Discriminant Analysis-Market research
-
Upload
calmchandan -
Category
Documents
-
view
223 -
download
0
Transcript of Discriminant Analysis-Market research
-
8/4/2019 Discriminant Analysis-Market research
1/29
Business Research Method
Discriminant Analysis
-
8/4/2019 Discriminant Analysis-Market research
2/29
Discriminant Analysis
Discriminant analysis helps in discriminating between two
or more sets of objects or people based on the knowledge ofsome of their characteristics
Discriminate between Bones or skeletons of males or
females Dividing people into potential buyers or non buyers
Classifying individuals as good or bad credit risk
Classifying companies as good or bad investment risks
Classifying consumers as brand loyal or brand switchers
-
8/4/2019 Discriminant Analysis-Market research
3/29
Similarities and Differences between
ANOVA, Regression, and Discriminant
Analysis ANOVA REGRESSION DISCRIMINANT ANALYSIS
Similarit iesNumber of One One OnedependentvariablesNumber ofindependent Multiple Mult iple Mult iplevariablesDifferencesNature of thedependent Metric Metric CategoricalvariablesNature of theindependent Categorical Metric Metricvariables
-
8/4/2019 Discriminant Analysis-Market research
4/29
Discriminant Analysis
Discriminant analysis is a technique for analyzing datawhen the criterion or dependent variable is categorical andthepredictor or independent variables are metricin nature.
The objectives of discriminant analysis are as follows:
Development ofdiscriminant functions, or linear
combinations of the predictor or independent variables,which will best discriminate between the categories of thecriterion or dependent variable (groups).
Classification of cases to one of the groups based on the
values of the predictor variables. Evaluation of the accuracy of classification.
-
8/4/2019 Discriminant Analysis-Market research
5/29
When the criterion variable has twocategories, 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.
Discriminant Analysis
-
8/4/2019 Discriminant Analysis-Market research
6/29
Discriminant Analysis Model
The discriminant analysis model involves linearcombinations of
the following form:
D = b0 + b1X1 + b2X2 + b3X3 + . . . + bkXk where
D = discriminant score
b 's = discriminant coefficient or weight
X's = predictor or independent variable
The coefficients, or weights (b), are estimated so
that the groups differ as much as possible on the
values of the discriminant function.
-
8/4/2019 Discriminant Analysis-Market research
7/29
Conducting Discriminant Analysis
Assess Validity of Discriminant Analysis
Estimate the Discriminant Function Coefficients
Determine the Significance of the Discriminant Function
Formulate the Problem
Interpret the Results
-
8/4/2019 Discriminant Analysis-Market research
8/29
Conducting Discriminant Analysis
Formulate Problem
Identify the objectives, the dependent variable, and the independentvariables.
The dependent variable must consist of two or more mutually exclusiveand collectively exhaustive categories. (Gender, Credit Risk,Investment Risk,)
The independent variables should be selected based on a theoretical
model or previous research, or the experience of the researcher. Collect data on independent variables for each category of criterion
variable
One part of the sample, called the estimation oranalysis sample, isused for estimation of the discriminant function.
The other part, called the holdoutorvalidation sample, is reserved forvalidating the discriminant function.
Often the distribution of the number of cases in the analysis andvalidation samples follows the distribution in the total sample.
