Day 9 Team I Discriminant analysis- assumptions and purpose Sanjay O S

Discriminant Function Analysis (DA) undertakes the same task as multiple linear regression

by predicting an outcome. However, multiple linear regression is limited to cases where the

dependent variable on the Y axis is an interval variable so that the combination of predictors

will, through the regression equation, produce estimated mean population numerical

Y values for given values of weighted combinations of X values.

DA is used when:

·         the dependent is categorical with the predictor IV’s at interval level such as age, income,

attitudes, perceptions, and years of education, although dummy variables can be used

as predictors as in multiple regression. Logistic regression IV’s can be of any level of measurement.

·         there are more than two DV categories, unlike logistic regression, which is limited to a

dichotomous dependent variable.

DA involves the determination of a linear equation like regression that will predict which

group the case belongs to. The form of the equation or function is:

  Where  D = discriminate function

              v = the discriminant coeffi cient or weight for that variable

            X = respondent’s score for that variable

            a = a constant

   i = the number of predictor variables

This function is similar to a regression equation or function. The v’s are unstandardized

discriminant coeffi cients analogous to the b’s in the regression equation. These v’s maximize

the distance between the means of the criterion (dependent) variable. Standardized

discriminant coeffi cients can also be used like beta weight in regression. Good predictors

tend to have large weights.

The major underlying assumptions of DA are:

•          the observations are a random sample;
•      each predictor variable is normally distributed;
•      each of the allocations for the dependent categories in the initial classification are correctly classified;

• there must be at least two groups or categories, with each case belonging to only one group so that the groups are mutually exclusive and collectively exhaustive (all cases can be placed in a group);
• each group or category must be well defined, clearly differentiated from any other group(s) and natural. Putting a median split on an attitude scale is not a natural way to form groups. Partitioning quantitative variables is only justifiable if there are easily identifiable gaps at the points of division;
•  for instance, three groups taking three available levels of amounts of housing loan;
•  the groups or categories should be defined before collecting the data;
•  the attribute(s) used to separate the groups should discriminate quite clearly between
• the groups so that group or category overlap is clearly non-existent or minimal;
•  group sizes of the dependent should not be grossly different and should be at least five times the number of independent variables.

There are several purposes of DA:

•          To investigate differences between groups on the basis of the attributes of the cases, indicating which attributes contribute most to group separation. The descriptive technique successively identifies the linear combination of attributes known as canonical discriminant functions (equations) which contribute maximally to group separation.
•       Predictive DA addresses the question of how to assign new cases to groups. The DA function uses a person’s scores on the predictor variables to predict the category to which the individual belongs.
•        To determine the most parsimonious way to distinguish between groups.
•        To classify cases into groups. Statistical significance tests using chi square enable you see how well the function separates the groups.
•         To test theory whether cases are classified as predicted.