2012 handout 7 - discriminant analysis

7/31/2019 2012 Handout 7 - Discriminant Analysis

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Discrim inant AnalysisIdentifying key variables

How can you answer thesequestions?

In pr odu ct management,

1. What makes consumers choose differentbrands?

2. What consumer/ product at tr ibutes make myconsumers need post-purchase services?

3. What consumer attributes are important forsuccessful new pr oduct in tr oducti on?

4. etc.


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In pr omotion management ,

1. What are key success factors in choosingmedia vehi cles?

2. What makes difference between successfuland inefficient sales reps?

3. What attribut es are important in consumersadopt ion of coupons?

4. What makes difference between efficient andinefficient messages?

5. What attribut es are important t o create asuccessful display?

6. etc.

In supply channel man agement,

1. What m akes big difference between retailers(wholesalers)?

2. What attribu tes are important in choosingsales territories?

3. What attribu tes do I have to consider inchoosing channel m embers?

4. etc.


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Discriminant Analysistechn ique for analyzing mark eti ng research datawhen th e dependent vari able is categorical and thein dependent vari ables are metric in natur e. Itdevelops a set of criteria that can be used toseparate objects int o groups such th at each objectis more lik e oth er objects in i ts group th an lik eobjects out side the grou p . Objects can be eit hervariables or observations.

Objectives

a) development of linear combinations of theindependent variables, which will bestdiscriminate between the categories of th edependent variable (groups);

b) examin ation of wh eth er signi ficant differencesexist am ong the grou ps , in t erm s of the

independent variables;


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c) determinat ion of which independent variablescont ribu te to most of the inter-groupdifferences;

d) classification of cases to one of the groupsbased on the values of the independentvariables;

e) evaluat ion of the accuracy of classification,etc.

Univariate Univariate Representation of Discriminant Z Scores Representation of Discriminant Z Scores

Discriminant Function Discriminant Function

Discriminant Function Discriminant Function

Z

Z

A B

BA


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Graphic I llustration of TwoGraphic I llustration of Two- -Group DiscriminantGroup DiscriminantAnalysisAnalysis

X 2

X 1

Z

B

Discriminant Discriminant Function Function

A

B

A

Two-Group Discriminant Analysis :when the dependent variable has twocategories, it derives only onediscriminant function;

Mul tiple Discrimin ant Analysis : whenth e dependent variable has th ree ormore categories, it derives more thanone discrimin ant fun ction.


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Examples: 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

KitchenAidKitchenAid Survey Results for theSurvey Results for the

Evaluation* of a New Consumer ProductEvaluation* of a New Consumer ProductXX33StyleStyle

Group 1Would purchase 1 8 9 6

2 6 7 53 10 6 34 9 4 45 4 8 2

Group Mean 7.4 6.8 4.0Group 2Would not purchase 6 5 4 7

7 3 7 28 4 5 59 2 4 3

10 2 2 2Group Mean 3.2 4.4 3.8

Difference between group means 4.2 2.4 0.2

Purchase I ntentionPurchase I ntention SubjectSubjectNumberNumber

XX11DurabilityDurability

XX22PerformancePerformance

** Evaluations made on a 0 (very poor) to 10 (excellent) ratingscale.


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Group 1 - Definitelyswitch

Respondent Price competitiveness Service level1 2 22 1 23 3 24 2 15 2 3

Group 2 - Undecided6 4 27 4 38 5 19 5 2

10 5 3

Group 3 - Definitely notswitch

11 2 612 3 613 4 614 5 6

15 5 7

Selection of dependent andSelection of dependent andindependent variables.independent variables.

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

Sample division for validation.Sample division for validation.

Research Design for Discriminant AnalysisResearch Design for Discriminant Analysis


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Converting Metric Variables toConverting Metric Variables to NonmetricNonmetric

Most common approachMost common approach = to use the metric scale= to use the metric scaleresponses to developresponses to develop nonmetricnonmetric categories. Forcategories. Forexample, use a question asking the typical numberexample, use a question asking the typical numberof soft drinks consumed per week and develop aof soft drinks consumed per week and develop athreethree--category variable of 0 drinks for noncategory variable of 0 drinks for non- -users, 1users, 1

5 for light users, and 5 or more for heavy users.5 for light users, and 5 or more for heavy users.

