A note on goodness-of-fit statistics for probit and logit models

In the case of models designed to explain the choice among a finite set of alternatives, a number of goodness-of-fit statistics have been reported. This paper is primarily concerned with the properties of one of these statistics, the likelihood ratio index.By comparing the likelihood ratio index with some of the other statistics and by examining its mathematical properties, it is concluded that the index has desirable properties in binary and multinomial situations. However, the way in which the likelihood ratio index has been applied in many recent studies has led to results which are possibly unexpected. In these cases, the index was a measure of the extent to which a hypothesized model improved upon the explanatory power of a model with all coefficients, including the constant or the coefficients of alternative-specific dummies, equal to zero. It is shown that the minimum value of this likelihood ratio index depends on the relative proportions of sampled individuals selecting the various alternatives, contrary to the expectation of a zero minimum value. The dependence on the sampled proportions also prevents comparison of indices resulting from different samples.A simple adjustment alleviates these difficulties. This new definition makes the likelihood ratio index a measure of the extent to which the hypothesized model improves upon the explanatory of a model with only a constant or alternative-specific dummies. It is recommended that this index is more appropriate for assessing the value of choice models.A list of symbols used is given on page 387.