Approximate sampling variances of maximum-likelihood probability estimates in a logit response function

Rege, J.E.O. 1997. Approximate sampling variances of maximum-likelihood probability estimates in a logit response function. IN: Duchateau, L. and Mwambi, H.G. (eds.), Proceedings of the fifth scientific conference of the East, Central and Southern Africa Network of the International Biometric Society, Mombasa, Kenya, 1997: 108-115.

The maximum likelihood parameters estimated in logistic analysis are in terms of a transformation of the original response variable and, although inferences are easily made about the sources of variation in the linearized model using standard procedures applied in regression analysis, the variance-covariance structure of the predicted probabilities obtained following back-transformation of logits is complex and estimation of sampling variances normally require inversion of matrices and taking derivatives of the inverse of the link function evaluated at each prediction point. This paper presents a method for estimating sampling variances of such predicted probabilities without the need to invert any matrix or take derivatives of the link function. The method is based on the assumption that the exponent of a linear function of the logits is lognormal. It is demonstrated by way of a numerical example that this approximation is not different from the more complex methods applied by software such as SAS and GENSTAT.