-
8/4/2019 Discriminant Analysis-Market research
9/29
Example To determine salient characteristics of families that visited a vacation
resort during last two years
Data were obtained from a sample of 42 families of which 30 wereincluded in analysis sample & 12 in validation sample
HHs that visited resort coded as 1 & those that did not as 2
Both analysis and hold out samples were balanced in terms of visits
Independent variables were
--- Family income (V1)
---Attitude towards travel measured on a 9 point scale (V2)
---Importance attached to family vacation measured on a 9 point scale(V3)
---HH Size(V4)
---Age of the head of the HH(V5)
-
8/4/2019 Discriminant Analysis-Market research
10/29
Information on Resort Visits: Analysis Sample
Annual Attitude Importance Household Age of AmountResort Family Toward Attached Size Head ofSpent on No. Visit Income Travelto Family Household Family($000) Vacation Vacation1 1 50.2 5 8 3 43 M (2)2 1 70.3 6 7 4 61 H (3)3 1 62.9 7 5 6 52 H (3)4 1 48.5 7 5 5 36 L (1)5 1 52.7 6 6 4 55 H (3)6 1 75.0 8 7 5 68 H (3)7 1 46.2 5 3 3 62 M (2)
8 1 57.0 2 4 6 51 M (2)9 1 64.1 7 5 4 57 H (3)10 1 68.1 7 6 5 45 H (3)11 1 73.4 6 7 5 44 H (3)12 1 71.9 5 8 4 64 H (3)13 1 56.2 1 8 6 54 M (2)14 1 49.3 4 2 3 56 H (3)
15 1 62.0 5 6 2 58 H (3)
-
8/4/2019 Discriminant Analysis-Market research
11/29
Annual Attitude Importance Household Age of AmountResort Family Toward Attached Size Head ofSpent on No. Visit Income Travelto Family Household Family($000) Vacation Vacation
16 2 32.1 5 4 3 58 L (1)17 2 36.2 4 3 2 55 L (1)18 2 43.2 2 5 2 57 M (2)19 2 50.4 5 2 4 37 M (2)20 2 44.1 6 6 3 42 M (2)21 2 38.3 6 6 2 45 L (1)22 2 55.0 1 2 2 57 M (2)23 2 46.1 3 5 3 51 L (1)
24 2 35.0 6 4 5 64 L (1)25 2 37.3 2 7 4 54 L (1)26 2 41.8 5 1 3 56 M (2)27 2 57.0 8 3 2 36 M (2)28 2 33.4 6 8 2 50 L (1)29 2 37.5 3 2 3 48 L (1)30 2 41.3 3 3 2 42 L (1)
Information on Resort Visits: Analysis Sample
-
8/4/2019 Discriminant Analysis-Market research
12/29
Information on Resort Visits:
Holdout Sample
Annual Attitude Importance Household Age ofAmount Resort Family Toward Attached Size Head ofSpent on No. Visit IncomeTravel to Family Household Family($000) Vacation Vacation
1 1 50.8 4 7 3 45 M(2)2 1 63.6 7 4 7 55 H (3)3 1 54.0 6 7 4 58 M(2)4 1 45.0 5 4 3 60 M(2)5 1 68.0 6 6 6 46 H (3)6 1 62.1 5 6 3 56 H (3)7 2 35.0 4 3 4 54 L (1)8 2 49.6 5 3 5 39 L (1)9 2 39.4 6 5 3 44 H (3)10 2 37.0 2 6 5 51 L (1)11 2 54.5 7 3 3 37 M(2)12 2 38.2 2 2 3 49 L (1)
-
8/4/2019 Discriminant Analysis-Market research
13/29
Conducting Discriminant Analysis
Estimate the Discriminant Function
Coefficients The direct method involves estimating the discriminant
function so that all the predictors are included
simultaneously.
In stepwise discriminant analysis, the predictor variables
are entered sequentially, based on their ability to
discriminate among groups.
-
8/4/2019 Discriminant Analysis-Market research
14/29
Results of Two-Group Discriminant
AnalysisGROUP MEANSVISIT INCOME TRAVEL VACATION HSIZE AGE
1 60.52000 5.40000 5.80000 4.33333 53.733332 41.91333 4.33333 4.06667 2.80000 50.13333Total 51.21667 4.86667 4.9333 3.56667 51.93333
Group Standard Deviations
1 9.83065 1.91982 1.82052 1.23443 8.77062
2 7.55115 1.95180 2.05171 .94112 8.27101Total 12.79523 1.97804 2.09981 1.33089 8.57395
Pooled Within-Groups Correlation MatrixINCOME TRAVEL VACATION HSIZE AGE
INCOME 1.00000TRAVEL 0.19745 1.00000VACATION 0.09148 0.08434 1.00000HSIZE 0.08887 -0.01681 0.07046 1.00000
AGE - 0.01431 -0.19709 0.01742 -0.04301 1.00000Wilks' (U-statistic) and univariate F ratio with 1 and 28 degrees of freedom
Variable Wilks' F Significance
INCOME 0.45310 33.800 0.0000TRAVEL 0.92479 2.277 0.1425VACATION 0.82377 5.990 0.0209HSIZE 0.65672 14.640 0.0007
AGE 0.95441 1.338 0.2572 Contd.
