Rejoinder to Kullback and Keegel note on the minimum discrimination information approach

Abstract

Kullback and Keegel [3] (K&K) espouse the minimum discriminant information (MDI) approach. For the problem considered by Brown and Fuchs [1] (B&F), K & K agree that the results of the MDI procedure are the same as the maximum likelihood estimates and the MDI statistic.., is the log-likelihood ratio statistic. Therefore, the two methods must yield identical results when the same model is being fitted to the same set of data. One method cannot be more correct than the other. K & K then demonstrate that by using their program, they obtain the same results using a Newton-Raphson (NR) algorithm as by using an iterative proportional fitting (IPF) algorithm (see Table 1). In B&F different results were obtained by the two different algorithms. The difference in B&F occurred because the data sets to which the model was being fit~ted differed between the two algorithms. K & K forced the same dataset to be used by both algorithms. A major conceptual issue which arises in several of the specific comments of K& K is whether the definition of the model also includes the definition of the observation space. B& F used the term model to refer to the generic form of the model which can be defined even before data are collected. This generic form is specified to a computer program which then reads the data matrix presented to it and computes the parameter estimates of the model. When the data presented are changed, the estimates change even if the generic model is the same. In B& F we considered that the data presented to the Newton-Raphson algorithm were that specified in our Table 1 (which excluded all the observed zeros) whereas the program using an IPF algorithm first completes the frequency table and in that process modifies the set of observations. Therefore, the two algorithms used by B & F were fitting the same generic model to two sets of observations which differed in their zero frequencies. On the other hand K & K assume that there is a dependent variable with two levels and add a zero frequency whenever one and only one of the levels is present