Sample Size for Logistic Regression with Small Response Probability

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Abstract
The Fisher information matrix for the estimated parameters in a multiple logistic regression can be approximated by the augmented Hessian matrix of the moment-generating function for the covariates. The approximation is valid when the probability of response is small. With its use one can obtain a simple closed-form estimate of the asymptotic covariance matrix of the maximum likelihood parameter estimates, and thus approximate sample sizes needed to test hypotheses about the parameters. The method is developed for selected distributions of a single covariate and for a class of exponential-type distributions of several covariates. It is illustrated with an example concerning risk factors for coronary heart disease.

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