Repeated Measures Design. Advertising Sales Example

Transcription

1 STAT:5201 Anaylsis/Applied Statistic II Repeated Measures Design Advertising Sales Example A company is interested in comparing the success of two different advertising campaigns. It has 10 test markets, and 5 test markets will be randomly assigned to each campaign. The sales for each test market will be recorded after 1 months, 2 months, and 3 months. Response: Sales. Factors: Campaign (1 or 2), Time (1,2,3) and Test Market (10 unique markets). SAS statements for data input and initial plot: proc import datafile="sales.csv" out=sales dbms=csv replace; proc print data=sales; Obs TestMarket Campaign Time Sales

2 /* One symbol for each test market.*/ symbol1 value=circle i=std1mj color=black line=1; symbol2 value=star i=std1mj color=black line=2; symbol3 value=diamond i=std1mj color=black line=3; symbol4 value=plus i=std1mj color=black line=4; symbol5 value=triangle i=std1mj color=black line=5; symbol6 value=y i=std1mj color=black line=6; symbol7 value=z i=std1mj color=black line=7; symbol8 value=_ i=std1mj color=black line=8; symbol9 value=> i=std1mj color=black line=9; symbol10 value=: i=std1mj color=black line=10; proc gplot data=sales; plot sales*time=testmarket/haxis=0.5 to 3.5; NOTE: We will deal with the lack of linearity in time later, but for now, we ll start with a simple model of linear in time... SAS statements for specifying Compound Symmetry (CS) correlation structure: This analysis (incorporating a compound symmetry variance which is not time dependent ) can be done using either the random statement as we have done with split-plot designs which models as CS, or using both the random and repeated statements and specifying the R matrix to be independent residuals (or variance components(vc)). /*Analysis with REPEATED statement and overall Compound symmetry covariance structure. Here, var(g) and var(e) have variance components structure (i.e.~diagonal matrices), but the overall variance-covariance structure of Y is a compound symmetry form (block diagonal). This model assumes linear in time (but has a different line for each campaign).*/ proc mixed data=sales plot(only)=residualpanel(conditional); model Sales=Campaign Time/ddfm=satterth residual; random Testmarket(Campaign); repeated / type=vc sub=testmarket(campaign); 2

3 Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.SALES Sales Variance Components TestMarket(Campaign) REML Parameter Model-Based Containment Class Level Information Class Levels Values TestMarket Campaign Dimensions Covariance Parameters 2 Columns in X 6 Columns in Z 10 Subjects 1 Max Obs Per Subject 30 Cov Parm Subject Estimate TestMarket(Campaign) Residual TestMarket(Campaign) Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Campaign Time Time*Campaign

4 /*Analyze as a split plot, which assumes compound symmetry correlation, we ll use the same fixed effects.*/ proc mixed data=sales; model Sales=Campaign Time; random TestMarket(Campaign); Dimensions Covariance Parameters 2 Columns in X 6 Columns in Z 10 Subjects 1 Max Obs Per Subject 30 Cov Parm Estimate TestMarket(Campaign) Residual Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Campaign Time Time*Campaign All the output is the same for the two ways of specifying the model because the model itself was exactly the same (same fixed effects and same correlation structure). 4

5 Let s consider the same fixed effects (Campaign, linear in time, and their interaction), but choose an AR(1) correlation structure for the residuals within a TestMarket, or for var(e i ). SAS statements for first order Autoregressive correlation structure: /*Analysis with an AR(1) covariance structure for residuals, and linear in time (different line for different campaigns).*/ proc mixed data=sales plot(only)=residualpanel(conditional); model Sales=Campaign Time/ddfm=satterth residual; random TestMarket(Campaign); repeated / type=ar(1) sub=testmarket(campaign); Model Information Data Set Dependent Variable Covariance Structures Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.SALES Sales Variance Components, Autoregressive TestMarket(Campaign) REML Profile Model-Based Containment Class Level Information Class Levels Values TestMarket Campaign Dimensions Covariance Parameters 3 Columns in X 6 Columns in Z 10 Subjects 1 Max Obs Per Subject 30 Cov Parm Subject Estimate TestMarket(Campaign) AR(1) TestMarket(Campaign) Residual

6 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Campaign Time Time*Campaign Comparing models (same fixed effects, different covariances) with fit statistics By looking at the AIC fit statistics (or the BIC) we can see that we gained by including the AR(1) structure because the split-plot model had an AIC of and the AR(1) had an AIC of Quadratic time effect As is apparent from the original plot of Sales vs. time, it looks like we should have included a quadratic time effect. Let s bring in the quadratic effect in time. /*Analysis with an AR(1) covariance structure, and separate quadratics for separate campaigns.*/ proc mixed data=sales plot(only)=residualpanel(conditional); model Sales=Campaign Time Time*Time Campaign Time*Time/ddfm=satterth residual; random TestMarket(Campaign); repeated / type=ar(1) sub=testmarket(campaign); /**** Diagnostics now look MUCH better *****/ Model Information Data Set Dependent Variable Covariance Structures Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.SALES Sales Variance Components, Autoregressive TestMarket(Campaign) REML Profile Model-Based Containment 6

7 Cov Parm Subject Estimate TestMarket(Campaign) AR(1) TestMarket(Campaign) Residual Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Campaign Time <.0001 Time*Campaign Time*Time <.0001 Time*Time*Campaign It appears that there is no significant Campaign effect, and Sales is quadratic in time (same quadratic for both Campaigns). 7

8 Final model: /*Sales quadratic in time, and AR(1) covariance structure.*/ proc mixed data=sales plot(only)=residualpanel(conditional); model Sales=Time Time*Time/solution outp=diags ddfm=satterth residual; random TestMarket(Campaign); repeated / type=ar(1) sub=testmarket(campaign); Model Information Data Set Dependent Variable Covariance Structures Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.SALES Sales Variance Components, Autoregressive TestMarket(Campaign) REML Profile Model-Based Containment Cov Parm Subject Estimate TestMarket(Campaign) AR(1) TestMarket(Campaign) Residual Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept Time <.0001 Time*Time <.0001 Time <.0001 Time*Time <