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Topic 27: Strategies of Analysis

Topic 27: Strategies of Analysis. 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?. An analytical strategy.

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Topic 27: Strategies of Analysis

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  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 dfe 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 • (μ21 + μ22)/2 = (μ11 + μ12 + μ31 + μ32)/4 • -.25μ11-.25μ12+.5μ21+.5μ22-.25μ31-.25μ32 =0 • Then translate the contrast into the factor effects model using μij = μ + αi + βj + (αβ)ij

  13. Contrast/Estimate • -.25μ11 = -.25(μ + α1 + β1 + (αβ)11) • -.25μ12 = -.25(μ + α1 + β2 + (αβ)12) • +.5μ21 = +.5 (μ + α2 + β1 + (αβ)21) • +.5μ22 = +.5 (μ + α2 + β2 + (αβ)22) • -.25μ31 = -.25(μ + α3 + β1 + (αβ)31) • -.25μ32 = -.25(μ + α3 + β2 + (αβ)32) • C = (-.5α1 + α2 - .5α3) + (-.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 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 • n = 1 observation in cell (i,j)

  21. Factor effects model • μij = μ + αi + βj • μ is the overall mean • αi is the main effect of A • βj 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 • Σiαi= 0 • Σjβj = 0 • SAS GLM Constraints • αa= 0 (1 constraint) • βb= 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) = σ2 • E(MSA) = σ2 + b(Σiαi2)/(a-1) • E(MSB) = σ2 + a(Σjβj2)/(b-1) • Here, αi and βj, 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 • μij = μ + αi + βj + θαiβj • We use one degree of freedom to estimate θ • There are other variations, eg θiβj

  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)

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