Loomis, 1995) and (b) respiratory health of employees in the European carbon black manufacturing industry in relation to exposure to carbon black dust (van Tongeren et al., 1997). In section 2, we present the bias equations for individual- and group-based assessments for both random and FGE models. Findings derived from the simulation studies are described in section 3. In section 4, we provide two examples; and, the findings are discussed in section 5.
Theoretical Study Theoretical studies were considered by assuming an additive measurement error model together with linear and logistic response models. For both individual- and group-based strategies, the conditional mean of the linear response model, given the observed exposure (Harville, 1977), was used to obtain the attenuation factor for the regression coefficient in the linear model. For the logistic model, at first, the expressions of the conditional mean and variance of the true exposure, given the observed values, were derived and subsequently, the expression of the attenuation in the response model was found. For the group-based strategy, we considered two exposure models: the RGE exposure model in which the group is regarded as a random component, and the FGE exposure model, in which the group is a fixed component. For both exposure models, the Berkson error was induced from a classical error structure under the assumptions that (1) the number of measurements in each group is sufficiently large to estimate the true group means closely and (2) the group means are not correlated with the measurement error in the Berkson error model. As this approximation of the Berkson error depends on the sample size and the covariance between the group mean and measurement error, we call this a quasiBerkson error model. In the presence of fixed unknown group means, we assume that the group means are fixed with different between-group variabilities (o2) for each distinct grouping scheme. However, in deriving the attenuation equation for logistic regression models, the assumption of normality of the exposure distribution is required, and the assumption fails when the between-group exposure variability (o2) is large for the FGE model. In a such situation, we have exposures being distributed as a mixture normal distribution with the number of components equal to the number of groups. Therefore, an expression for attenuation cannot be easily derived, and in this paper, we do not explore the theoretical behavior of the regression coefficient when o2 is large. Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
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Models We postulate a classical exposure measurement error model. For the setting of RGE, the measurement error model is Wgi ¼ m þ ng þ ggi þ Zgi ¼ Xgi þ Zgi ð1Þ and for FGEs, the measurement error model is Wgi ¼ mg þ ggi þ Zgi ¼ Xgi þ Zgi
ð2Þ
where Wgi represents the observed exposure on the ith subject from the gth group, Xgi and represents the true exposure of the subject; m is the common true mean; mg is the fixed group mean, g ¼ 1, y, G; ngBN(0,sg2) is a random effect due to group g, g ¼ 1, y, G; ggiBN(0,sb2) is a random effect due to subject i in each group, i ¼ 1, y, N; ZgiBN(0,s2Z) is a random effect due to measurement error and daily fluctuations in exposure that may arise in occupational settings from variability of daily tasks (e.g., day-to-day variability for full-shift measurement, Wgi), and the errors are mutually independent. For the association between exposure and response, we consider the linear and logistic regression models given, respectively, by Ygi ¼ b0 þ b1 Xgi þ egi where b0 and b1 are the intercept and the slope parameters, respectively, and egiBN(0,se2), and PðZgi ¼ 1jXgi Þ ¼ Lðb0 þ b1 Xgi Þ where Zgi is a binary variable for the health outcome and L(t) ¼ 1/(1 þ exp(t)). The conditional expectation of response Ygi given the ¯ g: group observed values, Wi ¼ (Wgi: individual values, W mean), for the linear models is ð3Þ E½Ygi jWi ¼ b0 þ b1 E½Xgi jWi and for logistic regression models is E½PðZgi ¼ 1jXgi ÞjWi E½F½cðb0 þ b1 Xgi ÞjWi 0
0
¼ Lðb0 þ b1 EðXgi jWi ÞÞ
ð4Þ
where c ¼ 0.588, b00 and b01 are functions of V(Xgi|Wi) by using the approximation to probit regression model (Reeves et al., 1998) and F(t) is the cumulative density function of the standard normal distribution. By obtaining E(Xgi|Wi) ¼ f(Wi) and V(Xgi|Wi) ¼ c(Wi), the bias factor can be formulated (Burr, 1988; Wang et al., 1998; Carroll et al., 2006).
