Discriminant Function Analysis (Two Groups): SPSS Output

Notes This example is from the SPSS 7.5 "Applications Guide" example for file "gss 93 subset.sav". The dependent is "vote92." The independents are age, educ, income91, sex, and polviews (which is a 7-point Likert scale from "Extremely liberal" to "extremely conservative").

To obtain this output:

File, Open, point to gss 93 subset.sav.
Restrict vote92 to 1's and 2's by choosing Data, Select Cases, "If condition is satisfied". Click the If button and enter vote92 <3. Click Continue, OK.
Statistics, Classify, Discriminant
Select vote92 as the "grouping variable" (the dependent). As independents, select age, educ, income91, sex, and polviews. Check "Enter independents together" (i.e., not stepwise).
Click on Statistics and check all Descriptives and all Function Coefficients.
Click on Classify and check Results (limit to first 10), Summary Table, and all plots.
To run, click OK.

Comments in blue are by the instructor and are not part of SPSS output.

Discriminant

First come several blocks of general processing and descriptive statistics information.

**Notes**
Output Created		02 Mar 98 14:11:35
Comments
Input	Data	Y:\PC\spss95\GSS93 subset.sav
	Filter	vote92 < 3 (FILTER)
	Weight	<none>
	Split File	<none>
	N of Rows in Working Data File	1452
Missing Value Handling	Definition of Missing	User-defined missing values are treated as missing in the analysis phase.
Missing Value Handling	Cases Used	In the analysis phase, cases with no user- or system-missing values for any predictor variable are used. Cases with user-, system-missing, or out-of-range values for the grouping variable are always excluded.
Syntax		DISCRIMINANT /GROUPS=vote92(1 2) /VARIABLES=sex age educ income91 polviews /ANALYSIS ALL /PRIORS EQUAL /STATISTICS=MEAN STDDEV UNIVF BOXM COEFF RAW TABLE /PLOT=COMBINED SEPARATE MAP /PLOT=CASES(10) /CLASSIFY=NONMISSING POOLED .
Resources	Elapsed Time	0:00:01.21

**Analysis Case Processing Summary**
Unweighted Cases		N	Percent
Valid		1345	92.6
Excluded	Missing or out-of-range group codes	0	.0
	At least one missing discriminating variable	107	7.4
	Both missing or out-of-range group codes and at least one missing discriminating variable	0	.0
	Total	107	7.4
Total		1452	100.0

**Group Statistics**
		Mean	Std. Deviation	Valid N (listwise)
Voting in 1992 Election		Mean	Std. Deviation	Unweighted	Weighted
voted	Respondent's Sex	1.55	.50	971	971.000
	Age of Respondent	47.56	16.73	971	971.000
	Highest Year of School Completed	13.64	2.97	971	971.000
	Total family Income	15.51	5.00	971	971.000
	Think of Self as Liberal or Conservative	4.19	1.41	971	971.000
did not vote	Respondent's Sex	1.57	.50	374	374.000
	Age of Respondent	41.64	17.34	374	374.000
	Highest Year of School Completed	11.84	2.84	374	374.000
	Total Family Income	12.60	5.77	374	374.000
	Think of Self as Liberal or Conservative	4.10	1.21	374	374.000
Total	Respondent's Sex	1.56	.50	1345	1345.000
	Age of Respondent	45.91	17.10	1345	1345.000
	Highest Year of School Completed	13.14	3.04	1345	1345.000
	Total family Income	14.70	5.39	1345	1345.000
	Think of Self as Liberal or Conservative	4.17	1.36	1345	1345.000

In the ANOVA table below, the smaller the Wilks's lambda, the more important the independent variable to the discriminant function. Wilks's lambda is significant by the F test for age, educ, and income92. We might consider dropping sex and polviews from the model.

**Tests of Equality of Group Means**
	Wilks' Lambda	F	df1	df2	Sig.
Respondent's Sex	1.000	.418	1	1343	.518
Age of Respondent	.976	33.197	1	1343	.000
Highest Year of School Completed	.930	101.620	1	1343	.000
Total family Income	.941	83.840	1	1343	.000
Think of Self as Liberal or Conservative	.999	1.423	1	1343	.233

Analysis 1

Box's Test of Equality of Covariance Matrices

The larger the log determinant in the table below, the more that group's covariance matrix differs. The "Rank" column indicates the number of independent variables -- 5 in this case. Since discriminant analysis assumes homogeneity of covariance matrices between groups, we would like to see the determinants be relatively equal. Box's M, next, tests the homogeneity of covariances assumption.

