Left truncation in structural nested accelerated failure time mod-

The analyses described in §3.1 and §3.2 both utilize age as the time scale. A second approach, using time-on-study as the time scale, was not used because of the strong dependence of carcinogenic processes on age. Further, use of age as the time scale has been recommended for epidemiologic studies due to potential residual bias from covariates that might vary with ageThiébaut and Bénichou(2004), but that the choice should be purpose-drivenCheung et al. (2003). Partly, use of this time scale is moti- vated by the idea that two individuals of equal age are more likely to have similar haz- ards for cancer than are two individuals who begin follow-up at the same time but are very different ages. As shown in Figure A.8, time-on-study roughly, but not perfectly, corresponds to calendar time. Thus, another motivating factor is the lack of clear in- terpretation of time-on-study as a time scale. Because we can model two out of the list comprising age, period, and birth cohort, age was a clear choice of time scale be- cause it has a clear interpretation and it is relevant to this list of time-related factors important to cancer trends.

Use of age as the time scale may be problematic if not everyone enters follow-up at the same age. This process is referred to as late entry or left truncation. If cohort members that enter the study at one age are not exchangeable with respect to the out- come under study (i.e. their hazards are identically distributed), then interpretability is compromised. This compromise occurs because we are interested in some marginal or conditional distribution of the age at lung cancer mortality, but the person-time distribution would no longer reflect our population of interest if individuals enter late, because entering into the study would depend on non-random processes. For exam- ple, if smoking miners tend to die much earlier, then the proportion of smokers in the cohort will underestimate the proportion in the target population. This exchangeabil-

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Figure A.8: Calendar time distribution of pre-follow-up employment time (black), em- ployment time under observation (medium gray) and post-employment time (light gray) in the miner data - each horizontal line represents a single miner. The lines are grouped according to vital status at exit from the study.

ity assumption is similar to that made with respect to censoring: individuals who are in the study represent those who have left the study (or have yet to enter the study). If this assumption holds within strata of measured covariates, then late entry does not change the interpretation of our results, provided we adjust for those covariates. We could, for example, assume that smokers who are in the study are exchangeable with smokers who died before entering the study, so that the population is exchangeable within strata of the measured covariates. In this case, we would say that late entry is ignorable in strata of smoking. This can generalize to other covariates.

In the structural models presented in §3.1 and §3.2, we may be sensitive to assump- tions about late entry. As shown in Figure A.9, there was variation in the age of starting the study. In the methods of §3.2, we adjusted the marginal structural model for base- line covariates. Assuming that late entry is ignorable given the baseline covariates, the marginal structural model is not subject to interpretation problems due to late entry. For the methods in §3.1, the structural nested model was marginal with respect to study covariates, and so remains subject to this problem with late entry. Because all struc- tural nested models are subject to this caveat, we expect that the time-ratio may incur some bias. However, left truncation could not explain our observation that healthy worker survivor bias may be present in analyses with the miner data.

There are no current examples of correcting for late entry in structural nested mod- els. We note that previous examples of structural nested models have circumvented this problem by using time-on-study as the time scaleWitteman et al.(1998);Hernán et al. (2005); Chevrier et al. (2012); Naimi et al. (2014a) Analogous issues arise with right-censoring in our study, in that we assume that censoring is ignorable, given the measured covariates. Censoring is handled in §3.1 and §3.2 using inverse probability of censoring weights. These weights correct for non-ignorable censoring by applying weights to individuals in the study so that they represent themselves and any similar individuals who have been censored. A similar approach may be possible with respect

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Figure A.9: Age distribution of pre-follow-up employment time (black), and time un- der observation (gray) - each horizontal line represents a single miner. The lines are grouped according to vital status at exit from the study.

to late entry, in which individuals on the study can be weighted in such a way that they represent both themselves and individuals who have yet to enter the study. Develop- ment of such a technique may prove valuable for both marginal structural models and structural nested models.

APPENDIX B: FURTHER DETAILS OF MARGINAL STRUCTURAL MODELS IN OCCUPATIONAL STUDIES

B.1 Why inverse probability weighted MSMs with non-positivity cannot be treated