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Example 1. A researcher has collected data on three psychological variables, four academic variables (standardized test scores) and gender for 600 college freshman. She is interested in how the set of psychological variables relates to the academic variables and gender. In particular, the researcher is interested in how many dimensions are necessary to understand the association between the two sets of variables.
We have included the data file, which can be obtained by clicking on mmreg.sav. The dataset has 600 observations on eight variables. The psychological variables are locus of control, self-concept and motivation. The academic variables are standardized tests in reading, writing, math and science. Additionally, the variable female is a zero-one indicator variable with the one indicating a female student.
Let's look at the data.
get file='d:\data\mmreg.sav' .
descriptives
variables=locus_of_control self_concept motivation
read write math science female
/statistics=mean stddev min max .
Descriptive Statistics
N Minimum Maximum Mean Std. Deviation
locus_of_control 600 -2.23 1.36 .0965 .67028
self_concept 600 -2.62 1.19 .0049 .70551
motivation 600 .00 1.00 .6608 .34273
read 600 28.30 76.00 51.9018 10.10298
write 600 25.50 67.10 52.3848 9.72645
math 600 31.80 75.50 51.8490 9.41474
science 600 26.00 74.20 51.7633 9.70618
female 600 .00 1.00 .5450 .49839
Valid N (listwise) 600
frequencies
variables=female .
female
Frequency Percent Valid Percent Cumulative Percent
Valid .00 273 45.5 45.5 45.5
1.00 327 54.5 54.5 100.0
Total 600 100.0 100.0
Here are the correlations among the variables in the analysis with the p-values and sample size edited out for better readability.
correlations /variables=locus_of_control self_concept motivation read write math science female . Correlations locus... self... moti... read write math science female locus_of_control Pearson Correlation 1 .171 .245 .374 .359 .337 .325 .113 self_concept Pearson Correlation .171 1 .289 .061 .019 .054 .070 -.126 motivation Pearson Correlation .245 .289 1 .211 .254 .195 .116 .098 read Pearson Correlation .374 .061 .211 1 .629 .679 .691 -.042 write Pearson Correlation .359 .019 .254 .629 1 .633 .569 .244 math Pearson Correlation .337 .054 .195 .679 .633 1 .650 -.048 science Pearson Correlation .325 .070 .116 .691 .569 .650 1 -.138 female Pearson Correlation .113 -.126 .098 -.042 .244 -.048 -.138 1
SPSS performs canonical correlation using the manova command. Don't look for manova in the point-and-click analysis menu, its not there. The manova command is one of SPSS's hidden gems that is often overlooked. Used with the discrim option, manova will compute the canonical correlation analysis.
Due to the length of the output, we will be making comments in several places along the way.
manova locus_of_control self_concept motivation with read write math science female / discrim all alpha(1) / print=sig(eigen dim) .
The output below begins with an overall multivariate test of the entire model using four different multivariate criteria. This is followed by the three canonical correlations and the multivariate tests of each of the dimensions. These results show that the first two of the three canonical correlations are statistically significant at the .05 level.
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The default error term in MANOVA has been changed from WITHIN CELLS to
WITHIN+RESIDUAL. Note that these are the same for all full factorial
designs.
* * * * * * A n a l y s i s o f V a r i a n c e * * * * * *
600 cases accepted.
0 cases rejected because of out-of-range factor values.
0 cases rejected because of missing data.
1 non-empty cell.
1 design will be processed.
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* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
EFFECT .. WITHIN CELLS Regression
Multivariate Tests of Significance (S = 3, M = 1/2, N = 295 )
Test Name Value Approx. F Hypoth. DF Error DF Sig. of F
Pillais .25425 11.00057 15.00 1782.00 .000
Hotellings .31430 12.37633 15.00 1772.00 .000
Wilks .75436 11.71573 15.00 1634.65 .000
Roys .21538
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Eigenvalues and Canonical Correlations
Root No. Eigenvalue Pct. Cum. Pct. Canon Cor. Sq. Cor
1 .274 87.336 87.336 .464 .215
2 .029 9.185 96.522 .168 .028
3 .011 3.478 100.000 .104 .011
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Dimension Reduction Analysis
Roots Wilks L. F Hypoth. DF Error DF Sig. of F
1 TO 3 .75436 11.71573 15.00 1634.65 .000
2 TO 3 .96143 2.94446 8.00 1186.00 .003
3 TO 3 .98919 2.16461 3.00 594.00 .091
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EFFECT .. WITHIN CELLS Regression (Cont.)
Univariate F-tests with (5,594) D. F.
Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F
locus_of .18062 .17372 9.72160 .37123 26.18789
self_con .01957 .01131 1.16669 .49212 2.37076
motivati .07874 .07098 1.10799 .10913 10.15338
Variable Sig. of F
locus_of .000
self_con .038
motivati .000
In general, the number of canonical dimensions is equal to the number of variables in the smaller set; however, the number of significant dimensions may be even smaller. Canonical dimensions, also known as canonical variates, are latent variables that are analogous to factors obtained in factor analysis. For this particular model there are three canonical dimensions of which only the first two are statistically significant. The first test of dimensions tests whether all three dimensions are significant (F = 11.72), the next test tests whether dimensions 2 and 3 combined are significant (F = 2.94). Finally, the last test tests whether dimension 3, by itself, is significant (F = 2.16). Therefore dimensions 1 and 2 are each significant while the third dimension is not.
SPSS presents the output separately for each of the two sets of variables. Within each set SPSS gives the raw canonical coefficients, standardized coefficients, correlations between observed variables and the canonical functions, and the percent of variance accounted for by canonical function. Below are the results for the three psychological variables.
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
Raw canonical coefficients for DEPENDENT variables
Function No.
Variable 1 2 3
locus_of 1.254 -.621 .662
self_con -.351 -1.188 -.827
motivati 1.262 2.027 -2.000
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Standardized canonical coefficients for DEPENDENT variables
Function No.
Variable 1 2 3
locus_of .840 -.417 .444
self_con -.248 -.838 -.583
motivati .433 .695 -.686
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Correlations between DEPENDENT and canonical variables
Function No.
Variable 1 2 3
locus_of .904 -.390 .176
self_con .021 -.709 -.705
motivati .567 .351 -.745
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Variance in dependent variables explained by canonical variables
CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO
1 37.980 37.980 8.180 8.180
2 25.910 63.889 .727 8.907
3 36.111 100.000 .391 9.297
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The next section has the results for the four academic variables along with the indicator for gender.
Raw canonical coefficients for COVARIATES
Function No.
COVARIATE 1 2 3
read .045 -.005 -.021
write .036 .042 -.091
math .023 .004 -.009
science .005 -.085 .110
female .632 1.085 1.795
* * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * *
Standardized canonical coefficients for COVARIATES
CAN. VAR.
COVARIATE 1 2 3
read .451 -.050 -.216
write .349 .409 -.888
math .220 .040 -.088
science .049 -.827 1.066
female .315 .541 .894
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Correlations between COVARIATES and canonical variables
CAN. VAR.
Covariate 1 2 3
read .840 -.359 -.135
write .877 .065 -.255
math .764 -.298 -.148
science .658 -.677 .230
female .364 .755 .543
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Variance in covariates explained by canonical variables
CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO
1 11.305 11.305 52.488 52.488
2 .701 12.006 24.994 77.482
3 .098 12.104 9.066 86.548
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The raw canonical coefficients are interpreted in a manner analogous to interpreting regression coefficients i.e., for the variable read, a one unit increase in reading leads to a .0446 increase in the first canonical variate of set 2 when all of the other variables are held constant. Here is another example: being female leads to a .6321 increase in the dimension 1 for set 2 with the other predictors held constant. When the variables in the model have very different standard deviations, the standardized coefficients allow for easier comparisons among the variables.
The raw canonical coefficients are followed by the standardized canonical coefficients shown below. The standardized canonical coefficients are interpreted in a manner analogous to interpreting standardized regression coefficients. For example, consider the variable read, a one standard deviation increase in reading leads to a 0.45 standard deviation increase in the score on the first canonical variate for set 2 when the other variables in the model are held constant.
Below the standardized canonical coefficients are correlations between observed variables and canonical functions which are known as the canonical loadings. The canonical functions may also be called canonical variates and are a type of latent variable.
Table 1: Tests of Canonical Dimensions
Canonical Mult.
Dimension Corr. F df1 df2 p
1 0.46 11.72 15 1634.7 0.000
2 0.17 2.94 8 1186 0.003
3 0.10 2.16 3 594 0.091
Table 2: Standardized Canonical Coefficients
Dimension
1 2
Psychological Variables
locus of control 0.84 -0.42
self-concept -0.25 -0.84
motivation 0.43 0.69
Academic Variables plus Gender
reading 0.45 -0.05
writing 0.35 0.41
math 0.22 0.04
science 0.05 -0.83
gender (female=1) 0.32 0.54
Tests of dimensionality for the canonical correlation analysis, as shown in Table 1, indicate that two of the three canonical dimensions are statistically significant at the .05 level. Dimension 1 had a canonical correlation of 0.46 between the sets of variables, while for dimension 2 the canonical correlation was much lower at 0.17.
Table 2 presents the standardized canonical coefficients for the first two dimensions across both sets of variables. For the psychological variables, the first canonical dimension is most strongly influenced by locus of control (.84) and for the second dimension self-concept (-.84) and motivation (.69). For the academic variables plus gender, the first dimension was comprised of reading (.45), writing (.35) and gender (.32). For the second dimension writing (.41), science (-.83) and gender (.54) were the dominating variables.UCLA Researchers are invited to our Statistical Consulting Services
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