Comparison of two treatments with skewed ordinal responses

In clinical studies, the proportional odds model is widely used to compare treatment efficacies when the responses are categorically ordered. However, this model has been shown to be inappropriate when the proportional odds assumption is invalid, mainly because it is unable to control the type I error rate in such circumstances. To remedy this problem, the latent normal model was recently promoted and has been demonstrated to be superior to the proportional odds model. However, the application of the latent normal model is limited to compare treatments with similar underlying distributions except possibly their means and variances. When the underlying distributions are very different in skewness, both of the aforementioned procedures suffer from the undesirable inflation of the type I error rate. To solve the problem for clinical studies with ordinal responses, we provide a viable solution that relies on the use of the latent Weibull distribution, which is a member of the log-location-scale family. The proposed model is able to control the type I error rate regardless of the degree of skewness of the treatment responses. In addition, the power of the test also outperforms that of the latent normal model. The testing procedure draws on newly developed theoretical results related to latent distributions from the location-scale family. The testing procedure is illustrated with two clinical examples.