1. Multilevel modelling: general ideas and uses
Kari Nissinen
Finnish Institute for
Educational Research

2. Hierarchical data
Data in question is organized in a hierarchical / multilevel manner
Units at lower level (1-5) are arranged into higher-level units (A, B) A B 1 2 3 4 5

3. Hierarchical data Examples Students within classes within schools
Employees within workplaces
Partners in couples
Residents within neighbourhoods
Nestlings within broods within populations…
Repeated measures within individuals

4. Hierarchical data The key issue is clustering
lower-level units within an upper-level unit tend to be more homogeneous than two arbitrary lower-level units
E.g. students within a class: intra-cluster correlation ICC (positive)
Repeated measures: autocorrelation (usually positive)

5. Hierarchical data
Clustering => lower-level units are not independent
In cross-sectional studies this is a problem
Two correlated observations provide less information than two independent observations (partial ’overlap’)
Efficient sample size smaller than nominal sample size => statistical inference falsely powerful

6. Clustering in cross-sectional studies
Basic statistical methods do not recognize the dependence of observations
Standard errors (variances) underestimated => confidence intervals too short, statistical tests too significant
Special methodology needed for correct variances…
Design-based approaches (variance estimation in cluster sampling framework)
Model-based approaches: multilevel models

7. Clustering in cross-sectional studies
Measure of ’inference error’ due to clustering: design effect (DEFF)
= ratio of correct variance to underestimated variance (no clustering assumed)
A function of ratio of nominal sample size to effective sample size and/or homogeneity within clusters (ICC)

8. Hierarchical data
Hierarchy is a property of population, which can carry over into the sample data
Cluster sampling: hierarchy is explicitly present in data collection => data possess the same hierarchy (and possible clustering) exactly
Simple random sampling (etc): clustering may or may not appear in the data
It is present but hidden, may be difficult to identify
Effect may be negligible

9. Hierarchical data
Hierarchy does not always lead to clustering: units within a cluster can be uncorrelated
Other side of the coin is heterogeneity between upper-level units: if no heterogeneity, then no homogeneity among lower-level units
Zero ICC => no need for special methodology
Clustering can affect some target variables, but not some others

10. Longitudinal data
Clustering = measurements on an individual are not independent
When analyzing change this is a benefit
Each units serves as its own ’control unit’ (’block design’) => ’true’ change
Autocorrelation ’carries’ this link from time point to another
Appropriate methods utilize this correlation => powerful statistical inference

11. Mixed models
An approach for handling hierarchical / clustered / correlated data
Typically regression or ANOVA models, which contain effects of explanatory variables, which can be (i) fixed, (ii) random or (iii) both
Linear mixed models: error distribution normal (Gaussian)
Generalized linear mixed models: error distribution binomial, Poisson, gamma, etc

12. Mixed models Variance component models
Random coefficient regression models
Multilevel models
Hierachical (generalized) linear models
All these are special cases of mixed models
Similar estimation procedures (maximum likelihood & its variants), etc

13. Fixed vs random effects
1-way ANOVA fixed effects model
Y(ij) = μ + α(i) + e(ij)
μ = fixed intercept, grand mean
α(i) = fixed effect of group i
e(ij) = random error (’random effect’) of unit ij
random, because it is drawn from a population
it has a probability distribution (often N(0,σ²))

14. Fixed vs random effects
Fixed effects determine the means of observations
E(Y(ij)) = μ + α(i), since E(e(ij))=0
Random effects determine the variances (& covariances/correlations) of observations
Var(Y(ij)) = Var(e(ij)) = σ²

15. Fixed vs random effects
1-way ANOVA random effects model
Y(ij) = μ + u(i) + e(ij)
μ = fixed intercept, grand mean
u(i) = random effect of group i
random when the group is drawn from a population of groups
has a probability distribution N(0,σ(u)²)
e(ij) = random error (’random effect’) of unit ij

16. Fixed vs random effects
Now the mean of observations is just
E(Y(ij)) = μ
Variance is
Var(Y(ij)) = Var(u(i) + e(ij))
= σ(u)² + σ²
Sum of two variance components => variance component model

17. Random effects and clustering
Random group => units ij and ik within group i are correlated:
Cov(Y(ij),Y(ik))
= Cov(u(i) + e(ij), u(i) + e(ik))
= Cov(u(i), u(i)) = σ(u)²
Positive intra-cluster correlation
ICC = Cov(Y(ij),Y(ik)) / Var(Y(ij))
= σ(u)² / (σ(u)² + σ²)

18. Mixed model Contains both fixed and random effects, e.g.
Y(ij) = μ + βX(ij) + u(i) + e(ij)
i = school, j = student
μ = fixed intercept
β = fixed regression coefficient
u(i) = random school effect (’school intercept’)
e(ij) = random error of student j in school i

19. Mixed model Y(ij) = μ + βX(ij) + u(i) + e(ij)
The mean of Y is modelled as a function of explanatory variable X through the fixed parameters μ and β
The variance of Y and within-cluster covariance (ICC) are modelled through the random effects u (’level 2’) and e (’level 1’)
This is the general idea; extends versatilely

20. Regression lines in variance component model: high ICC

21. Regression lines in variance component model: low ICC

22. An extension: random coefficient regression
Y(ij) = μ + βX(ij) + u(i) + v(i)X(ij) + e(ij)
v(i) = random school slope
Regression coefficient of X varies between schools: β + v(i)
A ’side effect’: the variance of Y varies along with X
one possible way to model unequal variances (as a function of X)

23. Random coefficient regression

24. Regression for repeated measures data
Y(it) = μ(t) + βX(it) + e(it)
t = time, μ(t) = intercept at time t
i = individual
The errors e(it) of individual i correlated: different (auto)correlation structures (e.g. AR(1)) can be fitted as well as different variance structures (unequal variances)

25. Thanks!