7-2 Factorial Experiments

7-2 Factorial Experiments

7-2 Factorial Experiments

7-9 Factorial Experiments with More than Two Levels


Two-Way Factorial Model

where

= A main effect, i = 1, = B main effect, i = 1,

= Interaction effect= Error term, which is assumed to be normally

distributed with constant variance

HypothesisA main Effect Hypothesis:

B main Effect Hypothesis:


Interaction

Notice that the main effects are in terms of marginal means (means average over the other factor). It makes sense to do this only if the relationship between the means of one factor are the same for all levels of the other factor. If this is true, the factors are said not to interact. Before we can interpret the main effect tests, we must verify that the factors do not interact. That is to test

Two-way Interaction test:

A graphical means of assessing interaction is to make an interaction (profile) plot. This consists plotting one of the factors along the horizontal axis and the the vertical axis. The points corresponding to the same level of the other factor are connected by a line. No interaction implies that the lines will be parallel.


𝑆𝑆𝑀𝑂𝐷=𝑆𝑆𝐴+𝑆𝑆𝐵+𝑆𝑆𝐴𝐵 h𝑤𝑖𝑡 𝑑𝑓 =𝑎𝑏−1








Model Adequacy


Model Adequacy


Model Adequacy


Computer Output


Example 7-11 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;4 5.4 5.6 5.8 3.8 5.5 4.5 4.9 4.96.1 3.7 5 4.3 5.6 5.4 6.3 4 5PROC GLM DATA=ex711; CLASS type method; MODEL force= type method type*method; MEANS type method type*method/snk; OUTPUT out=new p=phat r=resid;TITLE 'Two-way ANOVA';DATA MEANS;INPUT Dipping Spraying type @@;CARDS;4.26 5.30 1 5.30 6.07 2 3.83 5.17 3PROC PLOT data=means; PLOT Dipping*type='0' Spraying*type='1'/overlay vaxis=3.5 to 6.5 by 0.5; TITLE 'Interaction Plot';PROC PLOT data=new; PLOT resid*phat/vaxis=-0.4 to 0.35 by 0.05; /* Residual Plot */ PLOT resid*type/vaxis=-0.4 to 0.35 by 0.05; /* Residual by Primer Type */ PLOT resid*method/vaxis=-0.4 to 0.35 by 0.05; /* Residuals by Application Method */ TITLE ‘Residual Plot';RUN; QUIT;


Two-way ANOVA The GLM Procedure

Dependent Variable: force Sum of Source DF Squares Mean Square F Value Pr > F Model 5 9.73111111 1.94622222 23.67 <.0001 Error 12 0.98666667 0.08222222 Corrected Total 17 10.71777778

R-Square Coeff Var Root MSE force Mean 0.907941 5.747656 0.286744 4.988889

Source DF Type I SS Mean Square F Value Pr > F type 2 4.58111111 2.29055556 27.86 <.0001 method 1 4.90888889 4.90888889 59.70 <.0001 type*method 2 0.24111111 0.12055556 1.47 0.2693

Source DF Type III SS Mean Square F Value Pr > F type 2 4.58111111 2.29055556 27.86 <.0001 method 1 4.90888889 4.90888889 59.70 <.0001 type*method 2 0.24111111 0.12055556 1.47 0.2693



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 0.082222

Number of Means 2 3 Critical Range 0.3607064 0.4416697

Means with the same letter are not significantly different.

SNK Grouping Mean N type A 5.6833 6 2 B 4.7833 6 1 B 4.5000 6 3


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.


Number of Means 2 Critical Range 0.2945156


SNK Grouping Mean N method A 5.5111 9 Sprayin B 4.4667 9 Dipping



Level of Level of ------------force------------ type method N Mean Std Dev

1 Dipping 3 4.26666667 0.25166115 1 Sprayin 3 5.30000000 0.36055513 2 Dipping 3 5.30000000 0.36055513 2 Sprayin 3 6.06666667 0.25166115 3 Dipping 3 3.83333333 0.15275252 3 Sprayin 3 5.16666667 0.28867513


Interaction Plot

Dipping*type 도표 . 사용된 기호 : '0'. Spraying*type 도표 . 사용된 기호 : '1'.

