Chapter Eighteen - l n

Chapter TenCopyright © 2010 Pearson Education, Inc. publishing as Prentice Hall 18-1
Chapter Eighteen
Copyright © 2010 Pearson Education, Inc. publishing as Prentice Hall 18-2
Direct marketing scenario
• Every year, the DM company sends to one annual catalog, four seasonal catalogs, and a number of calalogs for holiday seasons.
• The company has a list of 10 million potential buyers.
• The response rate is on average 5%. • Who should receive a catalog? (mostly likely to
buy) • How are buyers different from nonbuyers? • Who are most likely to default?...
Similarities and Differences between ANOVA, Regression, and Discriminant Analysis
Table 18.1
ANOVA REGRESSION DISCRIMINANT/LOGIT Similarities Number of One One One dependent variables Number of independent Multiple Multiple Multiple variables
Differences Nature of the dependent Metric Metric Categorical variables Nature of the independent Categorical Metric Metric (or binary, dummies) variables
Discriminant Analysis
Discriminant analysis is a technique for analyzing data when the criterion or dependent variable is categorical and the predictor or independent variables are interval in nature.
The objectives of discriminant analysis are as follows: • Development of discriminant functions, or linear
combinations of the predictor or independent variables, which will best discriminate between the categories of the criterion or dependent variable (groups). (buyers vs. nonbuyers)
• Examination of whether significant differences exist among the groups, in terms of the predictor variables.
• Determination of which predictor variables contribute to most of the intergroup differences.
• Classification of cases to one of the groups based on the values of the predictor variables.
• Evaluation of the accuracy of classification.
• When the criterion variable has two categories, 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.
• The main distinction is that, in the two-group case, it is possible to derive only one discriminant function. In multiple discriminant analysis, more than one function may be computed. In general, with G groups and k predictors, it is possible to estimate up to the smaller of G - 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, has the second highest ratio, and so on. However, not all the functions may be statistically significant.
Geometric Interpretation
Fig. 18.1
2 2 2 2 2 2 2 2 2 2
2 1 1
1 1 1 1
Discriminant Analysis Model
the following form:
Where:
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.
• 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.
Statistics Associated with Discriminant Analysis
• Canonical correlation. Canonical correlation measures the extent of association between the discriminant scores and the groups. It is a measure of association between the single discriminant function and the set of dummy variables that define the group membership.
• Centroid. The centroid is the mean values for the discriminant scores for a particular group. There are as many centroids as there are groups, as there is one for each group. The means for a group on all the functions are the group centroids.
• Classification matrix. Sometimes also called confusion or prediction matrix, the classification matrix contains the number of correctly classified and misclassified cases.
• Discriminant function coefficients. The discriminant function coefficients (unstandardized) are the multipliers of variables, when the variables are in the original units of measurement.
• Discriminant scores. The unstandardized coefficients are multiplied by the values of the 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. Large Eigenvalues imply superior functions.
• F values and their significance. These are calculated from a one-way ANOVA, with the grouping variable serving as the categorical independent variable. Each predictor, in turn, serves as the metric dependent variable in the ANOVA.
• Group means and group standard deviations. These are computed for each predictor for each group.
• Pooled within-group correlation matrix. The pooled within-group correlation matrix is computed by averaging the separate covariance matrices for all the groups.
• Standardized discriminant function coefficients. The standardized discriminant function coefficients are the discriminant function coefficients and are used as the multipliers when the variables have been standardized to a mean of 0 and a variance of 1.
• Structure correlations. Also referred to as discriminant loadings, the structure correlations represent the simple correlations between the predictors and the discriminant function.
• Total correlation matrix. If the cases are treated as if they were from a single sample and the correlations computed, a total correlation matrix is obtained.
• Wilks' . Sometimes also called the U statistic, Wilks' for each predictor is the ratio of the within-group sum of squares to the total 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 seem to be different.
