Multiple discriminant analysis (MDA) is used to classify cases into groups when there are more than two categories. MDA derives multiple discriminant functions to discriminate between groups, with the first function accounting for the most variation between groups. The number of functions derived is usually equal to the number of groups minus one or the number of predictor variables, whichever is smaller. MDA outputs include standardized discriminant function coefficients, structure correlations, group centroids, and a classification matrix assessing prediction accuracy.

1. Multiple Discriminant Analysis(MDA)
Muhammad Hasrath

2. Multiple discriminant analysis
• Discriminant analysis techniques are described by the number of
categories possessed by the criterion variable. When the criterion
variable has two categories, the technique is known as two-group
discriminant analysis. When three or more categories are involved, the
technique is referred to as multiple discriminant analysis. The main
distinction is that in the two-group case it is possible to derive only one
discriminant function, but in multiple discriminant analysis more than
one function may be computed.

3. • The main distinction is that, in the two-group case, it is possible
to derive only one discriminant function. In multiple discriminant
analysis, more than one function may be computed. In general, with
G groups and k predictors, it is possible to estimate up to the smaller
of G - 1, or k , discriminant functions.
• The first function has the highest ratio of between-groups to within
groups sum of squares. The second function, be uncorrelated with the
first, has the second highest ratio, and so on. However, not all
the functions may be statistically significant.

4. 4
Multiple discriminant analysis(MDA)
• Example from the Trust data-set
• Dependent variable: Type of chicken purchased ‘in a
typical week’, choosing among four categories: value
(good value for money), standard, organic and luxury
• Predictors: age (q50), stated relevance of taste (q24a),
value for money (q24b) and animal welfare (q24k),
plus an indicator of income (q60)

5. 5
Multiple discriminant analysis(2)
• In this case there will be more than one discriminant function.
• The exact number of discriminant functions is equal to either (g-
1), where g is the number of categories in classification or to k,
the number of independent variables, whichever is the smaller
• Trust example: four groups and five explanatory variables, the
number of discriminant functions is three (that is g-1 which is
smaller than k=5).

6. 6
The output of MDS
• Similarities with factor (principal component) analysis
• the first discriminant function is the most relevant for discriminating across groups, the
second is the second most relevant, etc.
• the discriminant functions are also independent, which means that the resulting scores are
non-correlated.
• Once the coefficients of the discriminant functions are estimated and standardized, they are
interpreted in a similar fashion to the factor loadings.
• The larger the standardised coefficients (in absolute terms), the more relevant the respective
variables to discriminating between groups
• There is no single discriminant score in MDA
• group means are computed (centroids) for each of the discriminant functions to have a
clearer view of the classification rule

7. Canonical correlation
Canonical correlation measures the extent of association between the
discriminant scores and the groups. It is a measure of association between
the single discriminant function and the set of dummy variables that define
the group membership.
Centroid
The centroid is the mean values for the discriminant scores for a
particular group. There areas many centroids as there are groups, as there Is
one for each group. The means for a group on all the functions are the
group centroids
Classification matrix
Sometimes also called confusion or prediction matrix , the classification
matrix contains the number of correctly classified and misclassified cases.

8. Discriminant function coefficients
The discriminant function coefficients (un standardized)are the multipliers of
variables, when the variables are in the original units of measurement.
Discriminant scores.
The un standardized coefficients are multiplied by the values of the
variables. These products are summed and added to the constant term to
obtain the discriminant scores.
Eigen value
For each discriminant function, the Eigen value is the ratio of between-group
to within group sums of squares. Large Eigen values imply superior functions.

9. values and their significance
These are calculated from a one-way ANOVA, with the grouping variable
serving as the categorical independent variable. Each predictor, in
turn, serves as the metric dependent variable in the ANOVA.
Group means and group standard deviations
These are computed for each predictor for each group.
Pooled within-group correlation matrix
The pooled within-group correlation matrix is computed by averaging the
separate covariance matrices for all the groups.

10. Standardized discriminant function coefficients
The standardized discriminant function coefficients are the discriminant function coefficients and are
used as the multipliers when the variables have been standardized to a mean of 0 and
a variance of 1.
Structure correlations.
Also referred to as discriminant loadings , the structure correlations represent the simple correlations
between the predictors and the discriminant function.
Total correlation matrix
If the cases are treated as if they were from a single sample and the correlations computed, atotal
correlation matrix is obtained.
Wilks' .
Sometimes also called the U statistic, Wilks' for each predictor is the ratio of the within-group sum of
squares to the total sum of squares. Its value varies between 0 and
Large values of (near 1) indicate that group means do not
seem to be different. Small values of (near 0) indicate that the group means seem to be different.