Click here
for a pdf file depicting the 2007 slides for this lecture.In the
document, you will be referred back to the SPSS runs on this html page.
(Added May 21, 2007)
set width = 80 length = none header = off
data list notable free
/ subnr wend_no wday_no wend_cr wday_cr
begin data
1 6 10 4 15
2 3 8 6 17
3 5 11 5 11
4 3 5 1 21
5 4 5 6 20
6 7 12 7 17
7 5 10 8 19
8 5 6 2 20
9 6 7 7 24
10 4 10 6 11
11 5 8 5 15
12 5 8 6 13
end data
manova wend_no wday_no wend_cr wday_cr
/wsfact = crit(2) week(2)
/print = cellinfo(means) transf
/rename = grand main_crit main_week interact
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Cell Means and Standard Deviations
Variable .. WEND_NO
Mean Std. Dev. N
For entire sample 4.833 1.193 12
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Variable .. WDAY_NO
Mean Std. Dev. N
For entire sample 8.333 2.309 12
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Variable .. WEND_CR
Mean Std. Dev. N
For entire sample 5.250 2.050 12
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Variable .. WDAY_CR
Mean Std. Dev. N
For entire sample 16.917 4.078 12
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Orthonormalized Transformation Matrix (Transposed)
GRAND MAIN_CRI MAIN_WEE INTERACT
WEND_NO .500 -.500 -.500 .500
WDAY_NO .500 -.500 .500 -.500
WEND_CR .500 .500 -.500 -.500
WDAY_CR .500 .500 .500 .500
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Tests of Significance for GRAND using UNIQUE sums of squares
Source of Variation SS DF MS F Sig of F
WITHIN CELLS 65.17 11 5.92
CONSTANT 3745.33 1 3745.33 632.20 .000
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Source of Variation SS DF MS F Sig of F
WITHIN CELLS 93.50 11 8.50
CRIT 243.00 1 243.00 28.59 .000
WITHIN CELLS 32.42 11 2.95
WEEK 690.08 1 690.08 234.17 .000
WITHIN CELLS 112.42 11 10.22
CRIT BY WEEK 200.08 1 200.08 19.58 .001
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SPSS tests each effect against its own (slightly different) error
term. Alternatively, we could form a pooled error term and test each
effect against this one. We could do this by hand:
SSpooled = 93.5 + 32.42 + 112.42 = 238.34 dfpooled = 11 + 11 + 11 = 33 Hence, MSpooled = 7.22We could also request the pooled error term from SPSS: ------------------------------------------------------------------------
manova wend_no wday_no wend_cr wday_cr /wsfact = crit(2) week(2) /print = cellinfo(means) transf signif(aver) /rename = grand main_crit main_week interact /wsdesign = crit + week + crit by week [grand mean and multivariate output suppressed] AVERAGED Tests of Significance for MEAS.1 using UNIQUE sums of squares Source of Variation SS DF MS F Sig of F WITHIN+RESIDUAL 238.33 33 7.22 CRIT + WEEK + CRIT 1133.17 3 377.72 52.30 .000 BY WEEK - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
What if we stopped right here and said we are not interested in multivariate within-subject designs? Could we analyze our data in a univariate way? Indeed, this is called the "averaged F" approach (and you have probably used it already). Basically, you run your two (or more) univariate within-subject F tests as an average omnibus test and/or interpret the Fs separately as single-df contrasts. But this approach ignores covariances in your data (specifically, among your within-subject variables) and it is not one I highly recommend. Nonetheless, in some circumstances that's all you want, and if certain assumptions are met, it is also statistically permissible.
These considerations lead to the following decision tree (which also includes a between-subject factor to make the scheme applicable to a mixed multifactorial design):
In the output below, you can see the two tests (first Box's M test,
later Mauchly's sphericity test). They do not indicate violation, so
either solution is acceptable, and in fact all solutions arrive at
approximately equal p-values.
Multivariate and univariate ANOVA run
MANOVA TRIAL1 TO TRIAL3 BY GROUP (1,2)
/WSFACT = TRIALS(3)
/CONTR (TRIALs) = poly
/RENAME = const lin quad
/PRINT = TRANSFORM homog (bart boxm) signif (hf gg hypoth eigen) error(sscp)
/DISCRIM = stan corr
Summaries of TRIAL1
By levels of GROUP
Variable Value Label Mean Std Dev Cases
For Entire Population 24.7500 5.8493 8
GROUP 1.00 25.0000 5.7735 4
GROUP 2.00 24.5000 6.8069 4
Summaries of TRIAL2
By levels of GROUP
Variable Value Label Mean Std Dev Cases
For Entire Population 20.2500 4.9497 8
GROUP 1.00 19.5000 6.6081 4
GROUP 2.00 21.0000 3.4641 4
Summaries of TRIAL3
By levels of GROUP
Variable Value Label Mean Std Dev Cases
For Entire Population 15.5000 6.2106 8
GROUP 1.00 16.0000 8.4853 4
GROUP 2.00 15.0000 4.1633 4
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The above tests are not very useful, but SPSS provides them
whenever you ask for Box's M test.