λ λ
λ λ
Conducting Discriminant Analysis
Determine the Significance of the Discriminant Function
Formulate the Problem
Interpret the Results
Conducting Discriminant Analysis Formulate the Problem
• Identify the objectives, the criterion variable, and the independent variables.
• The criterion variable must consist of two or more mutually exclusive and collectively exhaustive categories.
• The predictor variables should be selected based on a theoretical model or previous research, or the experience of the researcher.
• One part of the sample, called the estimation or analysis sample, is used for estimation of the discriminant function.
• The other part, called the holdout or validation sample, is reserved for validating the discriminant function.
• Often the distribution of the number of cases in the analysis and validation samples follows the distribution in the total sample.
Information on Resort Visits: Analysis Sample
Table 18.2
Annual Attitude Importance Household Age of Amount Resort Family Toward Attached Size Head of Spent on
No. Visit Income Travel to Family Household Family ($000) Vacation Vacation
1 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)
Information on Resort Visits: Analysis Sample Ta
bl e
18 .2
, c on
t. Annual Attitude Importance Household Age of Amount Resort Family Toward Attached Size Head of Spent on
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: Holdout Sample Table 18.3
Annual Attitude Importance Household Age of Amount Resort Family Toward Attached Size Head of Spent on
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)
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.
Results of Two-Group Discriminant Analysis Ta
bl e
18 .4
GROUP MEANS VISIT INCOME TRAVEL VACATION HSIZE AGE
1 60.52000 5.40000 5.80000 4.33333 53.73333 2 41.91333 4.33333 4.06667 2.80000 50.13333 Total 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 8.27101 Total 12.79523 1.97804 2.09981 1.33089 8.57395
Pooled Within-Groups Correlation Matrix INCOME TRAVEL VACATION HSIZE AGE
INCOME 1.00000 TRAVEL 0.19745 1.00000 VACATION 0.09148 0.08434 1.00000 HSIZE 0.08887 -0.01681 0.07046 1.00000 AGE - 0.01431 -0.19709 0.01742 -0.04301 1.00000
Wilks' (U-statistic) and univariate F ratio with 1 and 28 degrees of freedom
Variable Wilks' F Significance
INCOME 0.45310 33.800 0.0000 TRAVEL 0.92479 2.277 0.1425 VACATION 0.82377 5.990 0.0209 HSIZE 0.65672 14.640 0.0007 AGE 0.95441 1.338 0.2572 Cont.
Results of Two-Group Discriminant Analysis Table 18.4, cont.
CANONICAL DISCRIMINANT FUNCTIONS
% of Cum Canonical After Wilks' Function Eigenvalue Variance % Correlation Function λ Chi-square df Significance
: 0 0 .3589 26.130 5 0.0001 1* 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 0.74301 TRAVEL 0.09611 VACATION 0.23329 HSIZE 0.46911 AGE 0.20922
Structure Matrix: Pooled within-groups correlations between discriminating variables & canonical discriminant functions (variables ordered by size of correlation within function)
FUNC 1
INCOME 0.82202 HSIZE 0.54096 VACATION 0.34607 TRAVEL 0.21337 AGE 0.16354 Cont.
Cont.
Unstandardized Canonical Discriminant Function Coefficients FUNC 1
INCOME 0.8476710E-01 TRAVEL 0.4964455E-01 VACATION 0.1202813 HSIZE 0.4273893 AGE 0.2454380E-01 (constant) -7.975476 Canonical discriminant functions evaluated at group means (group centroids)
Group FUNC 1 1 1.29118 2 -1.29118 Classification results for cases selected for use in analysis
Predicted Group Membership Actual Group No. of Cases 1 2
Group 1 15 12 3 80.0% 20.0%
Group 2 15 0 15 0.0% 100.0%
Percent of grouped cases correctly classified: 90.00%
Results of Two-Group Discriminant Analysis
Table 18.4, cont.
