7-2 Factorial Experiments

A significant interaction mask the significance of main effects. Consequently, when interaction is present, the main effects of the factors involved in the interaction may not have much meaning.

7-7Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels Sum of Squares partition: Degrees of freedom partition:

7-7 Factorial Experiments with More than Two Levels Mean Squares:

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

Model Adequacy 7-7 Factorial Experiments with More than Two Levels

Model Adequacy 7-7 Factorial Experiments with More than Two Levels

Model Adequacy 7-7 Factorial Experiments with More than Two Levels

Computer Output 7-7 Factorial Experiments with More than Two Levels

Example Factorial Experiments with More than Two Levels OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex711; DO obs= 1 to 3; DO type=1 to 3; DO method='Dipping', 'Spraying'; INPUT force OUTPUT; END; END;END; CARDS; ods graphics on; PROC GLM DATA=ex711 plots=all; CLASS type method; MODEL force= type method type*method; MEANS type method type*method/snk; OUTPUT out=new r=resid; TITLE 'Two-way ANOVA'; PROC PLOT DATA=NEW; PLOT RESID*TYPE; PLOT RESID*METHOD; run; QUIT;

Two-way ANOVA The GLM Procedure Dependent Variable: force Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE force Mean Source DF Type I SS Mean Square F Value Pr > F type <.0001 method <.0001 type*method Source DF Type III SS Mean Square F Value Pr > F type <.0001 method <.0001 type*method Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

Two-way ANOVA The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Number of Means 2 3 Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N type A B B Two-way ANOVA The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Number of Means 2 Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N method A Sprayin B Dipping 7-7 Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

Two-way ANOVA The GLM Procedure Level of Level of force type method N Mean Std Dev 1 Dipping Sprayin Dipping Sprayin Dipping Sprayin Factorial Experiments with More than Two Levels

Residual Plot resid*type 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | | A A A | | A A | A | A A | | A A | A | | A | B | | A A | | A A | type Residual Plot resid*method 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | | A A A | | A A | A | A A | | A A | A | | A | B | | A A | | A A | Dipping Sprayin method 7-7 Factorial Experiments with More than Two Levels

Speed MaterialFeed B V Thrust Forces in Drilling (3-Way Factorial) 7-7 Factorial Experiments with More than Two Levels

OPTIONS NOOVP NODATE NONUMBER LS=80; DATA threeway; INFILE 'C:\Users\user\Documents\Teaching\ 학부과목 \imen214-stats\ch07\sas\threeway.txt'; INPUT material$ feed speed thrust ods graphics on; PROC glm data=threeway plots=(diagnostics); CLASS material feed speed; MODEL thrust = material | feed | speed; MEANS material | feed | speed/snk; TITLE 'Three-way ANOVA'; DATA means1;INPUT b10 v10 feed PROC PLOT data=means1; PLOT b10*feed='B' v10*feed='V'/overlay; TITLE 'Interaction Plot for Material*Feed'; DATA means2; INPUT b10 v10 speed PROC PLOT data=means2; PLOT b10*speed='B' v10*speed='V'/overlay; TITLE 'Interaction Plot for Speed*Material'; DATA means3; INPUT fd1 fd2 fd3 speed PROC PLOT data=means3; PLOT fd1*speed='1' fd2*speed='2' fd3*speed='3'/overlay; TITLE 'Interaction Plot for Feed*Speed'; RUN; QUIT; 7-7 Factorial Experiments with More than Two Levels B B B B B B B B B B B B

Three-way ANOVA The ANOVA Procedure Dependent Variable: thrust Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE thrust Mean Source DF Anova SS Mean Square F Value Pr > F material <.0001 feed <.0001 material*feed speed material*speed feed*speed material*feed*speed Three-way ANOVA The ANOVA Procedure Class Level Information Class Levels Values material 2 B10 V10 feed speed Number of Observations Read 60 Number of Observations Used Factorial Experiments with More than Two Levels