CELL NUMBER
1 2
Variable
GROUP 1 2
Univariate Homogeneity of Variance Tests
Variable .. TRIAL1
Bartlett-Box F(1,108) = .06902, P = .793
Variable .. TRIAL2
Bartlett-Box F(1,108) = 1.00849, P = .318
Variable .. TRIAL3
Bartlett-Box F(1,108) = 1.21242, P = .273
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Cell Number .. 1
Determinant of Covariance matrix of dependent variables = 533.33333
LOG(Determinant) = 6.27915
- - - - - - - - - -
Cell Number .. 2
Determinant of Covariance matrix of dependent variables = 6165.33333
LOG(Determinant) = 8.72670
- - - - - - - - - -
Determinant of pooled Covariance matrix of dependent vars. = 30142.88889
LOG(Determinant) = 10.31370
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As you can see, the test calculated the determinant of each
subgroup's var-cov matrix and compares them. This can be understood
as comparing the amount of information in each submatrix.
Multivariate test for Homogeneity of Dispersion matrices
Boxs M = 16.86469
F WITH (6,260) DF = 1.22362, P = .294 (Approx.)
Chi-Square with 6 DF = 7.72965, P = .259 (Approx.)
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Orthonormalized Transformation Matrix (Transposed)
CONST LIN QUAD
TRIAL1 .577 -.707 .408
TRIAL2 .577 .000 -.816
TRIAL3 .577 .707 .408
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The transformation matrix is a crucial piece of information
whenever your within-subject factor has more than two levels. These
contrasts (LIN, QUAD) are the two dependent variables that SPSS
will analyze for the "TRIALS" effects. But first it analyzes the
simple between-subject main effect for GROUPS using the CONST as the
dependent variables, which is the measure that collapses across levels
of the within-subject factor.
Order of Variables for Analysis
Variates Covariates
CONST
1 Dependent Variable
0 Covariates
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Note.. TRANSFORMED variables are in the variates column.
These TRANSFORMED variables correspond to the
Between-subject effects.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Tests of Between-Subjects Effects.
Tests of Significance for CONST using UNIQUE sums of squares
Source of Variation SS DF MS F Sig of F
WITHIN CELLS 263.33 6 43.89
GROUP .00 1 .00 .00 1.000
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Now we are getting ready for the within-subject part:
Order of Variables for Analysis
Variates Covariates
LIN
QUAD
2 Dependent Variables
0 Covariates
Note: The three levels of TRIALS are transformed into two
contrasts, and these are the new variables analyzed---because they
are two, a multivariate analysis is indicated.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Tests involving 'TRIALS' Within-Subject Effect.
Mauchly sphericity test, W = .79852
Chi-square approx. = 1.12498 with 2 D. F.
Significance = .570
Greenhouse-Geisser Epsilon = .83231
Huynh-Feldt Epsilon = 1.00000
Lower-bound Epsilon = .50000
AVERAGED Tests of Significance that follow multivariate tests are equivalent to
univariate or split-plot or mixed-model approach to repeated measures.
Epsilons may be used to adjust d.f. for the AVERAGED results.
The total var-cov matrix, whose sphericity was just tested, is now
being decomposed into the error matrix and the hypothesis matrix. (The
first hypothesis matrix is the TRIALS x GROUP interaction.) As usual, the
ratio between the two, HE-1, is subjected to spectral
decomposition, and the resulting solution is the eigenvalue and
eigenvector of the "effect," tested for multivariate significance.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
WITHIN CELLS Sum-of-Squares and Cross-Products
LIN QUAD
LIN 159.500
QUAD 79.963 251.167
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
EFFECT .. GROUP BY TRIALS
Adjusted Hypothesis Sum-of-Squares and Cross-Products
LIN QUAD
LIN .250
QUAD 1.299 6.750
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Multivariate Tests of Significance (S = 1, M = 0, N = 1 1/2)
Test Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .02693 .06918 2.00 5.00 .934
Hotellings .02767 .06918 2.00 5.00 .934
Wilks .97307 .06918 2.00 5.00 .934
Roys .02693
Note.. F statistics are exact.