Polar extremes approachPolar extremes approach = compares only the= compares only theextreme two groups and excludes the middleextreme two groups and excludes the middlegroup(s).group(s).

Discriminant Analysis DesignDiscriminant Analysis Design

The dependent variable must be nonmetric, representing groupsof objects that are expected to differ on the independent variables.

Choose a dependent variable that:best represents group differences of interest,defines groups that are substantially different, andminimizes the number of categories while still meeting theresearch objectives.

In converting metric variables to a non-metric scale for use as thedependent variable, consider using extreme groups to maximizethe group differences.

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


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continued . . .continued . . .

The sample size must be large enough to:have at least one more observation per group than the number ofindependent variables, but striving for at least 20 cases per group.have 20 cases per independent variable, with a minimumrecommended level of 7 observations per variable.have a large enough sample to divide it into an estimation and holdoutsample, each meeting the above requirements.

Assess the equality of covariance matrices with the Boxs M test, but applya conservative significance level of .01.

Examine the independent variables for univariate normality.

Multicollinearity among the independent variables can markedly reduce theestimated impact of independent variables in the derived discriminantfunction(s).

Assumptions of Discriminant AnalysisAssumptions of Discriminant Analysis

Key AssumptionsKey Assumptions

Multivariate normality of theMultivariate normality of theindependent variables.independent variables.

Equal variance and covarianceEqual variance and covariancefor the groups.for the groups.


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Other AssumptionsOther Assumptions

Minimal multicollinearity amongMinimal multicollinearity amongindependent variables.independent variables.

Group sample sizes relatively equal.Group sample sizes relatively equal. Linear relationships.Linear relationships. Elimination of outliers.Elimination of outliers.

Assumptions of Discriminant AnalysisAssumptions of Discriminant Analysis

Model

Data Str uctur e

Each object is characterized by a set ofmeasur ement s. We also know to whichgroup each object belongs.

e.g., object 1: ( x 11 , x 12 , , x 1m ) Group I

object 2: ( x 21 , x 22 , , x 2m ) Group II


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Index Fu nction:

D = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 + + b m x m ,

where D is discriminant score ; b 's arediscriminant coefficients or weights; x 's arein dependent variables. Note th at D and b 's areNOT observed.

The method of findin g th e weights ( b j ): a set ofweights wi ll be generated in order to m aximi ze th e

rat io of between-group vari ation and within-groupvariation . Thu s, vari ation in D between th e twogroups is as large as possible, while the variationin D wit hi n t he group is as small as possible.

Related Qu estions?

how well does the discr im inan t fun ctionclassify the sample?is the discrim inant function statisticallysignificant ?what are th e relative import ance of the

independent variables? etc.


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1. How well does th e discrimi nant

function classify the sample?a) compu te the mean values of the independent

variables for each group;b) find the mean discrim inant score for each

grou p which i s th e sum of these mean valu esof the in dependent variables weighted by t hecorresponding estimated weights;

c) find the cut t ing score D cs which is t he averageof th e mean discrim in ant scores ;

21

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nn

Dn Dn D

cs

d) estimate the discriminant score D for eachrespondent;

e) compare an individual discriminant score D with th e cutt ing score D cs and assign therespondent to one of the two groups;

f) const ru ct t he confusion matrix (2 x 2 table) of

which the diagonal shows the hit rat e (th eproport ion corr ectly classified) and evaluateth e model.


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2. Is th e discr iminan t fu nction

stati sti cally significan t?Each variable: Look at Wilk s

The ratio of th e with in -groups sum of squares toth e total su m of squares

Corresponding F-StatHo: The Equalit y of Grou p MeansHa: Group m eans are not equal

3. What are th e relative im port anceof the predictor variables?

since independent variables may have differentun its, th e estimates of th e discrim inantcoefficients b j need t o be standardized. Then, rankb j * to obtain the relative importance of thein dependent vari ables.


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Other Competing Meth ods: LinearProbability, Probit, Logit type models.

Data Analysis in SPSS

2012 handout 7 - discriminant analysis

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