Dipping | 6.5 + | | | | 1 6.0 + | | | | 5.5 + | |1 0 | | 1 5.0 + | | | | 4.5 + | |0 | | 4.0 + | 0 | | | 3.5 + | -+----------------------------------+----------------------------------+ 1 2 3

type

Residual Plot

resid*phat 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid |0.35 + | A0.30 + B | 0.25 + | A A0.20 + | A 0.15 + | 0.10 + B | 0.05 + | A A0.00 + | A-0.05 + |-0.10 + | A-0.15 + | B-0.20 + |-0.25 + | A A-0.30 + |-0.35 +|-0.40 + B | -+-----------+-----------+-----------+-----------+-----------+-----------+ 3.5 4.0 4.5 5.0 5.5 6.0 6.5

phat


Residual Plot

resid*type 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid |0.35 + | A0.30 +A A |0.25 + | A A0.20 + | A 0.15 + | 0.10 +A A | 0.05 + | A A0.00 + | A-0.05 + |-0.10 + | A-0.15 + | B-0.20 + |-0.25 + | A A-0.30 + |-0.35 + |-0.40 +A A | -+-----------------------------------+-----------------------------------+ 1 2 3

type

Residual Plot

resid*method 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid | 0.35 + | A 0.30 + A A | 0.25 + | A A 0.20 + | A 0.15 + | 0.10 + A A | 0.05 + | A A 0.00 + | A -0.05 + | -0.10 + | A -0.15 + | B -0.20 + | -0.25 + | A A -0.30 + | -0.35 + | -0.40 + A A | ---+------------------------------------------------+-- Dipping Sprayin

method


Speed

Material Feed 100 220 475 715 870

B10 0.004 122110

10885

10860

6650

8060

0.008 332330

276310

248295

248275

276310

0.014 640500

612500

543450

612610

696610

V10 0.004 192170

136130

12285

10875

13675

0.008 386365

333330

318330

473350

499390

0.014 810725

779670

810750

893890

1820890

Thrust Forces in Drilling (3-Way Factorial)


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA threeway;INFILE 'C:\users\myung\Documents\Teaching\threeway.dat';INPUT material$ feed speed thrust @@;PROC ANOVA data=threeway; CLASS material feed speed; MODEL thrust = material | feed | speed; MEANS material | feed | speed/snk;TITLE 'Three-way ANOVA';

DATA means1; INPUT b10 v10 feed @@; CARDS;84.9 122.9 0.004 290 377.3 0.008 577.3 903.7 0.014PROC PLOT data=means1; PLOT b10*feed='B' v10*feed='V'/overlay;TITLE 'Interaction Plot for Material*Feed';

DATA means2; INPUT b10 v10 speed @@; CARDS;339 441.3 100 315.2 396.3 220 284 402.5 475 310.2 464.7 715 338.7 635 870PROC PLOT data=means2; PLOT b10*speed='B' v10*speed='V'/overlay;TITLE 'Interaction Plot for Speed*Material';

DATA means3; INPUT fd1 fd2 fd3 speed @@; CARDS;148.5 353.3 668.3 100 114.8 312.3 640.3 220 93.8 297.8 638.3 47574.8 336.3 751.3 715 87.8 368.8 1004 870PROC PLOT data=means3; PLOT fd1*speed='1' fd2*speed='2' fd3*speed='3'/overlay;TITLE 'Interaction Plot for Feed*Speed';RUN; QUIT;


Three-way ANOVA The ANOVA ProcedureDependent Variable: thrust Sum of Source DF Squares Mean Square F Value Pr > F Model 29 5343804.483 184269.120 11.30 <.0001 Error 30 489376.500 16312.550 Corrected Total 59 5833180.983

R-Square Coeff Var Root MSE thrust Mean 0.916105 32.52509 127.7206 392.6833

Source DF Anova SS Mean Square F Value Pr > F material 1 340054.817 340054.817 20.85 <.0001 feed 2 4157143.633 2078571.817 127.42 <.0001 material*feed 2 237956.433 118978.217 7.29 0.0026 speed 4 152472.233 38118.058 2.34 0.0781 material*speed 4 88303.433 22075.858 1.35 0.2735 feed*speed 8 255092.367 31886.546 1.95 0.0880 material*feed*speed 8 112781.567 14097.696 0.86 0.5565

Three-way ANOVA The ANOVA Procedure Class Level Information Class Levels Values material 2 B10 V10 feed 3 0.004 0.008 0.014 speed 5 100 220 475 715 870

Number of Observations Read 60 Number of Observations Used 60-------------------------------------------------------------------------------


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.


