1. ANOVA TABLE Factorial Experiment Completely Randomized Design

2. Anova table for the 3 factor Experiment SourceSSdfMSFp -value ASS A a - 1MS A MS A /MS Error BSS B b - 1MS B MS B /MS Error CSS C c - 1MS C MS C /MS Error ABSS AB (a - 1)(b - 1)MS AB MS AB /MS Error ACSS AC (a - 1)(c - 1)MS AC MS AC /MS Error BCSS BC (b - 1)(c - 1)MS BC MS BC /MS Error ABCSS ABC (a - 1)(b - 1)(c - 1)MS ABC MS ABC /MS Error ErrorSS Error abc(n - 1)MS Error

3. Sum of squares entries Similar expressions for SS B, and SS C. Similar expressions for SS BC, and SS AC.

4. Sum of squares entries Finally

5. The statistical model for the 3 factor Experiment

6. Anova table for the 3 factor Experiment SourceSSdfMSFp -value ASS A a - 1MS A MS A /MS Error BSS B b - 1MS B MS B /MS Error CSS C c - 1MS C MS C /MS Error ABSS AB (a - 1)(b - 1)MS AB MS AB /MS Error ACSS AC (a - 1)(c - 1)MS AC MS AC /MS Error BCSS BC (b - 1)(c - 1)MS BC MS BC /MS Error ABCSS ABC (a - 1)(b - 1)(c - 1)MS ABC MS ABC /MS Error ErrorSS Error abc(n - 1)MS Error

7. The testing in factorial experiments 1.Test first the higher order interactions. 2.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 3.The testing continues with lower order interactions and main effects for factors which have not yet been determined to affect the response.

8. Random Effects and Fixed Effects Factors

9. So far the factors that we have considered are fixed effects factors This is the case if the levels of the factor are a fixed set of levels and the conclusions of any analysis is in relationship to these levels. If the levels have been selected at random from a population of levels the factor is called a random effects factor The conclusions of the analysis will be directed at the population of levels and not only the levels selected for the experiment

10. Example - Fixed Effects Source of Protein, Level of Protein, Weight Gain Dependent –Weight Gain Independent –Source of Protein, Beef Cereal Pork –Level of Protein, High Low

11. Example - Random Effects In this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured. Dependent –Mileage Independent –Tire brand (A, B, C), Fixed Effect Factor –Driver (1, 2, 3, 4), Random Effects factor

12. The Model for the fixed effects experiment where ,  1,  2,  3,  1,  2, (  ) 11, (  ) 21, (  ) 31, (  ) 12, (  ) 22, (  ) 32, are fixed unknown constants And  ijk is random, normally distributed with mean 0 and variance  2. Note:

13. The Model for the case when factor B is a random effects factor where ,  1,  2,  3, are fixed unknown constants And  ijk is random, normally distributed with mean 0 and variance  2.  j is normal with mean 0 and variance and (  ) ij is normal with mean 0 and variance Note: This model is called a variance components model

14. The Anova table for the two factor model SourceSSdfMS ASS A a -1 SS A /(a – 1) BSS A b - 1 SS B /(a – 1) ABSS AB (a -1)(b -1) SS AB /(a – 1) (a – 1) ErrorSS Error ab(n – 1) SS Error /ab(n – 1)

15. The Anova table for the two factor model (A, B – fixed) SourceSSdfMSEMSF ASS A a -1MS A MS A /MS Error BSS A b - 1MS B MS B /MS Error ABSS AB (a -1)(b -1)MS AB MS AB /MS Error ErrorSS Error ab(n – 1)MS Error EMS = Expected Mean Square

16. The Anova table for the two factor model (A – fixed, B - random) SourceSSdfMSEMSF ASS A a -1MS A MS A /MS AB BSS A b - 1MS B MS B /MS Error ABSS AB (a -1)(b -1)MS AB MS AB /MS Error ErrorSS Error ab(n – 1)MS Error Note: The divisor for testing the main effects of A is no longer MS Error but MS AB.

