Bias in Cross-Sectional and Longitudinal Estimates of Pulmonary Function Decline: A Different Perspective

In this issue, Hendrick and colleagues [[1]] report an ambitious simulation study designed to investigate how differences between cohorts in highest attained FEV₁ during young adulthood (cohort effects) affect cross-sectional and longitudinal estimates of the effects of aging and environmental exposures on FEV₁. The investigators developed an unusually sophisticated simulation model in an effort to provide a realistic representation of the many factors that influence exposure, retention, and survival of members of a cohort. Among other features, the model assumes that the probability of beginning work in a dusty occupation, the probabilities of continuing smoking and dusty work, and the probability of survival all depend on an individual's level of FEV₁. Specifically, the probabilities of both beginning and continuing the exposures increase with the concurrent level of FEV₁. These assumptions are related to the “healthy worker effect” that has posed a substantial challenge in occupational epidemiology.

We welcome the authors' decision to make their program available on-line. When invited to write this editorial, we explored the simulation program and were impressed with its flexibility and capacity to simulate a wide range of scenarios. It is easy to use in SAS and other investigators interested in simulation studies may find it useful.

The simulations reported in the article demonstrate that cohort effects induce bias in the cross-sectional estimate of the (linear) effect of aging on pulmonary function but not the longitudinal estimate of the annual rate of decline. One result, however, puzzled us from the first reading of this manuscript. Our intuition suggested that, in the absence of selection effects, a fixed or random cohort effect would not induce bias in the cross-sectional estimates of the coefficients of any covariate other than age. Indeed, absent interindividual variability in the cohort effect, the variable representing the cohort effect would be completely collinear with age and thus would enter as an additional contribution of age in a linear model for FEV₁, but would not affect other parts of the model. Moreover, interindividual variability in the cohort effect would increase the standard errors of regression coefficients but would not generate additional bias. Thus, we spent considerable time exploring the source of the bias in the cross-sectional estimate of the coefficient of dust exposure.

We have concluded that the bias in the dust exposure coefficient arises not from the cohort effect per se but from the interplay between the cohort effect and the selection effects arising from the “healthy worker” features of the model. In the simulation program, these selection effects can be induced solely by increasing population variability. For example, if all values of the “base model” are taken as in the Appendix, with no cohort effect, but the variability of FEV₁ at age 25 is increased to 0.6, bias is induced in the cross-sectional estimates of the coefficients of age and the exposure variables. In the presence of moderate population heterogeneity of FEV₁ at age 25, variability in the cohort effect accentuates the selection bias arising from the “healthy worker” features of the model by differentially reducing the probability that individuals in earlier and later cohorts will, for example, “begin work.” The “base model” assumes that exposure and survival depend on level of FEV₁ in the four ways listed in our first paragraph. When all of these selection effects are eliminated from the model, we find no bias in the cross-sectional estimate of the coefficient of dust. Thus, we conclude that absent these healthy worker features, a cohort effect would not induce bias in the cross-sectional estimates of the coefficients of smoking and dust exposure.

The challenge of unraveling the healthy worker effect has stimulated important work on causal inference by Robins and colleagues over the past two decades [[2]]. These investigators have shown that “even in the absence of unmeasured confounders or model misspecification, standard methods for estimating the causal effect of a time-varying treatment [exposure] on the mean of a repeated measures outcome … may be biased when there are time-varying confounders (for example, previous FEV₁) that are simultaneously confounders of the effect of interest and are affected by previous treatment” [[3]]. In particular, when exposure (dust) influences outcome (FEV₁) AND outcome predicts exposure, both conventional cross-sectional and longitudinal analysis can produce biased estimates of the causal effect of the exposure on the outcome. This is essentially a restatement of the problem of healthy worker bias. More importantly, however, Robins and colleagues have developed a new class of models, called “Marginal Structural Models,” whose parameters can be consistently estimated by a new class of estimators known as inverse probability of treatment weighted (IPTW) estimators. These estimators will correct for the selection bias induced by the healthy worker effect and produce unbiased estimates of the causal effect of exposure on outcome. This work is essential reading for investigators seeking to quantify the causal effects of occupational and environmental exposures in settings subject to outcome-dependent selection. It is intriguing that the healthy worker features of the simulations reported by Hendrick et al. did not induce bias in the longitudinal estimates of the dust and smoking effects, and we are not yet able to explain that result.

Thus, we commend the authors for the scope of their investigation and encourage other investigators to explore the software they have made available online, but also caution readers about the complexity of the healthy worker problem in occupational research and suggest that the impact of the cohort effect on the coefficient for dust in the simulations reported by Hendrick et al. depends fundamentally on the selection effects in their model. We disagree with the authorsd' conclusion that the cohort effect alone influenced the cross-sectional estimates for dust. Rather, the combination of the cohort effect and the selection effects in the onset and continuation of exposure generated the bias.

Finally, we were pleased to see that these investigators chose to normalize FEV₁ by the square of height. In our analyses of data provided by adult participants in the Six Cities Study of Air Pollution and Health [[4]], we found compelling evidence that this normalization most effectively standardizes variance, removes the effects of height (in that the standardized variable is independent of height), and yields a variable that is well described by models that are linear in age, the square of age, and exposures. The same is true for the other measures of pulmonary function obtained in that study. Though we can appreciate the appeal of simpler models, such as a linear regression of FEV₁ on height or the square of height and other variables, we found such models to be demonstrably inferior and believe that they induce biases that may not be apparent in smaller studies but affect other parts of the analysis.