18-1 © 2007 Prentice Hall Chapter Eighteen 18-1 Discriminant and Logit Analysis

18-2 © 2007 Prentice Hall Chapter Outline 1) Overview 2) Basic Concept 3) Relation to Regression and ANOVA 4) Discriminant Analysis Model 5) Statistics Associated with Discriminant Analysis 6) Conducting Discriminant Analysis i. Formulation ii. Estimation iii. Determination of Significance iv. Interpretation v. Validation

18-3 © 2007 Prentice Hall Chapter Outline 7)Multiple Discriminant Analysis i.Formulation ii.Estimation iii.Determination of Significance iv.Interpretation v.Validation 8)Stepwise Discriminant Analysis

18-4 © 2007 Prentice Hall 9) The Logit Model i.Estimation ii.Model Fit iii.Significance Testing iv.Interpretation of Coefficients v.An Illustrative Application 10) Summary Chapter Outline

18-5 © 2007 Prentice Hall Similarities and Differences between ANOVA, Regression, and Discriminant Analysis Table 18.1 ANOVA REGRESSION DISCRIMINANT/LOGIT Similarities Number ofOneOneOne dependent variables Number of independentMultipleMultipleMultiple variables Differences Nature of the dependentMetricMetricCategorical variables Nature of the independentCategoricalMetricMetric variables

18-6 © 2007 Prentice Hall 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). 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.

18-7 © 2007 Prentice Hall 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. Discriminant Analysis

18-8 © 2007 Prentice Hall Geometric Interpretation Fig X 2 X 1 G1 G2 D G1 G

18-9 © 2007 Prentice Hall Discriminant Analysis Model The discriminant analysis model involves linear combinations of the following form: D = b 0 + b 1 X 1 + b 2 X 2 + b 3 X b k X k 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. 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.

18-10 © 2007 Prentice Hall 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. Statistics Associated with Discriminant Analysis

18-11 © 2007 Prentice Hall 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. Statistics Associated with Discriminant Analysis

18-12 © 2007 Prentice Hall 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. Statistics Associated with Discriminant Analysis

18-13 © 2007 Prentice Hall 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. Statistics Associated with Discriminant Analysis

18-14 © 2007 Prentice Hall Conducting Discriminant Analysis Fig Assess Validity of Discriminant Analysis Estimate the Discriminant Function Coefficients Determine the Significance of the Discriminant Function Formulate the Problem Interpret the Results

18-15 © 2007 Prentice Hall 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.

18-16 © 2007 Prentice Hall 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 M (2) H (3) H (3) L (1) H (3) H (3) M (2) M (2) H (3) H (3) H (3) H (3) M (2) H (3) H (3)

18-17 © 2007 Prentice Hall Information on Resort Visits: Analysis Sample Table 18.2, cont. 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 L (1) L (1) M (2) M (2) M (2) L (1) M (2) L (1) L (1) L (1) M (2) M (2) L (1) L (1) L (1)

18-18 © 2007 Prentice Hall 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 No. Visit Income Travel to Family Household Family ($000) Vacation Vacation M(2) H (3) M(2) M(2) H (3) H (3) L (1) L (1) H (3) L (1) M(2) L (1)

18-19 © 2007 Prentice Hall 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.

18-20 © 2007 Prentice Hall Results of Two-Group Discriminant Analysis Table 18.4 GROUP MEANS VISITINCOME TRAVEL VACATION HSIZE AGE Total Group Standard Deviations Total Pooled Within-Groups Correlation Matrix INCOMETRAVELVACATION HSIZE AGE INCOME TRAVEL VACATION HSIZE AGE Wilks' (U-statistic) and univariate F ratio with 1 and 28 degrees of freedom VariableWilks' F Significance INCOME TRAVEL VACATION HSIZE AGE Cont.

18-21 © 2007 Prentice Hall Results of Two-Group Discriminant Analysis Table 18.4, cont. CANONICAL DISCRIMINANT FUNCTIONS % of Cum Canonical After Wilks' FunctionEigenvalue Variance %Correlation Function Chi-square df Significance : * : * marks the 1 canonical discriminant functions remaining in the analysis. Standard Canonical Discriminant Function Coefficients FUNC 1 INCOME TRAVEL VACATION HSIZE AGE Structure Matrix: Pooled within-groups correlations between discriminating variables & canonical discriminant functions (variables ordered by size of correlation within function) FUNC 1 INCOME HSIZE VACATION TRAVEL AGE Cont.

