The Evaluation of Integrals of the form ∫^+∞_−∞f(t)exp(−t²) dt: Application to Logistic-Normal Models

Abstract

Logistic-normal distributions and related functions arise in a variety of statistical applications of current interest, including binary measurement-error models and the analysis of teratogenicity experiments. Analytic intractability has led to the development of numerous approximations to the desired forms, often with consequences that have not been well studied. A method is developed to compute these forms to arbitrary accuracy, and comparative calculations are made that show when the common numerical alternative, 20-point Gaussian quadrature, begins to fail. By using a simple matrix transformation, this method can be used with multiple covariate regression models of the logistic-normal form. We conducted a simulation study that compares the ability of 20-point Gaussian quadrature and our new method to obtain the maximum likelihood estimator of relative risk in the logistic-normal measurement-error model. Using standard subroutines to maximize the likelihood equations, 27 of 50 trials failed to converge with 20-point Gaussian quadrature, whereas the new method allowed convergence in all but one case.

Key Words:

• Covariate error
• Extra-binomial variation
• Logistic regression
• Logistic-normal distribution
• Numerical integration