Multivariate Analysis of Covariance
This example concerns a one way layout with 45 patients in 4 groups (defined by body type and obesity status). There are 4 biochemical measurements made on the urine of each subject: three are treated as response variables and one as a covariate, the specific gravity of the urine.
The issue is to look for group differences adjusting and not adjusting for the covariate specific gravity.
run;
data long;
infile 'dq5.1';
input group creatin chloride choline specgrav;
run;
proc print;
run;
proc glm;
class group;
model creatin chloride choline = group|specgrav;
manova h= group*specgrav / printh printe;
run;
proc glm;
class group;
model creatin chloride choline = group specgrav;
manova h= group / printh printe;
means group / bon scheffe tukey;
run;
proc glm;
class group;
model creatin chloride choline = group;
manova h= group / printh printe;
means group / bon scheffe tukey;
run;
Here are som conclusions based on the output below:
OBS GROUP CREATIN CHLORIDE CHOLINE SPECGRAV
1 1 17.6 5.15 7.5 24
2 1 13.4 5.75 7.1 32
3 1 20.3 4.35 2.3 17
4 1 22.3 7.55 4.0 30
5 1 20.5 8.50 2.0 30
6 1 18.5 10.25 2.0 27
7 1 12.1 5.95 16.8 25
8 1 12.0 6.30 14.5 30
9 1 10.1 5.45 0.9 28
10 1 14.7 3.75 2.0 24
11 1 14.8 5.10 0.4 26
12 1 14.4 4.05 3.8 27
13 2 18.1 9.00 14.5 31
14 2 19.7 5.30 12.5 23
15 2 16.9 9.85 8.0 32
16 2 23.7 3.60 4.9 20
17 2 19.2 4.05 0.2 18
18 2 18.0 4.40 3.6 23
19 2 14.8 7.15 12.0 31
20 2 15.6 7.25 5.2 28
21 2 16.2 5.30 10.2 21
22 2 14.1 3.10 8.5 20
23 2 17.5 2.40 9.6 15
24 2 14.1 4.25 6.9 26
25 2 19.1 5.80 4.7 24
26 2 22.5 1.55 3.5 16
27 3 17.0 4.55 1.9 18
28 3 12.5 2.65 0.7 10
29 3 21.5 6.50 8.3 33
30 3 22.2 4.85 9.3 25
31 3 13.0 8.75 13.0 35
32 3 13.0 5.20 18.3 33
33 3 10.9 4.75 10.5 31
34 3 12.0 5.85 14.5 34
35 3 22.8 2.85 3.3 16
36 3 16.5 6.55 6.3 31
37 3 18.4 6.60 4.9 28
38 4 12.5 2.90 22.5 32
39 4 8.7 3.00 19.5 25
40 4 9.4 3.40 1.3 28
41 4 15.0 5.40 20.0 27
42 4 12.9 4.45 1.0 23
43 4 12.1 4.30 5.0 25
44 4 13.2 5.00 3.0 26
45 4 11.5 3.40 5.1 34
General Linear Models Procedure
Class Level Information
Class Levels Values
GROUP 4 1 2 3 4
Number of observations in data set = 45
General Linear Models Procedure
Dependent Variable: CREATIN
Source DF Sum of Squares Mean Square F Value Pr > F
Model 7 219.55914002 31.36559143 2.53 0.0311
Error 37 458.27997109 12.38594516
Corrected Total 44 677.83911111
R-Square C.V. Root MSE CREATIN Mean
0.323910 22.14057 3.51936715 15.89555556
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 181.07035029 60.35678343 4.87 0.0059
SPECGRAV 1 33.55457614 33.55457614 2.71 0.1082
SPECGRAV*GROUP 3 4.93421359 1.64473786 0.13 0.9399
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 14.02852999 4.67617666 0.38 0.7697
SPECGRAV 1 15.99112366 15.99112366 1.29 0.2632
SPECGRAV*GROUP 3 4.93421359 1.64473786 0.13 0.9399
General Linear Models Procedure
Dependent Variable: CHLORIDE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 7 114.69580798 16.38511543 10.42 0.0001
Error 37 58.19619202 1.57287005
Corrected Total 44 172.89200000
R-Square C.V. Root MSE CHLORIDE Mean
