We consider another example of MANOVA and DISCRIM but this time on three populations. In this example, we go a bit beyond the basics of these procedures and feature a variety of output provided by SPSS, including a variety of coefficients generated by the discriminant functions. Consider again a version of the training data featured earlier, but this time having a grouping variable with three categories (1 = no training, 2 = some training, and 3 = extensive training):
Hypothetical Data on Quantitative and Verbal Ability as a Function of Training (1 = No training, 2 = Some training, 3 = Extensive training)
Subject Quantitative Verbal Training
1 2 3 4 5 6 7 8 9
5 2 6 9 8 7 9 10 10
2 1 3 7 9 8 8 10 9
1 1 1 2 2 2 3 3 3
To get the Box Test, select Box’s M in the Discriminant Analysis: Statistics window:
Test Results Box’s M
F Approx.
df1 df2 Sig.
Tests null hypothesis of equal population covariance matrices.
48.547 1.141 40 46378.676 .250
We would like to first run the MANOVA on the following function statement:
Quantitative Verbal as a function of Training Entered into SPSS, we have:
Multivariate Testsa
Hypothesis df Error df Sig. Partial Eta Squared .000
.000 .000 .000
.986 .986 .986 .986 .537 .763 .879 .935 .042
.004 .001 .000 Effect
Intercept
T
a. Design: Intercept + T b. Exact statistic
c. The statistic is an upper bound on F that yields a lower bound on the significance level.
Pillai’s Trace
Value F
.986 .014 70.218 70.218
1.074 3.477
2.000 5.000
5.000 5.000 5.000 12.000 10.000 8.000 6.000 2.000 2.000 2.000
2.000 4.000 4.000 4.000 8.055b
14.513 43.055c .056
175.545b 175.545b 175.545b 175.545b
14.513 14.352 Wilks’ Lambda
Hotelling’s Trace Roy’s Largest Root Pillai’s Trace Wilks’ Lambda Hotelling’s Trace Roy’s Largest Root
All multivariate significance tests suggest we reject the multivariate null hypothesis (p < 0.05). We can get the eigenvalues for our MANOVA using the following syntax:
Root No. Eigenvalue PCt. Cum. Pct. Canon Cor.
Eigenvalues and Canonical Correlations
1 2
14.35158 .16124
98.88896 1.11104
98.88896 100.00000
.96688 .37263
The total sum of the eigenvalues is 14.35158 + 0.16124 = 14.51282. The first discriminant function is quite important, since 14.35158/14.51282 = 0.989. The second discriminant function is quite a bit less important, since 0.16124/14.51282 = 0.01. When we square the canonical correlation of 0.96688 for the first function, we get 0.935, meaning that approximately 93% of the variance is accounted for by this first function. When we square the canonical correlation of 0.37263, we get 0.139, meaning that approximately 14% of the variance is accounted for by this second discriminant function. Recall that we could have also gotten these squared canonical correlations by 14.35158/(1 + 14.35158) = 0.935 and 0.16124/(1 + 0.16124) = 0.139.
We now obtain the corresponding discriminant analysis on these data and match up the eigenvalues with those of MANOVA, as well as obtain more informative output – ANALYZE → CLASSIFY → DISCRIMINANT – and then make the following selections:
We can see on the left that the eigenvalues and canoni- cal correlations for each discriminant function match those obtained via MANOVA in SPSS. We also see that Wilks’ Lambda for the first through the second discrimi- nant function is statistically significant (p = 0.003). The second discriminant function is not statistically signifi- cant (p = 0.365).
SPSS also provides us with the unstandardized discrimi- nant function coefficients (left), along with the constant for computing discriminant scores. To the right are the stand- ardized function coefficients ( usually recommended for interpreting the relative “importance” of the variables mak- ing up the function.)
We interpret these coefficients in a bit more detail:
1) Canonical Discriminant Function Coefficients – these are analogous to raw partial regression weights in regression. The constant value of −6.422 is the intercept for computing discrimi- nant scores. For function 1, the computation is Y = −6.422 + 0.030 (Q) + 0.979(V). For function 2, the computation is Y = −2.360 + 0.8 32(Q) − 0.590(V). SPSS prints the standardized coefficients automatically (discussed below), but you have to request the unstandardized ones (in the Statistics window, select Unstandardized under Function Coefficients).
2) Standardized Canonical Discriminant Function Coefficients – these are analogous to standardized Beta weights in multiple regression. They can be used as a measure of importance or rele- vance of each variable in the discriminant function. We can see that for function 1, “V” is a heavy contributor.
3) Structure Matrix – these are bivariate correlations between the variables with the given discriminant function. Rencher (1998)
guards against relying on these too heavily, as they represent the univariate contribution rather than the multivariate. Interpreting standardized coefficients is often preferable, though looking at both kinds of coefficients can be informative on “triangulating” on the nature of the extracted dimensions.
