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Biostatistics Advance Access published online on April 5, 2006
Biostatistics, doi:10.1093/biostatistics/kxj034
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© The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org
Received September 30, 2004
Revised March 16, 2006
Accepted March 29, 2006

Article

The logistic-transform for bounded outcome scores

Emmanuel Lesaffre 1 *, Dimitris Rizopoulos 1, and Roula Tsonaka 1

1 Biostatistical Centre, Catholic University of Leuven, U.Z. St. Rafaël, Kapucijnenvoer 35, B-3000 Leuven, Belgium

* To whom correspondence should be addressed.
Emmanuel Lesaffre, E-mail: emmanuel.lesaffre{at}med.kuleuven.be


  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 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 analogue scores, but all of 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 logit-normal distribution. Two cases are considered: (a) the bounded outcome score is a proportion where the true probabilities have a logit-normal distribution on (0,1), (b) the bounded outcome score on [0,1] is a coarsened version of a latent score with a logit-normal 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 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 ordinal probit regression. Finally, the different methods are illustrated on two data sets.

Keywords: Barthel index; compliance research; bounded outcome scores; logistic-transform; ordinal probit regression.
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