SUGI 27: Redesigning Experiments with Polychotomous Logistic Regression: A Power Computation Application
نویسندگان
چکیده
Power and sample size calculations for experiments modeled with binary logistic regression are becoming more common, and are even available as freeware (e.g. Ralph O’Brien’s UnifyPow application). Somewhat less common, if not altogether absent, is software that allows power analysis for multinomial logistic regression models. Using the algorithm introduced below, it is now possible to compute powers and sample sizes for arbitrary multinomial (ordinal) logistic regression models. In particular, this application has been designed to allow investigators to explore subsets of data from previous studies that will, in turn, allow them to design or redesign experiments with optimal power. This algorithm requires SAS/BASE, SAS/STAT and SAS/IML. The algorithm currently runs on PC SAS, but may run on any platform. Finally, a heuristic will be presented for the general programmer. A knowledge of random number generation is helpful. INTRODUCTION We will assume the reader is familiar with the general form for ordinal logistic regression: Pr( | , , ) exp( ( )) , response a X X X i k i m i t ≤ = + − + ≤ ≤ 1 1 1 1 K α β for where a1 < a2 <...< ak are k ordinal response levels, X1,...,Xm are m explanatory variables, β = [β1 ... βm] is the vector of slope parameters, and X = [X1 ...Xm] is the vector of explanatory variables. Geometrically, each of the k 1 cumulative linear predictors αi + βX forms an m-dimensional hyperplane. If we can quantify corresponding measures of dispersion, say, εi, about each of these hyperplanes from the observed data, then we can simulate response probabilities, and hence patient responses (see figure 1). In figure 1, we have, for ease of explanation, illustrated a trichotomous model for a given treatment group (placebo or drug) with one explanatory variable X1. In this case the hyperplanes are simply parallel lines. For a fixed cross-section of these parallel lines, the measures of dispersion εi about each line at x1 is given by [ ] [ ] $ $ , , . εi i t x V x i = ⋅ ⋅ = 1 1 1 2 1 1
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