We can see then that across the board of coefficients, it looks like “V” is most relevant in function 1, while Q is most relevant in function 2. Incidentally, we are not showing Box’s M test for these data since we have demonstrated the test before. Try it yourself and you’ll find it is not statistically significant (p = 0.532), which means we have no reason to doubt the assumption of equality of covariance matrices.
Summary of Canonical Discriminant Functions Eigenvalues
Wilks’ Lambda Function
Test of Function(s) Wilks’
Lambda Chi-square df Sig.
a. First 2 canonical discriminant functions were used in the analysis.
Eigenvalue % of Variance Cumulative % Canonical Correlation 1
2
98.9 1.1
98.9 100.0
.967 .373
.003 .365 4 1 15.844
.822 .056 .861 1 through 2
2
14.352a .161a
Canonical Discriminant Function Coefficients
Function 1
Unstandardized coefficients Q
V (Constant)
.030 .979 –6.422
.832 –.590 –2.360
2 Structure Matrix
V Q
.999*
.516 –.036 .857*
1 2
Function Standardized Canonical
Discriminant Function Coefficients
Q V
.041 .979
1.143 –.590
1 2
Function
Two discriminant functions were extracted, the first boasting a large measure of association (squared canonical correlation of 0.935), which was found to be statistically sig- nificant (Wilks’ Lambda = 0.056, p = 0.003). Canonical discrimi- nant function coefficients and their standardized counterparts both suggested that verbal was more relevant to function 1 and quantitative was more relevant to the second function. Structure coefficients likewise assigned a similar pattern of importance.
Discriminant scores were obtained and plotted, revealing that function 1 provided good discrimination between groups 1 vs. 2 and 3, while the second function provided minimal discriminatory power.
Since we requested SPSS to save discriminant scores, we show the 9 on each discriminant function:
How was each column computed? They were computed using the unstandardized coefficients. Let us compute a few of the scores for the first function and the second function (note: in what follows, we put the coefficient after the score, whereas we previously put the coefficient first – it does not matter which way you do it since either way, we are still weighting each variable appropriately):
Function case discriminant score1, 1 6 422. Q 0 030. V 0 979. 6 422 5 0 030 2 0 979 6 422 0 15 1 958 4 314
. . .
. . .
.
Function case discriminant score1, 2 6 422. Q 0 030. V 0 979. 6 422 2 0 030 1 0 979 6 422 0 06 0 979 5 383
. . .
. . .
.
Function case discriminant score2, 1 2 360. Q 0 832. V 0 590. 2 360 5 0 832 2 0 590 2 360 4 16 1 18
0 617
. . .
. . .
.
Function case discriminant score2, 2 2 360. Q 0 832. V 0 590. 2 360 2 0 832 1 0 590 2 360 1 664 0 590 1 287
. . .
. . .
.
We can see that our computations match up to those generated by SPSS for the first two cases on each function.
SPSS also provides us with the functions at group cen- troids (means):
We match up the above group centroids with the numbers in the plot:
Function 1:
● Mean of discriminant scores for T = 1 is equal to −4.334. We can confirm this by verifying with the discriminant scores we saved. Recall that those three values for T = 1 were − 4.31397, −5.38294,
−3.30467, for a mean of −4.33386, which matches that produced above by SPSS.
● Mean of discriminant scores for T = 2 is equal to 1.652. We can again confirm this by verifying with the discriminant scores we saved. Recall that those values for T = 2 were 0.70270, 2.63180, 1.62250, for a mean of 1.65233, which again matches that produced by SPSS.
● Mean of discriminant scores for T = 3 is equal to 2.682. This agrees with (1.68217 + 3.67094 + 2.69147) /3 = 2.6815.
Function 2:
● [(0.61733 + (−1.28702) + 0.85864)]/3 = 0.063.
● [(0.99239 + (−1.01952) + (−1.26084))]/3 = −0.429.
● [(0.40219 + 0.05331 + 0.64351)]/3 = 0.366.
Functions at Group Centroids
T 1.00 2.00 3.00
Unstandardized canonical discriminant functions evaluated at group means
–4.334 1.652 2.682
.063 –.429 .366
1 2
Function
–5.0
–5.0 –2.5 0.0 2.5 5.0
12 3Group Centroid
3 1 2
T
–2.5 0.0
Function 2
Function 1 Canonical Discriminant Functions
2.5 5.0
To appreciate what these are, consider the plot generated by SPSS (left).
We can get even more specific about the actual values in the plot by requesting SPSS to label each point (double‐click on the plot points to reveal the labels – right‐click, then scroll down to Show Data Labels):