Multilevel and Partial Pooling Models
Multilevel models let regression coefficients vary by group while tying them together through a population distribution, producing partially pooled estimates.
Definition
A multilevel model is a regression in which some coefficients are allowed to differ across groups and are themselves modeled as draws from a common distribution, so that group-level estimates are partially pooled toward the population pattern.
Scope
This topic covers varying-intercept and varying-slope structures, the population distribution that links group coefficients, the formula for the partial-pooling weight, and the relationship to classical mixed-effects and random-effects models.
Core questions
- How are varying-intercept and varying-slope models specified?
- What determines the amount of pooling for a given group?
- How do multilevel models relate to frequentist mixed-effects models?
- How are group-level predictors incorporated at the higher level?
Key concepts
- varying intercepts
- varying slopes
- population distribution
- pooling factor
- group-level predictors
- mixed-effects model
- random effects
Key theories
- Group-level population distribution
- Modeling group coefficients as exchangeable draws from a population distribution is what couples the groups and yields partial pooling determined by within- and between-group variances.
- Pooling factor
- The weight given to the population mean versus a group's own estimate depends on the ratio of sampling variance to group-level variance, so sparse or noisy groups are pooled more strongly.
Clinical relevance
Multilevel models handle clustered and longitudinal data such as patients within hospitals or repeated measures within subjects, giving stable group estimates and correct uncertainty when groups vary in size.
History
The Bayesian linear hierarchical model was set out by Lindley and Smith in 1972. The varying-intercept and varying-slope formulation popularized by Gelman and Hill in 2007 made multilevel modeling accessible to applied researchers across disciplines.
Key figures
- Dennis Lindley
- Adrian Smith
- Andrew Gelman
- Jennifer Hill
Related topics
Seminal works
- gelman2007
- lindley1972
Frequently asked questions
- When should I let slopes vary, not just intercepts?
- Allow a coefficient to vary by group when the effect of a predictor is expected to differ across groups; varying slopes capture this heterogeneity, while varying intercepts only adjust the baseline level.