1. Statistical Analysis Professor Richard F. Gunst Department of Statistical Science Lecture 16 Analysis of Data from Unbalanced Experiments

2. Mason, Gunst, & Hess: Table 8.1 2 The ANOVA Procedure Dependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 6 1682.400000 280.400000 46.73 0.0047 Error 3 18.000000 6.000000 Corrected Total 9 1700.400000 R-Square Coeff Var Root MSE yield Mean 0.989414 3.912923 2.449490 62.60000 Source DF Anova SS Mean Square F Value Pr > F temperature 1 1440.000000 1440.000000 240.00 0.0006 concentration 1 64.066667 64.066667 10.68 0.0469 temperatu*concentrat 1 41.666667 41.666667 6.94 0.0780 catalyst 1 194.400000 194.400000 32.40 0.0107 temperature*catalyst 1 0.000000 0.000000 0.00 1.0000 concentrati*catalyst 1 0.000000 0.000000 0.00 1.0000 temper*concen*cataly 0 0.000000... Compare with Table 8.1(c)

3. Mason, Gunst, & Hess: Table 8.1 6 The GLM Procedure Dependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 6 1682.400000 280.400000 46.73 0.0047 Error 3 18.000000 6.000000 Corrected Total 9 1700.400000 R-Square Coeff Var Root MSE yield Mean 0.989414 3.912923 2.449490 62.60000 Source DF Type I SS Mean Square F Value Pr > F temperature 1 1440.000000 1440.000000 240.00 0.0006 concentration 1 64.066667 64.066667 10.68 0.0469 temperatu*concentrat 1 41.666667 41.666667 6.94 0.0780 catalyst 1 21.878788 21.878788 3.65 0.1522 temperature*catalyst 1 114.502165 114.502165 19.08 0.0222 concentrati*catalyst 1 0.285714 0.285714 0.05 0.8413 temper*concen*cataly 0 0.000000... Source DF Type III SS Mean Square F Value Pr > F temperature 1 1120.666667 1120.666667 186.78 0.0008 concentration 1 40.333333 40.333333 6.72 0.0809 temperatu*concentrat 1 8.333333 8.333333 1.39 0.3236 catalyst 1 14.516129 14.516129 2.42 0.2177 temperature*catalyst 1 85.333333 85.333333 14.22 0.0326 concentrati*catalyst 1 0.285714 0.285714 0.05 0.8413 temper*concen*cataly 0 0.000000... Compare with Table 8.1(c) Order is Important

4. The GLM Procedure Dependent Variable: yield Sum of Source DF Squares Mean Square F Value Pr > F Model 6 1682.400000 280.400000 46.73 0.0047 Error 3 18.000000 6.000000 Corrected Total 9 1700.400000 R-Square Coeff Var Root MSE yield Mean 0.989414 3.912923 2.449490 62.60000 Source DF Type I SS Mean Square F Value Pr > F temperature 1 1440.000000 1440.000000 240.00 0.0006 concentration 1 64.066667 64.066667 10.68 0.0469 catalyst 1 16.351351 16.351351 2.73 0.1973 temperatu*concentrat 1 47.194103 47.194103 7.87 0.0676 temperature*catalyst 1 114.502165 114.502165 19.08 0.0222 concentrati*catalyst 1 0.285714 0.285714 0.05 0.8413 Source DF Type III SS Mean Square F Value Pr > F temperature 1 736.3333333 736.3333333 122.72 0.0016 concentration 1 40.0000000 40.0000000 6.67 0.0816 catalyst 1 2.5714286 2.5714286 0.43 0.5594 temperatu*concentrat 1 8.3333333 8.3333333 1.39 0.3236 temperature*catalyst 1 85.3333333 85.3333333 14.22 0.0326 concentrati*catalyst 1 0.2857143 0.2857143 0.05 0.8413 Compare with Table 8.1(c)

5. Fleet Fuel Comparisons CO 2 Composite Emissions (g/mi) Apparent Conclusion Very Large Difference in Average Emissions Between Conventional and California 150 ppm Sulfur Fuels Apparent Conclusion Very Large Difference in Average Emissions Between Conventional and California 150 ppm Sulfur Fuels

6. Fleet Fuel Comparisons CO 2 Composite Emissions (g/mi) Correct Conclusion Very Small Difference in Average Emissions Between Conventional and California 150 ppm Sulfur Fuels Correct Conclusion Very Small Difference in Average Emissions Between Conventional and California 150 ppm Sulfur Fuels

7. Multiple Comparisons Unweighted Averages Should be Avoided Some Averages Have More Data Values than Others Outliers Can be Very Influential LSMEANS for Nonmissing Averages Based on Population Marginal Means Adjust=Bon, Tukey Pdiff

8. The GLM Procedure Level of Level of ------------yield------------ temperature catalyst N Mean Std Dev 160 c1 1 59.0000000. 160 c2 4 48.5000000 4.43471157 180 c1 3 71.0000000 2.64575131 180 c2 2 80.0000000 1.41421356 Means Statement LSmeans Statement The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Bonferroni LSMEAN temperature catalyst yield LSMEAN Number 160 c1 55.0000000 1 160 c2 48.5000000 2 180 c1 70.5000000 3 180 c2 80.0000000 4 Least Squares Means for effect temperature*catalyst Pr > |t| for H0: LSMean(i)=LSMean(j) Dependent Variable: yield i/j 1 2 3 4 1 1.0000 0.1461 0.0485 2 1.0000 0.0088 0.0040 3 0.1461 0.0088 0.1529 4 0.0485 0.0040 0.1529

9. proc glm data=tabl0801; class temperature concentration catalyst; model yield = temperature concentration catalyst temperature*concentration temperature*catalyst concentration*catalyst; means temperature*catalyst / Bon ; lsmeans temperature*catalyst / adjust=Bon pdiff=all ; run;