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.