1. Topic 24: Two-Way ANOVA

2. Outline • Two-way ANOVA • Data • Cell means model • Parameter estimates • Factor effects model

3. Two-Way ANOVA • The response variable Y is continuous • There are two categorical explanatory variables or factors

4. Data for two-way ANOVA • Y is the response variable • Factor A with levels i = 1 to a • Factor B with levels j = 1 to b • Yijk is the kth observation in cell (i,j) • In Chapter 19, we assume equal sample size in each cell (nij=n)

5. KNNL Example • KNNL p 833 • Y is the number of cases of bread sold • A is the height of the shelf display, a=3 levels: bottom, middle, top • B is the width of the shelf display, b=2 levels: regular, wide • n=2 stores for each of the 3x2=6 treatment combinations (nT=12)

6. Read the data data a1; infile ‘../data/ch19ta07.txt'; input sales height width; proc print data=a1; run;

7. The data Obs sales height width 1 47 1 1 2 43 1 1 3 46 1 2 4 40 1 2 5 62 2 1 6 68 2 1 7 67 2 2 8 71 2 2 9 41 3 1 10 39 3 1 11 42 3 2 12 46 3 2

8. Notation • For Yijk we use • i to denote the level of the factor A • j to denote the level of the factor B • k to denote the kth observation in cell (i,j) • i = 1, . . . , a levels of factor A • j = 1, . . . , b levels of factor B • k = 1, . . . , n observations in cell (i,j)

9. Model • We assume that the response variable observations are • Normally distributed • With a mean that may depend on the levels of the factors A and B • With a constant variance • Independent

10. Cell Means Model • Yijk = μij + εijk • where μij is the theoretical mean or expected value of all observations in cell (i,j) • the εijk are iid N(0, σ2) • This means Yijk ~ N(μij, σ2), independent • The parameters of the model are • μij, for i = 1 to a and j = 1 to b • σ2

11. Estimates • Estimate μij by the mean of the observations in cell (i,j), • For each (i,j) combination, we can get an estimate of the variance • We need to combine these to get an estimate of σ2

12. Pooled estimate of σ2 • In general we pool the sij2, using weights proportional to the df, nij -1 • The pooled estimate is s2 = (Σ (nij-1)sij2) / (Σ(nij-1)) • Here, nij = n, so s2 = (Σsij2) / (ab), which is the average sample variance

13. Run proc glm proc glm data=a1; class height width; model sales= height width height*width; means height width height*width; run;

14. Output

15. Means statement height

16. Means statement width

17. Means statement ht*w

18. Code the factor levels data a1; set a1; if height eq 1 and width eq 1 then hw='1_BR'; if height eq 1 and width eq 2 then hw='2_BW'; if height eq 2 and width eq 1 then hw='3_MR'; if height eq 2 and width eq 2 then hw='4_MW'; if height eq 3 and width eq 1 then hw='5_TR'; if height eq 3 and width eq 2 then hw='6_TW';

19. Plot the data symbol1 v=circle i=none; proc gplot data=a1; plot sales*hw/frame; run;

20. The plot

21. Put the means in a2 proc means data=a1; var sales; by height width; output out=a2 mean=avsales; proc print data=a2; run;

22. Output Data Set Obs height width _TYPE_ _FREQ_ avsales 1 1 1 0 2 45 2 1 2 0 2 43 3 2 1 0 2 65 4 2 2 0 2 69 5 3 1 0 2 40 6 3 2 0 2 44

23. Plot the means symbol1 v=square i=join c=black; symbol2 v=diamond i=join c=black; proc gplot data=a2; plot avsales*height=width/frame; run;

24. The interaction plot

25. Questions to consider • Does the height of the display affect sales? If yes, compare top with middle, top with bottom, and middle with bottom • Does the width of the display affect sales? If yes, compare regular and wide

26. But wait!!! Are these factor level comparisons meaningful? • Does the effect of height on sales depend on the width? • Does the effect of width on sales depend on the height? • If yes, we have an interaction and we need to do some additional analysis