Number of Means 2 Critical Range 67.348663


SNK Grouping Mean N material A 467.97 30 V10 B 317.40 30 B10


Student-Newman-Keuls Test for thrust



Number of Means 2 3 Critical Range 82.48493 99.569343


SNK Grouping Mean N feed A 740.50 20 0.014 B 333.65 20 0.008 C 103.90 20 0.004



Level of Level of ------------thrust----------- material feed N Mean Std Dev B10 0.004 10 84.900000 25.631794 B10 0.008 10 290.000000 30.430248 B10 0.014 10 577.300000 75.938645 V10 0.004 10 122.900000 38.922858 V10 0.008 10 377.300000 62.241555 V10 0.014 10 903.700000 330.551408-----------------------------------------------------------------------------


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.


Number of Means 2 3 4 5 Critical Range 106.48759 128.54347 141.77902 151.24267


SNK Grouping Mean N speed A 486.83 12 870 A 390.17 12 100 A 387.42 12 715 A 355.75 12 220 A 343.25 12 475



Level of Level of ------------thrust----------- material speed N Mean Std Dev

B10 100 6 339.000000 207.933643 B10 220 6 315.166667 209.632456 B10 475 6 284.000000 188.258333 B10 715 6 310.166667 250.363269 B10 870 6 338.666667 264.809869 V10 100 6 441.333333 268.722658 V10 220 6 396.333333 271.457302 V10 475 6 402.500000 309.417356 V10 715 6 464.666667 362.389661 V10 870 6 635.000000 649.672533

Level of Level of ------------thrust----------- feed speed N Mean Std Dev

0.004 100 4 148.50000 38.897301 0.004 220 4 114.75000 23.200216 0.004 475 4 93.75000 27.183022 0.004 715 4 74.75000 24.459150 0.004 870 4 87.75000 33.270357 0.008 100 4 353.25000 27.097048 0.008 220 4 312.25000 26.234519 0.008 475 4 297.75000 36.206583 0.008 715 4 336.25000 100.260910 0.008 870 4 368.75000 99.113992 0.014 100 4 668.75000 132.185161 0.014 220 4 640.25000 116.339661 0.014 475 4 638.25000 169.800618 0.014 715 4 751.25000 161.953440 0.014 870 4 1004.00000 556.462637

Level of Level of Level of ------------thrust----------- material feed speed N Mean Std Dev

B10 0.004 100 2 116.00000 8.485281 B10 0.004 220 2 96.50000 16.263456 B10 0.004 475 2 84.00000 33.941125 B10 0.004 715 2 58.00000 11.313708 B10 0.004 870 2 70.00000 14.142136 B10 0.008 100 2 331.00000 1.414214 B10 0.008 220 2 293.00000 24.041631 B10 0.008 475 2 271.50000 33.234019 B10 0.008 715 2 261.50000 19.091883 B10 0.008 870 2 293.00000 24.041631 B10 0.014 100 2 570.00000 98.994949 B10 0.014 220 2 556.00000 79.195959 B10 0.014 475 2 496.50000 65.760931 B10 0.014 715 2 611.00000 1.414214 B10 0.014 870 2 653.00000 60.811183 V10 0.004 100 2 181.00000 15.556349 V10 0.004 220 2 133.00000 4.242641 V10 0.004 475 2 103.50000 26.162951 V10 0.004 715 2 91.50000 23.334524 V10 0.004 870 2 105.50000 43.133514 V10 0.008 100 2 375.50000 14.849242 V10 0.008 220 2 331.50000 2.121320 V10 0.008 475 2 324.00000 8.485281 V10 0.008 715 2 411.00000 86.267027 V10 0.008 870 2 444.50000 77.074639 V10 0.014 100 2 767.50000 60.104076 V10 0.014 220 2 724.50000 77.074639 V10 0.014 475 2 780.00000 42.426407 V10 0.014 715 2 891.50000 2.121320 V10 0.014 870 2 1355.00000 657.609307


