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

• 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

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

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

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

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

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

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

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

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

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

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)

Proc glm

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

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;

Output

Cont middle two vs all othersDF SS MS F Pr > F1 1536 1536 148.65 <.0001

Par middle two vs all others Estimate StErr t Pr > |t| 24 1.96 12.19 <.0001

Check with means

1 1 451 2 432 1 652 2 693 1 403 2 44(65+69)/2 – (45+43+40+44)/4 = 67 – 43 = 24

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

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

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)

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

Constraints

• Σi αi= 0• Σjβj = 0• SAS GLM Constraints

–αa= 0 (1 constraint)–βb = 0 (1 constraint)

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

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

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

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;

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

Proc glm

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

Output

Class Level InformationClass Levels Valuessizea 3 1_small 2_medium 3_largeregion 2 1 2Number of observations 6

Output

Source DF MS F PModel 3 3550 71.00 0.0139Error 2 50Total 5

Output

Source DF MS F Psizea 2 4650 93.00 0.0106region 1 1350 27.00 0.0351

Output solution

Parameter EstimateInt 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

Check vs predicted values (muhat)

reg sizea muhat1 1_s 135=195-90+30 2 1_s 105=195-901 2_m 210=195-15+30 2 2_m 180=195-151 3_l 225=195 +30 2 3_l 195=195

Multiple comparisons Size

Mean N sizea

A 210 2 3_largeAA 195 2 2_medium

B 120 2 1_small

Multiple comparisons Region

Mean N region

A 190 3 1

B 160 3 2

The anova results told us thatthese were different. No need for multiple comparison adjustment

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;

The plot

Possible interaction?

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;

The plot

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

Step 1: Get muhat (grand mean)

proc glm data=a2; model premium=; output out=aall p=muhat;proc print data=aall;run;

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

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;

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

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;

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

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;

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

Step 5: Run glm

proc glm data=a4; class size region; model premium=size region atimesb/solution; output out=a5 p=tukmuhat;run;

Output

Source DF F Value Pr > Fsize 2 360.37 0.0372region 1 104.62 0.0620atimesb 1 6.75 0.2339

Not significant…not enough evidence of this type of

interaction but really small study (1 df error)

Output

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

Does this look more like the means plot?

Last slide

• Read KNNL Chapters 19-20

• We used program topic27.sas to generate the output for today