1. 1 Multifactor ANOVA

2. 2 What We Will Learn Two-factor ANOVA K ij =1 Two-factor ANOVA K ij =1 –Interaction –Tukey’s with multiple comparisons –Concept of randomized blocked experiments –Random effects and mixed models Two-factor ANOVA K ij >1 Two-factor ANOVA K ij >1 –Interactions –Tukey’s –Mixed and random effects Three-factor ANOVA Three-factor ANOVA –Latin Squares 2 p Factorial Experiments 2 p Factorial Experiments –2 3 Experiments –2 p >3 –Concept of confounding

3. 3 2 Factor ANOVA Factor A consists of I levels Factor A consists of I levels Factor B consists of J levels Factor B consists of J levels IJ different pairs IJ different pairs Number of observations per each factor pair K ij =1 Number of observations per each factor pair K ij =1 Example - Tires Example - Tires

4. 4 Terminology X ij = the rv denoting the measurement when factor A is held at level i and factor B is held at level j X ij = the rv denoting the measurement when factor A is held at level i and factor B is held at level j x ij = actual observed value x ij = actual observed value The average when factor A is held at level i The average when factor A is held at level i The average when factor B is held at level j The average when factor B is held at level j The grand mean The grand mean

5. 5 An Additive Model I parameters  1,  2,  3 …  I J parameters  1,  2,  3, …  j ST  ij =  i +  j  ij is the sum of an effect due to factor A at level i (  I ) and an effect due to factor B at level j (  j ) Then X ij =  i +  j +  ij Number of model parameters I+J+1 One for the error  ij -  i'j = (  i +  j ) - (  i' +  j )=  i -  i’ The difference in mean responses for two levels of one of the factors is the same for all levels of the other factor  i and  j are not uniquely defined

6. 6 Additive Model X ij =  +  i +  j +  ij where and  ij ’s are independent and N(0,  2 ) Now  = 4,  1 = -.5,  2 =.5,  1 = -1.5,  2 = 1.5 The parameters are uniquely defined Have (I-1)+(J-1)+1 = I+J-1 Estimators

7. 7 Two Factor Hypothesis

8. 8 Two-way ANOVA

9. 9 ANOVA Table

10. 10 Problem 1

11. 11 Problem 1

12. 12 Problem 1

13. 13 SPSS Data Entry Analyze> General Linear Model (GLM)>Univariate

14. 14 Problem 1

15. 15 Your Turn The effects of four types of graphite coaters on light box readings are to be studied. As these readings might differ from day to day, observations are to be taken on each of the four types everyday for three days. The order of testing of the four types on any given day can be randomized. The effects of four types of graphite coaters on light box readings are to be studied. As these readings might differ from day to day, observations are to be taken on each of the four types everyday for three days. The order of testing of the four types on any given day can be randomized. Analyze these data as a randomized block design and state your conclusions. Analyze these data as a randomized block design and state your conclusions.

16. 16 Your Turn

17. 17 Your Turn

18. 18 Multiple Comparisons in ANOVA Tukey’s Procedure (T Method) Select  Select  Determine Q Determine Q –For A - Q ,I,(I-1)(J-1) –For B - Q ,J,(I-1)(J-1) Determine w for factors A and B Determine w for factors A and B List sample means in increasing order List sample means in increasing order Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Pairs not underlined are significantly different Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Pairs not underlined are significantly different

19. 19 Problem 1 Select  Select  Determine Q Determine Q –For A - Q ,I,(I-1)(J-1) –For B - Q ,J,(I-1)(J-1) Determine w for factors A and B Determine w for factors A and B List sample means in increasing order List sample means in increasing order Compute the difference in each  i -  j pair Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Underline those pairs that differ by less than w

20. 20 Problem 1

21. 21 Problem 1

22. 22 Problem 1

23. 23 Problem 1

24. 24 Your Turn Select  Select  Determine Q Determine Q –For A - Q ,I,(I-1)(J-1) –For B - Q ,J,(I-1)(J-1) Determine w for factors A and B Determine w for factors A and B List sample means in increasing order List sample means in increasing order Compute the difference in each  i -  j pair Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Underline those pairs that differ by less than w

25. 25 Concept of Randomized Block Designs 1-way ANOVA 1-way ANOVA –I treatments –IJ total observations or subjects –J observations per treatment selected randomly Observations or subjects can be heterogeneous WRT other variables Observations or subjects can be heterogeneous WRT other variables –Significance or non-significance due to the treatment or something else –Hence paired t-test pre to post exams - looking at the difference of an individual’s scores; know that the difference isn’t due to the subject pre to post exams - looking at the difference of an individual’s scores; know that the difference isn’t due to the subject When I>2 we want to perform a randomized block experiment When I>2 we want to perform a randomized block experiment

