Statistical Models for Count Data from Multiple Sclerosis Clinical Trials and their Applications

Multiple sclerosis (MS) is an autoimmune disease in which the body’s own immune system attacks the central nervous system. Relapsing remitting MS (RRMS) is an initial stage of the disease where the patient experiences distinct phases of relapse and remittance. Magnetic resonance imaging (MRI) is commonly used to monitor the RRMS disease progression. MRI scans of the brain are taken each month and the total number of new MRI lesions seen during the follow-up period is used as the response variable of interest. The Negative Binomial (NB) and the Poisson-Inverse Gaussian (P-IG) distributions have been shown to fit this over-dispersed data well. Currently, only nonparametric tests are being used to test for the treatment effect in RRMS trials, but the NB and P-IG distributions have been used for simulating the MRI data for the power analyses of these tests and determination of the associated sample sizes.

We consider three different trial designs in our study, namely parallel group (PG), baseline vs. treatment (BVT), and parallel group with a baseline correction (PGB). We identify the treatment effect by the parameter γ, with 1-γ representing the proportion reduction in the mean count of new lesions. For these designs we investigate the finite-sample properties of likelihood based parametric tests such as the likelihood ratio test (LRT) and Rao’s score test (RST) for γ, and Wald tests (WT) for g(γ) with g(γ) = γ, γ², √γ, and log(γ).

We use the NB and the P-IG models for PG trials and propose optimal likelihood based tests. Recently, tests based on the NB model have been proposed for PG trials; they rely on the chi-square approximation and do not maintain Type I error rates for small samples. We propose simulation based tests that maintain Type I error rates, and for the NB model we also consider the case of unequal dispersion parameters for the two groups. For BVT and PGB trials, assuming a bivariate NB (BNB) model, we investigate various parametric tests and compare them. We perform power analyses and sample size estimation using the simulated percentiles of the exact distribution of the preferred test statistics for all the above scenarios.

We compare the sample sizes of our recommended parametric tests with those of the nonparametric tests published in the literature. For the NB models the exact LRT, RST, and WT for log(γ) remained unbiased and generally did equally well for all the three designs. When compared to the corresponding nonparametric test, the LRT gave 30-45% reduction in sample sizes for the PG trials, 25-60% for the BVT trials, and 70-80% for the PGB trials. The WT for γ², though not unbiased, had the highest power for γ < 1 and provided a further reduction of around 10-20% over the LRT in terms of sample sizes. Hence, it is best suited for RRMS clinical trials. For the P-IG model for PG trials, the LRT provided a sample size reduction of 30-50% compared to the Wilcoxon Rank Sum test and the exact WT for γ provided a reduction of 40-50%.