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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.sas7bdat, 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.
options nocenter;
proc means n mean std min max data="c:\data\manova";
var useful difficulty importance;
run;
The MEANS Procedure
Variable N Mean Std Dev Minimum Maximum
USEFUL 33 16.3303030 3.2924615 11.8999996 24.2999992
DIFFICULTY 33 5.7151515 2.0175978 2.4000001 10.2500000
IMPORTANCE 33 6.4757576 3.9851309 0.2000000 18.7999992
proc freq data="c:\data\manova";
table group;
run;
The FREQ Procedure
Cumulative Cumulative
GROUP Frequency Percent Frequency Percent
1 11 33.33 11 33.33
2 11 33.33 22 66.67
3 11 33.33 33 100.00
proc means n mean std min max data="c:\data\manova";
class group;
var useful difficulty importance;
run;
The MEANS Procedure
N
GROUP Obs Variable N Mean Std Dev Minimum Maximum
1 11 USEFUL 11 18.1181817 3.9037974 13.0000000 24.2999992
DIFFICULTY 11 6.1909091 1.8997129 3.7500000 10.2500000
IMPORTANCE 11 8.6818181 4.8630890 3.3000000 18.7999992
2 11 USEFUL 11 15.5272729 2.0756162 12.8000002 19.7000008
DIFFICULTY 11 5.5818183 2.4342631 2.4000001 9.8500004
IMPORTANCE 11 5.1090909 2.5311873 0.2000000 8.5000000
3 11 USEFUL 11 15.3454545 3.1382682 11.8999996 19.7999992
DIFFICULTY 11 5.3727273 1.7590287 2.6500001 8.7500000
IMPORTANCE 11 5.6363637 3.5469065 0.7000000 10.3000002
proc corr data="c:\data\manova";
var useful difficulty importance;
run;
The CORR Procedure
3 Variables: USEFUL DIFFICULTY IMPORTANCE
Pearson Correlation Coefficients, N = 33
Prob > |r| under H0: Rho=0
USEFUL DIFFICULTY IMPORTANCE
USEFUL 1.00000 0.09783 -0.34112
0.5881 0.0520
DIFFICULTY 0.09783 1.00000 0.19782
0.5881 0.2698
IMPORTANCE -0.34112 0.19782 1.00000
0.0520 0.2698
We will use the SAS proc glm procedure to do the one-way manova.
proc glm data="c:\data\manova"; class group; model useful difficulty importance = group / ss3; contrast '1 vs 2&3' group 2 -1 -1; contrast '2 vs 3' group 0 1 -1; manova h=_all_; manova h=group m=(1 0 1); run; quit;
Although this is a multivariate analysis, we will begin with the output for the separate univariate anovas to get a feel for what is happening with the data.
The GLM Procedure
Class Level Information
Class Levels Values
GROUP 3 1 2 3
Number of Observations Read 33
Number of Observations Used 33
Dependent Variable: USEFUL
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 52.9242378 26.4621189 2.70 0.0835
Error 30 293.9654425 9.7988481
Corrected Total 32 346.8896803
R-Square Coeff Var Root MSE USEFUL Mean
0.152568 19.16873 3.130311 16.33030
Source DF Type III SS Mean Square F Value Pr > F
GROUP 2 52.92423783 26.46211891 2.70 0.0835
Contrast DF Contrast SS Mean Square F Value Pr > F
1 vs 2&3 1 52.74241913 52.74241913 5.38 0.0273
2 vs 3 1 0.18181870 0.18181870 0.02 0.8926
Dependent Variable: DIFFICULTY
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 3.9751512 1.9875756 0.47 0.6282
Error 30 126.2872767 4.2095759
Corrected Total 32 130.2624279
R-Square Coeff Var Root MSE DIFFICULTY Mean
0.030516 35.89975 2.051725 5.715152
Source DF Type III SS Mean Square F Value Pr > F
GROUP 2 3.97515121 1.98757560 0.47 0.6282
Contrast DF Contrast SS Mean Square F Value Pr > F
1 vs 2&3 1 3.73469643 3.73469643 0.89 0.3538
2 vs 3 1 0.24045478 0.24045478 0.06 0.8127
Dependent Variable: IMPORTANCE
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 81.8296936 40.9148468 2.88 0.0718
Error 30 426.3708962 14.2123632
Corrected Total 32 508.2005898
R-Square Coeff Var Root MSE IMPORTANCE Mean
0.161018 58.21603 3.769929 6.475758
Source DF Type III SS Mean Square F Value Pr > F
GROUP 2 81.82969356 40.91484678 2.88 0.0718
Contrast DF Contrast SS Mean Square F Value Pr > F
1 vs 2&3 1 80.30060224 80.30060224 5.65 0.0240
2 vs 3 1 1.52909132 1.52909132 0.11 0.7452
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 look at the overall the manova itself.
Multivariate Analysis of Variance
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SSCP Matrix for GROUP
E = Error SSCP Matrix
Characteristic Characteristic Vector V'EV=1
Root Percent USEFUL DIFFICULTY IMPORTANCE
0.89198790 99.42 0.06410227 -0.00186162 0.05375069
0.00524207 0.58 0.01442655 0.06888878 -0.02620577
0.00000000 0.00 -0.03149580 0.05943387 0.01270798
MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall GROUP Effect
H = Type III SSCP Matrix for GROUP
E = Error SSCP Matrix
S=2 M=0 N=13
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.52578838 3.54 6 56 0.0049
Pillai's Trace 0.47667013 3.02 6 58 0.0122
Hotelling-Lawley Trace 0.89722998 4.12 6 35.61 0.0031
Roy's Greatest Root 0.89198790 8.62 3 29 0.0003
Now that we have have determined that the overall multivariate test is significant, we will follow up with several post-hoc tests beginning with multivariate test of group 1 versus the average of group 2 and 3.
/* contrast '1 vs 2&3' group 2 -1 -1; manova h-_all_; */
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall 1 vs 2&3 Effect
H = Contrast SSCP Matrix for 1 vs 2&3
E = Error SSCP Matrix
S=1 M=0.5 N=13
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.52899035 8.31 3 28 0.0004
Pillai's Trace 0.47100965 8.31 3 28 0.0004
Hotelling-Lawley Trace 0.89039367 8.31 3 28 0.0004
Roy's Greatest Root 0.89039367 8.31 3 28 0.0004
Here is the multivariate test of group 2 versus group 3.
/* contrast '2 vs 3' group 0 1 -1; manova h-_all_; */
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall 2 vs 3 Effect
H = Contrast SSCP Matrix for 2 vs 3
E = Error SSCP Matrix
S=1 M=0.5 N=13
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.99321011 0.06 3 28 0.9785
Pillai's Trace 0.00678989 0.06 3 28 0.9785
Hotelling-Lawley Trace 0.00683631 0.06 3 28 0.9785
Roy's Greatest Root 0.00683631 0.06 3 28 0.9785
We know from the univariate tests above that difficulty by itself was clearly not significant. This next test does the multivariate test using the combination of useful and importance.
/* manova h=group m=(1 0 1); */
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall GROUP Effect
on the Variables Defined by the M Matrix Transformation
H = Type III SSCP Matrix for GROUP
E = Error SSCP Matrix
S=1 M=0 N=14
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.53598494 12.99 2 30 <.0001
Pillai's Trace 0.46401506 12.99 2 30 <.0001
Hotelling-Lawley Trace 0.86572405 12.99 2 30 <.0001
Roy's Greatest Root 0.86572405 12.99 2 30 <.0001
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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