Log Determinants

Voting in 1992 Election
Rank
Log Determinant

voted
5
10.006

did not vote
5
9.945

Pooled within-groups
5
10.019

The ranks and natural logarithms of determinants printed are those of the group covariance matrices.

**Log Determinants**
Voting in 1992 Election	Rank	Log Determinant
voted	5	10.006
did not vote	5	9.945
Pooled within-groups	5	10.019
The ranks and natural logarithms of determinants printed are those of the group covariance matrices.

**Test Results**

Box's M test tests the assumption of homogeneity of covariance matrices. This test is very sensitive to meeting also the assumption of multivariate normality. Discriminant function analysis is robust even when the homogeneity of variances assumption is not met, provided the data do not contain important outliers. For the data below, the test is significant so we conclude the groups do differ in their covariance matrices, violating an assumption of DA. Note that when n is large, as it is here, small deviations from homogeneity will be found significant, which is why Box's M must be interpreted in conjunction with inspection of the log determinants, above.
Box's M		40.399
F	Approx.	2.679
	df1	15
	df2	2102169.732
	Sig.	.000
Tests null hypothesis of equal population covariance matrices.

Summary of Canonical Discriminant Functions

The table below shows the eigenvalues. The larger the eigenvalue, the more of the variance in the dependent variable is explained by that function. Since the dependent in this example has only two categories, there is only one discriminant function. However, if there were more categories, we would have multiple discriminant functions and this table would list them in descending order of importance. The second column lists the percent of variance explained by each function. The third column is the cumulative percent of variance explained. The last column is the canonical correlation, where the squared canonical correlation is the percent of variation in the dependent discriminated by the independents in DA. Sometimes this table is used to decide how many functions are important (ex., eigenvalues over 1, percent of variance more than 5%, cumularive percentage of 75%, canonical correlation of .6). This issue does not arise here since there is only one discriminant function, though we may note its canonical correlation is not high.

**Eigenvalues**
Function	Eigenvalue	% of Variance	Cumulative %	Canonical Correlation
1	.164(a)	100.0	100.0	.376
a First 1 canonical discriminant functions were used in the analysis.

This second appearance of Wilks's lambda serves a different purpose than its use in the ANOVA table above. In the table below it tests the significance of the eigenvalue for each discriminant function. In this example there is only one, and it is significant.

**Wilks' Lambda**
Test of Function(s)	Wilks' Lambda	Chi-square	df	Sig.
1	.859	203.909	5	.000

The standardized discriminant function coefficients in the table below serve the same purpose as beta weights in multiple regression: they indicate the relative importance of the independent variables in predicting the dependent.

**Standardized Canonical Discriminant Function Coefficients**
	Function
	1
Respondent's Sex	.011
Age of Respondent	.657
Highest Year of School Completed	.712
Total family Income	.423
Think of Self as Liberal or Conservative	.018

The structure matrix table below shows the correlations of each variable with each discriminant function. In this case, there is only one discriminant function. However, when the dependent has more categories there will be more discriminant functions. In that case, there will be additional columns in the table, one for each function. The correlations then serve like factor loadings in factor analysis -- that is, by identifying the largest absolute correlations associated with each discriminant function the researcher gains insight into how to name each function.

Structure Matrix

Function

1

Highest Year of School Completed
.679

Total family Income
.616

Age of Respondent
.388

Think of Self as Liberal or Conservative
.080

Respondent's Sex
-.044

Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions
Variables ordered by absolute size of correlation within function.

**Structure Matrix**
	Function
1
Highest Year of School Completed	.679
Total family Income	.616
Age of Respondent	.388
Think of Self as Liberal or Conservative	.080
Respondent's Sex	-.044
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function.

The table below contains the unstandardized discriminant function coefficients. These would be used like unstandardized b (regression) coefficients in multiple regression -- that is, they are used to construct the actual prediciton equation which can be used to classify new cases.

**Canonical Discriminant Function Coefficients**
	Function
	1
Respondent's Sex	.021
Age of Respondent	.039
Highest Year of School Completed	.243
Total family Income	.081
Think of Self as Liberal or Conservative	.013
(Constant)	-6.253
Unstandardized coefficients

The table below is used to establish the cutting point for classifying cases. If the two groups are of equal size, the best cutting point is half way between the values of the functions at group centroids (that is, the average). If the groups are unequal, the optimal cutting point is the weighted average of the two values. Cases which evaluate on the function above the cutting point are classified as "did not vote," while those evaluating below the cutting point are evaluated as "Voted." Of course, the computer does the classification automatically, so these values are for informational purposes.

**Functions at Group Centroids**
	Function
Voting in 1992 Election	1
voted	.251
did not vote	-.653
Unstandardized canonical discriminant functions evaluated at group means

Classification Statistics

The table below just tells the researcher about the status of cases in terms of processing.