0.663396 23.90358 1.25414116 5.24666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 20.03772646 6.67924215 4.25 0.0112
SPECGRAV 1 68.81231628 68.81231628 43.75 0.0001
SPECGRAV*GROUP 3 25.84576523 8.61525508 5.48 0.0032
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 18.50717670 6.16905890 3.92 0.0158
SPECGRAV 1 22.66541545 22.66541545 14.41 0.0005
SPECGRAV*GROUP 3 25.84576523 8.61525508 5.48 0.0032
General Linear Models Procedure
Dependent Variable: CHOLINE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 7 382.31484760 54.61640680 1.75 0.1269
Error 37 1153.34515240 31.17149061
Corrected Total 44 1535.66000000
R-Square C.V. Root MSE CHOLINE Mean
0.248958 74.77424 5.58314343 7.46666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 103.80568182 34.60189394 1.11 0.3574
SPECGRAV 1 254.70756080 254.70756080 8.17 0.0070
SPECGRAV*GROUP 3 23.80160497 7.93386832 0.25 0.8576
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 17.32050967 5.77350322 0.19 0.9058
SPECGRAV 1 129.17662849 129.17662849 4.14 0.0490
SPECGRAV*GROUP 3 23.80160497 7.93386832 0.25 0.8576
General Linear Models Procedure
Multivariate Analysis of Variance
E = Error SS&CP Matrix
CREATIN CHLORIDE CHOLINE
CREATIN 458.27997109 63.446754032 -141.3962534
CHLORIDE 63.446754032 58.196192023 -30.41709806
CHOLINE -141.3962534 -30.41709806 1153.3451524
General Linear Models Procedure
Multivariate Analysis of Variance
Partial Correlation Coefficients from the Error SS&CP Matrix / Prob > |r|
DF = 37 CREATIN CHLORIDE CHOLINE
CREATIN 1.000000 0.388505 -0.194488
0.0001 0.0145 0.2355
CHLORIDE 0.388505 1.000000 -0.117406
0.0145 0.0001 0.4766
CHOLINE -0.194488 -0.117406 1.000000
0.2355 0.4766 0.0001
General Linear Models Procedure
Multivariate Analysis of Variance
H = Type III SS&CP Matrix for SPECGRAV*GROUP
CREATIN CHLORIDE CHOLINE
CREATIN 4.9342135916 -10.88755514 8.822735835
CHLORIDE -10.88755514 25.845765234 -15.74572547
CHOLINE 8.822735835 -15.74572547 23.801604973
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for SPECGRAV*GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
CREATIN CHLORIDE CHOLINE
0.6025909825 97.71 -0.02637071 0.13956526 -0.00315708
0.0141014522 2.29 0.01487276 0.02486753 0.02918876
0.0000178077 0.00 0.04151186 0.01358309 -0.00641201
General Linear Models Procedure
Multivariate Analysis of Variance
Manova Test Criteria and F Approximations for
the Hypothesis of no Overall SPECGRAV*GROUP Effect
H = Type III SS&CP Matrix for SPECGRAV*GROUP E = Error SS&CP Matrix
S=3 M=-0.5 N=16.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.61530177 2.0939 9 85.33141 0.0387
Pillai's Trace 0.38993364 1.8426 9 111 0.0683
Hotelling-Lawley Trace 0.61671024 2.3070 9 101 0.0211
Roy's Greatest Root 0.60259098 7.4320 3 37 0.0005
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
General Linear Models Procedure
Class Level Information
Class Levels Values
GROUP 4 1 2 3 4
Number of observations in data set = 45
General Linear Models Procedure
Dependent Variable: CREATIN
Source DF Sum of Squares Mean Square F Value Pr > F
