2. 7-2 Factorial Experiments

3. 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.

4. 7-7 Factorial Experiments with More than
Two Levels

5. 7-7 Factorial Experiments with More than Two Levels
Two-Way Factorial Model
𝑌 𝑖𝑗𝑘 =𝜇+ 𝛼 𝑖 + 𝛽 𝑗 + (𝛼𝛽) 𝑖𝑗 + 𝜀 𝑖𝑗𝑘 = 𝜇 𝑖𝑗 + 𝜀 𝑖𝑗𝑘
where
𝜇= 𝑖=1 𝑎 𝑗=1 𝑏 𝜇 𝑖𝑗 𝑎𝑏 =𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑚𝑒𝑎𝑛
𝛼 𝑖 = 𝜇 𝑖∙ −𝜇 = A main effect, i = 1, ∙∙∙∙, 𝑎
𝛽 𝑗 = 𝜇 ∙𝑗 −𝜇 = B main effect, i = 1, ∙∙∙∙, 𝑏
(𝛼𝛽) 𝑖𝑗 = 𝜇 𝑖𝑗 − 𝜇 𝑖∙ − 𝜇 ∙𝑗 +𝜇 = Interaction effect
𝜀 𝑖𝑗𝑘 = Error term, which is assumed to be normally
distributed with constant variance
Hypothesis
A main Effect Hypothesis:
𝐻 0 : 𝜇 1∙ = 𝜇 2∙ =∙∙∙= 𝜇 𝑎∙ 𝑜𝑟 𝐻 0 : 𝛼 1 = 𝛼 2 =∙∙∙= 𝛼 𝑎 =0
B main Effect Hypothesis:
𝐻 0 : 𝜇 ∙1 = 𝜇 ∙2 =∙∙∙= 𝜇 ∙𝑏 𝑜𝑟 𝐻 0 : 𝛽 1 = 𝛽 2 =∙∙∙= 𝛽 𝑏 =0

6. 7-7 Factorial Experiments with More than Two Levels
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 test
Two-way Interaction test:
𝐻 0 : (𝛼𝛽) 11 = (𝛼𝛽) 12 =∙∙∙= 𝛼𝛽 𝑎𝑏 =0
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 𝑋 𝑖𝑗 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.

7. 7-7 Factorial Experiments with More than
Two Levels

8. 7-7 Factorial Experiments with More than Two Levels
Sum of Squares partition:
Degrees of freedom partition:

9. 7-7 Factorial Experiments with More than Two Levels
Mean Squares:

10. 7-7 Factorial Experiments with More than Two Levels
𝑆𝑆𝑀𝑂𝐷=𝑆𝑆𝐴+𝑆𝑆𝐵+𝑆𝑆𝐴𝐵 𝑤𝑖𝑡ℎ 𝑑𝑓=𝑎𝑏−1

11. 7-7 Factorial Experiments with More than Two Levels
𝑆𝑆𝑇= 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 ( 𝑌 𝑖,𝑗,𝑘 − 𝑌 ) 2 = 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 2 − ( 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2
𝑆𝑆𝐴= 𝑖=1 𝑎 𝑛𝑏 ( 𝑌 𝑖,∙,∙ − 𝑌 ) 2 = 𝑖=1 𝑎 ( 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2 − ( 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2
𝑆𝑆𝐵= 𝑖=1 𝑎 𝑛𝑎 ( 𝑌 ∙,𝑗,∙ − 𝑌 ) 2 = 𝑗=1 𝑏 ( 𝑖=1 𝑎 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2 − ( 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2
𝑆𝑆𝐴𝐵=𝑆𝑆𝑀𝑂𝐷−𝑆𝑆𝐴−𝑆𝑆𝐵
𝑆𝑆𝑀𝑂𝐷= 𝑖=1 𝑎 𝑗=1 𝑏 𝑛 ( 𝑌 𝑖,𝑗,∙ − 𝑌 ) 2 = 𝑖=1 𝑎 𝑗=1 𝑏 ( 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2 − ( 𝑖=1 𝑎 𝑗=1 𝑏 𝑘=1 𝑛 𝑌 𝑖,𝑗,𝑘 ) 2
𝑆𝑆𝐸=𝑆𝑆𝑇−𝑆𝑆𝐴−𝑆𝑆𝐵−𝑆𝑆𝐴𝐵
Multiple comparison and contrasts follow the same formulas as in the one-way ANOVA. The difference is that the comparisons are made on the marginal means for factors A & B. The ni are replaced by the number of observations used in calculating the sample mean. Also, they are only meaningful if there is no interaction.

