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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 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.)

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Multilevel (Random Effects) Model

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Individual (e) and Group (u) Residuals in a Variance Components Model 0 y 42 e 42 u2u2 u1u1

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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 ρ 0 10 School 1 2 34 0 10 School 1 2 34 0 10 School 1 2 34 ρ 0 ρ 0.4 ρ 0.8 Note: Schools ranked according to mean of y

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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 Selected Countries In this case, there is little shrinkage because n j 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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