Topic 30: Random Effects

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 • Yij is the jth observation in cell i, j = 1 to ni • 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 differentapplicants 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 Key difference • Yij = μi + εij • the μi are iid N(μ, σμ2) • the εij are iid N(0, σ2) • μi and εij are independent • Yij ~ N(μ, σμ2 + σ2) • Two sources of variation • Observations with the same i are not independent, covariance is σμ2

14. Random effects model • This model is also called • Model II ANOVA • A variance components model • The components of variance are σμ2 and σ2 • The models that we previously studied are called fixed effects models

15. Random factor effects model • Yij = μ + i + εij • i ~ N(0, σμ2) ***** • εij ~ N(0, σ2) • Yij ~ N(μ, σμ2 + σ2)

16. Parameters • There are three parameters in these models • μ • σμ2 • σ2 • The cell means (or factor levels) are random variables, not parameters • Inference focuses on these variances

17. Primary Hypothesis • Want to know if H0: σμ2 = 0 • This implies all mi in model are equal but also all mi 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 σμ2 /(σμ2 +σ2) • This is the same as σμ2 /σY2 • 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) = σ2 + nσμ2 • E(MSE) = σ2 • We use MSE to estimate σ2 • Can use (MSA – MSE)/n to estimate σμ2 • Question: Why might it we want an alternative estimate for σμ2?

21. Main Hypotheses • H0: σμ2 = 0 • H1: σμ2 ≠ 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 σμ2 /(σμ2 +σ2), p1040 • CIs for σμ2 andσ2, p1041-1047 • Available using Proc Mixed

31. Applications • In the KNNL example we would like σμ2 /(σμ2 +σ2) 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