-
8/4/2019 Discriminant Analysis-Market research
15/29
Interpretation When Predictors are considered individually
only Income, Importance of vacation &HH sizesignificantly differentiate between those who
visited resort & those who did not( F ratio with
k=1 & n-k-1= 30-1-1=28 d.f )
Wilks lambda (also called U statistics) is ratioof within group SS to total SS. Its value varies
between 0 to 1
Small value of lambda indicate that groupmeans are different & a better discrimination
power of that variable
-
8/4/2019 Discriminant Analysis-Market research
16/29
Conducting Discriminant Analysis
Determine the Significance of Discriminant Function
The null hypothesis that, in thepopulation, the means of discriminant
functions in both groups are equal can
be statistically tested.
In SPSS this test is based on Wilks' .
If the null hypothesis is rejected,indicating significant discrimination,
one can proceed to interpret the results.
-
8/4/2019 Discriminant Analysis-Market research
17/29
Results of Two-Group Discriminant Analysis
cont.
CANONICAL DISCRIMINANT FUNCTIONS% of Cum Canonical After Wilks'Func tion Eigenvalue Va riance % Corr el at ion Funct ion Chi-square df Significance: 0 0 .3589 26.130 5 0.00011* 1.7862 100.00 100.00 0.8007 :
* marks the 1 canonical discriminant functions remaining in the analysis.Standard Canonical Discriminant Function Coefficients
FUNC 1 INCOME 0.74301TRAVEL 0.09611VACATION 0.23329HSIZE 0.46911AGE 0.20922Structure Matrix:
Pooled within-groups correlations between discriminating variables & canonical discriminant functions(variables ordered by size of correlation within function)FUNC 1 INCOME 0.82202HSIZE 0.54096VACATION 0.34607TRAVEL 0.21337AGE 0.16354
Contd.
-
8/4/2019 Discriminant Analysis-Market research
18/29
Results of Two-Group Discriminant Analysis
cont.
Unstandardized Canonical Discriminant Function CoefficientsFUNC 1INCOME 0.8476710E-01TRAVEL 0.4964455E-01VACATION 0.1202813HSIZE 0.4273893
AGE 0.2454380E-01(constant) -7.975476Canonical discriminant functions evaluated at group means (group centroids)Group FUNC 11 1.291182 -1.29118Classification results for cases selected for use in analysis
Predicted Group MembershipActual Group No. of Cases 1 2Group 1 15 12 380.0% 20.0%Group 2 15 0 150.0% 100.0%
Percent of grouped cases correctly classified: 90.00% Contd.
-
8/4/2019 Discriminant Analysis-Market research
19/29
Interpretation
Since 0.0001 is less than .05 we reject the nullhypothesis of equality of group means indicating
better discriminating power of the discriminantfunction
The unstandardised discriminant function is
D= -7.975476 +
+0.8476710E-01(INCOME)
+0.4964455E-01(TRAVEL)
+0.1202813(VACATION)+0.4273893(HSIZE)
+0.2454380E-01(AGE)
-
8/4/2019 Discriminant Analysis-Market research
20/29
Classification Using Model
Group Centroids are values of discriminant function at Group
Means The average of the two Group Centroids gives the cut off
point
The Group Centroids For the Resort Example are:
Group Centroid 1 1.29118
2 -1.29118
The average of two centroids is 0
Therefore any positive value of discriminant Score willlead to Classification as Resort Visit & negative valueto No Resort Visit
Classification Using Model
-
8/4/2019 Discriminant Analysis-Market research
21/29
Classification Using Model Annual Attitude Importance Household Age of
Resort Family Toward Attached Size Head of
No. Visit Income Travel to Family
Household
($000) Vacation
1 2 50.8 4 7 3 45
Value Of Dicriminant Function For the Above Individual Would Be:
D= -7.975476 +
+0.8476710E-01(50.8)
+0.4964455E-01(4)
+0.1202813(7)+0.4273893(3)
+0.2454380E-01(45)
= - 0.242
-
8/4/2019 Discriminant Analysis-Market research
22/29
Results of Two-Group Discriminant Analysiscont.