Model 4 214.62492643 53.65623161 4.63 0.0036
Error 40 463.21418468 11.58035462
Corrected Total 44 677.83911111
R-Square C.V. Root MSE CREATIN Mean
0.316631 21.40845 3.40299201 15.89555556
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 181.07035029 60.35678343 5.21 0.0039
SPECGRAV 1 33.55457614 33.55457614 2.90 0.0965
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 142.67640002 47.55880001 4.11 0.0124
SPECGRAV 1 33.55457614 33.55457614 2.90 0.0965
General Linear Models Procedure
Dependent Variable: CHLORIDE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 4 88.85004274 22.21251069 10.57 0.0001
Error 40 84.04195726 2.10104893
Corrected Total 44 172.89200000
R-Square C.V. Root MSE CHLORIDE Mean
0.513905 27.62706 1.44949955 5.24666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 20.03772646 6.67924215 3.18 0.0342
SPECGRAV 1 68.81231628 68.81231628 32.75 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 28.32353333 9.44117778 4.49 0.0083
SPECGRAV 1 68.81231628 68.81231628 32.75 0.0001
General Linear Models Procedure
Dependent Variable: CHOLINE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 4 358.51324262 89.62831066 3.05 0.0278
Error 40 1177.14675738 29.42866893
Corrected Total 44 1535.66000000
R-Square C.V. Root MSE CHOLINE Mean
0.233459 72.65384 5.42481971 7.46666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 103.80568182 34.60189394 1.18 0.3310
SPECGRAV 1 254.70756080 254.70756080 8.66 0.0054
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 111.34065943 37.11355314 1.26 0.3007
SPECGRAV 1 254.70756080 254.70756080 8.66 0.0054
General Linear Models Procedure
Multivariate Analysis of Variance
E = Error SS&CP Matrix
CREATIN CHLORIDE CHOLINE
CREATIN 463.21418468 52.559198888 -132.5735176
CHLORIDE 52.559198888 84.041957257 -46.16282353
CHOLINE -132.5735176 -46.16282353 1177.1467574
General Linear Models Procedure
Multivariate Analysis of Variance
Partial Correlation Coefficients from the Error SS&CP Matrix / Prob > |r|
DF = 40 CREATIN CHLORIDE CHOLINE
CREATIN 1.000000 0.266385 -0.179536
0.0001 0.0882 0.2553
CHLORIDE 0.266385 1.000000 -0.146767
0.0882 0.0001 0.3537
CHOLINE -0.179536 -0.146767 1.000000
0.2553 0.3537 0.0001
General Linear Models Procedure
Multivariate Analysis of Variance
H = Type III SS&CP Matrix for GROUP
CREATIN CHLORIDE CHOLINE
CREATIN 142.67640002 57.975991093 -25.81759754
CHLORIDE 57.975991093 28.323533332 -31.2952734
CHOLINE -25.81759754 -31.2952734 111.34065943
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
CREATIN CHLORIDE CHOLINE
0.4876676013 79.60 0.02827764 0.07153539 0.00101538
0.1204865877 19.67 0.02648038 -0.03977179 0.02501754
0.0044972823 0.73 -0.02957063 0.07905405 0.01614189
General Linear Models Procedure
Multivariate Analysis of Variance
Manova Test Criteria and F Approximations for the Hypothesis of no Overall GROUP Effect
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
S=3 M=-0.5 N=18
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.59722595 2.4280 9 92.63262 0.0159
Pillai's Trace 0.43981457 2.2905 9 120 0.0208
