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 The GLM Procedure

## Example 30.5: Three-Way Analysis of Variance with Contrasts

This example uses data from Cochran and Cox (1957, p. 176) to illustrate the analysis of a three-way factorial design with replication, including the use of the CONTRAST statement with interactions, the OUTSTAT= data set, and the SLICE= option in the LSMEANS statement.

The object of the study is to determine the effects of electric current on denervated muscle. The variables are

Rep
the replicate number, 1 or 2

Time
the length of time the current is applied to the muscle, ranging from 1 to 4

Current
the level of electric current applied, ranging from 1 to 4

Number
the number of treatments per day, ranging from 1 to 3

MuscleWeight
the weight of the denervated muscle

The following code produces Output 30.5.1 through Output 30.5.4.

```   data muscles;
do Rep=1 to 2;
do Time=1 to 4;
do Current=1 to 4;
do Number=1 to 3;
input MuscleWeight @@;
output;
end;
end;
end;
end;
datalines;
72 74 69 61 61 65 62 65 70 85 76 61
67 52 62 60 55 59 64 65 64 67 72 60
57 66 72 72 43 43 63 66 72 56 75 92
57 56 78 60 63 58 61 79 68 73 86 71
46 74 58 60 64 52 71 64 71 53 65 66
44 58 54 57 55 51 62 61 79 60 78 82
53 50 61 56 57 56 56 56 71 56 58 69
46 55 64 56 55 57 64 66 62 59 58 88
;

proc glm outstat=summary;
class Rep Current Time Number;
model MuscleWeight = Rep Current|Time|Number;
contrast 'Time in Current 3'
Time 1 0 0 -1 Current*Time 0 0 0 0 0 0 0 0 1 0 0 -1,
Time 0 1 0 -1 Current*Time 0 0 0 0 0 0 0 0 0 1 0 -1,
Time 0 0 1 -1 Current*Time 0 0 0 0 0 0 0 0 0 0 1 -1;
contrast 'Current 1 versus 2' Current 1 -1;
lsmeans Current*Time / slice=Current;
run;

proc print data=summary;
run;
```

The first CONTRAST statement examines the effects of Time within level 3 of Current. This is also called the simple effect of Time within Current*Time. Note that, since there are three degrees of freedom, it is necessary to specify three rows in the CONTRAST statement, separated by commas. Since the parameterization that PROC GLM uses is determined in part by the ordering of the variables in the CLASS statement, Current is specified before Time so that the Time parameters are nested within the Current*Time parameters; thus, the Current*Time contrast coefficients in each row are simply the Time coefficients of that row within the appropriate level of Current.

The second CONTRAST statement isolates a single degree of freedom effect corresponding to the difference between the first two levels of Current. You can use such a contrast in a large experiment where certain preplanned comparisons are important, but you want to take advantage of the additional error degrees of freedom available when all levels of the factors are considered.

The LSMEANS statement with the SLICE= option is an alternative way to test for the simple effect of Time within Current*Time. In addition to listing the LS-means for each current strength and length of time, it gives a table of F-tests for differences between the LS-means across Time within each Current level. In some cases, this can be a way to disentangle a complex interaction.

Output 30.5.1: Overall Analysis

 The GLM Procedure

 Class Level Information Class Levels Values Rep 2 1 2 Current 4 1 2 3 4 Time 4 1 2 3 4 Number 3 1 2 3

 Number of observations 96

 The GLM Procedure Dependent Variable: MuscleWeight

 Source DF Sum of Squares Mean Square F Value Pr > F Model 48 5782.916667 120.477431 1.77 0.0261 Error 47 3199.489583 68.074246 Corrected Total 95 8982.406250

 R-Square Coeff Var Root MSE MuscleWeight Mean 0.643805 13.05105 8.250712 63.21875

The output, shown in Output 30.5.2 and Output 30.5.3, indicates that the main effects for Rep, Current, and Number are significant (with p-values of 0.0045, <0.0001, and 0.0461, respectively), but Time is not significant, indicating that, in general, it doesn't matter how long the current is applied. None of the interaction terms are significant, nor are the contrasts significant. Notice that the row in the sliced ANOVA table corresponding to level 3 of current matches the "Time in Current 3" contrast.

