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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
data a1; infile 'c:\...\CH25TA01.DAT'; input rating officer;proc print data=a1; run;
The dataObs 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
The dataObs 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
Plot the data
title1 'Plot of the data';symbol1 v=circle i=none c=black;proc gplot data=a1; plot rating*officer/frame;run;
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;
Random effects model• 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
Key difference
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
Random factor effects model
• Yij = μ + i + εij
i ~ N(0, σμ2) *****
• εij ~ N(0, σ2)
• Yij ~ N(μ, σμ2 + σ2)
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
Primary Hypothesis
• Want to know if H0: σμ2 = 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
Alternative Hypothesis
• We are sometimes interested in estimating σμ
2 /(σμ2 + σ2)
• This is the same as σμ2 /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 + 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 σμ
2?
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
Run proc glm
proc glm data=a1; class officer; model rating=officer; random officer/test;run;
Model and error output
Source DF MS F PModel 4 394 5.39 0.0068Error 15 73Total 19
Random statement output
Source Type III Expected MS
officer Var(Error) + 4 Var(officer)
Proc varcomp
proc varcomp data=a1; class officer; model rating=officer;run;
OutputMIVQUE(0) Estimates
Variance Component ratingVar(officer) 80.41042Var(Error) 73.28333
Other methods are availablefor estimation, minque is the default
Proc mixed
proc mixed data=a1 cl; class officer; model rating=; random officer/vcorr;run;
Output
Covariance Parameter Estimates
Cov Parm Est Lower Upper
officer 80.4 24.4 1498Residual 73.2 39.9 175
80.4104/(80.4104+73.2833)=.5232
Output from vcorr
Row Col1 Col2 Col3 Col4 1 1.0000 0.5232 0.5232 0.52322 0.5232 1.0000 0.5232 0.52323 0.5232 0.5232 1.0000 0.52324 0.5232 0.5232 0.5232 1.0000
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 /(σμ
2 + σ2), p1040
• CIs for σμ2 and σ2, p1041-1047
–Available using Proc Mixed
Applications
• In the KNNL example we would like σμ
2 /(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?
Last slide
• Start reading KNNL Chapter 25
• We used program topic30.sas to generate the output for today