Hotelling-Lawley Trace 0.61265147 2.4960 9 110 0.0123
Roy's Greatest Root 0.48766760 6.5022 3 40 0.0011
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 11.58035
Critical Value of Studentized Range= 3.791
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.199 1.476 5.151
2 - 1 -1.659 1.930 5.518
2 - 4 1.866 5.909 9.952 ***
3 - 2 -5.151 -1.476 2.199
3 - 1 -3.354 0.454 4.261
3 - 4 0.195 4.433 8.671 ***
1 - 2 -5.518 -1.930 1.659
1 - 3 -4.261 -0.454 3.354
1 - 4 -0.184 3.979 8.143
4 - 2 -9.952 -5.909 -1.866 ***
4 - 3 -8.671 -4.433 -0.195 ***
4 - 1 -8.143 -3.979 0.184
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 11.58035
Critical Value of T= 2.77587
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.330 1.476 5.282
2 - 1 -1.786 1.930 5.646
2 - 4 1.722 5.909 10.096 ***
3 - 2 -5.282 -1.476 2.330
3 - 1 -3.489 0.454 4.397
3 - 4 0.044 4.433 8.822 ***
1 - 2 -5.646 -1.930 1.786
1 - 3 -4.397 -0.454 3.489
1 - 4 -0.332 3.979 8.291
4 - 2 -10.096 -5.909 -1.722 ***
4 - 3 -8.822 -4.433 -0.044 ***
4 - 1 -8.291 -3.979 0.332
General Linear Models Procedure
Scheffe's test for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 11.58035
Critical Value of F= 2.83875
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.525 1.476 5.477
2 - 1 -1.977 1.930 5.837
2 - 4 1.508 5.909 10.310 ***
3 - 2 -5.477 -1.476 2.525
3 - 1 -3.692 0.454 4.599
3 - 4 -0.181 4.433 9.047
1 - 2 -5.837 -1.930 1.977
1 - 3 -4.599 -0.454 3.692
1 - 4 -0.554 3.979 8.512
4 - 2 -10.310 -5.909 -1.508 ***
4 - 3 -9.047 -4.433 0.181
4 - 1 -8.512 -3.979 0.554
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 2.101049
Critical Value of Studentized Range= 3.791
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -0.9820 0.6398 2.2616
1 - 2 -0.7302 0.7982 2.3267
1 - 4 0.2579 2.0312 3.8046 ***
3 - 1 -2.2616 -0.6398 0.9820
3 - 2 -1.4070 0.1584 1.7239
3 - 4 -0.4139 1.3915 3.1968
2 - 1 -2.3267 -0.7982 0.7302
2 - 3 -1.7239 -0.1584 1.4070
2 - 4 -0.4889 1.2330 2.9550
4 - 1 -3.8046 -2.0312 -0.2579 ***
4 - 3 -3.1968 -1.3915 0.4139
4 - 2 -2.9550 -1.2330 0.4889
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 2.101049
Critical Value of T= 2.77587
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -1.0398 0.6398 2.3193
1 - 2 -0.7847 0.7982 2.3811
1 - 4 0.1947 2.0312 3.8678 ***
3 - 1 -2.3193 -0.6398 1.0398
3 - 2 -1.4627 0.1584 1.7796
3 - 4 -0.4781 1.3915 3.2611
2 - 1 -2.3811 -0.7982 0.7847
2 - 3 -1.7796 -0.1584 1.4627
2 - 4 -0.5502 1.2330 3.0163
4 - 1 -3.8678 -2.0312 -0.1947 ***
4 - 3 -3.2611 -1.3915 0.4781
4 - 2 -3.0163 -1.2330 0.5502
General Linear Models Procedure
Scheffe's test for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 2.101049
Critical Value of F= 2.83875
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -1.1259 0.6398 2.4055
1 - 2 -0.8659 0.7982 2.4623
1 - 4 0.1005 2.0312 3.9620 ***
3 - 1 -2.4055 -0.6398 1.1259
3 - 2 -1.5459 0.1584 1.8628
3 - 4 -0.5740 1.3915 3.3570
2 - 1 -2.4623 -0.7982 0.8659
2 - 3 -1.8628 -0.1584 1.5459
2 - 4 -0.6417 1.2330 3.1078
4 - 1 -3.9620 -2.0312 -0.1005 ***
4 - 3 -3.3570 -1.3915 0.5740