18-22 © 2007 Prentice Hall Cont. Results of Two-Group Discriminant Analysis Table 18.4, cont. Unstandardized Canonical Discriminant Function Coefficients FUNC 1 INCOME E-01 TRAVEL E-01 VACATION HSIZE AGE E-01 (constant) Canonical discriminant functions evaluated at group means (group centroids) GroupFUNC Classification results for cases selected for use in analysis PredictedGroup Membership Actual GroupNo. of Cases12 Group %20.0% Group %100.0% Percent of grouped cases correctly classified: 90.00%

18-23 © 2007 Prentice Hall Results of Two-Group Discriminant Analysis Table 18.4, cont. Classification Results for cases not selected for use in the analysis (holdout sample) PredictedGroup Membership Actual GroupNo. of Cases12 Group %33.3% Group %100.0% Percent of grouped cases correctly classified: 83.33%.

18-24 © 2007 Prentice Hall Conducting Discriminant Analysis Determine the Significance of Discriminant Function 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 several functions are tested simultaneously (as in the case of multiple discriminant analysis), the Wilks' statistic is the product of the univariate for each function. The significance level is estimated based on a chi-square transformation of the statistic. If the null hypothesis is rejected, indicating significant discrimination, one can proceed to interpret the results.

18-25 © 2007 Prentice Hall 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.

18-26 © 2007 Prentice Hall 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.

18-27 © 2007 Prentice Hall Results of Three-Group Discriminant Analysis Table 18.5 Group Means AMOUNTINCOMETRAVELVACATIONHSIZEAGE Total Group Standard Deviations Total Pooled Within-Groups Correlation Matrix INCOMETRAVELVACATIONHSIZEAGE INCOME TRAVEL VACATION HSIZE AGE Cont.

18-28 © 2007 Prentice Hall Results of Three-Group Discriminant Analysis Table 18.5, cont. Wilks' (U-statistic) and univariate F ratio with 2 and 27 degrees of freedom. Variable Wilks' Lambda FSignificance INCOME TRAVEL VACATION HSIZE AGE CANONICAL DISCRIMINANT FUNCTIONS % of Cum Canonical After Wilks' FunctionEigenvalue Variance %Correlation Function Chi-square df Significance : * : * : * marks the two canonical discriminant functions remaining in the analysis. Standardized Canonical Discriminant Function Coefficients FUNC 1 FUNC 2 INCOME TRAVEL VACATION HSIZE AGE Cont.

18-29 © 2007 Prentice Hall Results of Three-Group Discriminant Analysis Table 18.5, cont. Structure Matrix: Pooled within-groups correlations between discriminating variables and canonical discriminant functions (variables ordered by size of correlation within function) FUNC 1 FUNC 2 INCOME * HSIZE * VACATION * TRAVEL * AGE * Unstandardized canonical discriminant function coefficients FUNC 1 FUNC 2 INCOME E-01 TRAVEL VACATION E HSIZE AGE E E-01 (constant) Canonical discriminant functions evaluated at group means (group centroids) GroupFUNC 1 FUNC Cont.

18-30 © 2007 Prentice Hall Results of Three-Group Discriminant Analysis Table 18.5, cont. Classification Results: PredictedGroup Membership Actual Group No. of Cases 123 Group %10.0%0.0% Group %90.0%0.0% Group %20.0%80.0% Percent of grouped cases correctly classified: 86.67% Classification results for cases not selected for use in the analysis PredictedGroup Membership Actual Group No. of Cases123 Group %25.0%0.0% Group %75.0%25.0% Group %0.0%75.0% Percent of grouped cases correctly classified: 75.00%

18-31 © 2007 Prentice Hall All-Groups Scattergram Fig Across: Function 1 Down: Function * * * * indicates a group centroid

18-32 © 2007 Prentice Hall Territorial Map Fig Across: Function 1 Down: Function * * * * Indicates a group centroid

18-33 © 2007 Prentice Hall Stepwise Discriminant Analysis Stepwise discriminant analysis is analogous to stepwise multiple regression (see Chapter 17) in that the predictors are entered sequentially based on their ability to discriminate between the groups. An F ratio is calculated for each predictor by conducting a univariate analysis of variance in which the groups are treated as the categorical variable and the predictor as the criterion variable. The predictor with the highest F ratio is the first to be selected for inclusion in the discriminant function, if it meets certain significance and tolerance criteria. A second predictor is added based on the highest adjusted or partial F ratio, taking into account the predictor already selected.

