Pre-Post MANOVA

We have pre-intervention and post-intervention data for four variables:

  • AUDIT – a screening test for problem drinking (of ethanol)
  • BMI – body mass index
  • SUD – a measure of subjective distress
  • Weight – body weight

We wish to analyze all four of these variables simultaneously. First I conduct a Doubly Multivariate Repeated Measures ANOVA using SPSS. Here is the code:

manovaAUDIT_BaselineAUDIT_PostBMI_BaselineBMI_PostSUD_BaselineSUD_PostWeight_BaselineWeight_Post

/ wsfactors = time(2) /

measure = AUDIT BMI SUD Weight /

print=transform signif(univhypoth) error(sscp) / design .

Here are selected parts of the output:

EFFECT .. TIME

Multivariate Tests of Significance (S = 1, M = 1 , N = 6 )

Test Name Value Exact F Hypoth. DF Error DF Sig. of F

Pillais .88970 28.23142 4.00 14.00 .000

Hotellings 8.06612 28.23142 4.00 14.00 .000

Wilks .11030 28.23142 4.00 14.00 .000

Roys .88970 [This value is incorrect, it should be the same as Hotelling’s trace]

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The multivariate tests here show that there is a significant change from pre to post on a weighted linear combination of the four outcome variables.

EFFECT .. TIME (Cont.)

Univariate F-tests with (1,17) D. F.

Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F

T2 9.00000 17.00000 9.00000 1.00000 9.00000 .008

T4 .34517 .47444 .34517 .02791 12.36802 .003

T6 96.69444 20.80556 96.69444 1.22386 79.00801 .000

T8 15.08028 22.32472 15.08028 1.31322 11.48345 .003

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Since the multivariate test is significant, I am comfortable conducting univariate analyses, one on each outcome variable. The results here are significant for each outcome variable. These F tests are absolutely equivalent to correlated samples t tests.

Paired Samples Statistics
Mean / N / Std. Deviation / Std. Error Mean
Pair 1 / AUDIT_Baseline / 3.6111 / 18 / 2.30444 / .54316
AUDIT_Post / 2.6111 / 18 / 1.33456 / .31456
Pair 2 / BMI_Baseline / 33.6492 / 18 / 5.56643 / 1.31202
BMI_Post / 33.4533 / 18 / 5.58116 / 1.31549
Pair 3 / SUD_Baseline / 6.3333 / 18 / 1.45521 / .34300
SUD_Post / 3.0556 / 18 / .87260 / .20567
Pair 4 / Weight_Baseline / 207.19 / 18 / 42.866 / 10.104
Weight_Post / 205.9000 / 18 / 42.21192 / 9.94944
Comparison
Mean / Std. Deviation / t / df / Sig. (2-tailed)
Pair 1 / AUDIT_Baseline - AUDIT_Post / 1.00000 / 1.41421 / 3.000 / 17 / .008
Pair 2 / BMI_Baseline - BMI_Post / .19584 / .23626 / 3.517 / 17 / .003
Pair 3 / SUD_Baseline - SUD_Post / 3.27778 / 1.56452 / 8.889 / 17 / .000
Pair 4 / Weight_Baseline - Weight_Post / 1.29444 / 1.62063 / 3.389 / 17 / .003

Note that the values of t here are, when squared, exactly equal to the values of F above, and the values of p also identical to those from the F tests above.

I computed, for each outcome variable, a difference score, Pre minus Post. One-sample t tests (null is  = 0) are identical to the correlated t tests.

One-Sample Test
Test Value = 0
t / df / Sig. (2-tailed) / Mean Difference / 95% Confidence Interval of the Difference
Lower / Upper
AUDIT_Diff / -3.000 / 17 / .008 / -1.00000 / -1.7033 / -.2967
BMI_Diff / -3.517 / 17 / .003 / -.19584 / -.3133 / -.0784
SUD_Diff / -8.889 / 17 / .000 / -3.27778 / -4.0558 / -2.4998
Weight_Diff / -3.389 / 17 / .003 / -1.29444 / -2.1004 / -.4885

Technically, the analysis done first here is not a doubly multivariate analysis. There are only two levels for the repeated measures dimension, and thus only one contrast to define it. We could have simply done a standard MANOVA on the difference scores:

GLM AUDIT_DiffBMI_DiffSUD_DiffWeight_Diff

/METHOD=SSTYPE(3)

/INTERCEPT=INCLUDE

/CRITERIA=ALPHA(.05).