Classification Results for cases not selected for use in the analysis (holdout sample)
Predicted Group Membership Actual Group No. of Cases 1 2
Group 1 6 4 2 66.7% 33.3%
Group 2 6 0 6 0.0% 100.0%
Percent of grouped cases correctly classified: 83.33%.
Conducting Discriminant Analysis Interpret the Results
• The interpretation of the discriminant weights, or coefficients, is similar to that in multiple regression analysis.
• Given the multicollinearity in the predictor variables, there is no unambiguous measure of the relative importance of the predictors in discriminating between the groups.
• With this caveat in mind, we can obtain some idea of the relative importance of the variables by examining the absolute magnitude of the standardized 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 or discriminant loadings. These simple correlations between each predictor and the discriminant 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 by describing each group in terms of the group means for the predictor variables.
Conducting Discriminant Analysis Assess Validity of Discriminant Analysis
• Many computer programs, such as SPSS, offer a leave-one- out cross-validation option.
• The discriminant weights, estimated by using the analysis 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 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.
Results of Three-Group Discriminant Analysis Ta
bl e
18 .5
1 38.57000 4.50000 4.70000 3.10000 50.30000
2 50.11000 4.00000 4.20000 3.40000 49.50000
3 64.97000 6.10000 5.90000 4.20000 56.00000
Total 51.21667 4.86667 4.93333 3.56667 51.93333
Group Standard Deviations 1 5.29718 1.71594 1.88856 1.19722 8.09732
2 6.00231 2.35702 2.48551 1.50555 9.25263
3 8.61434 1.19722 1.66333 1.13529 7.60117
Total 12.79523 1.97804 2.09981 1.33089 8.57395
INCOME 1.00000
AGE -0.20939 -0.34022 -0.01326 -0.02512 1.00000 Cont.
All-Groups Scattergram
Fig. 18.3
4.0
0.0
Territorial Map
Fig. 18.4
4.0
0.0
1 1 2 3 1 1 2 2 3 3
1 1 2 2
1 1 1 2 2
1 1 2 21 1 2
21 1 1 2 2
1 1 2 2
1 1 2 2
1 1 2 2 2
2 2 3 2 3 3
2 2 3 3
The Logit Model
• The dependent variable is binary and there are several independent variables that are metric
• The binary logit model commonly deals with the issue of how likely is an observation to belong to each group (classification n prediction, buyer vs nonbuyer)
• It estimates the probability of an observation belonging to a particular group
Binary Logit Model Formulation
The probability of success may be modeled using the logit model as:
Or i
Model Formulation
)exp(1 )exp(
Properties of the Logit Model
• Although Xi may vary from to , P is constrained to lie between 0 and 1.
• When Xi approaches , P approaches 0.
• When Xi approaches , P approaches 1.
∞−
∞+
∞+
∞−
Estimation and Model Fit
• Both these measures are similar to R2 in multiple regression.
• The Cox & Snell R Square can not equal 1.0, even if the fit is perfect.
• This limitation is overcome by the Nagelkerke R Square.
• Compare predicted and actual values of Y to determine the percentage of correct predictions.
Significance Testing
The significance of the estimated coefficients is based on Wald’s statistic.
Wald = (ai / SEai) 2
ai = logistical coefficient for that predictor variable
SEai= standard error of the logistical coefficient
The Wald statistic is chi-square distributed with 1 degree of freedom if the
variable is metric and the number of categories minus 1 if the variable is
nonmetric.
The significance of the estimated coefficients is based on Wald’s statistic.
Wald = (ai / SEai)2
ai = logistical coefficient for that predictor variable
SEai= standard error of the logistical coefficient
The Wald statistic is chi-square distributed with 1 degree of freedom if the variable is metric and the number of categories minus 1 if the variable is nonmetric.
Interpretation of Coefficients
• If Xi is increased by one unit, the log odds will change by ai units, when the effect of other independent variables is held constant.
