Discriminant analysis group no. 4


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Discriminant analysis group no. 4

  2. 2. DISCRIMINANT ANALYSIS • Discriminant analysis is a statistical procedure which allows us to classify cases in separate categories to which they belong on the basis of a set of characteristic independent variables called predictors or discriminant variables • The target variable (the one determining allocation into groups) is a qualitative (nominal or ordinal) one, while the characteristics are measured by quantitative variables. • DA looks at the discrimination between two groups • Multiple discriminant analysis (MDA) allows for classification into three or more groups.
  3. 3. APPLICATIONS OF DA DA is especially useful to understand the differences and factors leading consumers to make different choices allowing them to develop marketing strategies which take into proper account the role of the predictors. Examples: • Determinants of customer loyalty • Shopper profiling and segmentation • Determinants of purchase and non-purchase
  4. 4. EXAMPLE ON THE TRUST DATA-SET • Purchasers of Chicken at the Butcher’s Shop • Respondents may belong to one of two groups • Those who purchase chicken at the butcher’s shop • Those who do not • Discrimination between these groups through a set of consumer characteristics • Expenditure on chicken in a standard week • Age of the respondent • Whether respondents agree (on a seven-point ranking scale) that butchers sell safe chicken • Trust (on a seven-point ranking scale) towards supermarkets • Does a linear combination of these four characteristics allow one to discriminate between those who buy chicken at the butcher’s and those who do not?
  5. 5. DISCRIMINANT ANALYSIS (DA) • Two groups only, thus a single discriminating value (discriminating score) • For each respondent a score is computed using the estimated linear combination of the predictors (the discriminant function) • Respondents with a score above the discriminating value are expected to belong to one group, those below to the other group. • When the discriminant score is standardized to have zero mean and unity variance it is called Z score • DA also provides information about the discriminating power of each of the original predictors
  6. 6. MULTIPLE DISCRIMINANT ANALYSIS (MDA) (1) Discriminant analysis may involve more than two groups, in which case it is termed multiple discriminant analysis (MDA). Example from the Trust data-set • Dependent variable: Type of chicken purchased ‘in a typical week’, choosing among four categories: value (good value for money), standard, organic and luxury • Predictors: age , stated relevance of taste , value for money and animal welfare , plus an indicator of income
  7. 7. MULTIPLE DISCRIMINANT ANALYSIS (2) • In this case there will be more than one discriminant function. • The exact number of discriminant functions is equal to either (g-1), where g is the number of categories in classification or to k, the number of independent variables, whichever is the smaller • Trust example: four groups and five explanatory variables, the number of discriminant functions is three (that is g-1 which is smaller than k=5).
  8. 8. THE OUTPUT OF MDS Similarities with factor (principal component) analysis • the first discriminant function is the most relevant for discriminating across groups, the second is the second most relevant, etc. • the discriminant functions are also independent, which means that the resulting scores are non-correlated. • Once the coefficients of the discriminant functions are estimated and standardized, they are interpreted in a similar fashion to the factor loadings. • The larger the standardised coefficients (in absolute terms), the more relevant the respective variables to discriminating between groups There is no single discriminant score in MDA • group means are computed (centroids) for each of the discriminant functions to have a clearer view of the classification rule
  9. 9. RUNNING DISCRIMINANT ANALYSIS (2 GROUPS) 9 0 1 1 2 2 3 3 4 4z x x x xα α α α α= + + + + Discriminant function (Target variable: purchasers of chicken at the butcher’s shop) Discriminant score Predictors • weekly expenditure on chicken • age • safety of butcher’s chicken • trust in supermarkets The α discriminant coefficients need to be estimated
