1. Topic 30: Random Effects

2. Outline One-way random effects model –Data –Model –Inference

3. Data for one-way random effects model Y, the response variable Factor with levels i = 1 to r Y ij is the j th observation in cell i, j = 1 to n i Almost identical model structure to earlier one-way ANOVA. Difference in level of inference

4. Level of Inference In one-way ANOVA, interest was in comparing the factor level means In random effects scenario, interest is in the pop of factor level means, not just the means of the r study levels Need to make assumptions about population distribution Will take “random” draw from pop of factor levels for use in study

5. KNNL Example KNNL p 1036 Y is the rating of a job applicant Factor A represents five different personnel interviewers (officers), r=5 n=4 different applicants were interviewed by each officer The interviewers were selected at random and the applicants were randomly assigned to interviewers

6. Read and check the data data a1; infile 'c:\...\CH25TA01.DAT'; input rating officer; proc print data=a1; run;

7. The data Obs rating officer 1 76 1 2 65 1 3 85 1 4 74 1 5 59 2 6 75 2 7 81 2 8 67 2 9 49 3 10 63 3

8. The data Obs rating officer 11 61 3 12 46 3 13 74 4 14 71 4 15 85 4 16 89 4 17 66 5 18 84 5 19 80 5 20 79 5

9. Plot the data title1 'Plot of the data'; symbol1 v=circle i=none c=black; proc gplot data=a1; plot rating*officer/frame; run;

11. Find and plot the means proc means data=a1; output out=a2 mean=avrate; var rating; by officer; title1 'Plot of the means'; symbol1 v=circle i=join c=black; proc gplot data=a2; plot avrate*officer/frame; run;

13. Random effects model Y ij = μ i + ε ij –the μ i are iid N(μ, σ μ 2 ) –the ε ij are iid N(0, σ 2 ) – μ i and ε ij are independent Y ij ~ N(μ, σ μ 2 + σ 2 ) Two sources of variation Observations with the same i are not independent, covariance is σ μ 2 Key difference

14. Random effects model This model is also called –Model II ANOVA –A variance components model The components of variance are σ μ 2 and σ 2 The models that we previously studied are called fixed effects models

15. Random factor effects model Y ij = μ +  i + ε ij  i ~ N(0, σ μ 2 ) ***** ε ij ~ N(0, σ 2 ) Y ij ~ N(μ, σ μ 2 + σ 2 )

16. Parameters There are three parameters in these models –μ –σ μ 2 –σ 2 The cell means (or factor levels) are random variables, not parameters Inference focuses on these variances

17. Primary Hypothesis Want to know if H 0 : σ μ 2 = 0 This implies all  i in model are equal but also all  i not selected for analysis are also equal. Thus scope is broader than fixed effects case Need the factor levels of the study to be “representative” of the population

18. Alternative Hypothesis We are sometimes interested in estimating σ μ 2 /( σ μ 2 + σ 2 ) This is the same as σ μ 2 / σ Y 2 In some applications it is called the intraclass correlation coefficient It is the correlation between two observations with the same I Also percent of total variation of Y

19. ANOVA table The terms and layout of the anova table are the same as what we used for the fixed effects model The expected mean squares (EMS) are different because of the additional random effects but F test statistics are the same Be wary that hypotheses being tested are different

20. EMS and parameter estimates E(MSA) = σ 2 + n σ μ 2 E(MSE) = σ 2 We use MSE to estimate σ 2 Can use (MSA – MSE)/n to estimate σ μ 2 Question: Why might it we want an alternative estimate for σ μ 2 ?

21. Main Hypotheses H 0 : σ μ 2 = 0 H 1 : σ μ 2 ≠ 0 Test statistic is F = MSA/MSE with r-1 and r(n-1) degrees of freedom, reject when F is large, report the P-value

22. Run proc glm proc glm data=a1; class officer; model rating=officer; random officer/test; run;

23. Model and error output Source DF MS F P Model 4 394 5.39 0.0068 Error 15 73 Total 19

24. Random statement output Source Type III Expected MS officer Var(Error) + 4 Var(officer)

25. Proc varcomp proc varcomp data=a1; class officer; model rating=officer; run;

26. Output MIVQUE(0) Estimates Variance Component rating Var(officer) 80.41042 Var(Error) 73.28333 Other methods are available for estimation, minque is the default

27. Proc mixed proc mixed data=a1 cl; class officer; model rating=; random officer/vcorr; run;

28. Output Covariance Parameter Estimates Cov Parm Est Lower Upper officer 80.4 24.4 1498 Residual 73.2 39.9 175 80.4104/(80.4104+73.2833)=.5232

29. Output from vcorr Row Col1 Col2 Col3 Col4 1 1.0000 0.5232 0.5232 0.5232 2 0.5232 1.0000 0.5232 0.5232 3 0.5232 0.5232 1.0000 0.5232 4 0.5232 0.5232 0.5232 1.0000

30. Other topics Estimate and CI for μ, p1038 –Standard error involves a combination of two variances –Use MSA instead of MSE → r-1 df Estimate and CI for σ μ 2 /( σ μ 2 + σ 2 ), p1040 CIs for σ μ 2 and σ 2, p1041-1047 –Available using Proc Mixed

31. Applications In the KNNL example we would like σ μ 2 /( σ μ 2 + σ 2 ) to be small, indicating that the variance due to interviewer is small relative to the variance due to applicants In many other examples we would like this quantity to be large, –e.g., Are partners more likely to be similar in sociability?

32. Last slide Start reading KNNL Chapter 25 We used program topic30.sas to generate the output for today