Variance Components Models

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Variance Components Models
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Aims of a Variance Components Analysis

• Estimate the amount of variation between groups
  (level 2 variance) relative to within groups
  (level 1 variance)
• How much variation is there in life expectancy
  between and within countries?
• How much of the variation in student exam scores
  is between schools? i.e. is there within-school
  clustering in achievement?
• Compare groups
• Which countries have particularly low and high
  life expectancies?
• Which schools have the highest proportion of
  students achieving grade A-C, and which the
  lowest?
• As a baseline for further analysis

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Revision of Fixed Effects Approach
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Limitations of the Fixed Effects Approach
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Multilevel (Random Effects) Model
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Individual (e) and Group (u) Residuals in a
Variance Components Model
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Partitioning Variance
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Model Assumptions
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Intra-class Correlation
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Examples of ?
? 0.8
? 0.4
? 0
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Example Between-Country Differences in Hedonism
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Testing for Group Effects
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Example Testing for Country Differences in
Hedonism
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Residuals in a Variance Components Model
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Level 2 Residuals
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Shrinkage
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Example Mean Raw Residuals vs. Shrunken
Residuals for Countries
In this case, there is little shrinkage because
nj is very large.
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Example Caterpillar Plot showing Country
Residuals and 95 CIs
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Residual Diagnostics

• Use normal Q-Q plots to check assumptions that
  level 1 and 2 residuals are normally distributed
• Nonlinearities suggest departures from normality
• Residual plots can also be used to check for
  outliers at either level
• Under normal distribution assumption, expect 95
  of standardised residuals to lie between -2 and
  2
• Can assess influence of a suspected outlier by
  comparing results after its removal

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Example Normal Plot of Individual (Level 1)
Residuals
Linearity suggests normality assumption is
reasonable
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Example Normal Plot of Country (Level 2)
Residuals
Some nonlinearity but only 20 level 2 units
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Estimation of a Multilevel Model
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The IGLS Algorithm