17. Rules for determining Expected Mean Squares (EMS) in an Anova Table 1.Schultz E. F., Jr. “Rules of Thumb for Determining Expectations of Mean Squares in Analysis of Variance,”Biometrics, Vol 11, 1955, 123-48. Both fixed and random effects Formulated by Schultz [1]

18. 1.The EMS for Error is  2. 2.The EMS for each ANOVA term contains two or more terms the first of which is  2. 3.All other terms in each EMS contain both coefficients and subscripts (the total number of letters being one more than the number of factors) (if number of factors is k = 3, then the number of letters is 4) 4.The subscript of  2 in the last term of each EMS is the same as the treatment designation.

19. 5.The subscripts of all  2 other than the first contain the treatment designation. These are written with the combination involving the most letters written first and ending with the treatment designation. 6.When a capital letter is omitted from a subscript, the corresponding small letter appears in the coefficient. 7.For each EMS in the table ignore the letter or letters that designate the effect. If any of the remaining letters designate a fixed effect, delete that term from the EMS.

20. 8.Replace  2 whose subscripts are composed entirely of fixed effects by the appropriate sum.

21. Example: 3 factors A, B, C – all are random effects SourceEMSF A B C AB AC BC ABC Error

22. Example: 3 factors A fixed, B, C random SourceEMSF A B C AB AC BC ABC Error

23. Example: 3 factors A, B fixed, C random SourceEMSF A B C AB AC BC ABC Error

24. Example: 3 factors A, B and C fixed SourceEMSF A B C AB AC BC ABC Error

25. Example - Random Effects In this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects at random b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured. Dependent –Mileage Independent –Tire brand (A, B, C), Fixed Effect Factor –Driver (1, 2, 3, 4), Random Effects factor

26. The Data

27. Asking SPSS to perform Univariate ANOVA

28. Select the dependent variable, fixed factors, random factors

29. The Output The divisor for both the fixed and the random main effect is MS AB This is contrary to the advice of some texts

30. The Anova table for the two factor model (A – fixed, B - random) SourceSSdfMSEMSF ASS A a -1MS A MS A /MS AB BSS A b - 1MS B MS B /MS Error ABSS AB (a -1)(b -1)MS AB MS AB /MS Error ErrorSS Error ab(n – 1)MS Error Note: The divisor for testing the main effects of A is no longer MS Error but MS AB. References Guenther, W. C. “Analysis of Variance” Prentice Hall, 1964

31. The Anova table for the two factor model (A – fixed, B - random) SourceSSdfMSEMSF ASS A a -1MS A MS A /MS AB BSS A b - 1MS B MS B /MS AB ABSS AB (a -1)(b -1)MS AB MS AB /MS Error ErrorSS Error ab(n – 1)MS Error Note: In this case the divisor for testing the main effects of A is MS AB. This is the approach used by SPSS. References Searle “Linear Models” John Wiley, 1964

32. Crossed and Nested Factors

33. The factors A, B are called crossed if every level of A appears with every level of B in the treatment combinations. Levels of B Levels of A

34. Factor B is said to be nested within factor A if the levels of B differ for each level of A. Levels of B Levels of A

35. Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant Plants Machines

36. Machines (B) are nested within plants (A) The model for a two factor experiment with B nested within A.

37. The ANOVA table SourceSSdfMSFp - value ASS A a - 1MS A MS A /MS Error B(A)B(A)SS B(A) a(b – 1)MS B(A) MS B(A) /MS Error ErrorSS Error ab(n – 1)MS Error Note: SS B(A ) = SS B + SS AB and a(b – 1) = (b – 1) + (a - 1)(b – 1)

38. Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant. Also we have n = 5 measurements of paper strength for each of the 24 machines

39. The Data

40. Anova Table Treating Factors (Plant, Machine) as crossed

41. Anova Table: Two factor experiment B(machine) nested in A (plant)

42. Graph Paper Strength