Goodness-of-fit testing using components based on marginal frequencies of multinomial data

The goodness-of-fit test based on Pearson's chi-squared statistic is sometimes considered to be an omnibus test that gives little guidance to the source of poor fit when the null hypothesis is rejected. It has also been recognized that the omnibus test can often be outperformed by focused or directional tests of lower order. In this paper, a test is considered for a model on a data table formed by the cross-classification of q dichotomous variables, and a score statistic on overlapping cells that correspond to the first-through qth-order marginal frequencies is presented. Then orthogonal components of the Pearson-Fisher statistic are defined on marginal frequencies. The orthogonal components may be used to form test statistics, and a log-linear version of an item response model is used to investigate the order and dilution of a test based on these components, as well as the projection of components onto the space of lower-order marginals. The advantage of the components in terms of power and detection of the source of poor fit is demonstrated. Overcoming the adverse effects of sparseness provides another motive for using components based on marginal frequencies because an asymptotic chi-squared distribution will be more reliable for a statistic formed on overlapping cells if expected frequencies in the joint distribution are small.