Discriminant Function Analysis Introductory Overview - Discriminant Functions for Multiple Groups

When there are more than two groups, we can estimate more than one discriminant function like the one presented above. For example, when there are three groups, we could estimate (1) a function for discriminating between group 1 and groups 2 and 3 combined, and (2) another function for discriminating between group 2 and group 3. We could have one function that discriminates between those high school graduates that go to college and those who do not (but rather get a job or go to a professional or trade school), and a second function to discriminate between those graduates that go to a professional or trade school versus those who get a job. The b coefficients in those discriminant functions could then be interpreted as before.

Canonical analysis
When actually performing a multiple group discriminant analysis, we do not have to specify how to combine groups so as to form different discriminant functions. Rather, STATISTICA automatically determines some optimal combination of variables so that the first function provides the most overall discrimination between groups, the second provides second most, and so on. Moreover, the functions will be independent or orthogonal, that is, their contributions to the discrimination between groups will not overlap. Computationally, STATISTICA performs a canonical correlation analysis (see also Canonical Correlation) that will determine the successive functions and canonical roots (the term root refers to the eigenvalues that are associated with the respective canonical function). The maximum number of functions that STATISTICA computes are equal to the number of groups minus one, or the number of variables in the analysis, whichever is smaller.
Interpreting the discriminant functions
As before, we get b (and standardized Beta) coefficients for each variable in each discriminant (now also called canonical) function, and they can be interpreted as usual: the larger the standardized coefficient, the greater is the contribution of the respective variable to the discrimination between groups. (Note that we could also interpret the structure coefficients; see below.) However, these coefficients do not tell us between which of the groups the respective functions discriminate. We can identify the nature of the discrimination for each discriminant (canonical) function by looking at the means for the functions across groups. We can also visualize how these two functions discriminate between groups by plotting the individual scores for the two discriminant functions.
Factor structure matrix
Another way to determine which variables "mark" or define a particular discriminant function is to look at the factor structure. The factor structure coefficients are the correlations between the variables in the model and the discriminant functions; if you are familiar with factor analysis (see Factor Analysis) you may think of these correlations as factor loadings of the variables on each discriminant function.

Some authors have argued that these structure coefficients should be used when interpreting the substantive "meaning" of discriminant functions. The reasons given by those authors are that (1) supposedly the structure coefficients are more stable, and (2) they allow for the interpretation of factors (discriminant functions) in the manner that is analogous to factor analysis. However, subsequent Monte Carlo research (Barcikowski & Stevens, 1975; Huberty, 1975) has shown that the discriminant function coefficients and the structure coefficients are about equally unstable, unless the n is fairly large (e.g., if there are 20 times more cases than there are variables). The most important thing to remember is that the discriminant function coefficients denote the unique (partial) contribution of each variable to the discriminant function(s), while the structure coefficients denote the simple correlations between the variables and the function(s). If you want to assign substantive "meaningful" labels to the discriminant functions (akin to the interpretation of factors in factor analysis), then the structure coefficients should be used (interpreted); if you want to learn what is each variable's unique contribution to the discriminant function, use the discriminant function coefficients (weights).

Significance of discriminant functions
One can test the number of roots that add significantly to the discrimination between groups. Only those found to be statistically significant should be used for interpretation; non-significant functions (roots) should be ignored.
Summary
To summarize, when interpreting multiple discriminant functions, which arise from analyses with more than two groups and more than one variable, you would first test the different functions for statistical significance, and only consider the significant functions for further examination. Next, you would look at the standardized b coefficients for each variable for each significant function. The larger the standardized b coefficient, the larger is the respective variable's unique contribution to the discrimination specified by the respective discriminant function. In order to derive substantive "meaningful" labels for the discriminant functions, you can also examine the factor structure matrix with the correlations between the variables and the discriminant functions. Finally, you would look at the means for the significant discriminant functions in order to determine between which groups the respective functions seem to discriminate.