Topic 30: Random Effects

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Topic 30 Random Effects
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Outline

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

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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

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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

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KNNL Example

• KNNL p 1036
• Y is the rating of a job applicant
• Factor A represents five different personnel
  interviewers (officers), r5
• n4 different applicants were interviewed by each
  officer
• The interviewers were selected at random and the
  applicants were randomly assigned to interviewers

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Read and check the data
data a1 infile 'c\...\CH25TA01.DAT' input
rating officer proc print dataa1 run
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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
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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
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Plot the data
title1 'Plot of the data' symbol1 vcircle
inone cblack proc gplot dataa1 plot
ratingofficer/frame run
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Find and plot the means
proc means dataa1 output outa2
meanavrate var rating by officer title1
'Plot of the means' symbol1 vcircle ijoin
cblack proc gplot dataa2 plot
avrateofficer/frame run
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Random effects model

• Yij µi eij
• the µi are iid N(µ, sµ2)
• the eij are iid N(0, s2)
• µi and eij are independent
• Yij N(µ, sµ2 s2)
• Two sources of variation
• Observations with the same i are not
  independent, covariance is sµ2

Key difference
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Random effects model

• This model is also called
• Model II ANOVA
• A variance components model
• The components of variance are sµ2 and s2
• The models that we previously studied are called
  fixed effects models

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Random factor effects model

• Yij µ ?i eij
• ?i N(0, sµ2)
• eij N(0, s2)
• Yij N(µ, sµ2 s2)

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Parameters

• There are three parameters in these models
• µ
• sµ2
• s2
• The cell means (or factor levels) are random
  variables, not parameters
• Inference focuses on these variances

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Primary Hypothesis

• Want to know if H0 sµ2 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

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Alternative Hypothesis

• We are sometimes interested in estimating sµ2
  /(sµ2 s2)
• This is the same as sµ2 /sY2
• 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

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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

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EMS and parameter estimates

• E(MSA) s2 nsµ2
• E(MSE) s2
• We use MSE to estimate s2
• Can use (MSA MSE)/n to estimate sµ2
• Question Why might it we want an alternative
  estimate for sµ2?

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Main Hypotheses

• H0 sµ2 0
• H1 sµ2 ? 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

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Run proc glm
proc glm dataa1 class officer model
ratingofficer random officer/test run
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Model and error output
Source DF MS F P Model 4 394 5.39
0.0068 Error 15 73 Total 19
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Random statement output
Source Type III Expected MS officer
Var(Error) 4 Var(officer)
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Proc varcomp
proc varcomp dataa1 class officer model
ratingofficer run
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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
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Proc mixed
proc mixed dataa1 cl class officer
model rating random officer/vcorr run
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Output
Covariance Parameter Estimates Cov Parm Est
Lower Upper officer 80.4 24.4
1498 Residual 73.2 39.9 175
80.4104/(80.410473.2833).5232
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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
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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 sµ2 /(sµ2 s2), p1040
• CIs for sµ2 and s2, p1041-1047
• Available using Proc Mixed

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Applications

• In the KNNL example we would like sµ2 /(sµ2 s2)
  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?

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Last slide

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