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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.dta, 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.
use http://www.ats.ucla.edu/stat/stata/dae/manova, clear
summarize difficulty useful importance
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
useful | 33 16.3303 3.292461 11.9 24.3
difficulty | 33 5.715152 2.017598 2.4 10.25
importance | 33 6.475758 3.985131 .2 18.8
tabulate group
group | Freq. Percent Cum.
------------+-----------------------------------
treatment | 11 33.33 33.33
control_1 | 11 33.33 66.67
control_2 | 11 33.33 100.00
------------+-----------------------------------
Total | 33 100.00
tabstat difficulty useful importance, by(group)
Summary statistics: mean
by categories of: group
group | useful diffic~y import~e
----------+------------------------------
treatment | 18.11818 6.190909 8.681818
control_1 | 15.52727 5.581818 5.109091
control_2 | 15.34545 5.372727 5.636364
----------+------------------------------
Total | 16.3303 5.715152 6.475758
correlate useful difficulty importance
(obs=33)
| useful diffic~y import~e
-------------+---------------------------
useful | 1.0000
difficulty | 0.0978 1.0000
importance | -0.3411 0.1978 1.0000
Although this is a multivariate analysis, we will begin with separate univariate anovas to get a feel for what is happening with the data.
foreach vname in difficulty useful importance {
anova `vname' group
}
/* useful */
Number of obs = 33 R-squared = 0.1526
Root MSE = 3.13031 Adj R-squared = 0.0961
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 52.9242378 2 26.4621189 2.70 0.0835
|
group | 52.9242378 2 26.4621189 2.70 0.0835
|
Residual | 293.965442 30 9.79884808
-----------+----------------------------------------------------
Total | 346.88968 32 10.8403025
/* difficulty */
Number of obs = 33 R-squared = 0.0305
Root MSE = 2.05173 Adj R-squared = -0.0341
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 3.97515121 2 1.9875756 0.47 0.6282
|
group | 3.97515121 2 1.9875756 0.47 0.6282
|
Residual | 126.287277 30 4.20957589
-----------+----------------------------------------------------
Total | 130.262428 32 4.07070087
/* importance */
Number of obs = 33 R-squared = 0.1610
Root MSE = 3.76993 Adj R-squared = 0.1051
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 81.8296936 2 40.9148468 2.88 0.0718
|
group | 81.8296936 2 40.9148468 2.88 0.0718
|
Residual | 426.370896 30 14.2123632
-----------+----------------------------------------------------
Total | 508.20059 32 15.8812684
While none of the three anovas were statistically significant at the alpha = .05 level, in particular, the anova for difficulty was less than 1.
Next, we will run the manova itself.
manova difficulty useful importance = group
Number of obs = 33
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
-----------+--------------------------------------------------
group | W 0.5258 2 6.0 56.0 3.54 0.0049 e
| P 0.4767 6.0 58.0 3.02 0.0122 a
| L 0.8972 6.0 54.0 4.04 0.0021 a
| R 0.8920 3.0 29.0 8.62 0.0003 u
|--------------------------------------------------
Residual | 30
-----------+--------------------------------------------------
Total | 32
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
Now that we have have determined that the overall multivariate test is significant, we will follow up with several post-hoc tests.
/* multivariate test of group 1 versus the average of group 2 & 3 */
matrix c1=(0,2,-1,-1)
manovatest, test(c1)
Test constraint
(1) 2 group[1] - group[2] - group[3] = 0
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
-----------+--------------------------------------------------
manovatest | W 0.5290 1 3.0 28.0 8.31 0.0004 e
| P 0.4710 3.0 28.0 8.31 0.0004 e
| L 0.8904 3.0 28.0 8.31 0.0004 e
| R 0.8904 3.0 28.0 8.31 0.0004 e
|--------------------------------------------------
Residual | 30
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
/* multivariate test of group 2 versus group 3 */
matrix c2=(0,0,1,-1)
manovatest, test(c2)
Test constraint
(1) group[2] - group[3] = 0
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
-----------+--------------------------------------------------
manovatest | W 0.9932 1 3.0 28.0 0.06 0.9785 e
| P 0.0068 3.0 28.0 0.06 0.9785 e
| L 0.0068 3.0 28.0 0.06 0.9785 e
| R 0.0068 3.0 28.0 0.06 0.9785 e
|--------------------------------------------------
Residual | 30
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
/* we know from the univariate tests above that difficulty by itself was clearly not significant */
/* this test does the multivariate test using the combination of useful and importance */
matrix y=(0,1,1)
manovatest group , ytransform(y)
Transformation of the dependent variables
(1) y1 + y3
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
-----------+--------------------------------------------------
group | W 0.5360 2 2.0 30.0 12.99 0.0001 e
| P 0.4640 2.0 30.0 12.99 0.0001 e
| L 0.8657 2.0 30.0 12.99 0.0001 e
| R 0.8657 2.0 30.0 12.99 0.0001 e
|--------------------------------------------------
Residual | 30
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
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 multivariate test 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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