Infinite Estimates with Fractional Factorial Experiments

When fractional factorial experiments are run and incomplete data (e.g. right censored from failure times, ordered categorical data or interval-censored data from imprecise measurements) result then the fitting of models by maximum likelihood to obtain estimates of effects can result in infinite estimates. The problem has been long recognized in contingency table references and recently techniques based on linear programming have been introduced to investigate whether infinite estimates generally exist. In this paper we introduce a simple analysis based on the binomial distribution and pseudodata simply constructed from the original data. By monitoring iterative weights in the fitting process it is demonstrated that it is straightforward to determine whether infinite estimates would exist if complicated models were to be fitted. We illustrate the ideas by analysing two examples which have been discussed extensively in the literature. Finally, we give some suggestions about how to proceed in the analysis of data and models giving rise to infinite maximum likelihood estimates.