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Eigenvalues and Canonical Correlations
Root No. Eigenvalue Pct. Cum. Pct. Canon Cor.
1 .028 100.000 100.000 .164
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
EFFECT .. GROUP BY TRIALS (Cont.)
>Note # 12188
>Because there are no functions significant at level alpha, MANOVA will not
>report any canonical discriminant or correlation analysis for this effect.
EFFECT .. GROUP BY TRIALS (Cont.)
Univariate F-tests with (1,6) D. F.
Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F
LIN .25000 159.50000 .25000 26.58333 .00940 .926
QUAD 6.75000 251.16667 6.75000 41.86111 .16125 .702
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Now the main effect for TRIALS forms another H matrix, and a new
PCA is run on the corresponding HE-1 matrix.
EFFECT .. TRIALS
Adjusted Hypothesis Sum-of-Squares and Cross-Products
LIN QUAD
LIN 342.250
QUAD 5.340 .083
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Multivariate Tests of Significance (S = 1, M = 0, N = 1 1/2)
Test Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .71658 6.32080 2.00 5.00 .043
Hotellings 2.52832 6.32080 2.00 5.00 .043
Wilks .28342 6.32080 2.00 5.00 .043
Roys .71658
Note.. F statistics are exact.
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Eigenvalues and Canonical Correlations
Root No. Eigenvalue Pct. Cum. Pct. Canon Cor.
1 2.528 100.000 100.000 .847
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EFFECT .. TRIALS (Cont.)
Standardized discriminant function coefficients
Function No.
Variable 1
LIN 1.091
QUAD -.424
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Correlations between DEPENDENT and canonical variables
Canonical Variable
Variable 1
LIN .921
QUAD .011
As usual, the multivariate function has to be interpreted. The
function consists almost exclusively of linear contributions (which
you can verify by plotting the means by hand).
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EFFECT .. TRIALS (Cont.)
Univariate F-tests with (1,6) D. F.
Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig
LIN 342.25000 159.50000 342.25000 26.58333 12.87461 .012
QUAD .08333 251.16667 .08333 41.86111 .00199 .966
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Tests involving 'TRIALS' Within-Subject Effect.
AVERAGED Tests of Significance for TRIAL using UNIQUE sums of squares
Source of Variation SS DF MS F Sig of F
WITHIN CELLS 410.67 12 34.22
(Greenhouse-Geisser) 9.99
(Huynh-Feldt) 12.00
(Lower bound) 6.00
TRIALS 342.33 2 171.17 5.00 .026
(Greenhouse-Geisser) 1.66 5.00 .036
(Huynh-Feldt) 2.00 5.00 .026
(Lower bound) 1.00 5.00 .067
GROUP BY TRIALS 7.00 2 3.50 .10 .904
(Greenhouse-Geisser) 1.66 .10 .871
(Huynh-Feldt) 2.00 .10 .904
(Lower bound) 1.00 .10 .760
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Note how little the corrections change the results (because Mauchly's test
showed no violation). Also note how similar the p-values are to the
corresponding multivariate tests.
Another way of interpreting either the multivariate or averaged-F
results is shown below: by looking at its constituents---the
transformed variables LIN and QUAD---one at a time ("trend analysis"
in T&F's terms). Because the contrasts we formed are orthogonal
(i.e., LIN and QUAD are uncorrelated), you can safely interpret their
effect size. But the p-values you see should be treated with caution
since they are not corrected for multiple testing. They should be
used as interpretation aids, the same way you should use the
discriminant functions.
Estimates for LIN
--- Individual univariate .9500 confidence intervals
TRIALS
Parameter Coeff. Std. Err. t-Value Sig. t Lower -95% CL- Upper
1 -6.5407377 1.82289 -3.58812 .01153 -11.00118 -2.08029
GROUP BY TRIALS
Parameter Coeff. Std. Err. t-Value Sig. t Lower -95% CL- Upper
2 .176776695 1.82289 .09698 .92590 -4.28367 4.63722
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Estimates for QUAD
--- Individual univariate .9500 confidence intervals
TRIALS
Parameter Coeff. Std. Err. t-Value Sig. t Lower -95% CL- Upper
1 -.10206207 2.28750 -.04462 .96586 -5.69936 5.49524
GROUP BY TRIALS
Parameter Coeff. Std. Err. t-Value Sig. t Lower -95% CL- Upper
2 .918558654 2.28750 .40156 .70192 -4.67874 6.51586
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