  10. 10. FISHER’S LINEAR DISCRIMINANT ANALYSIS The discriminate function is the starting point. Two key assumptions behind linear DA (a) the predictors are normally distributed; (b) the covariance matrices for the predictors within each of the groups are equal. Departure from condition (a) should suggest use of alternative methods. Departure from condition (b) requires the use of different discriminant techniques (usually quadratic discriminant functions). In most empirical cases, the use of linear DA is appropriate. 10
  11. 11. ESTIMATION The first step is the estimation of the α coefficients, also termed as discriminant coefficients or weights. Estimation is similar to factor analysis or PCA, as the coefficients are those which maximize the variability between groups In MDA, the first discriminating function is the one with the highest between-group variability, the second discriminating function is independent from the first and maximizes the remaining between-group variability and so on 11
  12. 12. SPSS – TWO GROUPS CASE 12 1. Choose the target variable 2. Define the range of the dependent variable 3.Select the predictors
  13. 13. COEFFICIENT ESTIMATES 13 Fisher’s and standardized estimates of the discriminant function coefficients need to be asked for Additional statistics and diagnostics
  14. 14. CLASSIFICATION OPTIONS 14 Decide whether prior probabilities are equal across groups or group sizes reflect different allocation probabilities These are diagnostic indicators to evaluate how well the discriminant function predict the groups
  15. 15. SAVE CLASSIFICATION 15 Create new variables in the data-set, containing the predicted group membership and/or the discriminant score for each case and each function
  16. 16. OUTPUT – COEFFICIENT ESTIMATES 16 Canonical Discriminant Function Coefficients .095 .454 -.297 .025 -2.515 In a typical week how much do you spend on fresh or frozen chicken (Euro)? From the butcher Supermarkets Age (Constant) 1 Function Unstandardized coefficients Standardized Canonical Discriminant Function Coefficients .378 .748 -.453 .394 In a typical week how much do you spend on fresh or frozen chicken (Euro)? From the butcher Supermarkets Age 1 Function Unstandardized coefficients depend on the measurement unit Standardized coefficients do not depend on the measurement unit Most important predictor Trust in supermarkets has a – sign (thus it reduces the discriminant score)
  17. 17. CENTROIDS 17 Prior Probabilities for Groups .660 277 277.000 .340 143 143.000 1.000 420 420.000 Butcher no yes Total Prior Unweighted Weighted Cases Used in Analysis Functions at Group Centroids -.307 .594 Butcher no yes 1 Function Unstandardized canonical discriminant functions evaluated at group means These are the means of the discriminant score for each of the two groups Thus, the group of those not purchasing chicken at the butcher’s shop have a negative centroid With two groups, the discriminating score is zero This can be computed by weighting the centroids with the initial probabilities From these prior probabilities it follows that the discriminating score is -0.307 x 0.66 + 0.594 x 0.34 = 0
  18. 18. OUTPUT – CLASSIFICATION SUCCESS 18 Classification Resultsa 244 33 277 88 55 143 1 1 2 88.1 11.9 100.0 61.5 38.5 100.0 50.0 50.0 100.0 Butcher no yes Ungrouped cases no yes Ungrouped cases Count % Original no yes Predicted Group Membership Total 71.2% of original grouped cases correctly classified.a. Using the discriminant function, it is possible to correctly classify 71.2% of original cases (244 no-no + 55 yes-yes)/420
  19. 19. DIAGNOSTICS (1) • Box’s M test. This tests whether covariances are equal across groups • Wilks’ Lambda (or U statistic) tests discrimination between groups. It is related to analysis of variance. • Individual Wilks’Lambda for each of the predictors in a discriminant function; univariate ANOVA (are there significant differences in the predictor’s means between the groups?), p-value from the F distribution. • Wilks’ Lambda for the function as a whole. Are there significant differences in the group means for the discriminant function p-value from the Chi-square distribution? • The overall Wilks’ Lambda is especially helpful in multiple discriminant analysis as it allows one to discard those functions which do not contribute towards explaining differences between groups. 19