12. 7-7 Factorial Experiments with More than
Two Levels

13. 7-7 Factorial Experiments with More than
Two Levels

14. 7-7 Factorial Experiments with More than
Two Levels

15. 7-7 Factorial Experiments with More than
Two Levels

16. 7-7 Factorial Experiments with More than Two Levels
Model Adequacy

17. 7-7 Factorial Experiments with More than Two Levels
Model Adequacy

18. 7-7 Factorial Experiments with More than Two Levels
Model Adequacy

19. 7-7 Factorial Experiments with More than Two Levels
Computer Output

20. 7-7 Factorial Experiments with More than Two Levels
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;
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;

21. 7-7 Factorial Experiments with More than
Two Levels

22. 7-7 Factorial Experiments with More than
Two Levels

23. 7-7 Factorial Experiments with More than
Two Levels

24. 7-7 Factorial Experiments with More than
Two Levels

25. 7-7 Factorial Experiments with More than
Two Levels

26. 7-7 Factorial Experiments with More than
Two Levels

27. 7-7 Factorial Experiments with More than
Two Levels

28. 7-7 Factorial Experiments with More than
Two Levels

29. 7-7 Factorial Experiments with More than
Two Levels

30. 7-7 Factorial Experiments with More than
Two Levels

31. 7-7 Factorial Experiments with More than Two Levels
Residual Plot
resid*type 도표. 범례: A = 1 관측치, B = 2 관측치, 등.
resid |
0.35 +
| A
0.30 +A A |
0.25 +
| A A
0.20 +
0.15 +
0.10 +A A
0.05 +
0.00 +
| B
A A
type
Residual Plot
resid*method 도표. 범례: A = 1 관측치, B = 2 관측치, 등.
resid |
0.35 +
| A
A A |
0.25 +
| A A
0.20 +
| A
0.15 +
A A
0.05 +
0.00 +
| B
| A A
A A
Dipping Sprayin
method

32. 7-7 Factorial Experiments with More than Two Levels
Exercise
7-49
OPTIONS NOOVP NODATE NONUMBER LS=80;
proc format;
value tm 1='20min' 2='25min' 3='30min';
DATA ex749;
INPUT paint time finish
format time tm.;
cards;
ods graphics on;
PROC GLM DATA=ex749 plots=(diagnostics);
CLASS paint time;
MODEL finish= paint time paint*time;
MEANS paint time paint*time/snk;
OUTPUT out=new p=phat r=resid;
TITLE 'Two-way ANOVA';
PROC PLOT data=new;
PLOT resid*(paint time); /* Residual plots */
TITLE1 'Residual Plot';
run; ods graphics off; QUIT;

33. 7-7 Factorial Experiments with More than
Two Levels

34. 7-7 Factorial Experiments with More than
Two Levels

35. 7-7 Factorial Experiments with More than
Two Levels

36. 7-7 Factorial Experiments with More than
Two Levels

37. 7-7 Factorial Experiments with More than
Two Levels

38. 7-7 Factorial Experiments with More than
Two Levels

39. The Latin Square Design
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.

40. The Latin Square Design

41. The Latin Square Design
Example 5-4

42. 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 -3
2 1 B C D E A 5
3 1 C D E A B -5
4 1 D E A B C 4
5 1 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; QUIT;

43. The Latin Square Design

44. The Latin Square Design

45. The Latin Square Design

46. The Latin Square Design

47. The Latin Square Design

48. The Latin Square Design

49. The Latin Square Design

50. The Latin Square Design

51. The Latin Square Design

52. The Latin Square Design
Residual plot
resid*treat 도표. 범례: A = 1 관측치, B = 2 관측치, 등.
resid | A
| A
4.5 +
| A
4.0 + |
3.5 +
3.0 +
2.5 + A
| A
1.5 + A
0.5 +
A B
| A A
-0.5 +
| A
A B
| A A
-1.5 + A
-2.5 +
| B A
| A
-3.5 +
A B C D E
treat
Residual plot
resid*operator 도표. 범례: A = 1 관측치, B = 2 관측치, 등.
resid | A
| A
4.5 +
| A
4.0 + |
3.5 +
3.0 +
| A
2.5 + A
1.5 + A
| A
0.5 +
A B
| A A
-0.5 +
B A
| B
-1.5 + A
-2.5 +
| A A A
| A
-3.5 +
operator
Residual plot
resid*batch 도표. 범례: A = 1 관측치, B = 2 관측치, 등.
resid | A
| A
4.5 +
| A
4.0 + |
3.5 +
3.0 +
| A
2.5 + A
| A
1.5 +
1.0 + A
| A
0.5 +
A A A
| B
-0.5 +
A A A
| A A
-1.5 + A
-2.5 +
| A A A
-3.5 +
batch