Optimized scorings for ordinal data for the general linear model

There are a number of possible statistical procedures that can be used for analyzing ordinal categorical data collected in a designed experiment. In this paper we present an approach which allows us to do a preliminary examination of the data set to see whether or not the statistical results are sensitive to the choice of procedure. In our approach we start by considering analyzing ordinal categorical data collected in a designed experiment considered in the context of a general linear model with scores assigned to the ordinal categories. The standard F-statistics for testing linear hypotheses concerning model parameters are considered. Since the increasing scores can be chosen arbitarily, two sets of scores may potentially lead to opposing analytical and statistical conclusions. To deal with such concerns we optimize the F-statistics as functions of the scores assigned to the categories. For reference purpose we suggest using the F-distribution, although there is the usual caution if sample sizes are small. In two cases, namely, when the maximized F is nonsignificant, or when the minimized F is significant, all scores lead to the same conclusions, respectively, either rejecting or accepting H0. For example, in a one way lay-out with C treatments and K categories a nonsignificant maximum F indicates that there would be no significant treatment effect no matter what scores are used. Methods for computing the maximum and the minimum F-statistics are presented. The methods suggested in the paper are exemplified. The relationship between the F-statistics used for testing the treatment effect in one-way design and certain monotone correlations is also established.