Output 30.5.2: Individual Effects and Contrasts

 The GLM Procedure Dependent Variable: MuscleWeight

 Source DF Type I SS Mean Square F Value Pr > F Rep 1 605.010417 605.010417 8.89 0.0045 Current 3 2145.447917 715.149306 10.51 <.0001 Time 3 223.114583 74.371528 1.09 0.3616 Current*Time 9 298.677083 33.186343 0.49 0.8756 Number 2 447.437500 223.718750 3.29 0.0461 Current*Number 6 644.395833 107.399306 1.58 0.1747 Time*Number 6 367.979167 61.329861 0.90 0.5023 Current*Time*Number 18 1050.854167 58.380787 0.86 0.6276

 Source DF Type III SS Mean Square F Value Pr > F Rep 1 605.010417 605.010417 8.89 0.0045 Current 3 2145.447917 715.149306 10.51 <.0001 Time 3 223.114583 74.371528 1.09 0.3616 Current*Time 9 298.677083 33.186343 0.49 0.8756 Number 2 447.437500 223.718750 3.29 0.0461 Current*Number 6 644.395833 107.399306 1.58 0.1747 Time*Number 6 367.979167 61.329861 0.90 0.5023 Current*Time*Number 18 1050.854167 58.380787 0.86 0.6276

 Contrast DF Contrast SS Mean Square F Value Pr > F Time in Current 3 3 34.83333333 11.61111111 0.17 0.9157 Current 1 versus 2 1 99.18750000 99.18750000 1.46 0.2334

Output 30.5.3: Simple Effects of Time

 The GLM Procedure Least Squares Means

 Current*Time Effect Sliced by Current for MuscleWeight Current DF Sum of Squares Mean Square F Value Pr > F 1 3 271.458333 90.486111 1.33 0.2761 2 3 120.666667 40.222222 0.59 0.6241 3 3 34.833333 11.611111 0.17 0.9157 4 3 94.833333 31.611111 0.46 0.7085

The SS, F statistics, and p-values can be stored in an OUTSTAT= data set, as shown in Output 30.5.4.

Output 30.5.4: Contents of the OUTSTAT= Data Set

 Obs _NAME_ _SOURCE_ _TYPE_ DF SS F PROB 1 MuscleWeight ERROR ERROR 47 3199.49 . . 2 MuscleWeight Rep SS1 1 605.01 8.8875 0.00454 3 MuscleWeight Current SS1 3 2145.45 10.5054 0.00002 4 MuscleWeight Time SS1 3 223.11 1.0925 0.36159 5 MuscleWeight Current*Time SS1 9 298.68 0.4875 0.87562 6 MuscleWeight Number SS1 2 447.44 3.2864 0.04614 7 MuscleWeight Current*Number SS1 6 644.40 1.5777 0.17468 8 MuscleWeight Time*Number SS1 6 367.98 0.9009 0.50231 9 MuscleWeight Current*Time*Number SS1 18 1050.85 0.8576 0.62757 10 MuscleWeight Rep SS3 1 605.01 8.8875 0.00454 11 MuscleWeight Current SS3 3 2145.45 10.5054 0.00002 12 MuscleWeight Time SS3 3 223.11 1.0925 0.36159 13 MuscleWeight Current*Time SS3 9 298.68 0.4875 0.87562 14 MuscleWeight Number SS3 2 447.44 3.2864 0.04614 15 MuscleWeight Current*Number SS3 6 644.40 1.5777 0.17468 16 MuscleWeight Time*Number SS3 6 367.98 0.9009 0.50231 17 MuscleWeight Current*Time*Number SS3 18 1050.85 0.8576 0.62757 18 MuscleWeight Time in Current 3 CONTRAST 3 34.83 0.1706 0.91574 19 MuscleWeight Current 1 versus 2 CONTRAST 1 99.19 1.4570 0.23344

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