Factorial Experiments

1. Factorial Experiments Analysis of Variance Experimental Design

2. Dependent variable Y • k Categorical independent variables A, B, C, … (the Factors) • Let • a = the number of categories of A • b = the number of categories of B • c = the number of categories of C • etc.

3. The Completely Randomized Design • We form the set of all treatment combinations – the set of all combinations of the k factors • Total number of treatment combinations • t = abc…. • In the completely randomized design n experimental units (test animals , test plots, etc. are randomly assigned to each treatment combination. • Total number of experimental units N = nt=nabc..

4. The treatment combinations can thought to be arranged in a k-dimensional rectangular block B 1 2 b 1 2 A a

5. C B A

6. Another way of representing the treatment combinations in a factorial experiment C B ... A ... D

7. Example In this example we are examining the effect of The level of protein A (High or Low) and The source of protein B (Beef, Cereal, or Pork) on weight gainsY (grams) in rats. We have n = 10 test animals randomly assigned to k = 6 diets

8. The k = 6 diets are the 6 = 3×2 Level-Source combinations • High - Beef • High - Cereal • High - Pork • Low - Beef • Low - Cereal • Low - Pork

9. Table Gains in weight (grams) for rats under six diets differing in level of protein (High or Low) and s ource of protein (Beef, Cereal, or Pork) Level of Protein High Protein Low protein Source of Protein Beef Cereal Pork Beef Cereal Pork Diet 1 2 3 4 5 6 73 98 94 90 107 49 102 74 79 76 95 82 118 56 96 90 97 73 104 111 98 64 80 86 81 95 102 86 98 81 107 88 102 51 74 97 100 82 108 72 74 106 87 77 91 90 67 70 117 86 120 95 89 61 111 92 105 78 58 82 Mean 100.0 85.9 99.5 79.2 83.9 78.7 Std. Dev. 15.14 15.02 10.92 13.89 15.71 16.55

10. Example – Four factor experiment Four factors are studied for their effect on Y (luster of paint film). The four factors are: 1) Film Thickness - (1 or 2 mils) 2) Drying conditions (Regular or Special) 3) Length of wash (10,30,40 or 60 Minutes), and 4) Temperature of wash (92 ˚C or 100 ˚C) Two observations of film luster (Y) are taken for each treatment combination

11. The data is tabulated below: Regular Dry Special Dry Minutes 92 C 100 C 92C 100 C 1-mil Thickness 20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.4 30 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.3 40 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.8 60 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9 2-mil Thickness 20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.5 30 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.8 40 5.5 5.6 30.5 30.2 5.5 5.8 32.6 27.4 60 7.2 6.0 31.4 29.6 8.0 9.9 33.5 29.5

12. Notation Let the single observations be denoted by a single letter and a number of subscripts yijk…..l The number of subscripts is equal to: (the number of factors) + 1 1st subscript = level of first factor 2nd subscript = level of 2nd factor … Last subsrcript denotes different observations on the same treatment combination

13. Notation for Means When averaging over one or several subscripts we put a “bar” above the letter and replace the subscripts by • Example: y241• •

14. Profile of a Factor Plot of observations means vs. levels of the factor. The levels of the other factors may be held constant or we may average over the other levels

15. Definition: A factor is said to not affect the response if the profile of the factor is horizontal for all combinations of levels of the other factors: No change in the response when you change the levels of the factor (true for all combinations of levels of the other factors) Otherwise the factor is said to affect the response:

16. Definition: • Two (or more) factors are said to interact if changes in the response when you change the level of one factor depend on the level(s) of the other factor(s). • Profiles of the factor for different levels of the other factor(s) are not parallel • Otherwise the factors are said to be additive . • Profiles of the factor for different levels of the other factor(s) are parallel.

17. If two (or more) factors interact each factor effects the response. • If two (or more) factors are additive it still remains to be determined if the factors affect the response • In factorial experiments we are interested in determining • which factors effect the response and • which groups of factors interact.

18. Factor A has no effect B A

19. Additive Factors B A

20. Interacting Factors B A

21. The testing in factorial experiments • Test first the higher order interactions. • If an interaction is present there is no need to test lower order interactions or main effects involving those factors. All factors in the interaction affect the response and they interact • The testing continues with for lower order interactions and main effects for factors which have not yet been determined to affect the response.