  20. 20. DIAGNOSTICS (2) DA returns one eigenvalue (or more eigenvalues for MDA) of the discriminant function. These can be interpreted as in principal component analysis In MDA (more than one discriminant function) eigenvalues are exploited to compute how each function contributes to explain variability The canonical correlation measures the intensity of the relationship between the groups and the single discriminant function 20
  21. 21. TRUST EXAMPLE: DIAGNOSTICS 21 Statistic P-value Box's M statistic 37.3 0.000 Overall Wilks' Lambda 0.85 0.000 Wilks Lambda for Expenditure 0.98 0.002 Age 0.97 0.001 Safer for Butcher 0.91 0.000 Trust in Supermarket 0.98 0.002 Eigenvalue 0.18 Canonical correlation 0.39 % OF CORRECT PREDICTIONS 71.2% Covariance matrices are not equal The overall discriminating power of the DF is good All of the predictors are relevant to discriminating between the two groups The eigenvalue is the ratio between variances between and variance within groups (the larger the better) Square root of the ratio between variability between and total variability
  22. 22. MDA 22 To run MDA in SPSS the only difference is that the range has more than two categories
  23. 23. PREDICTORS 23 Test Results 65.212 1.382 45 53286.386 .045 Box's M Approx. df1 df2 Sig. F Tests null hypothesis of equal population covariance matrices. Tests of Equality of Group Means .981 1.798 3 282 .148 .971 2.761 3 282 .042 .960 3.878 3 282 .010 .982 1.679 3 282 .172 .919 8.272 3 282 .000 Age Tasty food Value for money Animal welfare Please indicate your gross annual household income range Wilks' Lambda F df1 df2 Sig. Three predictors only appear to be relevant in discriminating among preferred types of chicken Null rejected at 95% c.l., but not at 99% c.l.
  24. 24. DISCRIMINANT FUNCTIONS 24 Eigenvalues .102a 61.0 61.0 .304 .051a 30.8 91.8 .221 .014a 8.2 100.0 .116 Function 1 2 3 Eigenvalue % of Variance Cumulative % Canonical Correlation First 3 canonical discriminant functions were used in the analysis. a. Three discriminant functions (four groups minus one) can be estimated Wilks' Lambda .851 45.098 15 .000 .938 17.904 8 .022 .986 3.818 3 .282 Test of Function(s) 1 through 3 2 through 3 3 Wilks' Lambda Chi-square df Sig. The first two discriminant functions have a significant discriminating power.
  25. 25. COEFFICIENTS 25 Discriminant functions’ coefficients Unstandardized Standardized 1 2 1 2 Value for money -.043 .603 -.053 .746 Age -.009 -.013 -.148 -.208 Tasty food .169 .416 .152 .374 Animal welfare .186 -.132 .313 -.222 Please indicate your gross annual household income range .652 -.033 .870 -.044 (Constant) -2.298 -4.868 Income is very relevant for the first function Value for money is very relevant for the second function
  26. 26. STRUCTURE MATRIX 26 Structure Matrix .929* -.021 .078 .390* -.206 .125 -.010 .891* .168 .241 .660* .273 -.217 -.204 .944* Please indicate your gross annual household income range Animal welfare Value for money Tasty food Age 1 2 3 Function Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function. Largest absolute correlation between each variable and any discriminant function *. The values in the structure matrix are the correlations between the individual predictors and the scores computed on the discriminant functions. For example, the income variable has a strong correlation with the scores of the first function The structure matrix help interpreting the functions Income Value and taste Age
  27. 27. CENTROIDS 27 Functions at Group Centroids -.673 -.262 -.040 .058 .156 -.065 .525 -.470 -.030 .003 .052 .242 In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? 'Value' chicken 'Standard' chicken 'Organic' chicken 'Luxury' chicken 1 2 3 Function Unstandardized canonical discriminant functions evaluated at group means The first function discriminates well between value and organic (income matters to organic buyers) The second allows some discrimination standard-organic, value-standard, organic-luxury (taste and value matter)
  28. 28. PLOT OF TWO FUNCTIONS 28 The ‘territorial map’ shows the scores for the first two functions considering all groups Tick ‘separate-groups’ to show graphs of the first two functions for each individual group
  29. 29. PLOTS: INDIVIDUAL GROUPS 29 Example: organic chicken Most cases tend to be relatively high on function 1 (income) Example: organic chicken Most cases tend to be relatively high on function 1 (income)
