；Principal Components Discriminant Function Analysis
The data for this example come from the research reported by
; Poulson, R. L., Braithwaite, R. L., Brondino, M. J., & Wuensch, K. L. (1997).
Mock jurors' insanity defense verdict selections: The role of evidence, attitudes,
and verdict options. Journal of Social Behavior and Personality, 12, 743-758.
Participants watched a simulated trial in which the defendant was accused
of murder and was pleading insanity. There was so little doubt about his having
killed the victim that none of the jurors voted for a verdict of not guilty. Aside
from not guilty, their verdict options were Guilty, NGRI (not guilty by reason of
insanity), and GBMI (guilty but mentally ill). Each mock juror filled out a
questionnaire, answering 21 questions (from which 8 predictor variables were
constructed) about e’s attitudes about crime control, the insanity defense, the
death penalty, the attorneys, and e’s assessment of the expert testimony, the
defendant’s mental status, and the possibility that the defendant could be
rehabilitated. To avoid problems associated with multicollinearity among the 8
predictor variables (they were very highly correlated with one another, and such
multicollinearity can cause problems in a multivariate analysis), the scores on the
8 predictor variables were subjected to a principal components analysis, with the
resulting orthogonal components used as predictors in a discriminant analysis.
The verdict choice (Guilty, NGRI, or GBMI) was the criterion variable.
The data are in the file PCA-DFA.dat, available at my StatData page. A SAS
program to conduct the analysis, PCA-DFA.sas, is available at my SAS Programs Page.
Download both and run the program.
There is not really anything of exceptional importance in the statistical output from Proc Factor, which was employed to produce the scores on eight principal components, repackaging all of the variance in the original variables. The principal components are not correlated with one another, solving the problem with multicollinearlity.
The output of the DFA shows that there are two significant discriminant functions. Take a look at the total sample canonical coefficients and then look back at the program. In the DATA CMPSCORE set I used these coefficients to compute, for each participant, scores on the two discriminant functions. I then correlated scores on those two discriminant functions with scores on the original variables. This gave me the structure of the discriminant functions (the loadings) with respect to the original variables.
Next I obtained means (centroids), by verdict, for the two discriminant functions. Please note that these match those reported in the output from Proc Discrim.
Finally, I used ANOVA to compare the three verdict groups on both discriminant functions and on the original variables. Note that I used Fisher’s procedure (LSD) to
； Copyright 2008 Karl L. Wuensch - All rights reserved.
make pairwise comparisons among means, including the means on the two discriminant functions.
Here is a brief summary of the results: The first function discriminated between
jurors choosing a guilty verdict and subjects choosing a NGRI verdict. Believing that the defendant was mentally ill, believing the defense’s expert testimony more than the prosecution’s, being receptive to the insanity defense, opposing the death penalty, believing that the defendant could be rehabilitated, and favoring lenient treatment were associated with rendering a NGRI verdict. Conversely, the opposite orientation on these factors was associated with rendering a guilty verdict. The second function
separated those who rendered a GBMI verdict from those choosing Guilty or NGRI. Distrusting the attorneys (especially the prosecution attorney), thinking rehabilitation likely, opposing lenient treatment, not being receptive to the insanity defense, and favoring the death penalty were associated with rendering a GBMI verdict rather than a guilty or NGRI verdict.
Presenting the Results of a Discriminant Function Analysis
The manner in which the results are presented depends in part on what the goals of the analysis were -- was the focus of the research developing a model with which to classify subjects into groups, or was the focus on determining how the groups differ on a set of continuous variables. In the behavioral sciences the focus is more often the latter.
You should pay attention to the example presentations in Tabachnick and Fidell. Here I supplement that material with an example from some research I did with Ron Poulson while he was here (see the Journal of Social Behavior and Personality, 12:
There was a problem with multicollinearity among the continuous variables in this study. We handled that by first conducting a principal components analysis and then doing the discriminant function analysis on the component scores. Also note that I presented group means on each of the two discriminant functions, and made pairwise comparisons on these means (as well as on the original means).
In order to determine which of the evaluative and attitudinal factors were important in producing the differences in verdict choice, we conducted a principal components discriminant function analysis. A discriminant function is a weighted linear combination of the predictor variables, with the weights chosen such that the criterion groups differ as much as possible on the resulting discriminant function. In our analysis, verdict choice served as the criterion variable. The predictor variables were the five attitudinal and three evaluative clusters of variables described in the Methods section
and in Appendix 1. To avoid problems associated with multicollinearity among the original variables, these variables were subjected to a principal components analysis, with the resulting orthogonal components used as the predictors in the discriminant analysis. The results of this analysis were then transformed back into a form interpretable in terms of the original variables by correlating the participants' raw scores on the original eight variables with the participants' scores on the two significant discriminant functions (DF). These correlations are given in the structure matrices displayed in Table 1.
Structure of the Discriminant Functions
Variable DF1 DF2
Mental status of defendant .87 -.19
Evaluation of expert testimony .75 .12
Receptivity to insanity defense .65 .28
Opposition to death penalty .54 .25
Favoring lenient treatment .34 .35
Believing rehabilitation unlikely -.41 .43
Trusting the prosecuting attorneys -.10 .50
Trusting the defense attorneys -.13 .29
Table 2 contains the classification means for the groups on each discriminant function as well as the group means on each of the eight original variables. The classification means indicate that the first function distinguishes between participants choosing a guilty verdict and participants returning an insanity verdict, F(16, 252) =
10.71, p < .001. Believing that the defendant was mentally ill, believing the defense’s expert testimony more than the prosecution’s, being receptive to the insanity defense,
opposing the death penalty, believing that the defendant could be rehabilitated, and favoring lenient treatment were associated with rendering a insanity verdict. Conversely, the opposite orientation on these factors was associated with rendering a guilty verdict. The second function separated those who rendered a guilty-ill verdict from those choosing guilty or insanity, F(7, 127) = 3.40, p < .003. Distrusting the attorneys
(especially the prosecution attorney), thinking rehabilitation likely, opposing lenient treatment, not being receptive to the insanity defense, and favoring the death penalty were associated with rendering a guilty-ill verdict rather than a guilty or insanity verdict.
Those who prefer univariate presentation of results should focus on the last eight rows of Table 2. Do note that on every variable, excepting the trust of the attorneys variables, the mean for the guilty-ill group is between that for the guilty group and that for the insanity group. Fisher’s procedure was used to make pairwise comparisons
among the groups on each of the variables, including the discriminant functions. It should be noted that when employed to make pairwise comparisons among three and only three groups, Fisher’s procedure has been found to hold familywise error at or
below the nominal rate and to have more power than commonly employed alternative procedures (Levin, Serlin, & Seaman, 1994).
Group Means on the Discriminant Functions and the Original Eight Variables
Variable Guilty Guilty-Ill Insanity
ABCDiscriminant Function 1 -1.29 -0.01 2.32
ABADiscriminant Function 2 0.48 -0.39 0.46
ABCMental status of defendant 2.86 4.61 6.98
ABCEvaluation of expert testimony -2.47 -0.71 3.43
AABReceptivity to insanity defense 2.01 2.22 3.02
AABOpposition to death penalty 1.56 1.81 2.82
AABFavoring lenient treatment 1.88 1.89 2.50
ABBBelieving rehabilitation unlikely 6.82 5.47 4.91
ABABTrusting the prosecuting attorneys 2.21 1.81 2.00
AAATrusting the defense attorneys 2.03 1.75 1.77
Note. Within each row, means having the same letter in their superscripts are not significantly different from each other at the .05 level.
Copyright 2008 Karl L. Wuensch - All rights reserved.