• The sign of ai will determine whether the probability increases (if the sign is positive) or decreases (if the sign is negative) by this amount.
Explaining Brand Loyalty
Table 18.6 No. Loyalty Brand Product Shopping 1 1 4 3 5 2 1 6 4 4 3 1 5 2 4 4 1 7 5 5 5 1 6 3 4 6 1 3 4 5 7 1 5 5 5 8 1 5 4 2 9 1 7 5 4
10 1 7 6 4 11 1 6 7 2 12 1 5 6 4 13 1 7 3 3 14 1 5 1 4 15 1 7 5 5 16 0 3 1 3 17 0 4 6 2 18 0 2 5 2 19 0 5 2 4 20 0 4 1 3 21 0 3 3 4 22 0 3 4 5 23 0 3 6 3 24 0 4 4 2 25 0 6 3 6 26 0 3 6 3 27 0 4 3 2 28 0 3 5 2 29 0 5 5 3 30 0 1 3 2
No.
Loyalty
Brand
Product
Shopping
1
1
4
3
5
2
1
6
4
4
3
1
5
2
4
4
1
7
5
5
5
1
6
3
4
6
1
3
4
5
7
1
5
5
5
8
1
5
4
2
9
1
7
5
4
10
1
7
6
4
11
1
6
7
2
12
1
5
6
4
13
1
7
3
3
14
1
5
1
4
15
1
7
5
5
16
0
3
1
3
17
0
4
6
2
18
0
2
5
2
19
0
5
2
4
20
0
4
1
3
21
0
3
3
4
22
0
3
4
5
23
0
3
6
3
24
0
4
4
2
25
0
6
3
6
26
0
3
6
3
27
0
4
3
2
28
0
3
5
2
29
0
5
5
3
30
0
1
3
2
Results of Logistic Regression
Table 18.7
Dependent Variable Encoding Original Value Internal Value Not Loyal 0 Loyal 1
Model Summary
Nagelkerke R Square
1 23.471(a) .453 .604 a Estimation terminated at iteration number 6 because parameter estimates changed by less than .001.
Dependent Variable Encoding
a Estimation terminated at iteration number 6 because parameter estimates changed by less than .001.
Results of Logistic Regression
Variable(s) entered on step 1: Brand, Product, Shopping.a.
1.274 .479 7.075 1 .008 3.575 .186 .322 .335 1 .563 1.205 .590 .491 1.442 1 .230 1.804
-8.642 3.346 6.672 1 .010 .000
Brand Product Shopping Constant
Classification Tablea
12 3 80.0 3 12 80.0
80.0
Step 1 Not Loyal Loyal Loyalty to the Brand Percentage
Correct
Predicted
Results of Three-Group Discriminant Analysis Multinomial Logistic regression
Ta bl
e 18
1 38.57000 4.50000 4.70000 3.10000 50.30000
2 50.11000 4.00000 4.20000 3.40000 49.50000
3 64.97000 6.10000 5.90000 4.20000 56.00000
Total 51.21667 4.86667 4.93333 3.56667 51.93333
Group Standard Deviations 1 5.29718 1.71594 1.88856 1.19722 8.09732
2 6.00231 2.35702 2.48551 1.50555 9.25263
3 8.61434 1.19722 1.66333 1.13529 7.60117
Total 12.79523 1.97804 2.09981 1.33089 8.57395
INCOME 1.00000
AGE -0.20939 -0.34022 -0.01326 -0.02512 1.00000
Cont.
Chapter Nineteen
Factor Analysis
The company asked 20 questions about casual dining and lifestyle. How are these questions related to one another? What are the
important dimensions or factors?
Factor Analysis
• Factor analysis is a general name denoting a class of procedures primarily used for data reduction and summarization.
• Factor analysis is an interdependence technique in that an entire set of interdependent relationships is examined without making the distinction between dependent and independent variables.
