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STAT 330: 95-3

SAS example: ANOVA for a two way layout

The data consist of casting hardnesses for 18 samples prepared under 3 levels of sand added and 3 levels of carbon fibre added. See Q 15 in Chapter 11. I use proc anova to test the hypotheses of no effect of either sand content or fibre content after first testing for interactions.

I ran the following SAS code:

 options pagesize=60 linesize=80;
  data plaster;
  infile 'plaster.dat';
  input sand fibre hardness strength;
  proc anova  data=plaster;
   class sand fibre;
   model hardness = sand|fibre;
   means sand fibre / tukey cldiff ;
  run;

The line labelled model says that I am interested in the effects of sand, fibre and interactions between the two. The line class sand fibre is required so that SAS knows which variables define the levels of the factors.

The output from proc anova is

                                 The SAS System                                1
                                                14:05 Tuesday, November 14, 1995

                         Analysis of Variance Procedure
                            Class Level Information

                           Class    Levels    Values

                           SAND          3    0 15 30

                           FIBRE         3    0 25 50


                    Number of observations in data set = 18


                                 The SAS System                                2
                                                14:05 Tuesday, November 14, 1995

                         Analysis of Variance Procedure

Dependent Variable: HARDNESS
                                     Sum of            Mean
Source                  DF          Squares          Square   F Value     Pr > F

Model                    8     202.77777778     25.34722222      3.10     0.0557

Error                    9      73.50000000      8.16666667

Corrected Total         17     276.27777778

                  R-Square             C.V.        Root MSE        HARDNESS Mean

                  0.733963         4.105290       2.8577380            69.611111


Source                  DF         Anova SS     Mean Square   F Value     Pr > F

SAND                     2     106.77777778     53.38888889      6.54     0.0176
FIBRE                    2      87.11111111     43.55555556      5.33     0.0297
SAND*FIBRE               4       8.88888889      2.22222222      0.27     0.8887



                                 The SAS System                                3
                                                14:05 Tuesday, November 14, 1995

                         Analysis of Variance Procedure

          Tukey's Studentized Range (HSD) Test for variable: HARDNESS

          NOTE: This test controls the type I experimentwise error rate.

              Alpha= 0.05  Confidence= 0.95  df= 9  MSE= 8.166667
                   Critical Value of Studentized Range= 3.948
                     Minimum Significant Difference= 4.6066

       Comparisons significant at the 0.05 level are indicated by '***'.

                            Simultaneous            Simultaneous
                                Lower    Difference     Upper
                 SAND        Confidence    Between   Confidence
              Comparison        Limit       Means       Limit

             30   - 15         -2.773       1.833       6.440
             30   - 0           1.227       5.833      10.440   ***

             15   - 30         -6.440      -1.833       2.773
             15   - 0          -0.607       4.000       8.607

             0    - 30        -10.440      -5.833      -1.227   ***
             0    - 15         -8.607      -4.000       0.607


                                 The SAS System                                4
                                                14:05 Tuesday, November 14, 1995

                         Analysis of Variance Procedure

          Tukey's Studentized Range (HSD) Test for variable: HARDNESS

          NOTE: This test controls the type I experimentwise error rate.

              Alpha= 0.05  Confidence= 0.95  df= 9  MSE= 8.166667
                   Critical Value of Studentized Range= 3.948
                     Minimum Significant Difference= 4.6066

       Comparisons significant at the 0.05 level are indicated by '***'.

                            Simultaneous            Simultaneous
                                Lower    Difference     Upper
                FIBRE        Confidence    Between   Confidence
              Comparison        Limit       Means       Limit

             50   - 25         -4.607       0.000       4.607
             50   - 0           0.060       4.667       9.273   ***

             25   - 50         -4.607       0.000       4.607
             25   - 0           0.060       4.667       9.273   ***

             0    - 50         -9.273      -4.667      -0.060   ***
             0    - 25         -9.273      -4.667      -0.060   ***

The conclusions are that both sand and fibre have an effect on hardness but that there is little evidence of an interaction between the two factors. The Tukey procedures show a clear difference between the 0% fibre and the other two levels but not between the last two levels in terms of effect on hardness. The high level of sand clearly differs from the low level but the intermediate level is not clearly distinguished from the other two.

