Discriminan t Analysis What Is It?  Analysis when the dependent variable is categorical or nominal and independent variables metric  Discriminate or Classify individuals into groups on the basis of independent variables  Involves deriving a variate which is a linear combination of the independent variables  This variate obtained by maximizing Between-Group Variance relative to Within-Variance Group  The linear combination is called Discriminant Function as follows: Z = a + W1X1 + W2X2 + + WnXn  Can also be used to test the hypothesis that group means of a set of independent variables for or more groups are equal Muhamad Jantan & T Ramayah Discriminant Analysis Objectives ♦ Profile Analysis ♦ Predictive Technique ♦ Test differences between groups on average score profiles of a set of variables ♦ Determine the impact of independent variables on the differences in average score profiles of two or more groups ♦ Classify units on the basis of their scores on a set of independent variables ♦ Establishing the number and composition of the dimensions of discrimination between groups formed from the set of independent variables Muhamad Jantan & T Ramayah Discriminant Analysis Assumptions ♦ Independent variables distributed as Multivariate Normal ♦ ♦ ♦ ♦ ♦ with unknown but equal Covariance matrices across the groups Non-normality affects estimation of the discriminant function Use logistic regression instead Inequality of Covariance Matrices across groups affects classification process If sample size is small, estimation process affected, where groups with larger covariance will be overclassified Rectify by using larger sample sizes or quadratic classification techniques Multicollinearity of Independent Variables: Especially when stepwise procedure is used Especially critical when used for explanation purposes When interpreting the results, always be aware of the level of collinearity Linearity of relationship Outliers may have substantial impact on results Muhamad Jantan & T Ramayah Discriminant Analysis Estimation And Assessment ♦ Estimation  Simultaneous Method, or  Stepwise Method ♦ Assessment  Statistical Significance of Discriminant Function  Wilk’s λ, Hotelling trace, Pillai Criteria - evaluate the discriminatory power of the function  Roy’s max root - evaluate the first discriminant function only  For Stepwise procedure: Mahalanobis D and Rao’s V ;  D2 - uses distances, adjusts for unequal covariances  When or more groups - evaluate overall significance and significance of individual function Muhamad Jantan & T Ramayah Discriminant Analysis SPSS Commands  Dividing the Sample into Estimation and Split/Holdout Sample: Random Selection Command: TRANSFORM ⇒ RANDOM NUMBER SEED TRANSFORM ⇒ COMPUTE Randz = UNIFORM(1) > 0.65 ⇒ will give ≈ 65% of respondent for estimation and the remainder for holdout sample  Estimating the Discriminant Function(s): SPSS ⇒ CLASSIFY ⇒ DISCRIMINANT:  This will give you a dialogue box for Discriminant: Select the grouping (dependent) variable and the independent variables Also need the SELECT option to identify units for estimation sample: In this case use Randz with SET VALUE at  Options Available: • Method: Stepwise or Simultaneous • Classify: Provide options for Prior Probabilities, Using VarCov Matrices, Plots and Display Muhamad Jantan & T Ramayah Discriminant Analysis SPSS: Discriminant Analysis SPSS Command: Analyze  Classify  Discriminant  Select Cases  Statistics  Classify Muhamad Jantan & T Ramayah Discriminant Analysis SPSS: Results for Two groups Wilks' Lambda Test of Funct ion(s) Wilks' Lambda 283 Chi-square 67.544 df Sig .000 Wilks’ is significant indicating that we have a significant discriminant function Muhamad Jantan & T Ramayah