4 - 2 -3.1078 -1.2330 0.6417
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 29.42867
Critical Value of Studentized Range= 3.791
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -5.354 1.402 8.159
4 - 2 -4.220 2.225 8.670
4 - 1 -2.237 4.400 11.037
3 - 4 -8.159 -1.402 5.354
3 - 2 -5.036 0.823 6.681
3 - 1 -3.072 2.998 9.067
2 - 4 -8.670 -2.225 4.220
2 - 3 -6.681 -0.823 5.036
2 - 1 -3.545 2.175 7.895
1 - 4 -11.037 -4.400 2.237
1 - 3 -9.067 -2.998 3.072
1 - 2 -7.895 -2.175 3.545
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 29.42867
Critical Value of T= 2.77587
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -5.595 1.402 8.399
4 - 2 -4.449 2.225 8.899
4 - 1 -2.473 4.400 11.273
3 - 4 -8.399 -1.402 5.595
3 - 2 -5.245 0.823 6.890
3 - 1 -3.288 2.998 9.284
2 - 4 -8.899 -2.225 4.449
2 - 3 -6.890 -0.823 5.245
2 - 1 -3.749 2.175 8.099
1 - 4 -11.273 -4.400 2.473
1 - 3 -9.284 -2.998 3.288
1 - 2 -8.099 -2.175 3.749
General Linear Models Procedure
Scheffe's test for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 40 MSE= 29.42867
Critical Value of F= 2.83875
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -5.954 1.402 8.758
4 - 2 -4.791 2.225 9.241
4 - 1 -2.826 4.400 11.626
3 - 4 -8.758 -1.402 5.954
3 - 2 -5.556 0.823 7.201
3 - 1 -3.611 2.998 9.606
2 - 4 -9.241 -2.225 4.791
2 - 3 -7.201 -0.823 5.556
2 - 1 -4.053 2.175 8.403
1 - 4 -11.626 -4.400 2.826
1 - 3 -9.606 -2.998 3.611
1 - 2 -8.403 -2.175 4.053
General Linear Models Procedure
Class Level Information
Class Levels Values
GROUP 4 1 2 3 4
Number of observations in data set = 45
General Linear Models Procedure
Dependent Variable: CREATIN
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 181.07035029 60.35678343 4.98 0.0049
Error 41 496.76876082 12.11631124
Corrected Total 44 677.83911111
R-Square C.V. Root MSE CREATIN Mean
0.267129 21.89825 3.48084921 15.89555556
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 181.07035029 60.35678343 4.98 0.0049
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 181.07035029 60.35678343 4.98 0.0049
General Linear Models Procedure
Dependent Variable: CHLORIDE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 20.03772646 6.67924215 1.79 0.1639
Error 41 152.85427354 3.72815301
Corrected Total 44 172.89200000
R-Square C.V. Root MSE CHLORIDE Mean
0.115897 36.80132 1.93084257 5.24666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 20.03772646 6.67924215 1.79 0.1639
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 20.03772646 6.67924215 1.79 0.1639
General Linear Models Procedure
Dependent Variable: CHOLINE
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 103.80568182 34.60189394 0.99 0.4067
Error 41 1431.85431818 34.92327605
Corrected Total 44 1535.66000000
R-Square C.V. Root MSE CHOLINE Mean
0.067597 79.14632 5.90959187 7.46666667
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 103.80568182 34.60189394 0.99 0.4067
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 103.80568182 34.60189394 0.99 0.4067
General Linear Models Procedure
Multivariate Analysis of Variance