Bias In this section, the conditional expectation and variance are calculated for both the RGE (Eq. (1)) and FGE (Eq. (2)) models and used to derive the bias factors for both linear and logistic regression models. Individual-Based Strategy With the RGE model, E(Xgi|Wgi) ¼ m(1l*) þ l*Wgi, where l* ¼ (sg2 þ sb2)/(sg2 þ sb2 þ s2Z) and the conditional variance is Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
given by V(Xgi|Wgi) ¼ V(Xgi)(1l*) ¼ s2Zl*. With the FGE (1l0) þ l0Wgi and V(Xgi|Wgi) ¼ model, E(Xgi|Wgi) ¼ m l20s2Z þ (1l0)2sb2, where l0 ¼ (o2 þ sb2)/(o2 þ sb2 þ s2Z), the P Þ2 =g ðmg m between-group variability is defined as o2 ¼ P ¼ mg =g. and m On the basis of Eqs (3) and (4), we obtained approximate equations that describe the relationship between the true regression coefficient, b1, and the observed regression coefficient, b*1, with the observed exposures, Wi. In the linear regression, b1 ¼ l b1 and b1 ¼ l0 b1 for RGE and FGE models, respectively. In the logistic regression context with a RGE model, l b1 ð5Þ b1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c b21 s2Z l þ 1 and with the FGE model when the between-group variability is small so that the exposures are approximately normally distributed, l0 b1 b1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ c2 b21 ½s2Z l20 þ ð1 l0 Þ2 s2b þ 1 The attenuation factors, l* ¼ (sg2 þ sb2)/(sg2 þ sb2 þ s2Z) 2 2 2 2 2 (l0 ¼ (o þ sb)/(o þ sb þ sZ)) for the linear and logistic models (Eqs (5) and (6)) go to l ¼ sb2/(sb2 þ s2Z) as the between-group variability decreases. There is attenuation as lrl*(l0)p1. There is less attenuation when the between-subject variability increases for a model with fixed between-group variability, ((sg2(o2) þ 2 2 2 2 )/(sg2(o2) þ sb,1 þ s2Z)r(sg2(o2) þ sb,2 )/(sg2(o2) þ sb,2 þ s2Z) sb,1 2 2 r1 if sb,1osb,2). In addition, when the measurement error variance increases, the attenuation increases.
Group-Based Strategy ¯ g) of the observed In the group-based strategy, an average (W measurements for a group g is taken to apply to all subjects in P g ¼ Ygi =n, the group (e.g., from the same job title); W where n is the number of subjects from a group of the total ¯ g is an size (N). For each subject, this group mean W approximation of his/her true exposure (Xgi), if the number of measurements is reasonably large. The conditional expectation of the true exposure given the g ¼ W gþ observed group mean in this case is E½Xgi jW ðm 1ÞðWg mg Þ, where m ¼ covðXgi ; Wg Þ=varðWg Þ. The derivation is made under the assumption of a classic measurement error model. If the number of subjects in each group is sufficient for the true mean and the estimated group g E½Xgi , then we have mean to be close in value, that is, W E½Xgi jWg Wg . By showing an approximate property, g ¼ W g Þ, we postulate a quasi-Berkson error ðE½Xgi jW ¯ g) and true exposures model with the assigned group mean (W (Xgi), that is, g þ egi ; E½egi jW g ¼ 0 Xgi ¼ W ð7Þ 3
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This approximation may depend on the sample selected, but we need only a moderately large sample size to obtain this property, wherein the true exposure of each worker randomly ¯ g, and this mean is varies about the group mean, W approximately the true group mean. This situation is analogous to the Berkson error model, in which the true exposure, given the observed exposure, has an expected value equal to the observed exposure. However, the model is not a ¯ g) is true Berkson error model unless the group mean (W independent of the error (egi) (Kim et al., 2006). With RGE, it is necessary to consider the possibility of correlation ¯ g and egi as the group means are random between W components correlated with the model error (egi) and egi may be correlated with the model error (egi). The Berkson error structure will be approximated if this covariance is small. The covariance can be either positive or negative and a g ; egi Þ ¼ xðsg Þ. function of sg, covðW ¯ g, egi) ¼ 0 and With FGE, one can derive that cov(W ¯ ¯ cov(Xgi, egi) ¼ V(Xgi|Wg)a0, as Wg can be considered a constant when the number of measurements for the group mean is large. The model, however, is not a truly Berkson ¯ g, egi) ¼ 0 and it does not imply that the error model as cov(W observed value and the error are independent, which is required for the Berkson error model. Applying the RGE model in the linear regression model (Eq. (3)) leads from non-differential error to differential error ¯ g þ egi, ¯ g and egi. As Xgi ¼ W if covariance exists between W * ¯ the linear model is expressed as Ygi ¼ b0 þ b1Wg þ egi, where g Þ ¼ covðb1 egi þ egi ; W gÞ e*gi ¼ b1egi þ egi. Thus, covðegi ; W b1 covðegi ; Wg Þ, that is, the model error (egi*) is correlated with ¯ g). With RGE, the quasi-Berkson error structure the covariate (W (Eq. (7)) and non-zero covariance, g ; egi Þ covðW ð8Þ b1 ¼ b1 þ s2g whereas with FGE, b1 b1