18-34 © 2007 Prentice Hall Each predictor selected is tested for retention based on its association with other predictors selected. The process of selection and retention is continued until all predictors meeting the significance criteria for inclusion and retention have been entered in the discriminant function. The selection of the stepwise procedure is based on the optimizing criterion adopted. The Mahalanobis procedure is based on maximizing a generalized measure of the distance between the two closest groups. The order in which the variables were selected also indicates their importance in discriminating between the groups. Stepwise Discriminant Analysis

18-35 © 2007 Prentice Hall 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 It estimates the probability of an observation belonging to a particular group

18-36 © 2007 Prentice Hall Binary Logit Model Formulation The probability of success may be modeled using the logit model as: Or i n i e X a P P          1 log i    i  0 X a X a X aa kk e P P        ... 1 log 22110

18-37 © 2007 Prentice Hall Model Formulation )exp(1 ) 0 0 X a X a i k i i i k i i P       Where: P = Probability of success X i = Independent variable i a i = parameter to be estimated.

18-38 © 2007 Prentice Hall Properties of the Logit Model Although X i may vary from to, P is constrained to lie between 0 and 1. When X i approaches, P approaches 0. When X i approaches, P approaches 1. When OLS regression is used, P is not constrained to lie between 0 and 1.

18-39 © 2007 Prentice Hall Estimation and Model Fit The estimation procedure is called the maximum likelihood method. Fit: Cox & Snell R Square and Nagelkerke R Square. 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.

18-40 © 2007 Prentice Hall Significance Testing

18-41 © 2007 Prentice Hall Interpretation of Coefficients If X i is increased by one unit, the log odds will change by a i units, when the effect of other independent variables is held constant. The sign of a i will determine whether the probability increases (if the sign is positive) or decreases (if the sign is negative) by this amount.

18-42 © 2007 Prentice Hall Explaining Brand Loyalty Table 18.6

18-43 © 2007 Prentice Hall Results of Logistic Regression Table 18.7

18-44 © 2007 Prentice Hall Results of Logistic Regression Table 18.7, cont. Variables in the Equation a Variable(s) entered on step 1: Brand, Product, Shopping. a Brand Product Shopping Constant Step 1 BS.E.WalddfSig.Exp(B) Classification Table a The cut value is.500 a Observed Not Loyal Loyal Loyalty to the Brand Overall Percentage Step 1 Not LoyalLoyal Loyalty to the Brand Percentage Correct Predicted

18-45 © 2007 Prentice Hall 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 … The run logit analysis or logistic regression using SPSS for Windows, click: Analyze > Regression>Binary Logistic 

18-46 © 2007 Prentice Hall SPSS Windows: Two-group Discriminant 1. Select ANALYZE from the SPSS menu bar. 2. Click CLASSIFY and then DISCRIMINANT. 3. Move “visit” in to the GROUPING VARIABLE box. 4. Click DEFINE RANGE. Enter 1 for MINIMUM and 2 for MAXIMUM. Click CONTINUE. 5. Move “income,” “travel,” “vacation,” “hsize,” and “age” in to the INDEPENDENTS box. 6. Select ENTER INDEPENDENTS TOGETHER (default option) 7. Click on STATISTICS. In the pop-up window, in the DESCRIPTIVES box check MEANS and UNIVARIATE ANOVAS. In the MATRICES box check WITHIN-GROUP CORRELATIONS. Click CONTINUE. 8. Click CLASSIFY.... In the pop-up window in the PRIOR PROBABILITIES box check ALL GROUPS EQUAL (default). In the DISPLAY box check SUMMARY TABLE and LEAVE-ONE-OUT CLASSIFICATION. In the USE COVARIANCE MATRIX box check WITHIN-GROUPS. Click CONTINUE. 9. Click OK.

18-47 © 2007 Prentice Hall SPSS Windows: Logit Analysis 1. Select ANALYZE from the SPSS menu bar. 2. Click REGRESSION and then BINARY LOGISTIC. 3. Move “Loyalty to the Brand [Loyalty]” in to the DEPENDENT VARIABLE box. 4. Move “Attitude toward the Brand [Brand},” “Attitude toward the Product category [Product},” and “Attitude toward Shopping [Shopping],” in to the COVARIATES(S box.) 5. Select ENTER for METHOD (default option) 6. Click OK.