  30. 30. PLOTS – ALL GROUPS 30
  31. 31. PREDICTION RESULTS 31 Classification Resultsa 3 38 0 0 41 2 154 1 0 157 1 30 4 0 35 1 51 1 0 53 0 51 3 0 54 7.3 92.7 .0 .0 100.0 1.3 98.1 .6 .0 100.0 2.9 85.7 11.4 .0 100.0 1.9 96.2 1.9 .0 100.0 .0 94.4 5.6 .0 100.0 In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? 'Value' chicken 'Standard' chicken 'Organic' chicken 'Luxury' chicken Ungrouped cases 'Value' chicken 'Standard' chicken 'Organic' chicken 'Luxury' chicken Ungrouped cases Count % Original 'Value' chicken 'Standard' chicken 'Organic' chicken 'Luxury' chicken Predicted Group Membership Total 56.3% of original grouped cases correctly classified.a. The functions do not predict well; most units are allocated to standard chicken – on average only 56.3% of the cases are allocated correctly
  32. 32. STEPWISE DISCRIMINANT ANALYSIS As for linear regression it is possible to decide whether all predictors should appear in the equation regardless of their role in discriminating (the Enter option) or a sub-set of predictors is chosen on the basis of their contribution to discriminating between groups (the Stepwise method) 32
  33. 33. THE STEP-WISE METHOD 1. A one-way ANOVA test is run on each of the predictors, where the target grouping variable determines the treatment levels. The ANOVA test provides a criterion value and tests statistics (usually the Wilks Lambda). According to the criterion value, it is possible to identify the predictor which is most relevant in discriminating between the groups 2. The predictor with the lowest Wilks Lambda (or which meets an alternative optimality criterion) enters the discriminating function, provided the p-value is below the set threshold (for example 5%). 3. An ANCOVA test is run on the remaining predictors, where the covariates are the target grouping variables and the predictors that have already entered the model. The Wilks Lambda is computed for each of the ANCOVA options. 4. Again, the criteria and the p-value determine which variable (if any) enter the discriminating function (and possibly whether some of the entered variables should leave the model). 5. The procedure goes back to step 3 and continues until none of the excluded variables have a p-value below the threshold and none of the entered variables have a p-value above the threshold (the stopping rule is met). 33
  34. 34. ALTERNATIVE CRITERIA Unexplained variance Smallest F ratio Mahalanobis distance Rao’s V 34
  35. 35. IN SPSS 35 The step-wise method allows selection of relevant predictors
  36. 36. OUTPUT OF THE STEP-WISE METHOD 36 Variables in the Analysis 1.000 8.272 1.000 8.241 .960 1.000 3.863 .919 Please indicate your gross annual household income range Please indicate your gross annual household income range Value for money Step 1 2 Tolerance F to Remove Wilks' Lambda Variables Not in the Analysis 1.000 1.000 1.798 .981 1.000 1.000 2.761 .971 1.000 1.000 3.878 .960 1.000 1.000 1.679 .982 1.000 1.000 8.272 .919 .988 .988 1.507 .905 .991 .991 2.437 .896 1.000 1.000 3.863 .883 .992 .992 1.052 .909 .987 .987 1.549 .868 .821 .821 .793 .875 .992 .992 1.057 .873 Age Tasty food Value for money Animal welfare Please indicate your gross annual household income range Age Tasty food Value for money Animal welfare Age Tasty food Animal welfare Step 0 1 2 Tolerance Min. Tolerance F to Enter Wilks' Lambda Only two predictors are kept in the model
  37. 37. APPLICATIONS IN MARKETING: After getting to know the Technical Aspect of this useful concept, we can conclude that DA has the following applications in the field of Marketing: • Discriminate analysis, a multivariate technique used for market segmentation and predicting group membership is often used for this type of problem because of its ability to classify individuals or experimental units into two or more uniquely defined populations.
  38. 38. • Product research – Distinguish between heavy, medium, and light users of a product in terms of their consumption habits and lifestyles. • Perception/Image research – Distinguish between customers who exhibit favorable perceptions of a store or company and those who do not. • Advertising research – Identify how market segments differ in media consumption habits. • Direct marketing – Identify the characteristics of consumers who will respond to a direct marketing campaign and those who will not.
  39. 39. THANK YOU