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The ANOVA Procedure

Example 23.5 Strip-Split Plot

In this example, four different fertilizer treatments are laid out in vertical strips, which are then split into subplots with different levels of calcium. Soil type is stripped across the split-plot experiment, and the entire experiment is then replicated three times. The dependent variable is the yield of winter barley. The data come from the notes of G. Cox and A. Rotti.

The input data are the 96 values of Y, arranged so that the calcium value (Calcium) changes most rapidly, then the fertilizer value (Fertilizer), then the Soil value, and, finally, the Rep value. Values are shown for Calcium (0 and 1); Fertilizer (0, 1, 2, 3); Soil (1, 2, 3); and Rep (1, 2, 3, 4). The following example produces Output 23.5.1, Output 23.5.2, Output 23.5.3, and Output 23.5.4. 


   title1 'Strip-split Plot';
   data Barley;
      do Rep=1 to 4;
         do Soil=1 to 3; /* 1=d 2=h 3=p */
            do Fertilizer=0 to 3;
               do Calcium=0,1;
                  input Yield @;
                  output;
               end;
            end;
         end;
      end;
      datalines;
   4.91 4.63 4.76 5.04 5.38 6.21 5.60 5.08
   4.94 3.98 4.64 5.26 5.28 5.01 5.45 5.62
   5.20 4.45 5.05 5.03 5.01 4.63 5.80 5.90
   6.00 5.39 4.95 5.39 6.18 5.94 6.58 6.25
   5.86 5.41 5.54 5.41 5.28 6.67 6.65 5.94
   5.45 5.12 4.73 4.62 5.06 5.75 6.39 5.62
   4.96 5.63 5.47 5.31 6.18 6.31 5.95 6.14
   5.71 5.37 6.21 5.83 6.28 6.55 6.39 5.57
   4.60 4.90 4.88 4.73 5.89 6.20 5.68 5.72
   5.79 5.33 5.13 5.18 5.86 5.98 5.55 4.32
   5.61 5.15 4.82 5.06 5.67 5.54 5.19 4.46
   5.13 4.90 4.88 5.18 5.45 5.80 5.12 4.42
   ;

   proc anova data=Barley;
      class Rep Soil Calcium Fertilizer;
      model Yield =
              Rep
              Fertilizer Fertilizer*Rep
              Calcium Calcium*Fertilizer Calcium*Rep(Fertilizer)
              Soil Soil*Rep
              Soil*Fertilizer Soil*Rep*Fertilizer
              Soil*Calcium Soil*Fertilizer*Calcium
              Soil*Calcium*Rep(Fertilizer);
      test h=Fertilizer                 e=Fertilizer*Rep;
      test h=Calcium calcium*fertilizer e=Calcium*Rep(Fertilizer);
      test h=Soil                       e=Soil*Rep;
      test h=Soil*Fertilizer            e=Soil*Rep*Fertilizer;
      test h=Soil*Calcium
             Soil*Fertilizer*Calcium    e=Soil*Calcium*Rep(Fertilizer);
      means Fertilizer Calcium Soil Calcium*Fertilizer;
   run;

Output 23.5.1 Class Level Information
Strip-split Plot

The ANOVA Procedure

Class Level Information
Class Levels Values
Rep 4 1 2 3 4
Soil 3 1 2 3
Calcium 2 0 1
Fertilizer 4 0 1 2 3

Number of Observations Read 96
Number of Observations Used 96

Output 23.5.2 ANOVA Table
Strip-split Plot

The ANOVA Procedure
 
Dependent Variable: Yield

Source DF Sum of Squares Mean Square F Value Pr > F
Model 95 31.89149583 0.33569996 . .
Error 0 0.00000000 .    
Corrected Total 95 31.89149583      