ð9Þ
In a logistic regression analysis (Eq. (4)), it is necessary to find a relationship among the variances in Eqs. (1) and (2) under the quasi-Berkson error model to derive the amount of attenuation in the response models. Using the RGE model, the error variance, se2, is obtained: g ; egi Þ 0 s2e ¼ V ðegi Þ ¼ s2b 2covðW g ; egi Þ 0:5s2 . This equation implies that the bias where covðW b with the RGE model depends on the between-subject variance, as well as the covariance between the group mean and the measurement error of the quasi-Berkson error structure: b1 b1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g Þ þ 1 c2 b21 ½s2b 2covðegi ; W
ð10Þ
Using the FGE model, by using a property of the Berkson error structure when the between-group variance is small, the error 4
variance is obtained by gÞ s2e ¼ V ðegi Þ ¼ V ðXgi W g Þ 2covðXgi ; W g Þ ¼ V ðXgi Þ s2 ¼ V ðXgi Þ þ V ðW b This equation implies that the bias with the FGE model depends on the between-subject variance when the number of sampled subjects (n) is sufficiently large: b1 ffi b1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 b21 s2b þ 1
ð11Þ
When grouping is used to assess exposure, the measurement error variances vanish as the sample size increases for the FGE model. However, for the RGE model, the correlation between the measurement error (egi) and the group mean leads to bias. That is, the group-based strategy reduces the effect of measurement error in the regression coefficient estimation in FGE, but not for RGE. From Eqs (8) and (10), the attenuation in both the linear and logistic models depends on the between-group and between-subject variability with the RGE model. As the between-subject variability increases, the attenuation increases, whereas when the between-group variability increases, the bias decreases. For the FGE model (Eq. (11)), the derived expression for bias does not include the between-group variance component. One reason for this is that we consider only small group variability, so that the distribution for the exposures has an approximate normal distribution. However, the simulation study (below) shows that the bias decreases as the groups are far away from each other, just as with the RGE model.
Sample Size in Group-based Exposure Assessment The extent to which this quasi-Berkson error model fits the data depends on (a) the variance of the group mean and (b) the covariance between the group mean and measurement ¯ g approaches error. It can be shown that the variance of W zero as the sample size, n, increases. For each group, the variance of group means can be expressed based on the sample size using a binary variable, which indicates that the observation is in the sample with probability of Nn : g Þ ¼ 1fðs2 þ s2 þ s2 Þ þ ð1 n Þm2 g þ ð1 1 Þs2 for V ðW g Z b n N N g RGE. The variance of the group mean depends on the between-group variance, regardless of the sample size. This variability together with the covariance affects the parameter estimation in the response models (Eqs (8) and (10)). Figure 1 shows how the variance changes as the sample size increases. For the FGE model, the variance of group g Þ ¼ 1fðs2 þ s2 Þ þ ð1 n Þm2 g, starts to apmeans, V ðW Z b n N g proach zero with a relatively small number of exposure measurements drawn from each group. This condition leads to the quasi-Berkson model being a good approximation of the Berkson error model, so that the bias in the slope parameter is negligible. Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
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4 N = 500; μ = 1; μg = 1; σb = 0.5
variance
3 2 RGE: σg = 1 1
RGE: σg = 0.25 FGE
0 0
5
10 15 sample size
20
25
Figure 1. Variance of group mean in relation to sample size.
Comparison between Individual and Group Strategies In a linear regression analysis with the RGE model, if the measurement error gets smaller, there is negligible attenuation in using the individual-based assessment, while there is bias depen¯ g, egi), and the betweending on the covariance term, cov(W 2 group variability (sg) with group-based assessment. However, with the FGE model, the group-based exposure assessment is always superior if the sample size is moderately large, because it leads to a quasi-Berkson error structure that gives no bias. In logistic regression analysis, for a given between-group variability, as the error variance (s2Z) gets smaller, there is negligible attenuation when using the individual-based assessment, while there is still bias depending on the error variability (se2) when the group-based assessment is used on the same data. Therefore, when the measurement error is small and the error variability, se2, is large, the estimates with individual-based assessment are expected to be less biased than that with group-based assessment (Eq. (5) versus Eq. (10) and Eq. (6) versus Eq. (11)).