26. 26 Concept of Randomized Block Designs The extra factor (block) divides the IJ units into J groups with I units within each group The extra factor (block) divides the IJ units into J groups with I units within each group –The I units are homogeneous WRT other factors (the block) –Within each block, the I treatments are randomly selected and assigned to I observations or subjects –When social scientists use this - repeated measures –Each subject undergoes each treatment; thus, acting as their own control –Could use time periods, locations, etc. –From a large population of subjects - random effects

27. 27 Problem 2

28. 28 Why Block? If the experimental units are heterogeneous If the experimental units are heterogeneous –Then there will be a larger variance in MSE –Blocking minimizes the random (MSE) variance by accounting some of the error to the effects due to the subjects/experimental units –Thus, MSE (the random error) will be smaller allowing us to determine if there is significance in the main effect or treatment. –So…. We may be able to determine if the null hypothesis should be rejected. –There is a cost…. The MSE will have fewer degrees of freedom since some of those df’s need to go to the Block error (the larger the df, the smaller the MSE)

29. 29 Models for Random Effects 1-way ANOVA - Random Effects Model 1-way ANOVA - Random Effects Model 2-way ANOVA - Mixed Effects Model 2-way ANOVA - Mixed Effects Model –X ij =  +  i + B j +  ij –Distributions of B j &  ij are N(0,  2,B ), N(0,  2, , ) respectively –Hypothesis Test

30. 30 2-Factor ANOVA K ij >1 Here, the responses are not additive Here, the responses are not additive There is something going on between factor A and factor B There is something going on between factor A and factor B When additivity doesn’t apply we have an interaction When additivity doesn’t apply we have an interaction Additivity allows us to obtain an unbiased estimator for MSE Additivity allows us to obtain an unbiased estimator for MSE Need to have more than one observation per cell to find a unbiased estimator for MSE when interactions may be present Need to have more than one observation per cell to find a unbiased estimator for MSE when interactions may be present K ij >1 K ij >1 –K- is a constant number per cell (each cell has same number of observations)

31. 31 Parameters for Fixed Effects Model w/Interaction  i =  i. -  = the effect of factor A at level I  i =  i. -  = the effect of factor A at level I  j = . j -  = the effect of factor B at level J  j = . j -  = the effect of factor B at level J  ij is the interaction parameters  ij is the interaction parameters  ij =  ij – (  +  i +  j )  ij =  ij – (  +  i +  j ) so the individual means are represented below  ij =  +  i +  j +  ij  ij =  +  i +  j +  ij

32. 32 Hypotheses The model is additive if all  ij =0 The model is additive if all  ij =0 Order of testing Order of testing –Interaction first –Main effects Sometimes results can be confusing Sometimes results can be confusing

33. 33 The Model with Interactions X ijk = the rv denoting the measurement when factor A is held at level i and factor B is held at level j given k observations for each ij levels X ijk = the rv denoting the measurement when factor A is held at level i and factor B is held at level j given k observations for each ij levels x ijk = actual observed value x ijk = actual observed value X ijk =  +  i +  j +  ij +  ij X ijk =  +  i +  j +  ij +  ij  ij are N(0,  2 )  ij are N(0,  2 )

34. 34 The Model with Interactions

35. 35 The ANOVA Table

36. 36 Problem 2

37. 37 Problem 2

38. 38 Multiple Comparisons in ANOVA Tukey’s Procedure (T Method) H oAB is not rejected and H oA and/or H oB is rejected H oAB is not rejected and H oA and/or H oB is rejected Select  Select  Determine Q Determine Q –For A - Q ,I,IJ(K-1) –For B - Q ,J,IJ(K-1) Determine w for factors A and B Determine w for factors A and B List sample means in increasing order List sample means in increasing order Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Pairs not underlined are significantly different Compute the difference in each  i -  j pair Underline those pairs that differ by less than w Pairs not underlined are significantly different

39. 39 Problem 2

40. 40 Problem 2

41. 41 Mixed and Random Effects Models 1-way ANOVA - Random Effects Model 1-way ANOVA - Random Effects Model 2-way ANOVA - Mixed Effects Model 2-way ANOVA - Mixed Effects Model –X ij =  +  i + B j + G ij +  ij –Distributions of B j, G ij &  ij are N(0,  2,B ), N(0,  2,G ), N(0,  2, , ) respectively –Hypothesis Test –Order of testing Interaction first Interaction first Main effects Main effects

42. 42