• Factor analysis is used in the following circumstances: • To identify underlying dimensions, or factors, that explain
the correlations among a set of variables. • To identify a new, smaller, set of uncorrelated variables to
replace the original set of correlated variables in subsequent multivariate analysis (regression or discriminant analysis).
• To identify a smaller set of salient variables from a larger set for use in subsequent multivariate analysis.
Factors Underlying Selected Psychographics and Lifestyles
Fig. 19.1 Factor 2
Plays Movies
Statistics Associated with Factor Analysis
• Bartlett's test of sphericity. Bartlett's test of sphericity is a test statistic used to examine the hypothesis that the variables are uncorrelated in the population. In other words, the population correlation matrix is an identity matrix; each variable correlates perfectly with itself (r = 1) but has no correlation with the other variables (r = 0).
• Correlation matrix. A correlation matrix is a lower triangle matrix showing the simple correlations, r, between all possible pairs of variables included in the analysis. The diagonal elements, which are all 1, are usually omitted.
• Communality. Communality is the amount of variance a variable shares with all the other variables being considered. This is also the proportion of variance explained by the common factors.
• Eigenvalue. The eigenvalue represents the total variance explained by each factor.
• Factor loadings. Factor loadings are simple correlations between the variables and the factors.
• Factor loading plot. A factor loading plot is a plot of the original variables using the factor loadings as coordinates.
• Factor matrix. A factor matrix contains the factor loadings of all the variables on all the factors extracted.
• Factor scores. Factor scores are composite scores estimated for each respondent on the derived factors.
• Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. The Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy is an index used to examine the appropriateness of factor analysis. High values (between 0.5 and 1.0) indicate factor analysis is appropriate. Values below 0.5 imply that factor analysis may not be appropriate.
• Percentage of variance. The percentage of the total variance attributed to each factor.
• Residuals are the differences between the observed correlations, as given in the input correlation matrix, and the reproduced correlations, as estimated from the factor matrix.
• Scree plot. A scree plot is a plot of the Eigenvalues against the number of factors in order of extraction.
Conducting Factor Analysis
RESPONDENT NUMBER V1 V2 V3 V4 V5 V6
1 7.00 3.00 6.00 4.00 2.00 4.00 2 1.00 3.00 2.00 4.00 5.00 4.00 3 6.00 2.00 7.00 4.00 1.00 3.00 4 4.00 5.00 4.00 6.00 2.00 5.00 5 1.00 2.00 2.00 3.00 6.00 2.00 6 6.00 3.00 6.00 4.00 2.00 4.00 7 5.00 3.00 6.00 3.00 4.00 3.00 8 6.00 4.00 7.00 4.00 1.00 4.00 9 3.00 4.00 2.00 3.00 6.00 3.00
10 2.00 6.00 2.00 6.00 7.00 6.00 11 6.00 4.00 7.00 3.00 2.00 3.00 12 2.00 3.00 1.00 4.00 5.00 4.00 13 7.00 2.00 6.00 4.00 1.00 3.00 14 4.00 6.00 4.00 5.00 3.00 6.00 15 1.00 3.00 2.00 2.00 6.00 4.00 16 6.00 4.00 6.00 3.00 3.00 4.00 17 5.00 3.00 6.00 3.00 3.00 4.00 18 7.00 3.00 7.00 4.00 1.00 4.00 19 2.00 4.00 3.00 3.00 6.00 3.00 20 3.00 5.00 3.00 6.00 4.00 6.00 21 1.00 3.00 2.00 3.00 5.00 3.00 22 5.00 4.00 5.00 4.00 2.00 4.00 23 2.00 2.00 1.00 5.00 4.00 4.00 24 4.00 6.00 4.00 6.00 4.00 7.00 25 6.00 5.00 4.00 2.00 1.00 4.00 26 3.00 5.00 4.00 6.00 4.00 7.00 27 4.00 4.00 7.00 2.00 2.00 5.00 28 3.00 7.00 2.00 6.00 4.00 3.00 29 4.00 6.00 3.00 7.00 2.00 7.00 30 2.00 3.00 2.00 4.00 7.00 2.00
Table 19.1
Correlation Matrix
Variables V1 V2 V3 V4 V5 V6 V1 1.000 V2 -0.530 1.000 V3 0.873 -0.155 1.000 V4 -0.086 0.572 -0.248 1.000 V5 -0.858 0.020 -0.778 -0.007 1.000 V6 0.004 0.640 -0.018 0.640 -0.136 1.000
Table 19.2
Conducting Factor Analysis: Determine the Method of Factor Analysis
• In principal components analysis, the total variance in the data is considered. The diagonal of the correlation matrix consists of unities, and full variance is brought into the factor matrix. Principal components analysis is recommended when the primary concern is to determine the minimum number of factors that will account for maximum variance in the data for use in subsequent multivariate analysis. The factors are called principal components.
