9 discriminant analysis

7/28/2019 9 Discriminant Analysis

1/18

Discriminantanalysis

Dr James Abdey

Overview

Discriminant analysis

Discriminant analysis model


statistics

Formulate the problem

Estimate the discriminant

function coefficients

Determine the significance

of discriminant functions

Interpret the results

Assess validity ofdiscriminant analysis

Applied Marketing(Market Research Methods)

Topic 9:


Dr James Abdey
http://find/


2/18


Dr James Abdey

Overview




statistics








Overview

We discuss the technique of discriminant analysis,

initially by examining its relationship to regression

analysis

Modelling of discriminant analysis is presented,along with formulation, estimation, significance,

interpretation and validation of results

Two-group and multiple group discriminantanalyses are introduced
http://find/


3/18


Dr James Abdey

Overview




statistics









Discriminant analysis is a technique for analysing

data when the dependent variable is categorical

and the independent variables are measurable

The main objectives of discriminant analysis are:

Development of discriminant functions, i.e. linearcombinations of the independent variables, which willbest discriminate between the categories (groups) ofthe dependent variable

Checking whether significant differences existamong the groups, in terms of the independentvariables
http://find/


4/18


Dr James Abdey

Overview




statistics









Determination of which independent variablescontribute to most of the intergroup differences

Classification of cases to one of the groups basedon the values of the independent variables, and

determining the accuracy of classification

When the dependent variable has two categories,the technique is called two-group discriminantanalysis

When three or more categories are involved, thetechnique is called multiple discriminant analysis
http://find/


5/18


Dr James Abdey

Overview




statistics









In the two-group case, it is possible to derive only

one discriminant function

In multiple case, more than one function may be

computed

In general, with M groups and k independent

variables, it is possible to estimate up to the smallerof M 1, or k, discriminant functions

The first function has the highest ratio of

between-groups to within-groups sum of squares

The second function, uncorrelated with the first, hasthe second highest ratio, and so on

However, not all the functions may be statistically

significant
http://find/


6/18


Dr James Abdey

Overview




statistics









The discriminant analysis model involves linear

combinations of the following form:

D= 0 + 1X1 + 2X2 + 3X3 + . . .+ kXk

where

D= discriminant score

s = discriminant coefficient Xs = independent variables

The coefficients, , are estimated so that the

groups differ as much as possible on the values of

the discriminant function This occurs when the ratio of between-group sum of

squares to within-group sum of squares for the

discriminant scores is at a maximum
http://find/


7/18


Dr James Abdey

Overview




statistics








Discriminant analysis statistics

Canonical correlation the extent of associationbetween the discriminant scores and the groups. It is

a measure of association between a discriminant

function and the set of dummy variables that define

the group membership

Centroid mean values for the discriminant scores

for a particular group. There are as many centroids

as there are groups with one centroid per group

Classification matrix contains the number of

correctly classified and misclassified cases
http://find/


8/18


Dr James Abdey

Overview




statistics









Discriminant function coefficients the

(unstandardised) multipliers of independentvariables, when the variables are in the original units

of measurement

Discriminant scores the discriminant function

coefficients are multiplied by the values of therespective independent variables. These products

are summed and added to the constant term to

obtain the discriminant scores

Eigenvalue For each discriminant function, the

eigenvalue is the ratio of between-group to

within-group sums of squares. Larger eigenvalues

indicate better functions
http://find/


9/18


Dr James Abdey

Overview




statistics









Group means and group standard deviations computed for each independent variable for each

group

Standardised discriminant function coefficients used as the multipliers when the independent

variables have been standardised, i.e. have a mean

of 0 and a variance of 1

Structure correlations simple correlationsbetween the predictors and the discriminant function
http://find/


10/18


Dr James Abdey

Overview




statistics









Total correlation matrix treating the cases as a

single sample, a total correlation matrix is obtained

Wilks for each independent variable, Wilks is

the ratio of the within-group sum of squares to thetotal sum of squares. Its value varies between 0 and

1. 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 do seem to bedifferent
http://find/


11/18


Dr James Abdey

Overview




statistics









Identify the objectives, the dependent variable

and the independent variables

The dependent variable must consist of two or more

mutually exclusive and collectively exhaustivecategories

The independent variables should be selected based

on a theoretical model or previous research, or theexperience of the researcher
http://find/


12/18


Dr James Abdey

Overview

Discriminant analysisDiscriminant analysis model


statistics









One part of the sample, called the estimation

sample, is used for estimation of the discriminant

function

The other part, called the validation sample, isreserved for validating the discriminant function

Often the distribution of the number of cases in the

estimation and validation samples follows thedistribution in the total sample
http://find/


13/18


Dr James Abdey

Overview



statistics








Estimate the discriminant function

coefficients

The direct method involves estimating the

discriminant function so that all the independent

variables are included simultaneously

In stepwise discriminant analysis, the independent

variables are entered sequentially, based on their

ability to discriminate among the groups
http://find/http://goback/


14/18


Dr James Abdey

Overview



statistics








Determine the significance of

discriminant functions

The null hypothesis that, in the population, the

means of all discriminant functions in all groups are

equal can be statistically tested

In SPSS this test is based on Wilks if severalfunctions are tested simultaneously (as in the case of

multiple discriminant analysis), the Wilks statistic

is the product of the univariate for each function

If the null hypothesis is rejected, indicating

significant discrimination, one can proceed to

interpret the results

Di i i
http://find/http://goback/


15/18


Dr James Abdey

Overview



statistics









The interpretation of the discriminant coefficients issimilar to that in multiple regression analysis

Given the multicollinearity in the independent

variables, there is no unambiguous measure of the

relative importance of the independent variables in

discriminating between the groups

Nevertheless, we can obtain some idea of the

relative importance of the variables by examiningthe absolute magnitude of the standardised

discriminant function coefficients

Di i i tI h l
http://find/


16/18


Dr James Abdey

Overview



statistics









Some idea of the relative importance of the

independent variables can also be obtained by

examining the structure correlations

These simple correlations between each

independent variable and the discriminant functionrepresent the variance that the independent variable

shares with the function

Another aid to interpreting discriminant analysis

results is to develop a characteristic profile for

each group by describing each group in terms of the

group means for the independent variables

DiscriminantA lidi f di i i
http://find/


17/18


Dr James Abdey

Overview



statistics







Assess validity of

discriminant analysis

Assess validity of discriminant

analysis

The discriminant coefficients, estimated by using the

estimation sample, are multiplied by the values of the

independent variables in the validation sample to

generate discriminant scores for the cases in thevalidation sample

The cases are then assigned to groups based on

their discriminant scores and an appropriatedecision rule

DiscriminantA lidit f di i i t
http://find/


18/18


Dr James Abdey

Overview



statistics







Assess validity of

discriminant analysis

Assess validity of discriminant

analysis

The hit ratio, or the percentage of cases correctly

classified, can then be determined by summing the

diagonal elements and dividing by the total number

of cases

It is helpful to compare the percentage of cases

correctly classified by discriminant analysis to the

percentage that would be obtained by chance

Classification accuracy achieved by discriminant

analysis should be at least 25% greater than that

obtained by chance
http://find/

9 discriminant analysis

Documents