Treatment Effects in a Logistic Model Involving the Box-Cox Transformation

Abstract

We consider a logistic model for binary response data that allows the possibility of power transformation of x; that is, log[p/(1 - p)] = α + βx ^(λ) + γI, where x is a continuous variable, x ^(λ) is the Box–Cox transformation, and I is a binary variable indicating treatment or group. This model is applicable to observational studies or randomized trials when a treatment effect is investigated after controlling for a confounding variable x. Our focus is on inference concerning γ, the treatment effect. In the analysis, a common approach might be to treat the estimated value of λ as fixed and ignore uncertainty associated with its estimation in inference about γ. Alternatively, we might perform an unconditional analysis in which λ is regarded as a parameter. We show that under the null hypothesis, γ = 0, these two approaches are asymptotically equivalent 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 γ ≠ 0 and when there is reasonable overlap between the two distributions of x given I, the two procedures differ asymptotically; however, the difference between them is extremely small. The asymptotic findings are supported by a simulation study.

Key Words:

• Conditional analysis
• Monte Carlo
• Power transformation
• Unconditional analysis
• Variance inflation