27. Factor effects model • For the one-way ANOVA model, we wrote μi = μ + αi • Here we use μij = μ + αi + βj + (αβ)ij • Under “common” formulation • μ (μ.. in KNNL) is the “overall mean” • αi is the main effect of A • βj is the main effect of B • (αβ)ij is the interaction between A and B

28. Factor effects model • μ = (Σij μij)/(ab) • μi. = (Σj μij)/b and μ.j = (Σi μij)/a • αi = μi. – μ and βj = μ.j - μ • (αβ)ij is difference between μij and μ + αi + βj • (αβ)ij = μij - (μ + (μi. - μ)+ (μ.j - μ)) = μij – μi. – μ.j + μ

29. Interpretation • μij = μ + αi + βj + (αβ)ij • μ is the “overall” mean • αi is an adjustment for level i of A • βj is an adjustment for level j of B • (αβ)ij is an additional adjustment that takes into account both i and j that cannot be explained by the previous adjustments

30. Constraints for this framework • α. = Σi αi= 0 • β. = Σjβj = 0 • (αβ).j = Σi (αβ)ij = 0 for all j • (αβ)i. = Σj (αβ)ij = 0 for all i

31. Estimates for factor effects model

32. SS for ANOVA Table

33. df for ANOVA Table • dfA = a-1 • dfB = b-1 • dfAB = (a-1)(b-1) • dfE = ab(n-1) • dfT = abn-1 = nT-1

34. MS for ANOVA Table • MSA = SSA/dfA • MSB = SSB/dfB • MSAB = SSAB/dfAB • MSE = SSE/dfE • MST = SST/dfT

35. Hypotheses for two-way ANOVA • H0A: αi = 0 for all i • H1A: αi ≠ 0 for at least one i • H0B: βj = 0 for all j • H1B: βj ≠ 0 for at least one j • H0AB: (αβ)ij = 0 for all (i,j) • H1AB: (αβ)ij ≠ 0 for at least one (i,j)

36. F statistics • H0A is tested by FA = MSA/MSE; df=dfA, dfE • H0B is tested by FB = MSB/MSE; df=dfB, dfE • H0AB is tested by FAB = MSAB/MSE; df=dfAB, dfE

37. ANOVA Table Source df SS MS F A a-1 SSA MSA MSA/MSE B b-1 SSB MSB MSB/MSE AB (a-1)(b-1) SSAB MSAB MSAB/MSE Error ab(n-1) SSE MSE _ Total abn-1 SSTO MST

38. P-values • P-values are calculated using the F(dfNumerator, dfDenominator) distributions • If P ≤ 0.05 we conclude that the effect being tested is statistically significant

39. KNNL Example • NKNW p 833 • Y is the number of cases of bread sold • A is the height of the shelf display, a=3 levels: bottom, middle, top • B is the width of the shelf display, b=2: regular, wide • n=2 stores for each of the 3x2 treatment combinations

40. PROC GLM proc glm data=a1; class height width; model sales= height width height*width; run;

41. Output Note that there are 6 cells in this design…(6-1)df for model

42. Output ANOVA Note Type I and Type III Analyses are the same because nij is constant

43. Other output Commonly do not consider R-sq when performing ANOVA…interested more in difference in levels rather than the models predictive ability

44. Results • The main effect of height is statistically significant (F=74.71; df=2,6; P<0.0001) • The main effect of width is not statistically significant (F=1.16; df=1,6; P=0.32) • The interaction between height and width is not statistically significant (F=1.16; df=2,6; P=0.37)

45. Interpretation • The height of the display affects sales of bread • The width of the display has no apparent effect • The effect of the height of the display is similar for both the regular and the wide widths

46. Plot of the means

47. Additional analyses • We will need to do additional analyses to explain the height effect (factor A) • There were three levels: bottom, middle and top • We could rerun the data with a one-way anova and use the methods we learned in the previous chapters • Use means statement with lines

48. Run Proc GLM proc glm data=a1; class height width; model sales= height width height*width; means height / tukey lines; lsmeans height / adjust=tukey; run;

49. MEANS Output

50. LSMEANS Output