Interaction Plot for Material*Feed b10*feed 도표 . 사용된 기호 : 'B'. v10*feed 도표 . 사용된 기호 : 'V'.b10 |900 + V | | |800 + | | |700 + | | |600 + | B | |500 + | | |400 + | V | |300 + | B | |200 + | | | V100 + | B | | 0 + | ---+-------------+-------------+-------------+-------------+-------------+-- 0.004 0.006 0.008 0.010 0.012 0.014

feed

Interaction Plot for Speed*Material b10*speed 도표 . 사용된 기호 : 'B'. v10*speed 도표 . 사용된 기호 : 'V'.b10 |650 + | V | |600 + | | |550 + | | |500 + | | | V |450 + | V | | |400 + V | V | | |350 + | B B | | B | B300 + | | B | |250 + ---+-------------+-------------+-------------+-------------+-------------+-- 0 200 400 600 800 1000

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 | |1000 + 3 | | 900 + | | 800 + | | 3700 + | 3 | 3 3600 + | | 500 + | | 400 + | 2 2 | 2 | 2 300 + 2 | | 200 + | 1 | 1 100 + 1 | 1 1 | | | 0 + | --+-------------+-------------+-------------+-------------+-------------+-- 0 200 400 600 800 1000

speed

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 9408 6x6 Latin squares.

The Latin Square Design


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA latin;INPUT operator batch treat$ force @@;CARDS;1 1 A -1 1 2 B -8 1 3 C -7 1 4 D 1 1 5 E -32 1 B -5 2 2 C -1 2 3 D 13 2 4 E 6 2 5 A 53 1 C -6 3 2 D 5 3 3 E 1 3 4 A 1 3 5 B -54 1 D -1 4 2 E 2 4 3 A 2 4 4 B -2 4 5 C 4 5 1 E -1 5 2 A 11 5 3 B -4 5 4 C -3 5 5 D 6PROC GLM data=latin; 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*(phat operator batch treat); Title 'Residual plot'; RUN; QUIT;


Example 5-4

Latin Square Design The GLM Procedure Class Level Information

Class Levels Values operator 5 1 2 3 4 5 batch 5 1 2 3 4 5 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 12 548.0000000 45.6666667 4.28 0.0089 Error 12 128.0000000 10.6666667 Corrected Total 24 676.0000000

R-Square Coeff Var Root MSE force Mean 0.810651 816.4966 3.265986 0.400000

Source DF Type I SS Mean Square F Value Pr > F operator 4 150.0000000 37.5000000 3.52 0.0404 batch 4 68.0000000 17.0000000 1.59 0.2391 treat 4 330.0000000 82.5000000 7.73 0.0025

Source DF Type III SS Mean Square F Value Pr > F operator 4 150.0000000 37.5000000 3.52 0.0404 batch 4 68.0000000 17.0000000 1.59 0.2391 treat 4 330.0000000 82.5000000 7.73 0.0025


Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force





SNK Grouping Mean N operator A 3.600 5 2 B A 1.800 5 5 B A 1.000 5 4 B A -0.800 5 3 B -3.600 5 1







SNK Grouping Mean N batch A 1.800 5 2 A 1.400 5 5 A 1.000 5 3 A 0.600 5 4 A -2.800 5 1






SNK Grouping Mean N treat A 4.800 5 D A 3.600 5 A B A 1.000 5 E B C -2.600 5 C C -4.800 5 B


resid*phat 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .resid |5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A A A | A A -0.5 + | A -1.0 + A A A | A A -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + | --+---------+---------+---------+---------+---------+---------+---------+- -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

phat

Residual plot resid*operator 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 . resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A B | A A -0.5 + | A -1.0 + B A | B -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + ---+--------------+--------------+--------------+--------------+-- 1 2 3 4 5

operator


Residual plot resid*batch 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A A A | B -0.5 + | A -1.0 + A A A | A A -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + ---+--------------+--------------+--------------+--------------+-- 1 2 3 4 5 batch

Residual plot resid*treat 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A B | A A -0.5 + | A -1.0 + A B | A A -1.5 + | A -2.0 + A | -2.5 + | B -3.0 + A | A -3.5 + | ---+--------------+--------------+--------------+--------------+-- A B C D E treat


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