Simulation study Simulations were performed to examine attenuation in the regression coefficient estimates in linear and logistic models with individual- and group-based exposure assessment and a disease with an expected risk of about 10% (PE0.1) and less than 10% (Po0.1). We considered a cohort with timeinvariant exposure that segregates into five exposure groups. We further assumed that disease risk depended only on exposure intensity and not on its duration. The measurements of exposure for a sample of n ¼ 20(100) workers were obtainable among exposure measurements of all 500 subjects (N) in each group. Each subject was measured only once, and it was assumed that all variance components were known (the between-group, between-subject and measurement error variance on each subject from the measurement error models). The mean of the 1000 sets of estimates and standard errors were calculated. In addition, the empirical Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
standard deviations of the estimates were calculated and the empirical mean square error (MSE) were obtained. The true regression coefficient was set to 0.3, and 4 (for Po0.1) or 2 (for PE0.1) were used as the intercept parameters for both regression models. The probability of disease, p, P(Zgi ¼ 1|Xgi) ¼ L(b0 þ b1(exposure: Xgi)), was calculated and used to assign binary disease status from a Bernoulli distribution. The exposures were assumed to be normally distributed with the common means of 0.1 and the between-group standard deviation of 0.3, 0.5 and 1 for the RGE model, and 0.1(0.3)1.3 (o ¼ 0.3), 0.1(0.5)2.1(o ¼ 0.5) and 0.1(1) 4.1(o ¼ 1) for the first group to fifth group for the FGE model. To see the impact of the between-subject standard deviation, we examined values that span a plausible range (Kromhout et al., 1993), that is, a small value of 0.7 (sb2 ¼ 0.49) and a large value of 1.414(sb2 ¼ 2). As the measurement error disappears with the grouping strategy (s2Z/sample size in each group), we considered only the measurement error standard deviation (s2Z) values of 1 for both the RGE and FGE models. For the group-based strategy, the estimated mean exposure for each group was assigned to all workers in a given group. The regression coefficients were estimated using the generalized linear model procedures of R software, which was developed by John Chambers and Hastie (1991) at Lucent Technologies. The corresponding author will make the R code used in the simulations available on request.
Individual-Based Strategy Bias depends on all variance components for the RGE and FGE models in both linear and logistic regression models. As the measurement error variance increases, the bias increases, as expected (not shown). When the measurement error variance is fixed as s2Z ¼ 1 (Table 1), the bias is reduced when the between-group variability and between-subject variability are large, which shows that the bias also depends on the variability of an unknown true covariate (sg2 þ sb2 for RGE or sb2 for FGE). With the RGE model, for example, when the between-group variability is sg ¼ 0.3 and the between-subject variability increases from sb ¼ 0.7 to 1.414, the bias decreases as the estimate (MSE) increases from 0.108 (0.037) to 0.202 (0.010) in a linear model and as the estimate (MSE) increases from 0.110 (0.039) to 0.202 (0.011) in the logistic model. Group-Based Strategy Tables 1, 2 and 3 show the results for the RGE and FGE models with the probability of disease of B10%, the condition assumed in the preceding theoretical developments. The tables also present the results for the analogous set of simulation parameters, except in the case of ‘‘rare’’, occurring in less than 10% of the subjects. Under the grouping, the bias depends on the between-group and between-subject variances, because the measurement error variance vanishes as the number of measurements increases. As the between5
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Table 1. Individual-based assessment with (a) b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (linear model); (b) b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (logistic model); (c) b0 ¼ 4, b1 ¼ 0.3 and sZ ¼ 1 (logistic model) sb ¼ 0.7 ^1 SE b (mean) (mean)
SD
sb ¼ 1.414 MSE
^1 b SE (mean) (mean)
sb ¼ 0.7 SD
sb ¼ 1.414
MSE ^1 SE b (mean) (mean)
(a) RGE (sg) 0.3 0.108 0.5 0.121 1 0.162
0.003 0.003 0.003
0.007 0.037 0.014 0.032 0.029 0.020
0.202 0.206 0.219
0.003 0.003 0.003
0.003 0.010 0.005 0.009 0.011 0.007
FGE (o) 0.3 0.121 0.5 0.149 1 0.214
0.003 0.003 0.003
0.003 0.032 0.003 0.023 0.002 0.007
0.206 0.214 0.240
0.003 0.003 0.002
0.003 0.009 0.003 0.007 0.002 0.004
(b) RGE (sg) 0.3 0.110 0.5 0.123 1 0.162
0.049 0.047 0.041
0.051 0.039 0.051 0.034 0.051 0.022
0.202 0.205 0.218
0.035 0.034 0.032
0.037 0.011 0.036 0.010 0.035 0.008
FGE (o) 0.3 0.122 0.5 0.150 1 0.212
0.044 0.039 0.027
0.044 0.034 0.039 0.024 0.027 0.008
0.205 0.213 0.237
0.032 0.030 0.023
0.031 0.010 0.029 0.008 0.023 0.004
(c) RGE (sg) 0.3 0.110 0.5 0.126 1 0.166
0.119 0.114 0.099
0.121 0.051 0.117 0.042 0.105 0.029
0.203 0.206 0.219
0.082 0.081 0.074
0.082 0.016 0.081 0.015 0.075 0.012
FGE (o) 0.3 0.122 0.5 0.148 1 0.210
0.104 0.090 0.059
0.106 0.043 0.092 0.032 0.058 0.011
0.204 0.213 0.239
0.074 0.067 0.048
0.074 0.015 0.068 0.012 0.048 0.006
sg (o) is the between-group standard deviation; sb is the between-subject standard deviation; and sZ is the measurement error standard deviation. SD and MSE are the empirical standard deviation and the mean squared error (1000 iterations).
