Use of alternative time scales in Cox proportional hazard models: implications for time-varying environmental exposures

Beth Griffin, Garnet Anderson, Regina Shih and Eric Whitsel have a new paper in Statistics in Medicine. The paper investigates the use of different time scales (e.g. other than either study time or patient age) in cohort studies analysed via Cox proportional hazard models. Of particular focus is the use of calendar time as an alternative time scale, with a motivation in the variation of environmental exposures over time. They perform a simulation study considering two scenarios for the relationship between a time varying environmental exposure variable and calendar year. In the first scenario these are made independent, while the second scenario assumes a linear relationship. As one might expect, when there is no correlation between calendar time and the time dependent environmental exposure, estimates are unbiased regardless of the choice of time scale. When a linear relationship exists then models that account for calendar time, either as the primary time scale or as additional covariates in the model. Again, this isn't necessarily surprising because the model is effectively attempting to include a year effect twice, once in the baseline hazard and again as a large component of the time dependent covariate, e.g. you are fitting a model with "mean environmental exposure in year t" and "environmental exposure" as covariates and expecting the latter to have the correct coefficient. The paper only gives a simulation study, I don't think it would have been that hard to have given some basic theoretical results in addition to the simulations.
The conclusion of the paper, albeit with caveats, is that attempting to adjust for calendar time because you suspect the environmental exposure may be correlated with time is not useful. Clearly if there are other reasons to suspect that calendar year may be important to the hazard in a study then there is an inherent lack of information in the study to establish whether the environmental exposure is directly affecting the hazard or whether it is an indirect effect due to the association with calendar time. Ideally, one would look for other calendar time dependent covariates (e.g. prevailing treatment policy regimes etc.) and perhaps try directly adjusting for them rather than calendar time itself.