The model can be run assuming the interactions are all 0 by using

 
  options pagesize=60 linesize=80;
  data plaster;
  infile 'plaster.dat';
  input sand fibre hardness strength;
  proc anova  data=plaster;
   class sand fibre;
   model hardness = sand fibre;
   means sand / tukey cldiff alpha=0.01;
   means fibre / tukey cldiff alpha=0.05;
  run;
which produces
                                 The SAS System                                1
                                                13:52 Tuesday, November 14, 1995

                         Analysis of Variance Procedure
                            Class Level Information

                           Class    Levels    Values

                           SAND          3    0 15 30

                           FIBRE         3    0 25 50


                    Number of observations in data set = 18


                         Analysis of Variance Procedure

Dependent Variable: HARDNESS
                                     Sum of            Mean
Source                  DF          Squares          Square   F Value     Pr > F

Model                    4     193.88888889     48.47222222      7.65     0.0021

Error                   13      82.38888889      6.33760684

Corrected Total         17     276.27777778

                  R-Square             C.V.        Root MSE        HARDNESS Mean

                  0.701790         3.616463       2.5174604            69.611111


Source                  DF         Anova SS     Mean Square   F Value     Pr > F

SAND                     2     106.77777778     53.38888889      8.42     0.0045
FIBRE                    2      87.11111111     43.55555556      6.87     0.0092



          Tukey's Studentized Range (HSD) Test for variable: HARDNESS

          NOTE: This test controls the type I experimentwise error rate.

              Alpha= 0.01  Confidence= 0.99  df= 13  MSE= 6.337607
                   Critical Value of Studentized Range= 4.964
                     Minimum Significant Difference= 5.1013

       Comparisons significant at the 0.01 level are indicated by '***'.

                            Simultaneous            Simultaneous
                                Lower    Difference     Upper
                 SAND        Confidence    Between   Confidence
              Comparison        Limit       Means       Limit

             30   - 15         -3.268       1.833       6.935
             30   - 0           0.732       5.833      10.935   ***

             15   - 30         -6.935      -1.833       3.268
             15   - 0          -1.101       4.000       9.101

             0    - 30        -10.935      -5.833      -0.732   ***
             0    - 15         -9.101      -4.000       1.101


                         Analysis of Variance Procedure

          Tukey's Studentized Range (HSD) Test for variable: HARDNESS

          NOTE: This test controls the type I experimentwise error rate.

              Alpha= 0.05  Confidence= 0.95  df= 13  MSE= 6.337607
                   Critical Value of Studentized Range= 3.734
                     Minimum Significant Difference= 3.8378

       Comparisons significant at the 0.05 level are indicated by '***'.

                            Simultaneous            Simultaneous
                                Lower    Difference     Upper
                FIBRE        Confidence    Between   Confidence
              Comparison        Limit       Means       Limit

             50   - 25         -3.838       0.000       3.838
             50   - 0           0.829       4.667       8.504   ***

             25   - 50         -3.838       0.000       3.838
             25   - 0           0.829       4.667       8.504   ***

             0    - 50         -8.504      -4.667      -0.829   ***
             0    - 25         -8.504      -4.667      -0.829   ***

In this case the multiple comparison procedures lead to the same conclusions, even using the harsher 0.01 level. The overall F-tests still conclude that both sand and fibre have an effect. Notice that the tests now have more degrees of freedom for error since no interaction effects are fitted; the corresponding interaction sum of squares has been rolled in with the error sum of squares.





Richard Lockhart
Mon Oct 21 23:29:16 PDT 1996