Topic 30 Random Effects Outline Oneway random effects

Topic 30: Random Effects

Outline • One-way random effects model – Data – Model – Inference

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

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

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

Read and check the data a 1; infile 'c: . . . CH 25 TA 01. DAT'; input rating officer; proc print data=a 1; run;

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

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

Plot the data title 1 'Plot of the data'; symbol 1 v=circle i=none c=black; proc gplot data=a 1; plot rating*officer/frame; run;

Find and plot the means proc means data=a 1; output out=a 2 mean=avrate; var rating; by officer; title 1 'Plot of the means'; symbol 1 v=circle i=join c=black; proc gplot data=a 2; plot avrate*officer/frame; run;

Random effects model • Yij = μi + εij Key – the μi are iid N(μ, σμ 2) difference – the εij are iid N(0, σ2) – μi and εij are independent • Yij ~ N(μ, σμ 2 + σ2) • Two sources of variation • Observations with the same i are not independent, covariance is σμ 2

Random effects model • This model is also called – Model II ANOVA – A variance components model • The components of variance are σμ 2 and σ2 • The models that we previously studied are called fixed effects models

Random factor effects model • Yij = μ + i + εij • i ~ N(0, σμ 2) ***** • εij ~ N(0, σ2) • Yij ~ N(μ, σμ 2 + σ2)

Parameters • There are three parameters in these models –μ – σμ 2 – σ2 • The cell means (or factor levels) are random variables, not parameters • Inference focuses on these variances

Primary Hypothesis • Want to know if H 0: σμ 2 = 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

Alternative Hypothesis • We are sometimes interested in estimating σμ 2 /(σμ 2 + σ2) • This is the same as σμ 2 /σY 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

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

EMS and parameter estimates • • • E(MSA) = σ2 + nσμ 2 E(MSE) = σ2 We use MSE to estimate σ2 Can use (MSA – MSE)/n to estimate σμ 2 Question: Why might it we want an alternative estimate for σμ 2?

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

Run proc glm data=a 1; class officer; model rating=officer; random officer/test; run;

Model and error output Source DF Model 4 Error 15 Total 19 MS 394 73 F P 5. 39 0. 0068

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

Proc varcomp proc varcomp data=a 1; class officer; model rating=officer; run;

Output MIVQUE(0) Estimates Variance Component Var(officer) Var(Error) rating 80. 41042 73. 28333 Other methods are available for estimation, minque is the default

Proc mixed proc mixed data=a 1 cl; class officer; model rating=; random officer/vcorr; run;

Output Covariance Parameter Estimates Cov Parm officer Residual Est Lower 80. 4 73. 2 24. 4 39. 9 Upper 1498 175 80. 4104/(80. 4104+73. 2833)=. 5232

Output from vcorr Row Col 1 1 1. 0000 2 0. 5232 3 0. 5232 4 0. 5232 Col 2 0. 5232 1. 0000 0. 5232 Col 3 0. 5232 1. 0000 0. 5232 Col 4 0. 5232 1. 0000

Other topics • Estimate and CI for μ, p 1038 – Standard error involves a combination of two variances – Use MSA instead of MSE → r-1 df • Estimate and CI for σμ 2 /(σμ 2 + σ2), p 1040 • CIs for σμ 2 and σ2, p 1041 -1047 – Available using Proc Mixed

Applications • In the KNNL example we would like σμ 2 /(σμ 2 + 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?

Last slide • Start reading KNNL Chapter 25 • We used program topic 30. sas to generate the output for today