• In common factor analysis, the factors are estimated based only on the common variance. Communalities are inserted in the diagonal of the correlation matrix. This method is appropriate when the primary concern is to identify the underlying dimensions and the common variance is of interest. This method is also known as principal axis factoring.
Results of Principal Components Analysis
Communalities Variables Initial Extraction V1 1.000 0.926 V2 1.000 0.723 V3 1.000 0.894 V4 1.000 0.739 V5 1.000 0.878 V6 1.000 0.790
Initial Eigen values Factor Eigen value % of variance Cumulat. % 1 2.731 45.520 45.520 2 2.218 36.969 82.488 3 0.442 7.360 89.848 4 0.341 5.688 95.536 5 0.183 3.044 98.580 6 0.085 1.420 100.000
Table 19.3
Extraction Sums of Squared Loadings
Factor Eigen value % of variance Cumulat. % 1 2.731 45.520 45.520 2 2.218 36.969 82.488
Factor Matrix Variables Factor 1 Factor 2 V1 0.928 0.253 V2 -0.301 0.795 V3 0.936 0.131 V4 -0.342 0.789 V5 -0.869 -0.351 V6 -0.177 0.871
Rotation Sums of Squared Loadings
Factor Eigenvalue % of variance Cumulat. % 1 2.688 44.802 44.802 2 2.261 37.687 82.488
Table 19.3, cont.
Factor
Factor
Eigenvalue
Rotated Factor Matrix Variables Factor 1 Factor 2 V1 0.962 -0.027 V2 -0.057 0.848 V3 0.934 -0.146 V4 -0.098 0.845 V5 -0.933 -0.084 V6 0.083 0.885
Factor Score Coefficient Matrix Variables Factor 1 Factor 2 V1 0.358 0.011 V2 -0.001 0.375 V3 0.345 -0.043 V4 -0.017 0.377 V5 -0.350 -0.059 V6 0.052 0.395
Table 19.3, cont.
Rotated Factor Matrix
Conducting Factor Analysis: Rotate Factors
• Although the initial or unrotated factor matrix indicates the relationship between the factors and individual variables, it seldom results in factors that can be interpreted, because the factors are correlated with many variables. Therefore, through rotation, the factor matrix is transformed into a simpler one that is easier to interpret.
• In rotating the factors, we would like each factor to have nonzero, or significant, loadings or coefficients for only some of the variables. Likewise, we would like each variable to have nonzero or significant loadings with only a few factors, if possible with only one.
• The rotation is called orthogonal rotation if the axes are maintained at right angles.
• The most commonly used method for rotation is the varimax procedure. This is an orthogonal method of rotation that minimizes the number of variables with high loadings on a factor, thereby enhancing the interpretability of the factors. Orthogonal rotation results in factors that are uncorrelated.
• The rotation is called oblique rotation when the axes are not maintained at right angles, and the factors are correlated. Sometimes, allowing for correlations among factors can simplify the factor pattern matrix. Oblique rotation should be used when factors in the population are likely to be strongly correlated.