group variability increases, the bias decreases, whereas as the between-subject variability increases, the bias increases in both response models. With the RGE model, when the sample size is 100, for example, and the between-group variability is sg ¼ 0.3 and the between-subject variability increases from sb ¼ 0.7 to 1.414, the bias increases as the estimate (MSE) decreases from 0.260(0.006) to 0.235(0.012) in the linear model and as the estimate (MSE) decreases from 0.261(0.092) to 0.224(0.091) in the logistic model (Table 2). When the between-group variability is large (sg ¼ 1) and the between-subject variability increases from sb ¼ 0.7 to 1.414, there is only a negligible change in both the estimate and MSE in the linear model, whereas the estimate decreases 6
Table 2. Group-based assessment with (a) n ¼ 100, b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (linear model); (b) n ¼ 100, b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (logistic model); (c) n ¼ 100, b0 ¼ 4, b1 ¼ 0.3 and sZ ¼ 1 (logistic model)
SD
MSE
^1 b SE (mean) (mean)
SD
MSE
(a) RGE (sg) 0.3 0.260 0.5 0.284 1 0.296
0.018 0.011 0.006
0.069 0.006 0.043 0.002 0.025 0.001
0.235 0.272 0.293
0.034 0.023 0.012
0.089 0.012 0.059 0.004 0.033 0.001
FGE (o) 0.3 0.289 0.5 0.296 1 0.299
0.010 0.006 0.003
0.034 0.001 0.022 0.001 0.011 0.0001
0.280 0.292 0.297
0.019 0.012 0.006
0.047 0.003 0.030 0.001 0.015 0.0002
(b) RGE (sg) 0.3 0.261 0.5 0.287 1 0.298
0.263 0.166 0.085
0.302 0.092 0.196 0.034 0.103 0.011
0.224 0.264 0.289
0.242 0.159 0.083
0.292 0.091 0.194 0.039 0.097 0.010
FGE (o) 0.3 0.287 0.5 0.293 1 0.296
0.133 0.077 0.036
0.140 0.020 0.080 0.007 0.038 0.001
0.272 0.282 0.288
0.129 0.076 0.036
0.138 0.020 0.081 0.007 0.039 0.002
(c) RGE (sg) 0.3 0.302 0.5 0.319 1 0.319
0.645 0.407 0.210
0.747 0.558 0.455 0.207 0.236 0.056
0.233 0.278 0.306
0.568 0.386 0.202
0.687 0.476 0.455 0.207 0.237 0.056
FGE (o) 0.3 0.284 0.5 0.293 1 0.298
0.318 0.182 0.080
0.329 0.108 0.188 0.035 0.083 0.007
0.274 0.289 0.297
0.305 0.176 0.078
0.322 0.104 0.184 0.034 0.081 0.007
sg (o) is the between-group standard deviation; sb is the between-subject standard deviation; and sZ is the measurement error standard deviation. SD and MSE are the empirical standard deviation and the mean squared error (1000 iterations).
