Hierarchical Model

A modelling approach representing nested data, such as patients within hospitals, allowing effects to vary at each level while borrowing statistical strength.

Source: Goldstein 2011

A hierarchical model represents data that come in nested layers, such as patients grouped within hospitals, by allowing effects to vary at each level rather than treating every observation as independent. It estimates variation both between individuals and between the groups they belong to, and it lets groups borrow strength from one another, so a small group's estimate is pulled toward the overall average where its own data are sparse. This gives more stable estimates and correctly reflects the layered structure. Modelling variation at each nested level is its purpose. Kirkwood and Sterne (2003) describe such models.

Source: Kirkwood & Sterne 2003

A hierarchical model works by including random effects for the higher-level units, so that, for example, each hospital has its own random deviation from the overall average, and the model estimates both the fixed effects of predictors and the variation between units at each level. This accounts for within-group correlation and allows information to be shared across groups. So a hierarchical model works by modelling variation at multiple levels through random effects, which handles the nested structure of the data, estimates how much outcomes vary between groups, and borrows strength so that estimates for small groups are informed by the overall pattern, improving their stability.

Source: Goldstein 2011

Hierarchical models are used because nested data violate the assumption of independence, since observations within the same group are correlated, and ignoring this leads to incorrect standard errors; hierarchical models account for the structure, giving valid inference. They also allow variation at each level to be examined and strength to be borrowed across groups. So hierarchical models are used to analyse nested data correctly and informatively, since they handle the within-group correlation, quantify variation at different levels, and produce more stable estimates for small groups, which is why they are standard for data organised in levels, such as patients within clinics or measurements within individuals.

Source: Goldstein 2011

Borrowing strength in a hierarchical model means that the estimate for each group is informed by that group's own data together with the data from other groups, through the shared distribution of the random effects, so that estimates for small or noisy groups are pulled toward the overall average, improving their stability. So borrowing strength refers to the way a hierarchical model combines group-specific and overall information, shrinking unreliable individual-group estimates toward the general pattern, which yields more stable and often more accurate estimates for groups with little data than analysing each group separately would, a key advantage of the multilevel approach.

Source: Goldstein 2011

Examples of hierarchical models include random intercept models, where each group has its own baseline level; random slope models, where the effect of a predictor also varies between groups; and models with more than two levels, such as patients within wards within hospitals. They apply to continuous, binary, and other outcomes through generalised forms. So hierarchical models encompass a range of specifications for nested data, from simple random intercepts to complex multilevel structures with varying effects, all sharing the feature of modelling variation at multiple levels, which allows them to be adapted to the particular hierarchy and outcome type of the data being analysed.

Source: Goldstein 2011