Factor Matrix Before and After Rotation
Factors
Fig. 19.5
Factors Variables
2
X
Conducting Factor Analysis: Interpret Factors
• A factor can then be interpreted in terms of the variables that load high on it.
• Another useful aid in interpretation is to plot the variables, using the factor loadings as coordinates. Variables at the end of an axis are those that have high loadings on only that factor, and hence describe the factor.
Chapter Twenty
Cluster Analysis
Chapter Outline
known groups) 3) Statistics Associated with Cluster Analysis 4) Conducting Cluster Analysis
i. Formulating the Problem ii. Selecting a Distance or Similarity Measure iii. Selecting a Clustering Procedure iv. Deciding on the Number of Clusters v. Interpreting and Profiling the Clusters vi. Assessing Reliability and Validity
Cluster Analysis
• Cluster analysis is a class of techniques used to classify objects or cases into relatively homogeneous groups called clusters. Objects in each cluster tend to be similar to each other and dissimilar to objects in the other clusters. Cluster analysis is also called classification analysis, or numerical taxonomy.
• Both cluster analysis and discriminant analysis are concerned with classification. However, discriminant analysis requires prior knowledge of the cluster or group membership for each object or case included, to develop the classification rule. In contrast, in cluster analysis there is no a priori information about the group or cluster membership for any of the objects. Groups or clusters are suggested by the data, not defined a priori.
A Practical Clustering Situation
An Ideal Clustering Situation
Statistics Associated with Cluster Analysis
• Agglomeration schedule. An agglomeration schedule gives information on the objects or cases being combined at each stage of a hierarchical clustering process.
• Cluster centroid. The cluster centroid is the mean values of the variables for all the cases or objects in a particular cluster.
• Cluster centers. The cluster centers are the initial starting points in nonhierarchical clustering. Clusters are built around these centers, or seeds.
• Cluster membership. Cluster membership indicates the cluster to which each object or case belongs.
Statistics Associated with Cluster Analysis
• Dendrogram. A dendrogram, or tree graph, is a graphical device for displaying clustering results. Vertical lines represent clusters that are joined together. The position of the line on the scale indicates the distances at which clusters were joined. The dendrogram is read from left to right. Figure 20.8 is a dendrogram.
• Distances between cluster centers. These distances indicate how separated the individual pairs of clusters are. Clusters that are widely separated are distinct, and therefore desirable.
Attitudinal Data For Clustering
Case No. V1 V2 V3 V4 V5 V6
1 6 4 7 3 2 3 2 2 3 1 4 5 4 3 7 2 6 4 1 3 4 4 6 4 5 3 6 5 1 3 2 2 6 4 6 6 4 6 3 3 4 7 5 3 6 3 3 4 8 7 3 7 4 1 4 9 2 4 3 3 6 3 10 3 5 3 6 4 6 11 1 3 2 3 5 3 12 5 4 5 4 2 4 13 2 2 1 5 4 4 14 4 6 4 6 4 7 15 6 5 4 2 1 4 16 3 5 4 6 4 7 17 4 4 7 2 2 5 18 3 7 2 6 4 3 19 4 6 3 7 2 7 20 2 3 2 4 7 2
Ta bl
e 20
Conducting Cluster Analysis: Select a Distance or Similarity Measure
• The most commonly used measure of similarity is the Euclidean distance or its square. The Euclidean distance is the square root of the sum of the squared differences in values for each variable. Other distance measures are also available. The city-block or Manhattan distance between two objects is the sum of the absolute differences in values for each variable. The Chebychev distance between two objects is the maximum absolute difference in values for any variable.
• If the variables are measured in vastly different units, the clustering solution will be influenced by the units of measurement. In these cases, before clustering respondents, we must standardize the data by rescaling each variable to have a mean of zero and a standard deviation of unity. It is also desirable to eliminate outliers (cases with atypical values).
• Use of different distance measures may lead to different clustering results. Hence, it is advisable to use different measures and compare the results.