Multivariate Testsa
Effect / Value / F / Hypothesis df / Error df / Sig.
Intercept / Pillai's Trace / .890 / 28.231b / 4.000 / 14.000 / .000
Wilks' Lambda / .110 / 28.231b / 4.000 / 14.000 / .000
Hotelling's Trace / 8.066 / 28.231b / 4.000 / 14.000 / .000
Roy's Largest Root / 8.066 / 28.231b / 4.000 / 14.000 / .000

Notice that the multivariate tests here are identical to those produced in the first analysis.

Tests of Between-Subjects Effects
Source / Dependent Variable / Type III Sum of Squares / df / Mean Square / F / Sig.
Intercept / AUDIT_Diff / 18.000 / 1 / 18.000 / 9.000 / .008
BMI_Diff / .690 / 1 / .690 / 12.368 / .003
SUD_Diff / 193.389 / 1 / 193.389 / 79.008 / .000
Weight_Diff / 30.161 / 1 / 30.161 / 11.483 / .003
Error / AUDIT_Diff / 34.000 / 17 / 2.000
BMI_Diff / .949 / 17 / .056
SUD_Diff / 41.611 / 17 / 2.448
Weight_Diff / 44.649 / 17 / 2.626

SAS. To do this analysis with SAS, you need to create a difference score for each outcome variable and then test an intercept-only model. Here is the code to test the intercept-only model:

ProcGLM; ModelAUDIT_DiffBMI_DiffSUD_DiffWeight_Diff = / SS1;

manova h=intercept; run; quit;

The output is the same as with SPSS, just formatted differently. First come the univariate tests, which are identical to correlated t-tests.

Dependent Variable: AUDIT_Diff

Source / DF / Type I SS / Mean Square / F Value / PrF
Intercept / 1 / 18.00000000 / 18.00000000 / 9.00 / 0.0081
Parameter / Estimate / Standard Error / tValue / Pr|t|
Intercept / -1.000000000 / 0.33333333 / -3.00 / 0.0081

Dependent Variable: BMI_Diff

Source / DF / Type I SS / Mean Square / F Value / PrF
Intercept / 1 / 0.69034508 / 0.69034508 / 12.37 / 0.0026
Parameter / Estimate / Standard Error / tValue / Pr|t|
Intercept / -.1958379543 / 0.05568610 / -3.52 / 0.0026

Dependent Variable: SUD_Diff

Source / DF / Type I SS / Mean Square / F Value / PrF
Intercept / 1 / 193.3888889 / 193.3888889 / 79.01 / <.0001
Parameter / Estimate / Standard Error / tValue / Pr|t|
Intercept / -3.277777778 / 0.36876012 / -8.89 / <.0001

Dependent Variable: Weight_Diff

Source / DF / Type I SS / Mean Square / F Value / PrF
Intercept / 1 / 30.16055556 / 30.16055556 / 11.48 / 0.0035
Parameter / Estimate / Standard Error / tValue / Pr|t|
Intercept / -1.294444444 / 0.38198589 / -3.39 / 0.0035

Then comes the multivariate test:

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect
H = Type I SSCP Matrix for Intercept
E = Error SSCP Matrix
S=1 M=1 N=6
Statistic / Value / F Value / NumDF / DenDF / PrF
Wilks' Lambda / 0.11030076 / 28.23 / 4 / 14 / <.0001
Pillai's Trace / 0.88969924 / 28.23 / 4 / 14 / <.0001
Hotelling-Lawley Trace / 8.06612028 / 28.23 / 4 / 14 / <.0001
Roy's Greatest Root / 8.06612028 / 28.23 / 4 / 14 / <.0001

Doubly Multivariate Analysis of Repeated Measures Designs

The Data

Karl L. Wuensch, June, 2018