These are data from a 2 by 4 factorial design. The variable y is the dependent variable. The variable a is an independent variable with two levels while b is an independent variable with four levels.data crf24; input y a b; cards; 3 1 1 4 1 2 7 1 3 7 1 4 1 2 1 2 2 2 5 2 3 10 2 4 6 1 1 5 1 2 8 1 3 8 1 4 2 2 1 3 2 2 6 2 3 10 2 4 3 1 1 4 1 2 7 1 3 9 1 4 2 2 1 4 2 2 5 2 3 9 2 4 3 1 1 3 1 2 6 1 3 8 1 4 2 2 1 3 2 2 6 2 3 11 2 4 ; run;
It is quite clear that there is a significant overall F for the independent variable b. Now let's devise some contrast that we can test:proc glm data = crf24; class b; model y = b; run; quit;The GLM Procedure Class Level Information Class Levels Values b 4 1 2 3 4 Number of observations 32 Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 3 194.5000000 64.8333333 44.28 <.0001 Error 28 41.0000000 1.4642857 Corrected Total 31 235.5000000 R-Square Coeff Var Root MSE y Mean 0.825902 22.51306 1.210077 5.375000 Source DF Type I SS Mean Square F Value Pr > F b 3 194.5000000 64.8333333 44.28 <.0001 Source DF Type III SS Mean Square F Value Pr > F b 3 194.5000000 64.8333333 44.28 <.0001proc means data = crf24 mean; class b; var y; run;The MEANS Procedure Analysis Variable : y N b Obs Mean ----------------------------------- 1 8 2.7500000 2 8 3.5000000 3 8 6.2500000 4 8 9.0000000 -----------------------------------
proc glm data = 'd:\crf24'; class b; model y = b; means b /deponly; contrast 'Compare 3rd & 4th grp' b 0 0 1 -1; contrast 'Compare 1st & 2nd with 3rd & 4th grp' b 1 1 -1 -1; contrast 'Compare 1st, 2nd & 3rd grps with 4th grp' b 1 1 1 -3; run; quit;The GLM Procedure Class Level Information Class Levels Values b 4 1 2 3 4 Number of observations 32 Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 3 194.5000000 64.8333333 44.28 <.0001 Error 28 41.0000000 1.4642857 Corrected Total 31 235.5000000 R-Square Coeff Var Root MSE y Mean 0.825902 22.51306 1.210077 5.375000 Source DF Type I SS Mean Square F Value Pr > F b 3 194.5000000 64.8333333 44.28 <.0001 Source DF Type III SS Mean Square F Value Pr > F b 3 194.5000000 64.8333333 44.28 <.0001 Level of --------------y-------------- b N Mean Std Dev 1 8 2.75000000 1.48804762 2 8 3.50000000 0.92582010 3 8 6.25000000 1.03509834 4 8 9.00000000 1.30930734 Dependent Variable: y Contrast DF Contrast SS Mean Square F Value Pr > F Compare 3rd & 4th grp 1 30.2500000 30.2500000 20.66 <.0001 Compare 1st & 2nd with 3rd & 4th grp 1 162.0000000 162.0000000 110.63 <.0001 Compare 1st, 2nd & 3rd grps with 4th grp 1 140.1666667 140.1666667 95.72 <.0001
proc glm data = 'd:\crf24'; class a b; model y = a b a*b; contrast 'Compare 3rd & 4th grp' b 0 0 1 -1; contrast 'Compare 1st & 2nd with 3rd & 4th grp' b 1 1 -1 -1; contrast 'Compare 1st, 2nd & 3rd grps with 4th grp' b 1 1 1 -3; run; quit;The GLM Procedure Class Level Information Class Levels Values a 2 1 2 b 4 1 2 3 4 Number of observations 32 Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 7 217.0000000 31.0000000 40.22 <.0001 Error 24 18.5000000 0.7708333 Corrected Total 31 235.5000000 R-Square Coeff Var Root MSE y Mean 0.921444 16.33435 0.877971 5.375000 Source DF Type I SS Mean Square F Value Pr > F a 1 3.1250000 3.1250000 4.05 0.0554 b 3 194.5000000 64.8333333 84.11 <.0001 a*b 3 19.3750000 6.4583333 8.38 0.0006 Source DF Type III SS Mean Square F Value Pr > F a 1 3.1250000 3.1250000 4.05 0.0554 b 3 194.5000000 64.8333333 84.11 <.0001 a*b 3 19.3750000 6.4583333 8.38 0.0006 Contrast DF Contrast SS Mean Square F Value Pr > F Compare 3rd & 4th grp 1 30.2500000 30.2500000 39.24 <.0001 Compare 1st & 2nd with 3rd & 4th grp 1 162.0000000 162.0000000 210.16 <.0001 Compare 1st, 2nd & 3rd grps with 4th grp 1 140.1666667 140.1666667 181.84 <.0001
Note that the F-ratios in these contrasts are larger than the F-ratios in the one-way ANOVA example. This is because the two-way ANOVA has a smaller mean square residual than the one-way ANOVA.
The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.