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Example 1. A researcher randomly assigns 33 subjects to one of three groups. The first group receives technical dietary information interactively from an on-line website. Group 2 receives the same information in from a nurse practitioner, while group 3 receives the information from a video tape made by the same nurse practitioner. The researcher looks at three different ratings of the presentation, difficulty, useful and importance, to determine if there is a difference in the modes of presentation. In particular, the researcher is interested in whether the interactive website is superior because that is the most cost-effective way of delivering the information.
We have a data file, manova.sav, with 33 observations on three response variables. The response variables are ratings of useful, difficulty and importance. Level 1 of the group variable is the treatment group, level 2 is control group 1 and level 3 is control group 2.
Let's look at the data.
get file='d:\data\manova.sav' . descriptives variables=difficulty useful importance /statistics=mean stddev min max . Descriptive Statistics | ------------------ | -- | ------- | ------- | ------- | -------------- | | | N | Minimum | Maximum | Mean | Std. Deviation | | ------------------ | -- | ------- | ------- | ------- | -------------- | | difficulty | 33 | 2.40 | 10.25 | 5.7152 | 2.01760 | | ------------------ | -- | ------- | ------- | ------- | -------------- | | useful | 33 | 11.90 | 24.30 | 16.3303 | 3.29246 | | ------------------ | -- | ------- | ------- | ------- | -------------- | | importance | 33 | .20 | 18.80 | 6.4758 | 3.98513 | | ------------------ | -- | ------- | ------- | ------- | -------------- | | Valid N (listwise) | 33 | | | | | | ------------------ | -- | ------- | ------- | ------- | -------------- | frequencies variables=group . group | ----- | --------------- | --------- | ------- | ------------- | ------------------ | | | | Frequency | Percent | Valid Percent | Cumulative Percent | | ----- | --------------- | --------- | ------- | ------------- | ------------------ | | Valid | 1.00 treatment | 11 | 33.3 | 33.3 | 33.3 | | | --------------- | --------- | ------- | ------------- | ------------------ | | | 2.00 control_1 | 11 | 33.3 | 33.3 | 66.7 | | | --------------- | --------- | ------- | ------------- | ------------------ | | | 3.00 control_2 | 11 | 33.3 | 33.3 | 100.0 | | | --------------- | --------- | ------- | ------------- | ------------------ | | | Total | 33 | 100.0 | 100.0 | | | ----- | --------------- | --------- | ------- | ------------- | ------------------ | means tables=difficulty useful importance by group /cells mean count stddev . Means | --------- | -------------- | ---------- | ------- | ---------- | | group | | difficulty | useful | importance | | --------- | -------------- | ---------- | ------- | ---------- | | 1.00 | Mean | 6.1909 | 18.1182 | 8.6818 | | treatment | -------------- | ---------- | ------- | ---------- | | | N | 11 | 11 | 11 | | | -------------- | ---------- | ------- | ---------- | | | Std. Deviation | 1.89971 | 3.90380 | 4.86309 | | --------- | -------------- | ---------- | ------- | ---------- | | 2.00 | Mean | 5.5818 | 15.5273 | 5.1091 | | control_1 | -------------- | ---------- | ------- | ---------- | | | N | 11 | 11 | 11 | | | -------------- | ---------- | ------- | ---------- | | | Std. Deviation | 2.43426 | 2.07562 | 2.53119 | | --------- | -------------- | ---------- | ------- | ---------- | | 3.00 | Mean | 5.3727 | 15.3455 | 5.6364 | | control_2 | -------------- | ---------- | ------- | ---------- | | | N | 11 | 11 | 11 | | | -------------- | ---------- | ------- | ---------- | | | Std. Deviation | 1.75903 | 3.13827 | 3.54691 | | --------- | -------------- | ---------- | ------- | ---------- | | Total | Mean | 5.7152 | 16.3303 | 6.4758 | | | -------------- | ---------- | ------- | ---------- | | | N | 33 | 33 | 33 | | | -------------- | ---------- | ------- | ---------- | | | Std. Deviation | 2.01760 | 3.29246 | 3.98513 | | --------- | -------------- | ---------- | ------- | ---------- | correlations variables=difficulty useful importance . Correlations | ---------- | ------------------- | ---------- | ------ | ---------- | | | | difficulty | useful | importance | | ---------- | ------------------- | ---------- | ------ | ---------- | | difficulty | Pearson Correlation | 1 | .098 | .198 | | | ------------------- | ---------- | ------ | ---------- | | | Sig. (2-tailed) | | .588 | .270 | | | ------------------- | ---------- | ------ | ---------- | | | N | 33 | 33 | 33 | | ---------- | ------------------- | ---------- | ------ | ---------- | | useful | Pearson Correlation | .098 | 1 | -.341 | | | ------------------- | ---------- | ------ | ---------- | | | Sig. (2-tailed) | .588 | | .052 | | | ------------------- | ---------- | ------ | ---------- | | | N | 33 | 33 | 33 | | ---------- | ------------------- | ---------- | ------ | ---------- | | importance | Pearson Correlation | .198 | -.341 | 1 | | | ------------------- | ---------- | ------ | ---------- | | | Sig. (2-tailed) | .270 | .052 | | | | ------------------- | ---------- | ------ | ---------- | | | N | 33 | 33 | 33 | | ---------- | ------------------- | ---------- | ------ | ---------- |
Although this is a multivariate analysis, we will begin by looking at the separate univariate anovas to get a feel for what is happening with the data. Please note: Not all of the SPSS output will be shown on this page and it will not necessarily be displayed in the order that it appears in the printout.
