## Wednesday, 31 August 2011

### Discriminant Function Analysis

The main purpose of a discriminant function analysis is to predict group membership based on a linear combination of the interval variables. The procedure begins with a set of observations where both group membership and the values of the interval variables are known. The end result of the procedure is a model that allows prediction of group membership when only the interval variables are known. A second purpose of discriminant function analysis is an understanding of the data set, as a careful examination of the prediction model that results from the procedure can give insight into the relationship between group membership and the variables used to predict group membership.

An example might predict whether patients recovered from a coma or not based on combinations of demographic and treatment variables. The predictor variables might include age, sex, general health, time between incident and arrival at hospital, various interventions, etc. In this case the creation of the prediction model would allow a medical practitioner to assess the chance of recovery based on observed variables. The prediction model might also give insight into how the variables interact in predicting recovery.

A single interval variable might discriminate between groups in an almost perfect fashion, not at all, or somewhere in between. For example, if one wished to differentiate adult males and females, one could collect information on how many bras the person owned, score on the last statistics test, and height. In the case of the number of bras, the discrimination would be very good, but not perfect (some women don't own any bras, some men do). In the case of the score on the last statistics test, little discrimination would be possible because males and females generally score about the same. In the case of height, some discrimination between adult males and females would be possible, but it would be far from perfect.

In general, the larger the difference between the means of the two groups relative to the within groups variability, the better the discrimination between the groups.

In marketing, discriminant analysis was once often used to determine the factors which distinguish different types of customers and/or products on the basis of surveys or other forms of collected data. Logistic regression or other methods are now more commonly used. The use of discriminant analysis in marketing can be described by the following steps:

1. Formulate the problem and gather data
2. Estimate the Discriminant Function Coefficients and determine the statistical significance and validity
3. Plot the results on a two dimensional map, define the dimensions, and interpret the results.

While discriminant analysis is often used in marketing research for marketing segmentation and predicting group membership, there are more powerful and accurate techniques available.