from 0.298 to 0.289 with a negligible change of MSE in the logistic model. With FGE, the estimates do not change much as the between-group variability (o) changes, but the MSE decreases as the between-group variability increases in the logistic model. If the covariance between the group means and model error exists in the measurement error model, it is suspected that the ordinary least squares estimate in linear regression models do not take the correlation into account, underestimating SE. With the RGE model, when sg ¼ 1 and sb ¼ 0.7, for example, the mean of standard error is 0.006 and the empirical standard deviation is 0.025 in the linear model, and the mean of standard error is 0.085, but the empirical standard deviation is 0.103 in the logistic model Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
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Table 3. Group-based assessment with (a) n ¼ 20, b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (linear model); (b) n ¼ 20, b0 ¼ 2, b1 ¼ 0.3 and sZ ¼ 1 (logistic model); (c) n ¼ 20, b0 ¼ 4, b1 ¼ 0.3 and sZ ¼ 1 (logistic model) sb ¼ 0.7 ^1 SE b (mean) (mean)
SD
sb ¼ 1.414 MSE
^1 b SE (mean) (mean)
SD
MSE
(a) RGE (sg) 0.3 0.169 0.5 0.236 1 0.280
0.015 0.011 0.006
0.103 0.028 0.092 0.013 0.056 0.004
0.123 0.192 0.264
0.024 0.019 0.011
0.100 0.041 0.098 0.021 0.072 0.006
FGE (o) 0.3 0.251 0.5 0.284 1 0.296
0.010 0.006 0.003
0.075 0.008 0.050 0.008 0.026 0.001
0.203 0.264 0.292
0.018 0.012 0.006
0.088 0.017 0.068 0.006 0.036 0.001
(b) RGE (sg) 0.3 0.163 0.5 0.232 1 0.275
0.211 0.150 0.083
0.256 0.084 0.211 0.049 0.114 0.014
0.124 0.189 0.258
0.170 0.132 0.078
0.214 0.077 0.177 0.044 0.115 0.015
FGE (o) 0.3 0.244 0.5 0.280 1 0.294
0.130 0.077 0.036
0.153 0.027 0.090 0.009 0.044 0.002
0.194 0.256 0.283
0.119 0.075 0.036
0.153 0.035 0.099 0.012 0.050 0.003
(c) RGE (sg) 0.3 0.152 0.5 0.215 1 0.274
0.522 0.341 0.203
0.559 0.334 0.429 0.191 0.264 0.070
0.106 0.180 0.256
0.415 0.320 0.189
0.463 0.252 0.376 0.155 0.277 0.049
FGE (o) 0.3 0.254 0.5 0.290 1 0.298
0.310 0.181 0.080
0.342 0.119 0.187 0.035 0.084 0.007
0.198 0.265 0.292
0.281 0.173 0.077
0.306 0.104 0.190 0.037 0.088 0.008
sg (o) is the between-group standard deviation; sb is the between-subject standard deviation; and sZ is the measurement error standard deviation. SD and MSE are the empirical standard deviation and the mean squared error (1000 iterations).
(Table 2). When the number of measurements is small, the measurement error structure may contain the classic error together with being a quasi-Berkson error (Table 3). As a result, all variance and covariance components affect the bias in the estimation. In linear and logistic models for both RGE and FGE, the error depends on all variance components and the covariance. With RGE, for example, when the betweengroup variability is sg ¼ 0.3 and the between-subject variability increases from sb ¼ 0.7 to 1.414, the bias increases as the estimate (MSE) decreases from 0.169(0.028) to 0.123(0.041) in the linear model and decreases from 0.163(0.084) to 0.124(0.077) in the logistic model (Table 3). Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
In addition, the mean of standard error was not consistent with the empirical standard deviation in the linear model. With RGE, when sg ¼ 1 and sb ¼ 0.7, for example, the mean of standard error is 0.006 and the empirical standard deviation is 0.056 in the linear model, and the mean of standard error is 0.083 and the empirical standard deviation is 0.114 in the logistic model. The overall pattern of bias in the estimates when the disease is rare is consistent with the results obtained under the condition that the prevalence of the disease in the population is between 0.1 and 0.9, while the MSEs increase.
Examples Association of Cancer with Electric Magnetic Field Exposure Data were obtained from a large historical cohort study of workers in five electric utility companies in the USA, which included a survey of occupational exposure to 60 Hz electric magnetic fields among randomly selected workers in 28 job categories (Loomis et al., 1994; Kromhout et al., 1995; Saviz and Loomis, 1995). The between- and within-group variance components were estimated, and the effect of different grouping strategies was assessed for subsequent estimation of exposure to be used in an exposure–response analysis of mortality data. Exposure measurements were log-transformed to ensure normality. Men employed full-time at any time between 1950 and 1986 and who had accrued a total of at least 6 months of continuous employment were enumerated through personnel records and a complete history of employment at the electrical utility companies was obtained. A total of 138,906 eligible men were included. Vital status as of December 1988, was determined for this cohort. The outcome of interest F cancer mortality F was defined based on the underlying cause of death coded to the international classification of disease (ICD); there were about 3.5% of workers who died due to cancer (ICD codes 8 and 9 between 140 and 208). The exposures in each occupational category were assigned to all workers, and the regression coefficient was estimated with their cancer mortality (Saviz and Loomis, 1995). A priori grouping schemes based on exposure level and occupational categories (OC) were compared. Using experience gained from preliminary surveys, the 28 OCs were aggregated into three ordinary levels of presumed magnetic exposures for the a priori grouping. The OC grouping method gave large between-group variability with comparably small between-worker variability. The use of OC is equivalent to the FGE model, whereas aggregation of these groups into three groups is more akin to the RGE model. Kromhout et al. (1999) showed that this makes little difference compared with the use of the usual job category. 7
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Table 4. Groupings with a priori and occupational categories (OC) Grouping
g
sg2
sb2
s2Z
^1 (SE) Crude: b
A priori OC
3 28
0.145 0.225
0.361 0.236
0.987 0.988
0.062* (0.05) 0.205* (0.04)
Table 5. Groupings with factory and job category: all individuals used in group-based exposure assessment Grouping
sg2,
estimate variance of the between-group g, number of groups; distribution of log-transformed exposures; sb2, estimate variance of the ^1, estimate of between-worker distribution of log-transformed exposures; b the regression coefficient in logistic regression model; *P-valueo0.05.
