1. Topic 27: Strategies of Analysis

2. Outline Strategy for analysis of two-way studies –Interaction is not significant –Interaction is significant What if levels of factor or factors is quantitative? What if cell size is n=1?

3. An analytical strategy Run the model with main effects and the two-way interaction Plot the data, the means and look at the normal quantile plot Check the significance test for the interaction

4. AB interaction not sig Possibly rerun the analysis without the interaction. Only consider if df e is small For a main effect that is not significant –No evidence to conclude that the levels of this factor are associated with different means of the response variable –Possibly rerun without this factor giving a one-way anova. Watch out for Type II errors that inflate MSE

5. AB interaction not sig If both main effects are not significant –Model could be run as Y=; –Basically assuming a one population model For a main effect with more than two levels that is significant, use the means or lsmeans statement with the tukey multiple comparison procedure Contrasts and linear combinations can also be examined using the contrast and estimate statements

6. AB interaction sig but not important Plots and a careful examination of the cell means may indicate that the interaction is not very important even though it is statistically significant Use the marginal means for each significant main effect to describe the important results You may need to qualify these results using the interaction

7. AB interaction sig but not important Use the methods described for the situation where the interaction is not significant but keep the interaction in the model Carefully interpret the marginal means as averages over the levels of the other factor

8. AB interaction is sig and important Options include –Treat as a one-way with ab levels; use tukey to compare means; contrasts and estimate can also be useful –Report that the interaction is significant; plot the means and describe the pattern –Analyze the levels of A for each level of B (use a BY statement) or vice versa

9. Specification of contrast and estimate statements When using the contrast statement always double check your results with the estimate statement The order of factors is determined by the order in the class statement, not the order in the model statement Contrast should come a priori from research questions

10. 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: regular, wide n=2 stores for each of the 3x2 treatment combinations

11. Contrast/Estimate Example Suppose we want to compare the average of the two height = middle cells with the average of the other four cells With this approach, the contrast should correspond to a research question formulated before examining the data

12. Contrast/Estimate First, formulate question as a contrast using the cell means model –(μ 21 + μ 22 )/2 = (μ 11 + μ 12 + μ 31 + μ 32 )/4 –-.25μ 11 -.25μ 12 +.5μ 21 +.5μ 22 -.25μ 31 -.25μ 32 =0 Then translate the contrast into the factor effects model using μ ij = μ + α i + β j + (αβ) ij

13. Contrast/Estimate -.25μ 11 = -.25(μ + α 1 + β 1 + (αβ) 11 ) -.25μ 12 = -.25(μ + α 1 + β 2 + (αβ) 12 ) +.5μ 21 = +.5 (μ + α 2 + β 1 + (αβ) 21 ) +.5μ 22 = +.5 (μ + α 2 + β 2 + (αβ) 22 ) -.25μ 31 = -.25(μ + α 3 + β 1 + (αβ) 31 ) -.25μ 32 = -.25(μ + α 3 + β 2 + (αβ) 32 ) C = (-.5α 1 + α 2 -.5α 3 ) + (-.25(αβ) 11 -.25(αβ) 12 +.5(αβ) 21 +.5(αβ) 22 -.25(αβ) 31 -.25(αβ) 32 )

14. Proc glm proc glm data=a1; class height width; model sales=height width height*width;

15. Contrast and estimate contrast 'middle two vs all others' height -.5 1 -.5 height*width -.25 -.25.5.5 -.25 -.25; estimate 'middle two vs all others' height -.5 1 -.5 height*width -.25 -.25.5.5 -.25 -.25; means height*width; run;

16. Output Cont middle two vs all others DF SS MS F Pr > F 1 1536 1536 148.65 <.0001 Par middle two vs all others Estimate StErr t Pr > |t| 24 1.96 12.19 <.0001

17. Check with means 1 1 45 1 2 43 2 1 65 2 2 69 3 1 40 3 2 44 (65+69)/2 – (45+43+40+44)/4 = 67 – 43 = 24

18. One Quantitative factor and one categorical Plot the means vs the quantitative factor for each level of the categorical factor Consider linear and quadratic terms for the quantitative factor (regression) Consider different relationships for the different levels of the categorical factor Lack of fit analysis can be useful

19. Two Quantitative factors Plot the means –vs A for each level B –vs B for each level A Consider linear and quadratic terms Consider products to allow for interaction Lack of fit analysis can be useful

20. One observation per cell For Y ijk we use –i to denote the level of the factor A –j to denote the level of the factor B –k to denote the k th observation in cell (i,j) i = 1,..., a levels of factor A j = 1,..., b levels of factor B n = 1 observation in cell (i,j)

21. Factor effects model μ ij = μ + α i + β j μ is the overall mean α i is the main effect of A β j is the main effect of B Because we have only one observation per cell, we do not have enough information to estimate the interaction in the usual way…it is confounded with error

22. Constraints Σ i α i = 0 Σ j β j = 0 SAS GLM Constraints –α a = 0 (1 constraint) –β b = 0 (1 constraint)

23. ANOVA Table Source df SS MS F A a-1 SSA MSA MSA/MSE B b-1 SSB MSB MSB/MSE Error (a-1)(b-1) SSE MSE _ Total ab-1 SST MST Would normally be df for interaction

