The purposes of discriminant analysis (DA):
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. But many interesting
variables are categorical, such as political party voting intention, migrant/non-migrant status,
making a profi t or not, holding a particular credit card, owning, renting or paying a mortgage
for a house, employed/unemployed, satisfi ed versus dissatisfi ed employees, which customers are likely to buy a product or not buy, what distinguishes Stellar Bean clients from
Gloria Beans clients, whether a person is a credit risk or not, etc.

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.

Discriminant analysis linear equation:
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:
D v X v X v X ........v X a
1 1 2 2 3 3 i i
= + + = +
  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. What you want this function to do is maximize the distance
between the categories, i.e. come up with an equation that has strong discriminatory power
between groups. After using an existing set of data to calculate the discriminant function
and classify cases, any new cases can then be classifi ed. The number of discriminant functions is one less the number of groups. There is only one function for the basic two group
discriminant analysis.


Assumptions of discriminant analysis:
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 classifi cation are
correctly classifi ed;
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 defi ned, 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 justifi able if there are easily
identifi able 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 defi ned 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 fi ve
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 identifi es 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 signifi cance tests using chi square enable you
to see how well the function separates the groups.
To test theory whether cases are classifi ed as predicted.


SPSS activity – discriminant analysis:
Please access SPSS Chapter 25 Data File A on the web page. You will now be taken through
a discriminant analysis using that data which includes demographic data and scores on
various questionnaires. ‘smoke’ is a nominal variable indicating whether the employee
smoked or not. The other variables to be used are age, days absent sick from work last year,
self-concept score, anxiety score and attitudes to anti-smoking at work score. The aim of
the analysis is to determine whether these variables will discriminate between those who
smoke and those who do not. This is a simple discriminant analysis with only two groups
in the DV. With three or more DV groupings a multiple discriminant analysis is involved,
but this follows the same process in SPSS as described below except there will be more
than one set of eigenvalues, Wilks’ Lambda’s and beta coeffi cients. The number of sets is
always one less than the number of DV groups.
1  Analyse >> Classify >> Discriminant
2 Select ‘smoke’ as your grouping variable and enter it into the Grouping Variable Box

3 Click Defi ne Range button and enter the lowest and highest code for your groups (here
it is 1 and 2) (Fig. 25.5).
4 Click Continue.
5 Select your predictors (IV’s) and enter into Independents box (Fig. 25.6) and select
Enter Independents Together. If you planned a stepwise analysis you would at this
point select Use Stepwise Method and not the previous instruction.
6 Click on Statistics button and select Means, Univariate Anovas, Box’s M, Unstandardized
and Within-Groups Correlation

7  Continue >> Classify. Select Compute From Group Sizes, Summary Table, Leave
One Out Classifi cation, Within Groups, and all Plots (Fig. 25.8).
8  Continue >> Save and select Predicted Group Membership and Discriminant Scores.
9  OK.

The initial case processing summary as usual indicates sample size and any missing data.

   Group statistics tables:               

In discriminant analysis we are trying to predict a group membership, so fi rstly we examine 

whether there are any signifi cant differences between groups on each of the independent 

variables using group means and ANOVA results data. The Group Statistics and Tests of 

Equality of Group Means tables provide this information. If there are no signifi cant group 

differences it is not worthwhile proceeding any further with the analysis. A rough idea 

of variables that may be important can be obtained by inspecting the group means and standard deviations.