R-Square Coeff Var Root MSE Yield Mean
1.000000 . . 5.427292

Source DF Anova SS Mean Square F Value Pr > F
Rep 3 6.27974583 2.09324861 . .
Fertilizer 3 7.22127083 2.40709028 . .
Rep*Fertilizer 9 6.08211250 0.67579028 . .
Calcium 1 0.27735000 0.27735000 . .
Calcium*Fertilizer 3 1.96395833 0.65465278 . .
Rep*Calcium(Fertili) 12 1.76705833 0.14725486 . .
Soil 2 1.92658958 0.96329479 . .
Rep*Soil 6 1.66761042 0.27793507 . .
Soil*Fertilizer 6 0.68828542 0.11471424 . .
Rep*Soil*Fertilizer 18 1.58698125 0.08816563 . .
Soil*Calcium 2 0.04493125 0.02246562 . .
Soil*Calcium*Fertili 6 0.18936042 0.03156007 . .
Rep*Soil*Calc(Ferti) 24 2.19624167 0.09151007 . .

Notice in Output 23.5.2 that the default tests against the residual error rate are all unavailable. This is because the Soil*Calcium*Rep(Fertilizer) term in the model takes up all the degrees of freedom, leaving none for estimating the residual error rate. This is appropriate in this case since the TEST statements give the specific error terms appropriate for testing each effect. Output 23.5.3 displays the output produced by the various TEST statements. The only significant effect is the Calcium*Fertilizer interaction.

Output 23.5.3 Tests of Effects
Tests of Hypotheses Using the Anova MS for Rep*Fertilizer as an Error Term
Source DF Anova SS Mean Square F Value Pr > F
Fertilizer 3 7.22127083 2.40709028 3.56 0.0604

Tests of Hypotheses Using the Anova MS for Rep*Calcium(Fertili) as an Error Term
Source DF Anova SS Mean Square F Value Pr > F
Calcium 1 0.27735000 0.27735000 1.88 0.1950
Calcium*Fertilizer 3 1.96395833 0.65465278 4.45 0.0255

Tests of Hypotheses Using the Anova MS for Rep*Soil as an Error Term
Source DF Anova SS Mean Square F Value Pr > F
Soil 2 1.92658958 0.96329479 3.47 0.0999

Tests of Hypotheses Using the Anova MS for Rep*Soil*Fertilizer as an Error Term
Source DF Anova SS Mean Square F Value Pr > F
Soil*Fertilizer 6 0.68828542 0.11471424 1.30 0.3063

Tests of Hypotheses Using the Anova MS for Rep*Soil*Calc(Ferti) as an Error Term
Source DF Anova SS Mean Square F Value Pr > F
Soil*Calcium 2 0.04493125 0.02246562 0.25 0.7843
Soil*Calcium*Fertili 6 0.18936042 0.03156007 0.34 0.9059

Output 23.5.4 Results of MEANS statement
Level of
Fertilizer		 N Yield
Mean Std Dev
0 24 5.18416667 0.48266395
1 24 5.12916667 0.38337082
2 24 5.75458333 0.53293265
3 24 5.64125000 0.63926801

Level of
Calcium		 N Yield
Mean Std Dev
0 48 5.48104167 0.54186141
1 48 5.37354167 0.61565219

Level of
Soil		 N Yield
Mean Std Dev
1 32 5.54312500 0.55806369
2 32 5.51093750 0.62176315
3 32 5.22781250 0.51825224

Level of
Calcium		 Level of
Fertilizer		 N Yield
Mean Std Dev
0 0 12 5.34666667 0.45029956
0 1 12 5.08833333 0.44986530
0 2 12 5.62666667 0.44707806
0 3 12 5.86250000 0.52886027
1 0 12 5.02166667 0.47615569
1 1 12 5.17000000 0.31826233
1 2 12 5.88250000 0.59856077
1 3 12 5.42000000 0.68409197

Output 23.5.4 shows the results of the MEANS statement, displaying for various effects and combinations of effects, as requested. You can examine the Calcium*Fertilizer means to understand the interaction better.

In this example, you could reduce memory requirements by omitting the
Soil*Calcium*Rep(Fertilizer) effect from the model in the MODEL statement. This effect then becomes the ERROR effect, and you can omit the last TEST statement in the statements shown earlier. The test for the Soil*Calcium effect is then given in the Analysis of Variance table in the top portion of output. However, for all other tests, you should look at the results from the TEST statement. In large models, this method might lead to significant reductions in memory requirements.

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