Discriminant Function Analysis

In Chapter 4, we discussed multivariate analysis of variance (MANOVA) and saw how it could be used to tackle research questions concerned with examining group differences when at least two dependent variables (DVs) are considered. If the underlying null hypothesis of mean vector equality across several groups is not rejected, one can conclude that the groups do not differ from one another with respect to the set of studied response measures. We also indicated in Chapter 4 that in the case of nonsignificant mean differences, it would not be appropriate in general to examine group differences on any of the dependent measures considered separately from the others (assuming of course that the research question was of a multivariate nature to begin with). While this is a fairly straightforward approach to follow when there are

no mean differences, how to handle the alternative situation when the above mentioned null hypothesis is rejected appears to be less clear-cut. As one possibility, we could examine separately for group differences each of the dependent measures, but since they are interrelated in general we will be using in our respective statistical conclusions partly overlapping pieces of sample information (because these univariate tests are as a consequence also related to one another). In addition, it may actually be the case that substantively meaningful combinations of the DVs may exhibit interesting group differences, and we will miss them if we remain concerned only with univariate follow-up analyses after a significant MANOVA finding. The method of discriminant function analysis (sometimes for short referred to as discriminant analysis, DA) allows us to examine linear combinations of dependent measures that exhibit potentially very important group differences. In addition, DA will enable us to utilize these variable combinations subsequently to classify other individuals into appropriate groups. In fact, when the intent is to describe group differences on the basis of observed variables, DA may be used as a primarily descriptive approach; alternatively, when the intent is to classify individuals on the basis of studied dependent variables, DA may be used

ion in which a university admission officer is trying to determine what criteria to use in the process of admitting applicants to graduate school. Discriminant analysis could be used then in its descriptive capacity to find the most appropriate criterion to utilize from a large set of variables under consideration. Once the criterion is decided upon, DA could be used in its predictive capacity to classify new applicants into admit or do not admit groups. To formally begin the discussion of DA, let us suppose that a set of DVs

is given, denoted as x¼ (x1, x2, . . . , xp)0 (p> 1), which have been observed across several samples (or groups) from multiple populations of interest. Assume also that in a MANOVA we have found that there were significant group differences, that is, the respective mean vectors differed more than what could be explained by chance factors only. The question that we will be mainly interested in the rest of this chapter is, ‘‘What is the meaning of these differences?’’ In other words, how can we interpret the finding of significant mean differences in a way that combines appropriately the group difference information across the DVs, and at the same time highlights important subject-matter aspects of the phenomenon under investigation? We stress that the finding of groups differing overall, which is all a significant MANOVA result tells us, may not be especially revealing. Instead, one would typically wish to be in a position of coming up with more substantively interpretable ways for making sense of the group differences. In particular, a social or behavioral researcher would usually want to know how the groups differ with regard to the dependent measures while accounting for all available information, including that about their interrelationships. He or she would then be especially interested in obtaining insightful combined information about group differences that is contained in the response variables.