The Bayesian Multiple Logistic Random Effects Model for Analysis of Clinical Trial Data

Abstract

A prospective, multi-institutional and randomized surgical trial involving 724 early stage melanoma patients was conducted to determine whether excision margins for intermediate-thickness melanomas (1.0 to 4.0 mm) could be safely reduced from the standard 4-cm radius. Patients with 1- to 4-mm-thick melanomas on the trunk or proximal extremities were ran- domly assigned to receive either a 2- or 4-cm surgical margin with or without immediate node dissection (i.e. immediate vs. later -within 6 months). The median follow-up time was 6 years. Recurrence rates did not correlate with surgical margins, even among stratified thickness groups. The hospital stay was shortened from 7.0 days for patients receiving 4-cm surgical margins to 5.2 days for those receiving 2-cm margins (p = 0.0001). This reduction was largely due to reduced need for skin grafting in the 2cm group. The overall conclusion was that the narrower margins significantly reduced the need for skin grafting and shortened the hospital stay. Due to the adequacy of subject follow up, recently a statistical focus was on what prognostics factors usually called covariates actually determined recurrence. As was an- ticipated, the thickness of the lesion (p = 0.0091) and whether or not the lesion was ulcerated (p = 0.0079), were determined to be significantly asso- ciated with recurrence events using the logistic regression model. This type of fixed effect analysis is rather a routine. The authors have determined that a Bayesian consideration of the results would afford a more coherent interpretation of the effect of the model as- suming a random effect of the covariates of thickness and ulceration. Thus, using a Markov Chain Monte Carlo method of parameter estimation with non informative priors, one is able to obtain the posterior estimates and credible regions of estimates of these effects as well as their interaction on recurrence outcome. Graphical displays of convergence history and poste- rior densities affirm the stability of the results. We demonstrate how the model performs under relevant clinical conditions. The conditions are all tested using a Bayesian statistical approach allowing for the robust testing of the model parameters under various recursive partitioning conditions of the covariates and hyper parameters which we introduce into the model. The convergence of the parameters to stable values are seen in trace plots which follow the convergence patterns This allows for precise estimation for de- termining clinical conditions under which the response pattern will change. 496 Karan P. Singh et al. We give a numerical example of our results. The major platform for the theoretical development follows the Bayesian methodology and the multiple parameter logistic model with random effects having carefully chosen hyper parameters. We have done the basic infrastructure for the analysis using the commercially available WinBugs software employing the Markov Chain Monte Carlo (MCMC) methodology. The BUGS language allows a con- cise expression of the parametric model to denote stochastic (probabilistic) relationships and deterministic (logical) relationships.