22. Example: Diet Example Summary Table of Cell means Source of Protein Level of Protein Beef Cereal Pork Overall High 100.00 85.90 99.50 95.13 Low 79.20 83.90 78.70 80.60 Overall 89.60 84.90 89.10 87.87

23. Profiles of Weight Gain for Source and Level of Protein

24. Profiles of Weight Gain for Source and Level of Protein

25. Models for factorial Experiments Single Factor: A – a levels yij = m + ai + eiji = 1,2, ... ,a; j = 1,2, ... ,n Random error – Normal, mean 0, std-dev. s Overall mean Effect on y of factor A when A = i

26. Levels of A 1 2 3 a y11 y12 y13 y1n y21 y22 y23 y2n y31 y32 y33 y3n ya1 ya2 ya3 yan observations Normal dist’n m1 m2 m3 ma Mean of observations m + a1 m + a2 m + a3 m + aa Definitions

27. Two Factor: A (a levels), B (b levels yijk= m+ ai+ bj+ (ab)ij+ eijk i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,n Overall mean Interaction effect of A and B Main effect of A Main effect of B

28. Table of Means

29. Table of Effects – Overall mean, Main effects, Interaction Effects

30. Three Factor:A (a levels), B (b levels), C (c levels) yijkl =m+ ai+bj+(ab)ij+gk+(ag)ik+(bg)jk+(abg)ijk+eijkl = m+ai+bj+gk+(ab)ij+ (ag)ik+ (bg)jk +(abg)ijk+eijkl i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n Main effects Two factor Interactions Three factor Interaction Random error

31. mijk = the mean of y when A = i, B = j, C = k = m+ai+bj+gk+(ab)ij+ (ag)ik+ (bg)jk +(abg)ijk i = 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,n Two factor Interactions Overall mean Main effects Three factor Interaction

32. No interaction Levels of C Levels of B Levels of B Levels of A Levels of A

33. A, B interact, No interaction with C Levels of C Levels of B Levels of B Levels of A Levels of A

34. A, B, C interact Levels of C Levels of B Levels of B Levels of A Levels of A

35. Four Factor: yijklm = m + ai + bj+ (ab)ij + gk + (ag)ik + (bg)jk+ (abg)ijk + dl+ (ad)il + (bd)jl+ (abd)ijl + (gd)kl + (agd)ikl + (bgd)jkl+ (abgd)ijkl + eijklm = m +ai + bj+ gk + dl + (ab)ij + (ag)ik + (bg)jk + (ad)il + (bd)jl+ (gd)kl +(abg)ijk+ (abd)ijl + (agd)ikl + (bgd)jkl + (abgd)ijkl + eijklm i= 1,2, ... ,a ; j = 1,2, ... ,b ; k = 1,2, ... ,c; l = 1,2, ... ,d; m = 1,2, ... ,n where0 = S ai = S bj= S (ab)ij= Sgk = S (ag)ik = S(bg)jk= S (abg)ijk = Sdl= S (ad)il = S (bd)jl = S (abd)ijl = S (gd)kl = S (agd)ikl = S (bgd)jkl = S (abgd)ijkl and S denotes the summation over any of the subscripts. Overall mean Two factor Interactions Main effects Three factor Interactions Four factor Interaction Random error

36. Estimation of Main Effects and Interactions • Estimator of Main effect of a Factor = Mean at level i of the factor - Overall Mean • Estimator of k-factor interaction effect at a combination of levels of the k factors = Mean at the combination of levels of the k factors - sum of all means at k-1 combinations of levels of the k factors +sum of all means at k-2 combinations of levels of the k factors - etc.

37. Example: • The main effect of factor B at level j in a four factor (A,B,C and D) experiment is estimated by: • The two-factor interaction effect between factors B and C when B is at level j and C is at level k is estimated by:

38. The three-factor interaction effect between factors B, C and D when B is at level j, C is at level k and D is at level l is estimated by: • Finally the four-factor interaction effect between factors A,B, C and when A is at level i, B is at level j, C is at level k and D is at level l is estimated by:

39. Anova Table entries • Sum of squares interaction (or main) effects being tested = (product of sample size and levels of factors not included in the interaction) × (Sum of squares of effects being tested) • Degrees of freedom = df = product of (number of levels - 1) of factors included in the interaction.

40. Analysis of Variance (ANOVA) Table Entries (Two factors – A and B)

41. The ANOVA Table

42. Analysis of Variance (ANOVA) Table Entries (Three factors – A, B and C)

43. The ANOVA Table

44. The Completely Randomized Design is called balanced • If the number of observations per treatment combination is unequal the design is called unbalanced. (resulting mathematically more complex analysis and computations) • If for some of the treatment combinations there are no observations the design is called incomplete. (some of the parameters - main effects and interactions - cannot be estimated.)

45. Example: Diet example Mean = 87.867

46. Main Effects for Factor A (Source of Protein) Beef Cereal Pork 1.733 -2.967 1.233

47. Main Effects for Factor B (Level of Protein) High Low 7.267 -7.267

48. AB Interaction Effects Source of Protein Beef Cereal Pork Level High 3.133 -6.267 3.133 of Protein Low -3.133 6.267 -3.133

50. Example 2 Paint Luster Experiment