Treatment effects in a logistic model involving the box-cox transformation

We consider a logistic model for binary response data that allows the possi bility of power transformation of x; that is, log[p/(1-p)] = alpha+beta x(( lambda)) +gamma I, where x is a continuous variable, x((lambda)) is the Box -Cox transformation, and I is a binary variable indicating treatment or gro up. This model is applicable to observational studies or randomized trials when a treatment effect is: investigated after controlling for a confoundin g variable x. Our focus is on inference concerning gamma, the treatment eff ect. In the analysis, a common approach might be to treat the estimated val ue of lambda as tired and ignore uncertainty associated with its estimation in inference about gamma. Alternatively, we might perform an unconditional analysis in which lambda is regarded as a parameter. We show that under th e null hypothesis. gamma = 0, these two approaches are asymptotically equiv alent if the two groups have the same distribution of x and the same sample size. This result also holds for the situation of multiple covariates each with their own transformation. Furthermore, we find that when gamma not eq ual 0 and when there is reasonable overlap between the two distributions of x given I, the two procedures differ asymptotically; however, the differen ce between them is extremely small. The asymptotic findings are supported b y a simulation study.