# Discriminant Analysis

Discriminant analysis builds a predictive model for group membership. The model is composed of a discriminant function (or, for more than two groups, a set of discriminant functions) based on linear combinations of the predictor variables that provide the best discrimination between the groups. The functions are generated from a sample of cases for which group membership is known; the functions can then be applied to new cases that have measurements for the predictor variables but have unknown group membership.

Note: The grouping variable can have more than two values. The codes for the grouping variable must be integers, however, and you need to specify their minimum and maximum values. Cases with values outside of these bounds are excluded from the analysis.

Example. On average, people in temperate zone countries consume more calories per day than people in the tropics, and a greater proportion of the people in the temperate zones are city dwellers. A researcher wants to combine this information into a function to determine how well an individual can discriminate between the two groups of countries. The researcher thinks that population size and economic information may also be important. Discriminant analysis allows you to estimate coefficients of the linear discriminant function, which looks like the right side of a multiple linear regression equation. That is, using coefficients a, b, c, and d, the function is:

``D = a * climate + b * urban + c * population + d * gross domestic product per capita``

If these variables are useful for discriminating between the two climate zones, the values of D will differ for the temperate and tropic countries. If you use a stepwise variable selection method, you may find that you do not need to include all four variables in the function.

Statistics. For each variable: means, standard deviations, univariate ANOVA. For each analysis: Box's M, within-groups correlation matrix, within-groups covariance matrix, separate-groups covariance matrix, total covariance matrix. For each canonical discriminant function: eigenvalue, percentage of variance, canonical correlation, Wilks' lambda, chi-square. For each step: prior probabilities, Fisher's function coefficients, unstandardized function coefficients, Wilks' lambda for each canonical function.

Discriminant Analysis Data Considerations

Data. The grouping variable must have a limited number of distinct categories, coded as integers. Independent variables that are nominal must be recoded to dummy or contrast variables.

Assumptions. Cases should be independent. Predictor variables should have a multivariate normal distribution, and within-group variance-covariance matrices should be equal across groups. Group membership is assumed to be mutually exclusive (that is, no case belongs to more than one group) and collectively exhaustive (that is, all cases are members of a group). The procedure is most effective when group membership is a truly categorical variable; if group membership is based on values of a continuous variable (for example, high IQ versus low IQ), consider using linear regression to take advantage of the richer information that is offered by the continuous variable itself.

To Obtain a Discriminant Analysis

This feature requires the Statistics Base option.