Table 4 shows the estimates of the regression coefficient in a logistic regression model for each grouping strategy with a large sample size (450). The estimates conformed to expectations derived from our simulation studies: more attenuation was observed with small between-group variability and large between-worker variability under an approximated hypothesized true regression coefficient of 0.1–0.2. In keeping with theory and simulations, grouping with greater contrast resulted in more precise estimates.
Respiratory Symptoms and Exposure to Carbon Black A number of repeated cross-sectional studies with measures of respiratory health of employees in the European carbon black manufacturing industry, in relation to exposure to carbon black dust, were conducted (Gardiner et al., 1993, 2001). In the second survey, exposure to inhalable dust in 19 factories in 8 European countries was determined among 1870 workers, resulting in 3290 measurements (Gardiner et al., 1996). There were 8 job categories within 19 factories, and workers from each job title in each factory were selected randomly for monitoring of exposures. In addition, repeated measurements on the same worker were collected at random intervals to allow the estimation of the between- and withinworker variances. Inhalable dust measurements were logtransformed to satisfy the assumption of normality. All participants completed self-administered questionnaires to determine the prevalence of respiratory symptoms: cough, sputum production, cough with sputum production and chronic bronchitis. The prevalence of chronic bronchitis was 4% (see Gardiner et al. (2001) for details). Two exposure groupings schemes were formed for inhalable dust: factory and job category (FGE). Exposure models were fitted with both homogenous and heterogeneous between-subject variances among groups to check the assumption of the homogenous between-subject variability for all groupings. Comparison with Bayesian information criterion values in mixed-effect modeling indicated that there was no evidence for heterogeneous between-subject variability in the data (van Tongeren et al., 2006). As the groupbased assessment depends on the estimated group means that are assigned to all subjects in a group, the difference in the number of subjects drawn for exposure monitoring from each group should not matter, as long as it is sufficiently large to 8
Individual-based (no grouping) Factory Job-category
g
sg2
sb2
s2Z
^1 (SE) Crude: b
F
F
F
F
0.039 (0.08)
19 8
0.20 0.36
0.51 0.41
0.97 0.92
0.188 (0.26) 0.414* (0.17)
g, number of groups; sg2, estimate variance of the between-group distribution of log-transformed exposures; sb2, estimate variance of the ^1, estimate of between-worker distribution of log-transformed exposures; b the regression coefficient in logistic regression model; *P-valueo0.05.
approximate the true group means. In this paper, we used a subset of the original data in which (a) each subject within a group had exposure measurements (allowing us to use the total number of observations to calculate the group means) and (b) health outcomes were determined on every subject with at least one exposure measurement. We compare an individual-based and two grouping methods in Table 5. The individual-based strategy led to a non-significant estimate, whereas grouping by job category gave significant estimates in the expected direction. With the group-based strategy, the job category grouping had the largest between-group variability (sg2 ¼ 0.36), and the estimate was 0.414 (unadjusted). According to our theory and simulation studies, the latter estimate is least likely to be affected by measurement error.
