Green and Salkind (2017) described discriminant analysis as a tool that can classify people into groups or distinguish groups from one another, based on a variety of measures. For instance, job applicants could be grouped based on a psychological profile test, school GPA, years of experience, and other factors. In this post I will show a hypothetical example of a discriminant analysis as it pertains to quantitative research. The study referenced below was assigned with variables that I could derive from a quantitative consultation with any company or organization. The data points can change but the analysis gives clear actionable data upon which to make informed decisions. So, without further delay, let’s dig into it.
Research Question
Will the publications, grant funding, teaching rating, and committees served be enough to predict the type of professor the applicant will be?
Hypotheses
H0: Publications, grant funding, teaching rating, and committees served will not be enough to predict the type of professor the applicant will be.
H1: Publications, grant funding, teaching rating, and committees served will be enough to predict the type of professor the applicant will be.
Results
In this subheading, I will present descriptive statistics, discuss testing of the assumptions, present inferential statistic results, and conclude with a concise summary. Descriptive Statistics
A total of 25 professors participated in the study. Table 1 depicts descriptive statistics for the study variables for each group. Figure 1 depicts a bar graph of research scientist type and Figure 2 depicts the teaching mogul type.
Table 1
Means and Standard Deviations for Quantitative Study Variables
Type of professor
Variable
M
SD
Research Scientist
Number of publications in last 2 years
2.9200
.81240
Grant funding (in $10000) over the last 5 years
4.3692
2.47581
Mean teaching rating for last 3 semesters
2.8782
.60132
Number of committees served on during last 5 years
2.6400
2.34307
Teaching Mogul
Number of publications in last 2 years
3.4800
.82260
Grant funding (in $10000) over the last 5 years
4.5471
3.17875
Mean teaching rating for last 3 semesters
3.5122
.42025
Number of committees served on during last 5 years
3.4800
2.23830
Total
Number of publications in last 2 years
3.200
.85714
Grant funding (in $10000) over the last 5 years
4.4581
2.82125
Mean teaching rating for last 3 semesters
3.1952
.60511
Number of committees served on during last 5 years
3.0600
2.30713
Inferential Results
One discriminant function was found to be statistically significant (Wilks’s Λ = .609, (4) = 22.849, p = .000 for discriminant function 1). The first discriminant function explains 64.3% of variance and the second discriminant function explains the rest of variance. Canonical correlation is .626 for both discriminant functions 1, indicating that 63% of variances were explained by the relationship between predictors and group membership by DF1 and DF2, respectively. Discriminant function 1 has the largest relationship with grant funding, followed by number of committees, number of publications, and teaching rating. Pairwise comparison showed only one pair. DF1 scores indicates that differences on DF1 were found in the following: Research Scientist (M = -.786) vs. Teaching Mogul (M = .786).
References
Green, S. B., & Salkind, N. J. (2017). Using SPSS for Windows and Macintosh: Analyzing and understanding data (8th ed.). Pearson..
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Appendix – Discriminant Analysis SPSS Output
Analysis Case Processing Summary
Unweighted Cases
N
Percent
Valid
50
100.0
Excluded
Missing or out-of-range group codes
0
.0
At least one missing discriminating variable
0
.0
Both missing or out-of-range group codes and at least one missing discriminating variable
0
.0
Total
0
.0
Total
50
100.0
Group Statistics
Type of professor
Mean
Std. Deviation
Valid N (listwise)
Unweighted
Weighted
Research Scientist
Number of publications in last 2 years
2.9200
.81240
25
25.000
Grant funding (in $10000) over the last 5 years
4.3692
2.47581
25
25.000
Mean teaching rating for last 3 semesters
2.8782
.60132
25
25.000
Number of committees served on during last 5 years
2.6400
2.34307
25
25.000
Teaching Mogul
Number of publications in last 2 years
3.4800
.82260
25
25.000
Grant funding (in $10000) over the last 5 years
4.5471
3.17875
25
25.000
Mean teaching rating for last 3 semesters
3.5122
.42025
25
25.000
Number of committees served on during last 5 years
3.4800
2.23830
25
25.000
Total
Number of publications in last 2 years
3.2000
.85714
50
50.000
Grant funding (in $10000) over the last 5 years
4.4581
2.82125
50
50.000
Mean teaching rating for last 3 semesters
3.1952
.60511
50
50.000
Number of committees served on during last 5 years
3.0600
2.30713
50
50.000
Tests of Equality of Group Means
Wilks’ Lambda
F
df1
df2
Sig.