Classification Results for cases not selected for use in the analysis (holdout sample)Predicted Group Membersh ipActual Group No. of Cases 1 2
Group 1 6 4 266.7% 33.3%Group 2 6 0 60.0% 100.0%
Percent of grouped cases correctly classified: 83.33%.
-
8/4/2019 Discriminant Analysis-Market research
23/29
Interpretation of Results
Find out the percentage of cases correctly
classified by the model
Find out variables which are relatively
better in discriminating between groups
How to classify a new subject into one of
the groups
-
8/4/2019 Discriminant Analysis-Market research
24/29
Conducting Discriminant Analysis
Interpret the Results
We can obtain some idea of the relative importance of thevariables by examining the absolute magnitude of thestandardized discriminant function coefficients.
Some idea of the relative importance of the predictors can
also be obtained by examining the structure correlations,also called canonical loadings ordiscriminant loadings.These simple correlations between each predictor and thediscriminant function represent the variance that the
predictor shares with the function. Another aid to interpreting discriminant analysis results is
to develop a characteristic profile for each group bydescribing each group in terms of the group means for the
predictor variables.
-
8/4/2019 Discriminant Analysis-Market research
25/29
Conducting Discriminant Analysis
Access Validity of Discriminant Analysis
The discriminant weights, estimated by using theanalysis sample, are multiplied by the values of the
predictor variables in the holdout sample to generate
discriminant scores for the cases in the holdout
sample.
The cases are then assigned to groups based on
their discriminant scores and an appropriate decision
rule. The hit ratio, or the percentage of casescorrectly classified, can then be determined by
summing the diagonal elements and dividing by the
total number of cases.
-
8/4/2019 Discriminant Analysis-Market research
26/29
Stepwise Discriminant Analysis
Stepwise discriminant analysis is analogous to stepwise
multiple regression in that the predictors are enteredsequentially based on their ability to discriminate betweenthe groups.
AnFratio is calculated for each predictor by conducting aunivariate analysis of variance in which the groups are
treated as the categorical variable and the predictor as thecriterion variable.
The predictor with the highestFratio is the first to beselected for inclusion in the discriminant function, if it
meets certain significance and tolerance criteria. A second predictor is added based on the highest adjusted orpartialFratio, taking into account the predictor alreadyselected.
-
8/4/2019 Discriminant Analysis-Market research
27/29
Each predictor selected is tested for retention based on itsassociation with other predictors selected.
The process of selection and retention is continued until all
predictors meeting the significance criteria for inclusion andretention have been entered in the discriminant function.
The selection of the stepwise procedure is based on theoptimizing criterion adopted. The Mahalanobis procedureis based on maximizing a generalized measure of thedistance between the two closest groups.
The order in which the variables were selected alsoindicates their importance in discriminating between thegroups.
Stepwise Discriminant Analysis
-
8/4/2019 Discriminant Analysis-Market research
28/29
SPSS Windows
The DISCRIMINANT program performs both two-group
and multiple discriminant analysis. To select this procedure
using SPSS for Windows click:
Analyze>Classify>Discriminant
-
8/4/2019 Discriminant Analysis-Market research
29/29
Eigen Value =
Between S S/ Within SS
Wilk,s Lamda= WithinSS/TotalSS
Canonical R = Correlation beween
estimated Y & actual Y