Discriminant Function Analysis Introductory Overview - Computational Approach
Computationally, discriminant function analysis is very similar to analysis of variance (ANOVA). Let us consider a simple example. Suppose we measure height in a random sample of 50 males and 50 females. Females are, on the average, not as tall as males, and this difference will be reflected in the difference in means (for the variable Height). Therefore, variable height allows us to discriminate between males and females with a better than chance probability: if a person is tall, then he is likely to be a male, if a person is short, then she is likely to be a female.
We can generalize this reasoning to groups and variables that are less "trivial." For example, suppose we have two groups of high school graduates: Those who choose to attend college after graduation and those who do not. We could have measured students' stated intention to continue on to college one year prior to graduation. If the means for the two groups (those who actually went to college and those who did not) are different, then we can say that intention to attend college as stated one year prior to graduation allows us to discriminate between those who are and are not college bound (and this information may be used by career counselors to provide the appropriate guidance to the respective students).
To summarize the discussion so far, the basic idea underlying discriminant function analysis is to determine whether groups differ with regard to the mean of a variable, and then to use that variable to predict group membership (e.g., of new cases).
- Analysis of Variance
- Stated in this manner, the discriminant function problem can be rephrased as a one-way analysis of variance (ANOVA) problem. Specifically, one can ask whether or not two or more groups are significantly different from each other with respect to the mean of a particular variable. To learn more about how one can test for the statistical significance of differences between means in different groups you may want to read the Introductory Overview section to
ANOVA/MANOVA,. However, it should be clear that, if the means for a variable are significantly different in different groups, then we can say that this variable discriminates between the groups.
In the case of a single variable, the final significance test of whether or not a variable discriminates between groups is the F-test. As described in Elementary Concepts and ANOVA/MANOVA, F is essentially computed as the ratio of the between-groups variance in the data over the pooled (average) within-group variance. If the between-group variance is significantly larger then there must be significant differences between means.
- Multiple Variables
- Usually, you include several variables in a study in order to see which one(s) contribute to the discrimination between groups. In that case, we have a matrix of total variances and covariances; likewise, we have a matrix of pooled within-group variances and covariances. We can compare those two matrices via multivariate F-tests in order to determine whether or not there are any significant differences (with regard to all variables) between groups. This procedure is identical to multivariate analysis of variance or MANOVA. As in MANOVA, one could first perform the multivariate test, and, if statistically significant, proceed to see which of the variables have significantly different means across the groups. Thus, even though the computations with multiple variables are more complex, the principal reasoning still applies, namely, that we are looking for variables that discriminate between groups, as evident in observed mean differences. In fact, you may perform discriminant function analysis with the ANOVA/MANOVA module; however, different types of statistics are customarily computed and interpreted in discriminant analysis (as described later).