E = Error SS&CP Matrix
CREATIN CHLORIDE CHOLINE
CREATIN 496.76876082 4.5074756494 -225.0213636
CHLORIDE 4.5074756494 152.85427354 86.226818182
CHOLINE -225.0213636 86.226818182 1431.8543182
General Linear Models Procedure
Multivariate Analysis of Variance
Partial Correlation Coefficients from the Error SS&CP Matrix / Prob > |r|
DF = 41 CREATIN CHLORIDE CHOLINE
CREATIN 1.000000 0.016358 -0.266807
0.0001 0.9171 0.0837
CHLORIDE 0.016358 1.000000 0.184312
0.9171 0.0001 0.2367
CHOLINE -0.266807 0.184312 1.000000
0.0837 0.2367 0.0001
General Linear Models Procedure
Multivariate Analysis of Variance
H = Type III SS&CP Matrix for GROUP
CREATIN CHLORIDE CHOLINE
CREATIN 181.07035029 40.036857684 -66.72530303
CHLORIDE 40.036857684 20.037726461 -41.37181818
CHOLINE -66.72530303 -41.37181818 103.80568182
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
CREATIN CHLORIDE CHOLINE
0.4336849273 74.76 0.03815241 0.03717808 -0.00338557
0.1409699628 24.30 0.02655774 -0.05607206 0.02113111
0.0054156726 0.93 -0.00409794 0.04773387 0.01799753
General Linear Models Procedure
Multivariate Analysis of Variance
Manova Test Criteria and F Approximations for the Hypothesis of no Overall GROUP Effect
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
S=3 M=-0.5 N=18.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.60803198 2.3959 9 95.06636 0.0171
Pillai's Trace 0.43143592 2.2956 9 123 0.0203
Hotelling-Lawley Trace 0.58007056 2.4277 9 113 0.0147
Roy's Greatest Root 0.43368493 5.9270 3 41 0.0019
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 12.11631
Critical Value of Studentized Range= 3.787
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.279 1.476 5.231
2 - 1 -1.737 1.930 5.596
2 - 4 1.778 5.909 10.040 ***
3 - 2 -5.231 -1.476 2.279
3 - 1 -3.437 0.454 4.344
3 - 4 0.102 4.433 8.764 ***
1 - 2 -5.596 -1.930 1.737
1 - 3 -4.344 -0.454 3.437
1 - 4 -0.275 3.979 8.233
4 - 2 -10.040 -5.909 -1.778 ***
4 - 3 -8.764 -4.433 -0.102 ***
4 - 1 -8.233 -3.979 0.275
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 12.11631
Critical Value of T= 2.77235
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.412 1.476 5.364
2 - 1 -1.867 1.930 5.726
2 - 4 1.632 5.909 10.186 ***
3 - 2 -5.364 -1.476 2.412
3 - 1 -3.574 0.454 4.482
3 - 4 -0.051 4.433 8.917
1 - 2 -5.726 -1.930 1.867
1 - 3 -4.482 -0.454 3.574
1 - 4 -0.426 3.979 8.384
4 - 2 -10.186 -5.909 -1.632 ***
4 - 3 -8.917 -4.433 0.051
4 - 1 -8.384 -3.979 0.426
General Linear Models Procedure
Scheffe's test for variable: CREATIN
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 12.11631
Critical Value of F= 2.83275
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
2 - 3 -2.612 1.476 5.564
2 - 1 -2.062 1.930 5.922
2 - 4 1.412 5.909 10.406 ***
3 - 2 -5.564 -1.476 2.612
3 - 1 -3.782 0.454 4.690
3 - 4 -0.282 4.433 9.148
1 - 2 -5.922 -1.930 2.062
1 - 3 -4.690 -0.454 3.782
1 - 4 -0.652 3.979 8.611
4 - 2 -10.406 -5.909 -1.412 ***
4 - 3 -9.148 -4.433 0.282
4 - 1 -8.611 -3.979 0.652
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 3.728153
Critical Value of Studentized Range= 3.787
