Statistical Analysis

Professor Richard F. Gunst

Department of Statistical Science

Lecture 16

Analysis of Data from Unbalanced Experiments

Mason, Gunst, & Hess: Table 8.1 2  The ANOVA Procedure Dependent Variable: yield  Sum ofSource 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 0.0006concentration 1 64.066667 64.066667 10.68 0.0469temperatu*concentrat 1 41.666667 41.666667 6.94 0.0780catalyst 1 194.400000 194.400000 32.40 0.0107temperature*catalyst 1 0.000000 0.000000 0.00 1.0000concentrati*catalyst 1 0.000000 0.000000 0.00 1.0000temper*concen*cataly 0 0.000000 . . .

Compare withCompare withTable 8.1(c)Table 8.1(c)

Mason, Gunst, & Hess: Table 8.1 6  The GLM Procedure Dependent Variable: yield  Sum ofSource 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 0.0006concentration 1 64.066667 64.066667 10.68 0.0469temperatu*concentrat 1 41.666667 41.666667 6.94 0.0780catalyst 1 21.878788 21.878788 3.65 0.1522temperature*catalyst 1 114.502165 114.502165 19.08 0.0222concentrati*catalyst 1 0.285714 0.285714 0.05 0.8413temper*concen*cataly 0 0.000000 . . .  Source DF Type III SS Mean Square F Value Pr > F temperature 1 1120.666667 1120.666667 186.78 0.0008concentration 1 40.333333 40.333333 6.72 0.0809temperatu*concentrat 1 8.333333 8.333333 1.39 0.3236catalyst 1 14.516129 14.516129 2.42 0.2177temperature*catalyst 1 85.333333 85.333333 14.22 0.0326concentrati*catalyst 1 0.285714 0.285714 0.05 0.8413temper*concen*cataly 0 0.000000 . . .

Compare withCompare withTable 8.1(c)Table 8.1(c)

Order isOrder isImportantImportant

The GLM Procedure Dependent Variable: yield  Sum ofSource 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 0.0006concentration 1 64.066667 64.066667 10.68 0.0469catalyst 1 16.351351 16.351351 2.73 0.1973temperatu*concentrat 1 47.194103 47.194103 7.87 0.0676temperature*catalyst 1 114.502165 114.502165 19.08 0.0222concentrati*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 0.0016concentration 1 40.0000000 40.0000000 6.67 0.0816catalyst 1 2.5714286 2.5714286 0.43 0.5594temperatu*concentrat 1 8.3333333 8.3333333 1.39 0.3236temperature*catalyst 1 85.3333333 85.3333333 14.22 0.0326concentrati*catalyst 1 0.2857143 0.2857143 0.05 0.8413

Compare withCompare withTable 8.1(c)Table 8.1(c)

Fleet Fuel Comparisons

Model Fuel C3 Fuel S1 DifferenceTaurus 413.6 (7) 416.0 (4) Escort 315.3 (4) 319.1 (4) Civic 291.1 (4) 290.4 (4) Sentra 311.7 (4) 309.6 (5) Camry 358.6 (4) 359.8 (4) GeoMetro 281.1 (4) 280.3 (5) Average 338.0 326.6 11.4

CO2 Composite Emissions (g/mi)

Apparent ConclusionVery Large Difference in Average Emissions

Between Conventional and California150 ppm Sulfur Fuels

Apparent ConclusionVery Large Difference in Average Emissions

Between Conventional and California150 ppm Sulfur Fuels

Fleet Fuel Comparisons

Model Fuel C3 Fuel S1 DifferenceTaurus 413.6 (7) 416.0 (4) -2Escort 315.3 (4) 319.1 (4) -4Civic 291.1 (4) 290.4 (4) 1Sentra 311.7 (4) 309.6 (5) 2Camry 358.6 (4) 359.8 (4) -1GeoMetro 281.1 (4) 280.3 (5) 1Average 338.0 326.6 11.4LS Mean 328.6 329.2 -0.6Avg. of Differences -0.6

CO2 Composite Emissions (g/mi)

Correct ConclusionVery Small Difference in Average Emissions

Between Conventional and California150 ppm Sulfur Fuels

Correct ConclusionVery Small Difference in Average Emissions

Between Conventional and California150 ppm Sulfur Fuels

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

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 StatementMeans Statement

LSmeans StatementLSmeans 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

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;