Discriminant Analysis SPSS: Results for Two groups Eigenvalues Funct ion Eigenvalue 2.534 a % of Variance 100.0 Cumulat ive % 100.0 Canonical Correlat ion 847 a First canonical discriminant funct ions were used in t he analysis Indicates that (0.847)2 = 72% of variance in the dependent variable is explained by the independent variables Muhamad Jantan & T Ramayah Discriminant Analysis Descriptives Group St at ist ics Art iculat ion Of Needs Specificat ion Buying Tot al Value Analysis Tot al Muhamad Jantan & T Ramayah Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Mean 2.404 2.946 6.707 5.343 2.646 2.693 8.421 24 2.048 8.613 5.123 3.132 2.587 6.023 3.369 2.475 7.708 5.227 2.902 2.637 7.161 Discriminant Analysis St d Deviat ion 1.0772 1.1971 8602 9155 9913 6248 8883 1.0810 1.04 91 1.14 33 1.3732 5369 8628 1.2672 1.4149 1.2004 1.3935 1.1738 8163 7547 1.6302 Valid N (list wise) Unweight ed Weight ed 28 28.000 28 28.000 28 28.000 28 28.000 28 28.000 28 28.000 28 28.000 31 31.000 31 31.000 31 31.000 31 31.000 31 31.000 31 31.000 31 31.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 10 SPSS Results: Assessment St ruct ure Mat rix Funct ion Product Qualit y -.693 Price Flexibilit y 597 Delivery Speed 544 Price Level -.256 Service 197 Manufact urer Image -.060 Salesforce Image -.044 Pooled wit hin-groups correlat ions bet ween discriminat ing variables and st andardized canonical discriminant funct ions Variables ordered by absolut e size of correlat ion wit hin funct ion Linear Correlation between PQ and the discriminant function Muhamad Jantan & T Ramayah Discriminant Analysis 14 SPSS Results: Assessment Canonical Discriminant Funct ion Coefficient s Funct ion Delivery Speed 419 Price Level 116 Price Flexibilit y 560 Manufact urer Image -.049 Service -.141 Salesforce Image 342 Product Qualit y -.623 (Const ant ) -1.788 Unst andardized coefficient s Coefficients used in the discriminant function to calculate the Discriminant scores which is used to classify the individuals Discriminant Function: Z = -1.788 - 419X1 + 116X2 + 560X3 - 049X4 - 141X5 + 342X6 - 623X7 Muhamad Jantan & T Ramayah Discriminant Analysis 15 SPSS Results: Assessment Funct ions at Group Cent roids Art iculat ion Of Needs Specificat ion Buying Tot al Value Analysis Funct ion -1.646 1.487 Unst andardized canonical discriminant funct ions evaluat ed at group means Discriminant scores evaluated at the means of (x1,x2,x3,x4,x5,x6,x7) for the two groups Cutting Score: This means all respondents with Z 31( −1.646) + 28(1.487) ZCU = = − - 0.15915 scores less than 28 + 31 -0.15915 will be classified into Note: No in Group (Specification) Specification Buying = 27+1=28 and Group (Total) = and Total Value 27+4=31 Analysis otherwise Muhamad Jantan & T Ramayah Discriminant Analysis 16 Interpretation Of Results  Relative importance of each of the independent variable in discriminating the groups • Discrimination weight (or coefficient): relative contribution of the variable to the function; equivalent to beta of regression; but weight is unstable - thus caution  Discriminant Loading (or structure correlation): measures the simple linear correlation between the independent and the discriminant function  Partial F-values: Only when stepwise procedure is used Large F values indicate large contribution  Potency Index: Relative measure amongst variables; Composite contribution of a variable to all the significant discriminant function; Used when more than one significant discriminant function Muhamad Jantan & T Ramayah Discriminant Analysis 17 SPSS Results – groups Interpretation ♦ Use Low, Moderate and High Satisfaction ♦ How to assess the results?  