The use of regression models for medians when observed outcomes may be modified by interventions

Some outcomes used in epidemiological studies and clinical trials are prone to modification by interventions, for example, individuals with high blood pressure are likely to receive antihypertensive medication. Often the scientific interest is in the relationship of covariates (exposures or randomized treatment) to the outcomes that would have been observed in the absence of intervention. We compare three approaches to the analysis of such data: ignoring the intervention; excluding individuals who receive the intervention, and assuming that individuals who receive the intervention have underlying outcomes above the median. The latter approach requires comparison of median outcomes between groups. In many situations it is plausible that neither the probability of intervention nor the effect of intervention depend on the covariates. In this case we show that analysis of medians is unbiased in general, that the other approaches are biased towards the null but that ignoring the intervention typically has the greatest power for detecting an effect. In other situations, ignoring the intervention and excluding individuals who receive the intervention may be biased towards or away from the null, and we recommend analysis of medians. We illustrate practical analysis of medians in a study of the association between adult blood pressure and birth weight. Adjustment for confounders is performed by median regression. We show that the significance levels are comparable to those derived from logistic regression. We also discuss the effect of grouping of blood pressure and the need for bootstrap standard errors.