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -1.5183 0.6398 2.7979
1 - 2 -1.2357 0.7982 2.8321
1 - 4 -0.3286 2.0312 4.3911
3 - 1 -2.7979 -0.6398 1.5183
3 - 2 -1.9246 0.1584 2.2415
3 - 4 -1.0108 1.3915 3.7938
2 - 1 -2.8321 -0.7982 1.2357
2 - 3 -2.2415 -0.1584 1.9246
2 - 4 -1.0584 1.2330 3.5244
4 - 1 -4.3911 -2.0312 0.3286
4 - 3 -3.7938 -1.3915 1.0108
4 - 2 -3.5244 -1.2330 1.0584
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 3.728153
Critical Value of T= 2.77235
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -1.5947 0.6398 2.8742
1 - 2 -1.3076 0.7982 2.9041
1 - 4 -0.4120 2.0312 4.4745
3 - 1 -2.8742 -0.6398 1.5947
3 - 2 -1.9983 0.1584 2.3152
3 - 4 -1.0958 1.3915 3.8788
2 - 1 -2.9041 -0.7982 1.3076
2 - 3 -2.3152 -0.1584 1.9983
2 - 4 -1.1394 1.2330 3.6055
4 - 1 -4.4745 -2.0312 0.4120
4 - 3 -3.8788 -1.3915 1.0958
4 - 2 -3.6055 -1.2330 1.1394
General Linear Models Procedure
Scheffe's test for variable: CHLORIDE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 3.728153
Critical Value of F= 2.83275
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
1 - 3 -1.7098 0.6398 2.9893
1 - 2 -1.4161 0.7982 3.0125
1 - 4 -0.5379 2.0312 4.6004
3 - 1 -2.9893 -0.6398 1.7098
3 - 2 -2.1094 0.1584 2.4263
3 - 4 -1.2240 1.3915 4.0069
2 - 1 -3.0125 -0.7982 1.4161
2 - 3 -2.4263 -0.1584 2.1094
2 - 4 -1.2616 1.2330 3.7277
4 - 1 -4.6004 -2.0312 0.5379
4 - 3 -4.0069 -1.3915 1.2240
4 - 2 -3.7277 -1.2330 1.2616
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 34.92328
Critical Value of Studentized Range= 3.787
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -5.950 1.402 8.755
4 - 2 -4.788 2.225 9.238
4 - 1 -2.822 4.400 11.622
3 - 4 -8.755 -1.402 5.950
3 - 2 -5.553 0.823 7.198
3 - 1 -3.607 2.998 9.603
2 - 4 -9.238 -2.225 4.788
2 - 3 -7.198 -0.823 5.553
2 - 1 -4.050 2.175 8.400
1 - 4 -11.622 -4.400 2.822
1 - 3 -9.603 -2.998 3.607
1 - 2 -8.400 -2.175 4.050
General Linear Models Procedure
Bonferroni (Dunn) T tests for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 34.92328
Critical Value of T= 2.77235
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -6.210 1.402 9.015
4 - 2 -5.036 2.225 9.486
4 - 1 -3.078 4.400 11.878
3 - 4 -9.015 -1.402 6.210
3 - 2 -5.778 0.823 7.424
3 - 1 -3.841 2.998 9.837
2 - 4 -9.486 -2.225 5.036
2 - 3 -7.424 -0.823 5.778
2 - 1 -4.270 2.175 8.620
1 - 4 -11.878 -4.400 3.078
1 - 3 -9.837 -2.998 3.841
1 - 2 -8.620 -2.175 4.270
General Linear Models Procedure
Scheffe's test for variable: CHOLINE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey's for all
pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 41 MSE= 34.92328
Critical Value of F= 2.83275
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
GROUP Confidence Between Confidence
Comparison Limit Means Limit
4 - 3 -6.603 1.402 9.407
4 - 2 -5.410 2.225 9.860
4 - 1 -3.463 4.400 12.263
3 - 4 -9.407 -1.402 6.603
3 - 2 -6.118 0.823 7.764
3 - 1 -4.193 2.998 10.189
2 - 4 -9.860 -2.225 5.410
2 - 3 -7.764 -0.823 6.118
2 - 1 -4.602 2.175 8.952
1 - 4 -12.263 -4.400 3.463
1 - 3 -10.189 -2.998 4.193
1 - 2 -8.952 -2.175 4.602