24. Expected Mean Squares Based on model which has no interaction E(MSE) = σ 2 E(MSA) = σ 2 + b( Σ i α i 2 )/(a-1) E(MSB) = σ 2 + a( Σ j β j 2 )/(b-1) Here, α i and β j, are defined with the usual factor effects constraints

25. KNNL Example KNNL p 882 Y is the premium for auto insurance A is the size of the city, a=3 levels: small, medium and large B is the region, b=2: East, West n=1 the response is the premium charged by a particular company

26. The data data a2; infile 'c:\...\CH20TA02.txt'; input premium size region; if size=1 then sizea='1_small '; if size=2 then sizea='2_medium'; if size=3 then sizea='3_large '; proc print data=a1; run;

27. Output Obs premium size region sizea 1 140 1 1 1_small 2 100 1 2 1_small 3 210 2 1 2_medium 4 180 2 2 2_medium 5 220 3 1 3_large 6 200 3 2 3_large

28. Proc glm proc glm data=a2; class sizea region; model premium= sizea region/solution; means sizea region / tukey; output out=a3 p=muhat; run;

29. Output Class Level Information Class Levels Values sizea 3 1_small 2_medium 3_large region 2 1 2 Number of observations 6

30. Output Source DF MS F P Model 3 3550 71.00 0.0139 Error 2 50 Total 5

31. Output Source DF MS F P sizea 2 4650 93.00 0.0106 region 1 1350 27.00 0.0351

32. Output solution Parameter Estimate Int 195 B sizea 1_small -90 B sizea 2_medium -15 B sizea 3_large 0 B region 1 30 B region 2 0 B

33. Check vs predicted values (muhat) reg sizea muhat 1 1_s 135=195-90+30 2 1_s 105=195-90 1 2_m 210=195-15+30 2 2_m 180=195-15 1 3_l 225=195 +30 2 3_l 195=195

34. Multiple comparisons Size Mean N sizea A 210 2 3_large A A 195 2 2_medium B 120 2 1_small

35. Multiple comparisons Region Mean N region A 190 3 1 B 160 3 2 The anova results told us that these were different. No need for multiple comparison adjustment

36. Plot the data symbol1 v='E' i=join c=blue; symbol2 v='W' i=join c=green; title1 'Plot of the data'; proc gplot data=a3; plot premium*sizea= region; run;

37. The plot Possible interaction?

38. Plot the estimated model symbol1 v='E' i=join c=blue; symbol2 v='W' i=join c=green; title1 'Plot of the model estimates'; proc gplot data=a2; plot muhat*sizea=region; run;

39. The plot

40. Tukey test for additivity Can’t look at usual interaction but can add one additional term to the model (θ) to look at specific type of interaction μ ij = μ + α i + β j + θα i β j We use one degree of freedom to estimate θ There are other variations, eg θ i β j

41. Step 1: Get muhat (grand mean) proc glm data=a2; model premium=; output out=aall p=muhat; proc print data=aall; run;

42. Output Obs premium size region muhat 1 140 1 1 175 2 100 1 2 175 3 210 2 1 175 4 180 2 2 175 5 220 3 1 175 6 200 3 2 175

43. Step 2: Get muhatA (factor A means) proc glm data=a2; class size; model premium=size; output out=aA p=muhatA; proc print data=aA; run;

44. Output Obs premium size region muhatA 1 140 1 1 120 2 100 1 2 120 3 210 2 1 195 4 180 2 2 195 5 220 3 1 210 6 200 3 2 210

45. Step 3: Find muhatB (factor B means) proc glm data=a2; class region; model premium=region; output out=aB p=muhatB; proc print data=aB; run;

46. Output Obs remium size region muhatB 1 140 1 1 190 2 100 1 2 160 3 210 2 1 190 4 180 2 2 160 5 220 3 1 190 6 200 3 2 160

47. Step 4: Combine and compute data a4; merge aall aA aB; alpha=muhatA-muhat; beta=muhatB-muhat; atimesb=alpha*beta; proc print data=a4; var size region atimesb; run;

48. Output Obs size region atimesb 1 1 1 -825 2 1 2 825 3 2 1 300 4 2 2 -300 5 3 1 525 6 3 2 -525

49. Step 5: Run glm proc glm data=a4; class size region; model premium=size region atimesb/solution; output out=a5 p=tukmuhat; run;

50. Output Source DF F Value Pr > F size 2 360.37 0.0372 region 1 104.62 0.0620 atimesb 1 6.75 0.2339 Not significant…not enough evidence of this type of interaction but really small study (1 df error)

51. Output Par Est SE t P Int 195 B 2.9 66.49 0.0096 size1 -90 B 3.5 -25.05 0.0254 size2 -15 B 3.5 -4.18 0.1496 size3 0 B... region1 30 B 2.9 10.23 0.0620 region2 0 B... atimesb -0.006 0.002 -2.60 0.2339

52. Does this look more like the means plot?

53. Last slide Read KNNL Chapters 19-20 We used program topic27.sas to generate the output for today