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Aims of a Variance Components Analysis

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?

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Aims of a Variance Components Analysis

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  1. 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

  2. Revision of Fixed Effects Approach

  3. Limitations of the Fixed Effects Approach • When number of groups is large, there will be many extra parameters to estimate. (Only one in ML model.) • For groups with small sample sizes, the estimated group effects may be unreliable. (In a ML model residual estimates for such groups ‘shrunken’ towards zero.) • Fixed effects approach originated in experimental design where number of groups is small (e.g. treatment vs. control) and all groups sampled. More generally, our groups may be a sample from a population. (The multilevel approach allows inferences to this population.)

  4. Multilevel (Random Effects) Model

  5. Individual (e) and Group (u) Residuals in a Variance Components Model y42 e42 u2 0 u1

  6. Partitioning Variance

  7. Model Assumptions

  8. Intra-class Correlation

  9. 10 10 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 School School School Examples of ρ ρ ≈ 0.8 ρ ≈ 0.4 ρ≈ 0 Note: Schools ranked according to mean of y

  10. Example: Between-Country Differences in Hedonism

  11. Testing for Group Effects

  12. Example: Testing for Country Differences in Hedonism

  13. Residuals in a Variance Components Model

  14. Level 2 Residuals

  15. Shrinkage

  16. Example: Mean Raw Residuals vs. Shrunken Residuals for Selected Countries In this case, there is little shrinkage because nj is very large.

  17. Example: Caterpillar Plot showing Country Residuals and 95% CIs

  18. 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

  19. Example: Normal Plot of Individual (Level 1) Residuals Linearity suggests normality assumption is reasonable

  20. Example: Normal Plot of Country (Level 2) Residuals Some nonlinearity but only 20 level 2 units

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