Hierarchical Model

Hierarchical Model: Concept Architecture

Keep linked observations together

A hierarchical model represents data or parameters at more than one related level, such as patients within hospitals or studies within an evidence synthesis. This page shows the group structure, how partial pooling uses information across groups, and how the model can support cost-effectiveness decisions without erasing genuine differences.

Identify the levels and the outcome

Specify the observation unit and the grouping units before fitting a model. Patients treated at the same hospital may share staffing, prices, clinical practice, and case mix; their outcomes are therefore not necessarily independent. Likewise, treatment-effect estimates from different trials have study-level variation in addition to their sampling error.

For a continuous patient outcome $Y_{ij}$ for patient $i$ in hospital $j$, a simple random-intercept model can be written as:

$$Y_{ij}=\alpha_j+\beta x_{ij}+\varepsilon_{ij}, \qquad \alpha_j\sim N(\mu,\tau^2), \qquad \varepsilon_{ij}\sim N(0,\sigma^2).$$

Here $x_{ij}$ is a measured patient characteristic, $\beta$ its conditional association, $\alpha_j$ a hospital-specific intercept, $\tau^2$ between-hospital variance, and $\sigma^2$ residual within-hospital variance. The normal distributions are illustrative modelling choices, not requirements of every hierarchical model. Binary events can instead use an appropriate link and likelihood; skewed positive costs may require another outcome family. Research on multicentre economic evaluation explains why patient, centre, and country structure can affect costs and outcomes.

See how partial pooling works

Estimating each hospital wholly separately can make a small hospital's estimate very noisy. Treating every hospital as identical can hide genuine differences. Partial pooling estimates a shared distribution of hospital effects and draws imprecise group estimates toward the overall mean, with the amount determined by within-group information and between-group variation.

For a normal-means teaching case with known overall mean $\mu$, between-group variance $\tau^2$, common patient-level variance $\sigma^2$, group sample size $n_j$, and group sample mean $\bar y_j$, the posterior or empirical-Bayes point estimate under the stated assumptions is:

$$\widehat\alpha_j =\mu+w_j(\bar y_j-\mu), \qquad w_j=\frac{\tau^2}{\tau^2+\sigma^2/n_j}.$$

When $n_j$ is small, $\sigma^2/n_j$ is larger and the estimate receives more shrinkage toward $\mu$. When between-group differences $\tau^2$ are large relative to sampling error, the group data retain more influence. The expression illustrates the mechanism; real analyses must estimate uncertain variances and account for covariates and outcome distributions. An applied hospital performance study explains why low-volume hospital estimates can shrink most.

Work through two hospitals

Suppose a hypothetical outcome has an overall mean of 5, a known between-hospital variance of 1, and a patient-level variance of 25. Hospital A has 10 patients and an unadjusted mean of 10; its weight is $1/(1+25/10)=2/7\approx0.286$, producing $5+(2/7)(10-5)\approx6.43$. Hospital B has 100 patients and an unadjusted mean of 7; its weight is $1/(1+25/100)=0.8$, producing $5+0.8(7-5)=6.6$.

A's apparently higher raw mean is more heavily pulled toward the common mean because it comes from fewer observations. The example is arithmetic for an assumed normal model, not a hospital ranking or a quality judgment. A real comparison would adjust for patient risk, uncertainty in the shared parameters, sampling and outcome definitions, and the costs of false classifications.

Choose the hierarchy needed for the decision

A patient–hospital hierarchy can estimate clustering, contextual effects, and variation in costs or health outcomes. A study-level random-effects meta-analysis is also hierarchical: each study estimates its own underlying effect, while those effects are related through a distribution. A Bayesian implementation specifies prior distributions for its hyperparameters; frequentist mixed-effects approaches also represent multiple levels without using the same prior formulation.

The levels need not be a simple nesting. Patients may receive care at several providers, or repeated outcomes may be nested within people who are also grouped by country. Cross-classified, longitudinal, or multivariate models may be needed when the data structure demands them. A cluster-trial cost-effectiveness application shows how a hierarchy can jointly represent costs and effects; multinational cost-data research illustrates a non-normal cost model.

Diagnose the model rather than trusting shrinkage

Inspect group sizes, residual patterns, influential groups, the estimated between-group variance, and whether predicted observations resemble the actual data. Test sensitivity to the chosen random-effects distribution, prior when Bayesian, patient covariates, and whether group effects or treatment effects are allowed to vary. A model can pool biased observations efficiently and still be biased.

Shrinkage is not evidence that a small hospital truly performs near average. If important groups are unlike the population used for pooling, exchangeability may fail; separate strata or explanatory group characteristics may be more defensible. Uncertainty around group estimates and group rankings should be presented, especially when decisions affect payments, access, or reputations.

Carry group differences into economic decisions

For cost-effectiveness, decide whether the question concerns an average across all sites or a specific local population. A hierarchical analysis can propagate variation and uncertainty in costs, outcomes, treatment effects, and incremental net benefit across centres. It should separate between-centre heterogeneity from uncertainty about a pooled mean and from within-patient variation, then use appropriate local prices and practice patterns.

Report the levels, likelihood, link, covariates, varying terms, distribution or priors, group sizes, fit checks, and target population. Do not mistake statistical levels for a clinical state-transition hierarchy or an ontology parent-child relationship. The value of the model comes from matching its structure and assumptions to the actual evidence and decision.