Multiple discriminant analysis

Detailed discription about Multi discriminal Analysis (MDA) Technique used in Marketing Research.

1. Multiple Discriminant Analysis(MDA ) Muhammad Hasrath

2. M ultiple discriminant analysis Discriminant analysis techniques are described by the number of categories possessed by the criterion variable. 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, but in multiple discriminant analysis more than one function may be computed .

3. 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, be uncorrelated with the first, has the second highest ratio, and so on. However, not all the functions may be statistically significant.

4. 4 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 (q50), stated relevance of taste (q24a), value for money (q24b) and animal welfare (q24k), plus an indicator of income (q60)

5. 5 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 ).

6. 6 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

7. Canonical correlation . Canonical  correlationmeasures the extent of association between thediscriminant  scores and the groups. It is a measureof association between the single discriminantfunction and the set of dummy variables that definethe group membership.  Centroid . The centroid  is the mean values for  thediscriminant  scores for a particular group. There areas many centroids as there are groups, as there isone for each group. The means for a group on allthe functions are the group centroids   .  Classification matrix . Sometimes also called confusion    or   prediction matrix  , the classificationmatrix contains the number of correctly classified andmisclassified cases. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                             18-9  Discriminant function coefficients . Thediscriminant function coefficients ( unstandardized )are the multipliers of variables, when the variablesare in the original units of measurement.  Discriminant scores. The unstandardizedcoefficients are multiplied by the values of thevariables . These products are summed and added  tothe constant term to obtain the discriminant scores.  Eigenvalue . For each discriminant function,  theEigenvalue is the ratio of between-group to within-group sums of squares. Large  Eigenvalues   implysuperior functions. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                             18-10  F   values and their significance . These  arecalculated from a one-way ANOVA, with the groupingvariable serving as the categorical independentvariable . Each predictor, in turn, serves as the  metricdependent variable in the ANOVA.  Group means and group standard deviations .These are computed for each predictor for eachgroup .  Pooled within-group correlation matrix . Thepooled within-group correlation matrix is computedby averaging the separate covariance matrices for allthe groups. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                           Canonical correlation . Canonical  correlationmeasures the extent of association between thediscriminant  scores and the groups. It is a measureof association between the single discriminantfunction and the set of dummy variables that definethe group membership.  Centroid . The centroid  is the mean values for  thediscriminant  scores for a particular group. There areas many centroids as there are groups, as there isone for each group. The means for a group on allthe functions are the group centroids   .  Classification matrix . Sometimes also called confusion    or   prediction matrix  , the classificationmatrix contains the number of correctly classified andmisclassified cases. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                             18-9  Discriminant function coefficients . Thediscriminant function coefficients ( unstandardized )are the multipliers of variables, when the variablesare in the original units of measurement.  Discriminant scores. The unstandardizedcoefficients are multiplied by the values of thevariables . These products are summed and added  tothe constant term to obtain the discriminant scores.  Eigenvalue . For each discriminant function,  theEigenvalue is the ratio of between-group to within-group sums of squares. Large  Eigenvalues   implysuperior functions. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                             18-10  F   values and their significance . These  arecalculated from a one-way ANOVA, with the groupingvariable serving as the categorical independentvariable . Each predictor, in turn, serves as the  metricdependent variable in the ANOVA.  Group means and group standard deviations .These are computed for each predictor for eachgroup .  Pooled within-group correlation matrix . Thepooled within-group correlation matrix is computedby averaging the separate covariance matrices for allthe groups. 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 areas 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.

8. Discriminant function coefficients The discriminant function coefficients (un standardized)are the multipliers of variables, when the variables are in the original units of measurement. Discriminant scores. The un standardized coefficients are multiplied by the values of the variables. These products are summed and added to the constant term to obtain the discriminant scores. Eigen value For each discriminant function, the Eigen value is the ratio of between-group to within group sums of squares. Large Eigen values imply superior functions.

9. F   values and their significance . These  arecalculated from a one-way ANOVA, with the groupingvariable serving as the categorical independentvariable . Each predictor, in turn, serves as the  metricdependent variable in the ANOVA.  Group means and group standard deviations .These are computed for each predictor for eachgroup .  Pooled within-group correlation matrix . Thepooled within-group correlation matrix is computedby averaging the separate covariance matrices for allthe groups. Statistics Associated with Discriminant Analysis                                                                                                                                                                                                             18-11  Standardized discriminant function coefficients . Thestandardized discriminant function coefficients are thediscriminant function coefficients and are used as the multiplierswhen 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 simplecorrelations between the predictors and the  discriminantfunction .  Total correlation matrix . If the cases are treated as if they were from a single sample and the correlations computed, atotal correlation matrix is obtained.  Wilks ' . Sometimes also called the U  statistic,  Wilks '  foreach 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  notseem  to be different. Small values of (near 0) indicate  thatthe group means seem to be different. 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.

10. 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, atotal 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 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.

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