Results and Discussion In this paper, we developed models of bias due to measurement error in epidemiological studies using individual- and group-based estimates of exposure and analyses by linear or logistic regression. Exposures were assumed to be normally distributed with known variance components (the between-group and between-subject variances and the measurement error variance, sg, sb and sZ, respectively) and no differential misclassification. Both fixed- and random-effects models were considered for group-based exposure estimates. We used simulated and empirical data to explore the impact of measurement error structures and modeling approaches on bias and precision in estimated regression coefficients. Error in exposure estimates derived through an individualbased approach leads to bias in estimating exposure–disease associations, usually in the form of attenuation of the association. The magnitude of the bias depends on the between-group and between-subject variability and the measurement error variance, and the attenuation becomes more severe as the between-subject variability decreases. On the basis of our simulation study, the bias decreases when the assumed true variance of exposures increases. Group-based exposure assessment is often used when measuring exposure of all subjects in a study is not feasible. Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
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With this approach to exposure assessment, bias in exposure– response associations depends on between-group and between-subject variability. If the sample size is sufficiently large to accurately estimate the true group means and the measurement error is independent of true exposure, a quasiBerkson error structure is induced. This error structure depends on the covariance between the group means and the measurement error in the exposure model, which is repostulated from a classical model. The overall behavior of estimates is similar with both RGE and FGE designs: the estimates of regression coefficients were less attenuated when the sample size was large, the between-subject variability was small and the spread between-group means was large. We showed, however, that a true Berkson model, giving minimal bias in the estimated coefficient, is approximated when the exposure groups are treated as fixed, whereas no such approximation is assured when exposure groups are treated as random factors. In a random-effects model, the group means are assigned as random components, which may be correlated with both model errors and measurement error. This implies a different feature of RGE in linear models in contrast to the theoretical work of Tielemans et al. (1998) when the number of measurements in calculating the group means is large, wherein the estimate is derived using the ordinary least square estimate under the assumption that errors in the models are mutually exclusive. Although they predicted that attenuation bias becomes negligible as the number of measured workers per group increases, we show that, regardless of sample size, the bias (either positive or negative) in estimates of slope is present when groups are formed through a random process. Bias due to measurement error is negligible in both linear and logistic models when a fixed-effects structure is employed. When a random-effects model is used, however, the degree of bias in the estimate of association increases as the between-group variance decreases in both linear and logistic regression models. With both linear and logistic regression analysis, applying RGE leads to measurement error structures with differential error that can cause either over or underestimation of the regression coefficient. Fixedeffects exposure assignments consistently performed better than random-effects schemes, which generally produced higher variance and greater bias. Thus, fixed exposure groups, which may be based on narrowly targeted inferences (e.g., only job titles present in a particular cohort or factory), seem to be less vulnerable to bias when group-based exposure assessment is employed. When the sample size is small, the error structure retains features of both classical and Berkson error structures. In such cases, the attenuation is more severe than with a quasiBerkson error structure achieved with a larger sample size. Even when all measurements in each group are available, the group-based assessment with the total group mean may give Journal of Exposure Science and Environmental Epidemiology (2010), 1–10
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a more valid estimate of the regression coefficient than the individual-based assessment. This also implies that identical results are obtained with equal and different sample sizes from each group, as long as group means are estimated with sufficient precision. If the measurement error is severe, then the group-based strategy would give good estimates for the regression coefficient rather than using a more computationally intensive measurement error adjustment method with an individual-based strategy (Wang et al., 2000). We show that assignment of exposure based on RGE models is a mechanism by which non-differential measurement error can become differential. Another well-known situation in which a similar phenomenon arises is in dichotomization of a miss-measured continuous variable (Gustafson, 2003). Thus, our proposed quasi-Berkson error structure emphasizes the fact that the independence of measurement error and outcome may not be retained on transformation of a mismeasured covariate. Whether mechanisms other than grouping lead to a quasi-Berkson error structure or loss of non-differential error should be further investigated. One setting in which such an investigation may be warranted may be in studies that assess exposures on the basis of contaminant quantification in pooled biological samples that are stratified on health status, as in a case-referent study (Weinberg and Umbach, 1999). Our findings are largely in line with recommendations for bias reduction in the recently published book by Rappaport and Kupper (2008), but contain a more in-depth theoretical exploration of the impact of individual- and group-based strategies on bias in both linear and logistic regression in the presence of either random- or fixed-group means. In conclusion, a group-based exposure assignment can be an effective and versatile approach to estimate the relationship between exposure and disease when data on exposure are not available for all subjects in a study. However, loss of precision is expected, especially when the between-group variability is small. This general conclusion is in accordance with the principles that guide occupational exposure assessment, giving it greater theoretical credence, while emphasizing previously unanticipated complications with making inference beyond studied groups, when group-based exposure assessment is employed. In such a setting, it is natural to assume that the assigned group means are random effects. The existence of this additional uncertainty and associated bias are in agreement with the intuition that a penalty is to be incurred for drawing conclusions about unobserved situations, especially when there are gaps in the observed data (e.g., exposures were not measured for every subject).
Conflict of interest The authors declare no conflict of interest. 9
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Acknowledgements Hyang-Mi Kim is thankful to David Richardson and Dana Loomis for their hospitality during her stay at the University of North Carolina, Chapel Hill, USA. Drs Richardson and Loomis were supported by grant R01-CA117841 from the National Cancer Institute, National Institutes of Health. Igor Burstyn was supported by salary awards from the Canadian Institutes for Health Research and the Alberta Heritage Foundation for Medical Research.
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