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1/18
Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
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:
Discriminant analysis
Dr James Abdey
http://find/7/28/2019 9 Discriminant Analysis
2/18
Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis model
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis 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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis 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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Discriminant analysis 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
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11/18
Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysis
Discriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Formulate the problem
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Formulate the problem
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Interpret the results
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/7/28/2019 9 Discriminant Analysis
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
Assess validity ofdiscriminant analysis
Interpret the results
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
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
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Discriminantanalysis
Dr James Abdey
Overview
Discriminant analysisDiscriminant analysis model
Discriminant analysis
statistics
Formulate the problem
Estimate the discriminant
function coefficients
Determine the significance
of discriminant functions
Interpret the results
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
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