17-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

17-2 Chapter 17 Multivariate Analysis

17-3 The Multivariate Normal Distribution Discriminant Analysis Principal Components and Factor Analysis Using the Computer Multivariate Analysis 17

17-4 Describe a multivariate normal distribution Explain when a discriminant analysis could be conducted Interpret the results of a discriminant analysis Explain when a factor analysis could be conducted Differentiate between principal components and factors Interpret factor analysis results LEARNING OUTCOMES 17 After studying this chapter, you should be able to:

17-5 A k-dimensional (vector) random variable X: X = (X 1, X 2, X 3..., X k ) A realization of a k-dimensional random variable X: x = (x 1, x 2, x 3..., x k ) A joint cumulative probability distribution function of a k-dimensional random variable X: F(x 1, x 2, x 3..., x k ) = P(X 1  x 1, X 2  x 2,..., X k  x k ) 17-2 The Multivariate Normal Distribution

17-6 The Multivariate Normal Distribution

17-7 f(x 1,x 2 ) x1x1 x2x2 Picturing the Bivariate Normal Distribution

17-8 In a discriminant analysis, observations are classified into two or more groups, depending on the value of a multivariate discriminant function. X2X2 X1X1 Group 1 Group 2 11 22 Line L As the figure illustrates, it may be easier to classify observations by looking at them from another direction. The groups appear more separated when viewed from a point perpendicular to Line L, rather than from a point perpendicular to the X 1 or X 2 axis. The discriminant function gives the direction that maximizes the separation between the groups Discriminant Analysis

17-9 Group 1Group 2 C Cutting Score The form of the estimated predicted equation: D = b 0 +b 1 X 1 +b 2 X b k X k where the b i are the discriminant weights. b 0 is a constant. The intersection of the normal marginal distributions of two groups gives the cutting score, which is used to assign observations to groups. Observations with scores less than C are assigned to group 1, and observations with scores greater than C are assigned to group 2. Since the distributions may overlap, some observations may be misclassified. The model may be evaluated in terms of the percentages of observations assigned correctly and incorrectly. The Discriminant Function

17-10 Discriminant 'Repay' 'Assets' 'Debt' 'Famsize'. Group 0 1 Count Summary of Classification Put into....True Group.... Group Total N N Correct Proport N = 32 N Correct = 23 Prop. Correct = Linear Discriminant Function for Group 0 1 Constant Assets Debt Famsize Discriminant 'Repay' 'Assets' 'Debt' 'Famsize'. Group 0 1 Count Summary of Classification Put into....True Group.... Group Total N N Correct Proport N = 32 N Correct = 23 Prop. Correct = Linear Discriminant Function for Group 0 1 Constant Assets Debt Famsize Discriminant Analysis: Example 17-1 (Minitab)

17-11 Summary of Misclassified Observations Observation True Pred Group Sqrd Distnc Probability Group Group 4 ** ** ** ** ** ** ** ** ** Summary of Misclassified Observations Observation True Pred Group Sqrd Distnc Probability Group Group 4 ** ** ** ** ** ** ** ** ** Example 17-1: Misclassified Observations

set width 80 2 data list free / assets income debt famsize job repay 3 begin data 35 end data 36 discriminant groups = repay(0,1) 37 /variables assets income debt famsize job 38 /method = wilks 39 /fin = 1 40 /fout = 1 41 /plot 42 /statistics = all Number of cases by group Number of cases REPAY Unweighted Weighted Label Total set width 80 2 data list free / assets income debt famsize job repay 3 begin data 35 end data 36 discriminant groups = repay(0,1) 37 /variables assets income debt famsize job 38 /method = wilks 39 /fin = 1 40 /fout = 1 41 /plot 42 /statistics = all Number of cases by group Number of cases REPAY Unweighted Weighted Label Total Example 17-1: SPSS Output (1)

D I S C R I M I N A N T A N A L Y S I S On groups defined by REPAY Analysis number 1 Stepwise variable selection Selection rule: minimize Wilks' Lambda Maximum number of steps Minimum tolerance level Minimum F to enter … Maximum F to remove Canonical Discriminant Functions Maximum number of functions Minimum cumulative percent of variance Maximum significance of Wilks' Lambda Prior probability for each group is D I S C R I M I N A N T A N A L Y S I S On groups defined by REPAY Analysis number 1 Stepwise variable selection Selection rule: minimize Wilks' Lambda Maximum number of steps Minimum tolerance level Minimum F to enter … Maximum F to remove Canonical Discriminant Functions Maximum number of functions Minimum cumulative percent of variance Maximum significance of Wilks' Lambda Prior probability for each group is Example 17-1: SPSS Output (2)

Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda ASSETS INCOME DEBT FAMSIZE JOB * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * At step 1, ASSETS was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda ASSETS INCOME DEBT FAMSIZE JOB * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * At step 1, ASSETS was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Example 17-1: SPSS Output (3)

Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME DEBT FAMSIZE JOB At step 2, DEBT was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME DEBT FAMSIZE JOB At step 2, DEBT was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Example 17-1: SPSS Output (4)

Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS DEBT Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME FAMSIZE JOB At step 3, FAMSIZE was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS DEBT Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME FAMSIZE JOB At step 3, FAMSIZE was included in the analysis. Degrees of Freedom Signif. Between Groups Wilks' Lambda Equivalent F Example 17-1: SPSS Output (5)

Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS DEBT FAMSIZE Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME JOB Summary Table Action Vars Wilks' Step Entered Removed in Lambda Sig. Label 1 ASSETS DEBT FAMSIZE Variables in the Analysis after Step Variable Tolerance F to Remove Wilks' Lambda ASSETS DEBT FAMSIZE Variables not in the Analysis after Step Minimum Variable Tolerance Tolerance F to Enter Wilks' Lambda INCOME JOB Summary Table Action Vars Wilks' Step Entered Removed in Lambda Sig. Label 1 ASSETS DEBT FAMSIZE Example 17-1: SPSS Output (6)

17-18 Classification function coefficients (Fisher's linear discriminant functions) REPAY = 0 1 ASSETS DEBT FAMSIZE (Constant) Unstandardized canonical discriminant function coefficients Func 1 ASSETS DEBT FAMSIZE (Constant) Classification function coefficients (Fisher's linear discriminant functions) REPAY = 0 1 ASSETS DEBT FAMSIZE (Constant) Unstandardized canonical discriminant function coefficients Func 1 ASSETS DEBT FAMSIZE (Constant) Example 17-1: SPSS Output (7)

17-19 Case Mis Actual Highest Probability 2nd Highest Discrim Number Val Sel Group Group P(D/G) P(G/D) Group P(G/D) Scores ** ** ** ** ** ** ** ** ** Case Mis Actual Highest Probability 2nd Highest Discrim Number Val Sel Group Group P(D/G) P(G/D) Group P(G/D) Scores ** ** ** ** ** ** ** ** ** Example 17-1: SPSS Output (8)

17-20 Classification results - No. of Predicted Group Membership Actual Group Cases Group % 28.6% Group % 72.2% Percent of "grouped" cases correctly classified: 71.88% Classification results - No. of Predicted Group Membership Actual Group Cases Group % 28.6% Group % 72.2% Percent of "grouped" cases correctly classified: 71.88% Example 17-1: SPSS Output (9)

17-21 All-groups Stacked Histogram Canonical Discriminant Function | | F | | r e | 2 | q | 2 | u | 2 | e n | | c | | y | | | | X X out out Class Centroids 2 1 All-groups Stacked Histogram Canonical Discriminant Function | | F | | r e | 2 | q | 2 | u | 2 | e n | | c | | y | | | | X X out out Class Centroids 2 1 Example 17-1: SPSS Output (10)

17-22 First Component Second Component x y Total Variance Remaining After Extraction of First Second Third Component 17-4 Principal Components and Factor Analysis

17-23 The k original X i variables written as linear combinations of a smaller set of m common factors and a unique component for each variable: X 1 = b 11 F 1 + b 12 F b 1m F m + U 1 X 1 = b 21 F 1 + b 22 F b 2m F m + U 2. X k = b k1 F 1 + b k2 F b km F m + U k The F j are the common factors. Each U i is the unique component of variable X i. The coefficients b ij are called the factor loadings. Total variance in the data is decomposed into the communality, the common factor component, and the specific part. The k original X i variables written as linear combinations of a smaller set of m common factors and a unique component for each variable: X 1 = b 11 F 1 + b 12 F b 1m F m + U 1 X 1 = b 21 F 1 + b 22 F b 2m F m + U 2. X k = b k1 F 1 + b k2 F b km F m + U k The F j are the common factors. Each U i is the unique component of variable X i. The coefficients b ij are called the factor loadings. Total variance in the data is decomposed into the communality, the common factor component, and the specific part. Factor Analysis

17-24 Rotation of Factors

17-25 Factor Loadings Satisfaction with: Communality Information Variety Closure Pay Factor Loadings Satisfaction with: Communality Information Variety Closure Pay Factor Analysis of Satisfaction Items