Significance of Discriminant Function  Wilk’s λ, Hotelling trace, Pillai Criteria - evaluate the discriminatory power of the function ♦ Predictive Accuracy: Classification Table – summary and individual  Hits Ratio  Classification Results;  Determinant Function  Cutoff Points – Territorial Map ♦ Relative Importance of Variables  Discriminant Weights  Discriminant Loadings  Potency Index; Muhamad Jantan & T Ramayah Discriminant Analysis 18 Discriminant Analysis – group example Profile Analysis: Who are the companies that treats each purchase from HATCO as a straight rebuy, modified rebuy, and new task Method: Enter Muhamad Jantan & T Ramayah Discriminant Analysis 19 SPSS: Results for groups Eige nvalue s Funct ion Eigenvalue 3.952a 948a % of Variance 80.7 19.3 Cumulat ive % 80.7 100.0 Canonical Correlat ion 893 698 a First canonical discriminant funct ions were used in t he analysis Wilks' Lambda Test of Funct ion(s) t hrough 2 Wilks' Lambda 104 513 Chi-square 120.131 35.342 df 14 Sig .000 000 Both discriminant functions are significant Muhamad Jantan & T Ramayah Discriminant Analysis 20 Descriptives Group St at ist ics Type of Buying Sit uat ion New Task Modified Rebuy St raight Rebuy Tot al Muhamad Jantan & T Ramayah Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Delivery Speed Price Level Price Flexibilit y Manufact urer Image Service Salesforce Image Product Qualit y Mean 2.213 2.183 6.84 991 2.14 2.591 7.983 3.371 3.879 7.007 5.814 3.607 2.886 7.900 577 1.886 9.059 5.100 3.24 2.527 5.832 3.369 2.4 75 7.708 5.227 2.902 2.637 7.161 Discriminant Analysis St d Deviat ion 9593 8239 9154 1.0004 5720 6230 1.2561 9770 1.1081 1.2332 1.0399 6522 74 61 1.2521 9904 8593 6967 1.334 3996 8751 1.3275 1.4 14 1.2004 1.3935 1.1738 8163 754 1.6302 Valid N (list wise) Unweight ed Weight ed 23 23.000 23 23.000 23 23.000 23 23.000 23 23.000 23 23.000 23 23.000 14 14 000 14 14 000 14 14 000 14 14 000 14 14 000 14 14 000 14 14 000 22 22.000 22 22.000 22 22.000 22 22.000 22 22.000 22 22.000 22 22.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 59 59.000 21 SPSS Results: Assessment Classificat ion Result sa ,b Cases Select ed Original Count % Cases Not Selected Original Count % Type of Buying Sit uat ion New Task Modified Rebuy St raight Rebuy New Task Modified Rebuy St raight Rebuy New Task Modified Rebuy St raight Rebuy New Task Modified Rebuy St raight Rebuy Predict ed Group Membership Modified St raight New Task Rebuy Rebuy 21 1 11 20 91.3 4.3 4.3 78.6 21.4 9.1 90.9 2 0 12 63.6 18.2 18.2 16.7 38.9 44.4 0 100.0 Tot al 23 14 22 100.0 100.0 100.0 11 18 12 100.0 100.0 100.0 a 88.1% of select ed original grouped cases correct ly classified b 63.4% of unselect ed original grouped cases correct ly classified Hits ratio = % of correct classification COMPARED TO Muhamad Jantan & T Ramayah For Hold-out Sample (Unselected Cases): Maximum Chance Criterion: 43.90% Proportional Chance Criterion: 35.10% [41 - (26 * 3]2 Press Q = = 35.10 41(3 - 1) Discriminant Analysis 22 SPSS Results: Assessment Note: St ruct ure Mat rix Funct ion Delivery Speed 545* -.081 Price Level -.040 915* Service 487 703* Price Flexibility 520 -.523* Product Quality -.365 395* Manufacturer Image 032 297* Salesforce Image -.012 196* Pooled within-groups correlations between discriminat ing variables and st andardized canonical discriminant funct ions Variables ordered by absolut e size of correlation within function * Largest absolut e correlation between each variable and any discriminant function Muhamad Jantan & T Ramayah Discriminant Analysis • The “*” indicate the variables that likely to dominate the particular function • There are functions because there are groups 23 SPSS Results: Assessment Canonical Discriminant Funct ion Coefficient s Function Delivery Speed -.324 Price Level -.431 Price Flexibility 900 Manufacturer Image 285 Service 2.259 Salesforce Image -.323 Product Quality -.255 (Constant ) -10.145 Unstandardized coefficient s 1.055 1.770 -.144 162 -1.463 -.149 111 -3.830 Coefficients used in the discriminant function to calculate the Discriminant scores which is used to classify the individuals Discriminant Function: Z1 = -10.145 - 324X1 - 431X2 + 900X3 + 285X4 + 2.259X5 – 0.323X6 - 255X7 Z2 = -3.830 + 1.055X1 + 1.770X2 - 144X3 + 162X4 - 1.463X5 – 149X6 + 111X7 Muhamad Jantan & T Ramayah Discriminant Analysis 24 SPSS Results: Assessment Centroid Territorial Map Canonical Discriminant Function -6.0 -4.0 -2.0 2.0 4.0  6.0  12 23  12 23  12 23 4.0   12    23   12 23  12 23 2.0   12  23   12 * 23  12 23    12  23    * 12 23 *  12 23 -2.0    12  23    12 23  1223 -4.0    123    13  13 -6.0  13  -6.0 -4.0 -2.0 2.0 4.0 Canonical Discriminant Function Group Group Muhamad Jantan & T Ramayah Group Discriminant Analysis 6.0                    6.0 25 SPSS Results: Interpretation St ruct ure Mat rix Funct ion Delivery Speed 54 5* -.081 Price Level -.04 915* Service 87 703* Price Flexibilit y 520 -.523* Product Qualit y -.365 395* Manufact urer Image 032 297* Salesforce Image -.012 196* Pooled wit hin-groups correlat ions bet ween discriminat ing variables and st andardized canonical discriminant funct ions Variables ordered by absolut e size of correlat ion wit hin funct ion * Largest absolut e correlat ion bet ween each variable and any discriminant funct ion Since there are two discriminant functions, to identify the impact of each variable on the dependent, we need to calculate the potency index Muhamad Jantan & T Ramayah Discriminant Analysis 26 SPSS Results: Interpretation   Discriminant Function     Discriminant Function Variable Loading Squared Relative Potency Loading Squared Relative Potency Potency   (L) Loading Eigenvalue Value (L) Loading Eigenvalue Value Index 0.545 0.297 0.807 0.240 -0.081 0.007 0.193 0.001 0.241 -0.040 0.002 0.807 0.001 0.915 0.838 0.193 0.162 0.163 Service 0.487 0.237 0.807 0.191 0.703 0.494 0.193 0.095 0.286 Price Flexibility 0.520 0.270 0.807 0.218 -0.523 0.274 0.193 0.053 0.271 Product Quality -0.365 0.133 0.807 0.107 0.395 0.156 0.193 0.030 0.137 0.032 0.001 0.807 0.001 0.297 0.088 0.193 0.017 0.018 -0.012 0.000 0.807 0.000 0.196 0.038 0.193 0.007 0.008 Delivery Speed Price Level Manufacturer Image Salesforce Image       Thus service has the highest discriminatory power, followed by price flexibility, delivery speed, etc Muhamad Jantan & T Ramayah Discriminant Analysis 27 SPSS Results: Interpretation Test s of Equalit y of Group Means Delivery Speed Price Level Price Flexibilit y Manufacturer Image Service Salesforce Image Product Quality Wilks' Lambda 459 555 430 919 416 964 598 F 33.047 22.429 37.157 2.463 39.301 1.036 18.861 df1 df2 2 2 2 56 56 56 56 56 56 56 Sig .000 000 000 094 000 362 000 This test the equality of means across the three groups Clearly manufacturers image and sales force image cannot DISCRIMINATE the groups; which reinforces the potency values Muhamad Jantan & T Ramayah Discriminant Analysis 28 ... Jantan & T Ramayah Discriminant Analysis SPSS: Discriminant Analysis SPSS Command: Analyze  Classify  Discriminant  Select Cases  Statistics  Classify Muhamad Jantan & T Ramayah Discriminant Analysis... Map ♦ Relative Importance of Variables  Discriminant Weights  Discriminant Loadings  Potency Index; Muhamad Jantan & T Ramayah Discriminant Analysis 18 Discriminant Analysis – group example Profile... canonical discriminant funct ions Variables ordered by absolut e size of correlat ion wit hin funct ion Linear Correlation between PQ and the discriminant function Muhamad Jantan & T Ramayah Discriminant