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:

1. The group by specific gravity interaction appears to be marginally significant so that adjusting for specific gravity comparisons is difficult. In what follows we ignore the possibility of any such interaction.
2. Adjusted for the specific gravity of the urine the overall P-value for the test of no group effect is about 1 to 2 percent so that there is fairly clearly a difference between groups.
3. Without adjusting for the specific gravity the overall P-value is about the same.
4. Natural parameters to examine with confidence intervals are all of the 3x(4 choose 2) = 18 differences in two means. The natural simultaneous intervals are either Bonferroni or Roy type. SAS produces univariate simultaneous confidence intervals easily but not the Roy type intervals. For Bonferroni intervals the individual intervals are t type based on df=N-p=45-4=41 unadjusted for specific gravity or 40 degrees of freedom adjusted for specific gravity.

```                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```

Richard Lockhart
Tue Feb 17 22:46:33 PST 1998