A Classification of Clustering Procedures
Fig. 20.4
Other Agglomerative Clustering Methods
Results of Hierarchical Clustering
Stage cluster Clusters combined first appears
Stage Cluster 1 Cluster 2 Coefficient Cluster 1 Cluster 2 Next stage 1 14 16 1.000000 0 0 6 2 6 7 2.000000 0 0 7 3 2 13 3.500000 0 0 15 4 5 11 5.000000 0 0 11 5 3 8 6.500000 0 0 16 6 10 14 8.160000 0 1 9 7 6 12 10.166667 2 0 10 8 9 20 13.000000 0 0 11 9 4 10 15.583000 0 6 12 10 1 6 18.500000 6 7 13 11 5 9 23.000000 4 8 15 12 4 19 27.750000 9 0 17 13 1 17 33.100000 10 0 14 14 1 15 41.333000 13 0 16 15 2 5 51.833000 3 11 18 16 1 3 64.500000 14 5 19 17 4 18 79.667000 12 0 18 18 2 4 172.662000 15 17 19 19 1 2 328.600000 16 18 0
Agglomeration Schedule Using Ward’s Procedure Table 20.2
Conducting Cluster Analysis: Decide on the Number of Clusters
• Theoretical, conceptual, or practical considerations may suggest a certain number of clusters.
• In hierarchical clustering, the distances at which clusters are combined can be used as criteria. This information can be obtained from the agglomeration schedule or from the dendrogram.
• In nonhierarchical clustering, the ratio of total within-group variance to between-group variance can be plotted against the number of clusters. The point at which an elbow or a sharp bend occurs indicates an appropriate number of clusters.
• The relative sizes of the clusters should be meaningful.
Conducting Cluster Analysis: Interpreting and Profiling the Clusters
• Interpreting and profiling clusters involves examining the cluster centroids. The centroids enable us to describe each cluster by assigning it a name or label.
• It is often helpful to profile the clusters in terms of variables that were not used for clustering. These may include demographic, psychographic, product usage, media usage, or other variables.
Cluster Distribution
N % of
Combined % of Total 1 6 30.0% 30.0% 2 6 30.0% 30.0% 3 8 40.0% 40.0%
Cluster
Cluster Profiles
Fun Bad for Budget Eating Out
Mean Std. Deviation Mean Std. Deviation Mean Std. Deviation 1 1.67 .516 3.00 .632 1.83 .753 2 3.50 .548 5.83 .753 3.33 .816 3 5.75 1.035 3.63 .916 6.00 1.069
Cluster
Combined 3.85 1.899 4.10 1.410 3.95 2.012
Best Buys Don't Care Compare Prices Mean Std. Deviation Mean Std. Deviation Mean Std. Deviation
3.50 1.049 5.50 1.049 3.33 .816 6.00 .632 3.50 .837 6.00 1.549 3.13 .835 1.88 .835 3.88 .641 4.10 1.518 3.45 1.761 4.35 1.496
Fun
Slide Number 1
Direct marketing scenario
Geometric Interpretation
Conducting Discriminant Analysis
Conducting Discriminant Analysis Estimate the Discriminant Function Coefficients
Conducting Discriminant Analysis Interpret the Results
Conducting Discriminant Analysis Assess Validity of Discriminant Analysis
Results of Three-Group Discriminant Analysis
All-Groups Scattergram
Territorial Map
Estimation and Model Fit
Slide Number 39
Conducting Factor Analysis
Factor Matrix Before and After Rotation
Conducting Factor Analysis: Interpret Factors
Slide Number 55
Slide Number 56
Attitudinal Data For Clustering
A Classification of Clustering Procedures
Other Agglomerative Clustering Methods
Results of Hierarchical Clustering
Conducting Cluster Analysis:Interpreting and Profiling the Clusters
Cluster Distribution
Cluster Profiles

Chapter Eighteen - l n

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