Multilevel Modelling

Definition

Multilevel modelling, also known as hierarchical linear modelling or mixed-effects modelling, is a regression technique designed for data with a nested structure — such as students within classrooms, patients within hospitals, or repeated measures within individuals. It partitions variance into between-group and within-group components, allowing intercepts and slopes to vary across higher-level units.

Why It Matters

Standard regression assumes independence of observations, which is violated when data are clustered. Ignoring this clustering produces underestimated standard errors, inflated Type I error rates, and misleading conclusions. Multilevel modelling corrects for these issues while also enabling research questions about how group-level characteristics (such as school funding or hospital size) influence individual-level outcomes.

Example

An education study examines reading achievement in 800 students nested within 40 schools. A multilevel model reveals that 15% of the variance in reading scores lies between schools (the intraclass correlation). School-level average teacher experience predicts school-average achievement (β = 0.30, p = 0.02), while student-level prior achievement predicts individual scores (β = 0.55, p < 0.001). A standard regression would have incorrectly treated all 800 students as independent observations.

Related Terms

• Regression Analysis
• Random Sampling
• Cluster Sampling
• ANOVA (random-effects ANOVA is a special case of multilevel modelling)

Software Notes

• SPSS: Analyze > Mixed Models > Linear. Define the subject variable (e.g., school ID) and place predictors in either the fixed-effects or random-effects boxes. Check Statistics for covariance parameter estimates.
• R: lme4::lmer(y ~ x1 + x2 + (1 | group), data = df) for random intercepts. lmer(y ~ x1 + x2 + (1 + x1 | group), data = df) for random slopes. summary(model) and lmerTest::ranova(model) provide p-values.
• Stata: mixed y x1 x2 || group: for random intercepts. mixed y x1 x2 || group: x1 for random slopes. estat icc reports the intraclass correlation. predict re*, reffects extracts random effects.