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Chapter 5Multiple Discriminant Analysis


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Multiple discriminant analysis . . . is an appropriatetechnique when the dependent variable is categorical (nominal ornonmetric) and the independent variables are metric. The singledependent variable can have two, three or more categories.

Discriminant Analysis DefinedDiscriminant Analysis Defined

Ex amples: Gender Male vs. Female Heavy Users vs. Light Users Purchasers vs. Non-purchasers Good Credit Risk vs. Poor Credit Risk Member vs. Non-Member Attorney, Physician or Professor


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WHAT IS MULTIPLE DISCRIMINANT

ANALYSIS? Discriminant analysis is a dependence technique that forms variates (linearcombinations of metric independent variables), which are used to predict theclassification of a categorical dependent variable.

Classification is accomplished by a statistical procedure, which derives discriminantfunctions, or variates of the predictor variables, which ma x imize the between-group variance and minimize the within-group variance on the discriminantfunction score(s).

The null hypothesis is that the two or more group means are equal on thediscriminant function(s), thus a statistically significant model would indicate thatthe group means are not equal.

Researchers use multiple discriminant analysis to help them understand: group differences on a set of independent variables the ability to correctly classify statistical units into groups or classes the relative importance of independent variables in the classification process


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Multiple discriminant analysis may beconsidered a type of profile analysis or ananalytical predictive technique which is mostappropriate when there is a single categoricaldependent variable and multiple metricindependent variables.


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The discriminant function, or variates isderived so as to ma x imize the between-groupvariance and minimize the within-groupvariance.

From the linear function a Z score is calculated

for each observation. By averaging thesescores one arrive at the centroid or groupmean


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G raphic Illustration of G raphic Illustration of TwoTwo--G roup Discriminant AnalysisG roup Discriminant Analysis

XX 22

XX 11

ZZ

Discriminant Discriminant FunctionFunction

A

B

A

B


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S tage 1: Objectives of Discriminant AnalysisS tage 1: Objectives of Discriminant Analysis

1. Determine if statistically significant differences e x ist between the two (ormore) a priori defined groups.

2. Identify the relative importance of each of the independent variables inpredicting group membership.

3. Develop procedures for classifying objects (individuals, firms, products, etc.)into groups, and then e xamining the predictive accuracy ( ie. hit ratio) of thediscriminant function to see if it is acceptable (> 25% increase).


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Selection of dependent andSelection of dependent and

independent variables.independent variables.

Sample size (total & per variable).Sample size (total & per variable).

Sample division for validation.Sample division for validation.

S tage 2: Research Design for Discriminant AnalysisS tage 2: Research Design for Discriminant Analysis


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S election of dependent andS election of dependent andindependent variablesindependent variables

The dependent variable must be nonmetric, representing groups of objects that are e x pected to differ on the independent variables.

Choose a dependent variable that:

best represents group differences of interest,

defines groups that are substantially different, and

minimizes the number of categories while still meeting the researchobjectives.

In converting metric variables to a nonmetric scale for use as thedependent variable, consider using e x treme groups to ma x imize thegroup differences.

Independent variables must identify differences between at least twogroups to be of any use in discriminant analysis.


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Key AssumptionsKey Assumptions

Multivariate normality of the independent variables.

Equal variance and covariance for the groups.

Other AssumptionsOther Assumptions

Minimal multicollinearity among independent variables.

Linear relationships.E limination of outliers.

S tage 3: Assumptions of S tage 3: Assumptions of Discriminant Discriminant AnalysisAnalysis


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3 Stages of discriminant analysis

Deriving the discriminant functionCalibration

interpretation

S tage 4: Estimation of theS tage 4: Estimation of the Discriminant Discriminant Model andModel andAssessing Overall Fit Assessing Overall Fit


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S tage 4: Estimation of theS tage 4: Estimation of the Discriminant Discriminant

Model and Assessing Overall Fit Model and Assessing Overall Fit

Step 1:Step 1: SelectingSelecting An Estimation Method . . .An Estimation Method . . .

1.1. Simultaneous E stimationSimultaneous E stimation all independent variables areall independent variables areconsideredconsidered at the same time.at the same time.

2.2. Stepwise EstimationStepwise Estimation independent variables are entered intoindependent variables are entered intothethe discriminantdiscriminant function one at a timefunction one at a time ..

Step 2Step 2: Assess the function statistical significance: Assess the function statistical significanceC riteria: Wilks' lambda, Hotelling's trace, Pilliai's criteria, Roy'sgreatest characteristic root, Mahalanobis' distance, and Rao's Vmeasures.S ignificance l eve l : The conventional criterion of .05 or beyond is

most often used.


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Step 3: Construct classification matri x to assessoverall fit of discriminant function

CalculatingCalculating discriminantdiscriminant Z scores for each observation see pgZ scores for each observation see pg259259Evaluating group differences on theEvaluating group differences on the discriminantdiscriminant Z scoresZ scoresAssessing group membership prediction accuracy.Assessing group membership prediction accuracy.

The statistical and practical rational for developing classification matricesThe statistical and practical rational for developing classification matricesThe cutting score determinationThe cutting score determinationConstruction of the classification matricesConstruction of the classification matricesStandards for assessing classification accuraStandards for assessing classification accura cycy

Step 4:Compare the ratio to ascertain the predictive accuracy

of the modela) chance criteriab) Press Q statistic


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S tage 5: Interpretation of the ResultsS tage 5: Interpretation of the Results

Three Methods . . .Three Methods . . .

1.1. Standardized discriminant weights,Standardized discriminant weights,2.2. Discriminant loadings (structureDiscriminant loadings (structure

correlations), andcorrelations), and

3.3. Partial F values.Partial F values.


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Interpretation of the ResultsInterpretation of the Results

Two or More Functions . . .Two or More Functions . . .

1.1. Rotation of discriminant functionsRotation of discriminant functions

2.2. Potency inde xPotency inde x


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S tage 6: Validation of the ResultsS tage 6: Validation of the Results

Utilizing a Holdout SampleUtilizing a Holdout Sample

CrossCross--ValidationValidation

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