What to do when accumulated exposure affects health but only its duration was measured? A case of linear regression

Background: We considered a problem of inference in epidemiology when cumulative exposure is the true dose metric for disease, but investigators are only able to measure its duration on each subject. Methods: We undertook theoretical analysis of the problem in the context of a continuous response caused by cumulative exposure, when duration and intensity of exposure follow log-normal distributions, such that analysis by linear regression is natural. We present a Bayesian method to adjust duration-only analysis to incorporate partial knowledge about the relationship between duration and intensity of exposure and illustrate this method in the context of association of smoking and lung function. Results: We derive equations that (a) describe under what circumstances bias arises when duration of exposure is used as a proxy of cumulative exposure, (b) quantify the degree of such bias and loss of precision, and (c) describe how knowledge about relationship of duration and intensity of exposure can be used to recover an estimate of the effect of cumulative exposure when only duration was observed on every subject. Conclusions: Under our assumptions, when duration and intensity of exposure are either independent or positively correlated, we can be more confident in qualitatively interpreting the direction of effects that arise from use of duration of exposure per se. We can use external information on the relationship between duration and intensity of exposure (namely: Correlation and variance of intensity), even if intensity of exposure is not available at the individual level, to make reliable inferences about the magnitude of effect of cumulative exposure on the outcome.