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The ANOVA Procedure

Example 17.5: 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 17.5.1, Output 17.5.2, and Output 17.5.3. 

   title '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;
      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 17.5.1: Class Level Information and ANOVA Table

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 96


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 . .

As the model is completely specified by the MODEL statement, the entire top portion of output (Output 17.5.1) should be ignored. Look at the following output produced by the various TEST statements.

Output 17.5.2: Tests of Effects

Strip-split Plot

The ANOVA Procedure
Dependent Variable: Yield

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


The only significant effect is the Calcium*Fertilizer interaction.

Output 17.5.3: Results of MEANS statement

Strip-split Plot

The ANOVA Procedure

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


The final portion of output shows the results of the MEANS statement. This portion shows means for various effects and combinations of effects, as requested. Because no multiple comparison procedures are requested, none are performed. 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 code 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 may lead to significant reductions in memory requirements.

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