Number of publications in last 2 years
.891
5.865
1
48
.019
Grant funding (in $10000) over the last 5 years
.999
.049
1
48
.826
Mean teaching rating for last 3 semesters
.720
18.675
1
48
.000
Number of committees served on during last 5 years
.966
1.680
1
48
.201
Pooled Within-Groups Matricesa
Number of publications in last 2 years
Grant funding (in $10000) over the last 5 years
Mean teaching rating for last 3 semesters
Number of committees served on during last 5 years
Covariance
Number of publications in last 2 years
.668
.067
-.077
.219
Grant funding (in $10000) over the last 5 years
.067
8.117
-.043
3.907
Mean teaching rating for last 3 semesters
-.077
-.043
.269
-.057
Number of committees served on during last 5 years
.219
3.907
-.057
5.250
Correlation
Number of publications in last 2 years
1.000
.029
-.181
.117
Grant funding (in $10000) over the last 5 years
.029
1.000
-.029
.599
Mean teaching rating for last 3 semesters
-.181
-.029
1.000
-.048
Number of committees served on during last 5 years
.117
.599
-.048
1.000
a. The covariance matrix has 48 degrees of freedom.
Covariance Matricesa
Type of professor
Number of publications in last 2 years
Grant funding (in $10000) over the last 5 years
Mean teaching rating for last 3 semesters
Number of committees served on during last 5 years
Research Scientist
Number of publications in last 2 years
.660
.426
-.004
.137
Grant funding (in $10000) over the last 5 years
.426
6.130
-.155
3.185
Mean teaching rating for last 3 semesters
-.004
-.155
.362
.108
Number of committees served on during last 5 years
.137
3.185
.108
5.490
Teaching Mogul
Number of publications in last 2 years
.677
-.292
-.149
.302
Grant funding (in $10000) over the last 5 years
-.292
10.104
.068
4.630
Mean teaching rating for last 3 semesters
-.149
.068
.177
-.222
Number of committees served on during last 5 years
.302
4.630
-.222
5.010
Total
Number of publications in last 2 years
.735
.091
.016
.335
Grant funding (in $10000) over the last 5 years
.091
7.959
-.014
3.866
Mean teaching rating for last 3 semesters
.016
-.014
.366
.080
Number of committees served on during last 5 years
.335
3.866
.080
5.323
a. The total covariance matrix has 49 degrees of freedom.
Log Determinants
Type of professor
Rank
Log Determinant
Research Scientist
4
1.638
Teaching Mogul
4
.859
Pooled within-groups
4
1.542
The ranks and natural logarithms of determinants printed are those of the group covariance matrices.
Test Results
Box’s M
14.104
F
Approx.
1.283
df1
10
df2
11015.139
Sig.
.234
Tests null hypothesis of equal population covariance matrices.
Eigenvalues
Function
Eigenvalue
% of Variance
Cumulative %
Canonical Correlation
1
.643a
100.0
100.0
.626
a. First 1 canonical discriminant functions were used in the analysis.
Wilks’ Lambda
Test of Function(s)
Wilks’ Lambda
Chi-square
df
Sig.
1
.609
22.849
4
.000
Standardized Canonical Discriminant Function Coefficients
Function
1
Number of publications in last 2 years
.567
Grant funding (in $10000) over the last 5 years
-.118
Mean teaching rating for last 3 semesters
.890
Number of committees served on during last 5 years
.280
Structure Matrix
Function
1
Mean teaching rating for last 3 semesters
.778
Number of publications in last 2 years
.436
Number of committees served on during last 5 years
.233
Grant funding (in $10000) over the last 5 years
.040
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function.
Canonical Discriminant Function Coefficients
Function
1
Number of publications in last 2 years
.694
Grant funding (in $10000) over the last 5 years
-.041
Mean teaching rating for last 3 semesters
1.716
Number of committees served on during last 5 years
.122
(Constant)
-7.891
Unstandardized coefficients
Functions at Group Centroids
Type of professor
Function
1
Research Scientist
-.786
Teaching Mogul
.786
Unstandardized canonical discriminant functions evaluated at group means
Classification Processing Summary
Processed
50
Excluded
Missing or out-of-range group codes
0
At least one missing discriminating variable
0
Used in Output
50
Prior Probabilities for Groups
Type of professor
Prior
Cases Used in Analysis
Unweighted
Weighted
Research Scientist
.500
25
25.000
Teaching Mogul
.500
25
25.000
Total
1.000
50
50.000
Classification Function Coefficients
Type of professor
Research Scientist
Teaching Mogul
Number of publications in last 2 years
5.743
6.834
Grant funding (in $10000) over the last 5 years
.570
.505
Mean teaching rating for last 3 semesters
12.417
15.113
Number of committees served on during last 5 years
-.027
.166
(Constant)
-28.157
-40.560
Fisher’s linear discriminant functions
Classification Resultsa,c
Type of professor
Predicted Group Membership
Total
Research Scientist
Teaching Mogul
Original
Count
Research Scientist
20
5
25
Teaching Mogul
3
22
25
%
Research Scientist
80.0
20.0
100.0
Teaching Mogul
12.0
88.0
100.0
Cross-validatedb
Count
Research Scientist
18
7
25
Teaching Mogul
4
21
25
%
Research Scientist
72.0
28.0
100.0
Teaching Mogul
16.0
84.0
100.0
a. 84.0% of original grouped cases correctly classified.
b. Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case.
c. 78.0% of cross-validated grouped cases correctly classified.
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