Treatment Effects in a Logistic Model Involving the Box-Cox Transformation
研究了在逻辑回归模型中，当对连续变量x进行Box-Cox变换后，如何推断二元处理变量I的效应γ，并比较了将变换参数λ视为固定或参数两种方法的渐近性质。
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.