**Classification Processing Summary**
Processed		1452
Excluded	Missing or out-of-range group codes	0
Excluded	At least one missing discriminating variable	107
Used in Output		1345

Prior Probabilities below are used in classification. The default is using observed group sizes (marginals) in your sample to determine the prior probabilities of membership in the groups formed by the dependent, and this is necessary if you have different group sizes. If each group is of the same size, as an alternative you could specify equal prior probabilities for all groups.
Prior Probabilities for Groups

Prior
Cases Used in Analysis

Voting in 1992 Election

Unweighted
Weighted

voted
.500
971
971.000

did not vote
.500
374
374.000

Total
1.000
1345
1345.000

**Prior Probabilities for Groups**
	Prior	Cases Used in Analysis
Voting in 1992 Election	Unweighted	Weighted
voted	.500	971	971.000
did not vote	.500	374	374.000
Total	1.000	1345	1345.000

The table below is the result of checking "Fisher's" under "Function Coefficients" in the "Statistics" option of discriminant analysis. Two sets (one for each dependent group) of unstandardized linear discriminant coefficients are calculated, which can be used to classify cases. This is the classical method of classification, though now little used.

**Classification Function Coefficients**
	Voting in 1992 Election
	voted	did not vote
Respondent's Sex	7.048	7.029
Age of Respondent	.250	.215
Highest Year of School Completed	1.884	1.664
Total family Income	.319	.246
Think of Self as Liberal or Conservative	2.187	2.175
(Constant)	-32.013	-26.541
Fisher's linear discriminant functions

The table below results from checking "Casewise results" in the "Classify" options of discriminant function analysis. The table lists the actual group, the predicted group based on largest posterior probabilities, the prior probability (the probability of the observed group score given membership in the predicted group), the posterior probability (the chance the case belongs to the predicted group, based on the independents), the Mahalanobis distance squared of the case to the group centroid (large scores indicate outliers), and the discriminant score for the case. The case is classified based on the discriminant score in relation to the cutoff (not shown). Misclassified cases are marked with asterisks. The "Second Highest Group" columns show the posterior probabilities and Mahalanobis distances for the case had the case been classed based on the second highest posterior probability. Since there are only two groups in this example, the "second highest" is equivalent to the "other" group.

**Casewise Statistics**
		Actual Group	Highest Group					Second Highest Group			Discriminant Scores
			Predicted Group	P(D>d \| G=g)		P(G=g \| D=d)	Squared Mahalanobis Distance to Centroid	Group	P(G=g \| D=d)	Squared Mahalanobis Distance to Centroid	Function 1
	Case Number		Predicted Group	p	df	P(G=g \| D=d)	Squared Mahalanobis Distance to Centroid	Group	P(G=g \| D=d)	Squared Mahalanobis Distance to Centroid	Function 1
Original	1	1	2(**)	.774	1	.537	.082	1	.463	.381	-.366
	2	1	1	.561	1	.718	.339	2	.282	2.208	.833
	3	1	1	.528	1	.727	.399	2	.273	2.358	.883
	4	1	1	.445	1	.750	.583	2	.250	2.780	1.015
	5	1	1	.015	1	.932	5.941	2	.068	11.165	2.689
	6	1	1	.622	1	.701	.243	2	.299	1.951	.744
	7	1	1	.878	1	.634	.024	2	.366	1.119	.405
	8	2	1(**)	.835	1	.645	.044	2	.355	1.238	.460
	9	1	1	.390	1	.766	.738	2	.234	3.107	1.110
	10	1	1	.430	1	.755	.624	2	.245	2.870	1.041
** Misclassified case

Separate-Groups Graphs

The tables below result from checking "Combined-groups" and "Separate-groups" under "Plots" in the "Classify" options of discriminant analysis. If there were two or more discriminant functions, the charts below would be scatterplots showing the relation of the first two discriminant functions. As the dependent in this example has only one discriminant function, bar charts are displayed instead. In a good discriminant function, the bar chart will have most cases near the mean, with small tails.

Voting in 1992 election = voted

The table below is used to assess how well the discriminant function works, and if it works equally well for each group of the dependent variable. Here it correctly classifies about two-thirds of the cases, making about the same proportion of mistakes for both categories. This would normally not be considered a satisfactory level of discrimination and the researcher would seek to test other models.
Classification Results(a)

Predicted Group Membership
Total

Voting in 1992 Election
voted
did not vote

Original
Count
voted
649
322
971

did not vote
121
253
374

%
voted
66.8
33.2
100.0

did not vote
32.4
67.6
100.0

a 67.1% of original grouped cases correctly classified.