7-7 Factorial Experiments with More than Two Levels

Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square Number of Means 2 Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N material A V10 B B10 Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square Number of Means 2 3 Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N feed A B C Factorial Experiments with More than Two Levels

Three-way ANOVA The ANOVA Procedure Level of Level of thrust material feed N Mean Std Dev B B B V V V Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N speed A A A A A Factorial Experiments with More than Two Levels

Three-way ANOVA The ANOVA Procedure Level of Level of thrust material speed N Mean Std Dev B B B B B V V V V V Level of Level of thrust feed speed N Mean Std Dev Level of Level of Level of thrust material feed speed N Mean Std Dev B B B B B B B B B B B B B B B V V V V V V V V V V V V V V V Factorial Experiments with More than Two Levels

Interaction Plot for Material*Feed b10*feed 도표. 사용된 기호 : 'B'. v10*feed 도표. 사용된 기호 : 'V'. b10 | V | | | | B | | | V | | B | | | V | B | 0 + | feed Interaction Plot for Speed*Material b10*speed 도표. 사용된 기호 : 'B'. v10*speed 도표. 사용된 기호 : 'V'. b10 | | V | | | | | V | | V | V | V | | B B | | B | | B | speed Interaction Plot for Feed*Speed fd1*speed 도표. 사용된 기호 : '1'. fd2*speed 도표. 사용된 기호 : '2'. fd3*speed 도표. 사용된 기호 : '3'. fd1 | | | | | | | 3 | | | | 2 2 | | | | 1 1 | 0 + | speed 7-7 Factorial Experiments with More than Two Levels

The Latin square design is used to eliminate two nuisance sources of variability; that is, it systemically allows blocking two dimensions. Thus, the rows and columns actually represent two restrictions on randomization. In a Latin square design, there are p treatments and p levels of each of the two blocking variables. Each treatment level appears in each row and column once. The arrangement should be randomly selected from all possible arrangement. For instance, there is only 1 3x3, 4 4x4, 56 5x5, and x6 Latin squares. The Latin Square Design

OPTIONS NOOVP NODATE NONUMBER LS=80; DATA latin; INPUT operator batch treat$ force CARDS; 1 1 A B C D E B C D E A C D E A B D E A B C E A B C D 6 ods graphics on; PROC GLM data=latin plots=(diagnostics); CLASS operator batch treat; MODEL force = operator batch treat; MEANS operator batch treat/snk; output out=new p=phat r=resid; TITLE 'Latin Square Design'; proc plot data=new; plot resid*(operator batch treat)/vaxis= -3.5 to 5.0 by 0.5; Title 'Residual plot'; RUN; ods graphics off; QUIT; The Latin Square Design Example 5-4

Latin Square Design The GLM Procedure Class Level Information Class Levels Values operator batch treat 5 A B C D E Number of Observations Read 25 Number of Observations Used 25 Latin Square Design The GLM Procedure Dependent Variable: force Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE force Mean Source DF Type I SS Mean Square F Value Pr > F operator batch treat Source DF Type III SS Mean Square F Value Pr > F operator batch treat The Latin Square Design

7-7 Factorial Experiments with More than Two Levels

Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N operator A B A B A B A B The Latin Square Design Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N batch A A A A A

7-7 Factorial Experiments with More than Two Levels

Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. SNK Grouping Mean N treat A D A A B A E B C C C B The Latin Square Design

Residual plot resid*batch 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | A | A | A | | | A | A | A | A | A | A A A | B | A A A A | A A | A A | | A A A | A batch Residual plot resid*treat 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | A | A | A | | | A | A | A | A | A | A B | A A | A A B | A A | A A | | B A | A | A B C D E treat The Latin Square Design Residual plot resid*operator 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | A | A | A | | | A | A | A | A | A | A B | A A | A B A | B | A A | | A A A | A operator