While none of the three anovas were statistically significant at the alpha = .05 level, in particular, the anova for difficulty was less than 1.
manova difficulty useful importance by group(1,3) . EFFECT .. group Univariate F-tests with (2,30) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F difficul 3.97515 126.28728 1.98758 4.20958 .47216 .628 useful 52.92424 293.96544 26.46212 9.79885 2.70053 .083 importan 81.82969 426.37090 40.91485 14.21236 2.87882 .072
Next, we will look at the manova itself.
EFFECT .. group Multivariate Tests of Significance (S = 2, M = 0, N = 13 ) Test Name Value Approx. F Hypoth. DF Error DF Sig. of F Pillais .47667 3.02483 6.00 58.00 .012 Hotellings .89723 4.03753 6.00 54.00 .002 Wilks .52579 3.53823 6.00 56.00 .005 Roys .47146 Note.. F statistic for WILKS' Lambda is exact.
Now that we have have determined that the overall multivariate test is significant, we will follow up with several post-hoc tests. In particular, we wiil look at group 1 versus the average of groups 2 and 3 and also group2 versus group 3. These comparisons are performed using the contrast statement. The matrix needs as many rows as there are groups. The first row of the matrix needs to be all one's. The design statement is also needed to display the contrasts.
/manova difficulty useful importance by group(1,3)
/contrast(group) = special (1 1 1
2 -1 -1
0 1 -1)
/design = group(1) group(2) .
EFFECT .. GROUP(1)
Multivariate Tests of Significance (S = 1, M = 1/2, N = 13 )
Test Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .47101 8.31034 3.00 28.00 .000
Hotellings .89039 8.31034 3.00 28.00 .000
Wilks .52899 8.31034 3.00 28.00 .000
Roys .47101
Note.. F statistics are exact.
EFFECT .. GROUP(2)
Multivariate Tests of Significance (S = 1, M = 1/2, N = 13 )
Test Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .00679 .06381 3.00 28.00 .979
Hotellings .00684 .06381 3.00 28.00 .979
Wilks .99321 .06381 3.00 28.00 .979
Roys .00679
Note.. F statistics are exact.
We know from the univariate tests above that difficulty by itself was clearly not significant so we will look at a test using the combination of useful and importance. This analysis is performed using the transform statement. Once again the matrix needs three rows, although we are only interested in the second row which will be labed as T2 in the output. The first row of the matrix needs to be all one's.
manova difficulty useful importance by group(1,3)
/transform (useful difficulty importance) = special(1 1 1
1 0 1
1 0 -1)
/print signif(mult) .
EFFECT .. group (Cont.)
Univariate F-tests with (2,30) D. F.
Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F
T2 263.85695 304.78182 131.92848 10.15939 12.98586 .000
T1 329.74240 512.54273 164.87120 17.08476 9.65019 .001
T3 5.65091 1135.89086 2.82546 37.86303 .07462 .928
There is a lot of variation in the write-ups of multivariate analysis of variance. The write-up below is fairly minimal, more detail may be required for most instances.
The multivariate test of differences between groups using the Wilks Lambda criteria was statistically significant (F(6, 56) = 3.54; p=0.0049). Follow-up multivariate comparisons showed that the treatment group was significantly different from the average of control 1 and control 2 (F(3,28) = 8.31; p=0.0004). Further, it was determined that control 1 and control 2 were not significant different (F(3,28) = 0.06; p=0.9785). Each of the F-ratio transformations of the Wilks criteria were exact.
None of the separate univariate anovas were statistically significant. In particular, the univariate test for difficulty has an F less than 1, so the analysis was rerun using the combination of useful and importance, which was statistically significant (F(2,30) = 12.99; p<0.0001).
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