Stepwise Procedures In Discriminant Analysis

DEFINITION OF TERMS

Discriminant Function

This is a latent variable which is created as a linear combination of discriminating variables, such that

Y =      L1X1 + L2X2 +          + Lp Xp

where the L’s are the discriminant coefficients, the x’s are the discriminating variables.

The eigenvalue: This is the ratio of importance of the dimensions which classifies cases of the dependent variables. There is one eigenvalue for each discriminant function. With more than one discriminant function, the first eigenvalue will be the largest and the most important in explanatory power, while the last eigenvalue will be the smallest and the least important in eXplanatory power.

Relative importance is assessed by eigenvalues since they reflect the percents of variance eXplained in the dependent variable, cumulating to 100% for all functions. Eigenvalues are part of the default of output in SPSS (Analysis, Classify, Discrimination).

The Discriminant Score

This is the value obtained from applying a discriminant function formula to the data for a given case. For standardized data, Z score is the discriminant score.

Cutoff

When group sizes are equal, the mean of the two centroids for two- groups discriminant analysis is the cut off. The cut off is the weighted mean if the groups are unequal. A case is classed as 0 if the discriminant score of the discriminant function is less than or equal to the cut off or classed as 1 if above it.

The Relative Percentage

This is equal to the eigenvalue of a function divided by the sum of all eigenvalues of all discriminant functions in the model. It is the percent of discriminating power for the model associated with a particular discriminant function. It tells us how many functions are important. The ratio of eigenvalues indicates the relative discriminating power of the discriminant

functions.

The Canonical Correlation, R

This measures the association between the groups formed by the dependent and the given discriminant function. A large canonical correlation

indicates high correlation between the discriminant functions and the groups.

An R of 1.0 shows that all of the variability in the discriminant scores can be accounted for by that dimension. The relative percentage and R do not have to be correlated. Canonical Correlation, R , also shows how much each function is useful in determining group differences.

Mahalanobis Distances

This is the distance between a case and the centroid for each group (of the dependent variables) in attribute space (a dimensional space defined by n variables). There is one mahalanobis distance for each group of case, and it will be classified as belonging to the group with the smallest mahalanobis distance. This means that the closer the case to the group centriod, the smaller the mahalanobis distance. Mahalanobis distance is measured in terms of standard deviations from the centroid.

The Classification Table

This is a table in which the rows are observed categories of the dependent and the columns are the predicted categories of the dependent. All cases lie on the diagonal at perfect prediction.

Hit Ratio

This is the percentage of cases on the diagonal of a confusion matrix. It is the percentage of correct classifications. The higher the hit ratio the less the error of misclassification, also the less the hit ratio the higher the error rate.

Tolerance

This is the proportion of the variation in the independent variables that is not explained by the variables already in the model. Zero tolerance means that the independent variable under consideration is a perfect linear combination of other variables already in the model. A tolerance of 1 implies that the predictor variables are completely independent of other predictor variables already in the model. Most computer packages set the minimum tolerance at 0.01 as the default option.