Discriminant Analysis• Discriminant Analysis may be used for two objectives: – Either we want to assess the adequacy of classification, given the group memberships of the objects under study; or – we wish to assign objects to one of a number of (known) groups of objects.• Discriminant Analysis may thus have a descriptive or a predictive objective
Discriminant analysis• Discriminant analysis is used to analyze relationships between a non-metric dependent variable and metric or dichotomous independent variables.• Discriminant analysis attempts to use the independent variables to distinguish among the groups or categories of the dependent variable.• The usefulness of a discriminant model is based upon its accuracy rate, or ability to predict the known group memberships in the categories of the dependent variable.
Discriminant scores• Discriminant analysis works by creating a new variable called the discriminant function score which is used to predict to which group a case belongs.• The discriminant function is similar to a regression equation in which the independent variables are multiplied by coefficients and summed to produce a score.
Number of functions• If the dependent variable defines two groups, one statistically significant discriminant function is required to distinguish the groups; if the dependent variable defines three groups, two statistically significant discriminant functions are required to distinguish among the three groups; etc.• If a discriminant function is able to distinguish among groups, it must have a strong relationship to at least one of the independent variables.• The number of possible discriminant functions in an analysis is limited to the smaller of the number of independent variables or one less than the number of groups defined by the dependent variable.
Discriminant Function Zi = b1 X1 + b2 X2 + b3 X3 + ... + bn XnWhere Z = discriminant score b = discriminant weights X = predictor (independent) variables http://www.drvkumar.com/mr9/ 6
Determination of Significance• Null Hypothesis: In the population, the group means the discriminant function are equal Ho : μ A = μ B• Generally, predictors with relatively large standardized coefficients contribute more to the discriminating power of the function• Discriminant loadings show the variance that the predictor shares with the function http://www.drvkumar.com/mr9/ 7
Uses of Discriminant Analysis• 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
Steps in Discriminant Analysis 1. Form groups 2. Estimate discriminant function 3. Determine significance of function and variables 4. Interpret the discriminant function 5. Perform classification and validation http://www.drvkumar.com/mr9/ 9
Example X2 Back Yard BurgerIncome ($) Customers Other Fast-Food Restaurants X1 Lifestyle-Eating Nutritious Meals
Classification of Multivariate Methods Dependence One Number of None Interdependence Methods Dependent Variables Methods (Nonmetric) (Metric) Dependent Variable Interval • Factor Analysis Nominal Level of Measurement or Ratio • Cluster Analysis • Perceptual Mapping Ordinal • Multiple Regression • Discriminant • ANOVA Analysis • Spearman’s Rank • MANOVA • Conjoint Correlation • Conjoint