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Example 30.5: Three-Way Analysis of Variance with Contrasts

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.

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