The logistic transform for bounded outcome scores

Abstract

The logistic transformation, originally suggested by Johnson (1949), is applied to analyze responses that are restricted to a finite interval (e.g. (0,1)), so-called bounded outcome scores. Bounded outcome scores often have a non-standard distribution, e.g. J- or U-shaped, precluding classical parametric statistical approaches for analysis. Applying the logistic transformation on a normally distributed random variable, gives rise to a logit-normal (LN) distribution. This distribution can take a variety of shapes on (0,1). Further, the model can be extended to correct for (baseline) covariates. Therefore, the method could be useful for comparative clinical trials. Bounded outcomes can be found in many research areas, e.g. drug compliance research, quality-of-life studies, and pain (and pain relief) studies using visual analog scores, but all these scores can attain the boundary values 0 or 1. A natural extension of the above approach is therefore to assume a latent score on 0,1) having a LN distribution. Two cases are considered: (a) the bounded outcome score is a proportion where the true probabilities have a LN distribution on (0,1) and (b) the bounded outcome score on [0,1] is a coarsened version of a latent score with a LN distribution on (0,1). We also allow the variance (on the transformed scale) to depend on treatment. The usefulness of our approach for comparative clinical trials will be assessed in this paper. It turns out to be important to distinguish the case of equal and unequal variances. For a bounded outcome score of the second type and with equal variances, our approach comes close to ordinal probit (OP) regression. However, ignoring the inequality of variances can lead to highly biased parameter estimates. A simulation study compares the performance of our approach with the two-sample Wilcoxon test and with OP regression. Finally, the different methods are illustrated on two data sets.