Clustered Data and Multilevel Studies

Overview

Clustered data arise when participants are grouped within higher-level units such as clinics, schools, households, communities or study sites.

Participants from the same cluster may share:

• population characteristics;
• staff and protocols;
• geography and socioeconomic conditions;
• measurement procedures;
• exposure or intervention.

Multilevel models represent this hierarchy explicitly.

In cluster-randomised trials, treatment is allocated to clusters rather than individuals, so the independent treatment information is driven primarily by the number of clusters.

Why this matters

A study with 24 clinics and 18 participants per clinic contains 432 participant records but only 24 independently allocated clinics.

Ignoring clinic clustering can produce narrow confidence intervals and inflated type I error.

Adding many participants to a small number of clinics often contributes less information than recruiting additional clinics.

When to use this method

Use a multilevel or cluster-aware approach when observations are nested within naturally occurring or randomised clusters.

Examples include:

• patients within clinicians;
• clinicians within hospitals;
• students within schools;
• residents within communities;
• repeated visits within patients within clinics.

Research question and variables

The two-level random-intercept model is:

The intervention coefficient compares adjusted mean outcomes between clinic groups.

Data structure

Two-level clustered data

simulated data

Table 1. Study levels

Level	Unit	Example variables
Level 2	Clinic	Intervention, clinic type, random intercept
Level 1	Participant	Age, sex, systolic blood pressure

Note: The intervention is constant within every clinic.

A treatment value that changes within a clinic would contradict the stated cluster-randomised design unless crossover or stepped-wedge timing is represented.

Statistical assumptions

Intraclass correlation

For a continuous random-intercept model:

$$ICC= \frac{\sigma_u^2} {\sigma_u^2+\sigma_\varepsilon^2}$$

The ICC is outcome-, population- and design-specific. A small ICC can have a large design effect when clusters are large [1].

Worked example

Worked analysis in R

make_cluster_data <- function() {
  records <- vector("list", 24 * 18)
  row_number <- 1

  for (clinic in 1:24) {
    intervention <- as.integer(clinic %% 2 == 0)

    clinic_effect <-
      2.5 * sin(clinic * 0.73) +
      1.2 * cos(clinic * 0.31)

    for (person in 1:18) {
      index <- (clinic - 1) * 18 + person
      age <- 35 + ((7 * index) %% 41)
      male <- as.integer(index %% 2 == 0)

      residual <-
        5 * sin(index * 1.07) +
        2 * cos(person * 0.47)

      sbp <-
        145 -
        5.2 * intervention +
        0.18 * (age - 55) +
        1.5 * male +
        clinic_effect +
        residual

      records[[row_number]] <- data.frame(
        clinic = factor(clinic),
        person = person,
        intervention = factor(
          intervention,
          levels = c(0, 1),
          labels = c("Usual care", "Clinic intervention")
        ),
        age = age,
        male = male,
        sbp = sbp
      )

      row_number <- row_number + 1
    }
  }

  cluster_data <- do.call(rbind, records)
  rownames(cluster_data) <- NULL
  cluster_data
}

cluster_data <- make_cluster_data()

library(lme4)
library(lmerTest)

cluster_fit <- lmer(
  sbp ~ intervention + age + male + (1 | clinic),
  data = cluster_data,
  REML = TRUE
)

summary(cluster_fit)
confint(
  cluster_fit,
  parm = "beta_",
  method = "Wald"
)

variance_components <- as.data.frame(
  VarCorr(cluster_fit)
)

clinic_variance <- variance_components[
  variance_components$grp == "clinic",
  "vcov"
]

residual_variance <- sigma(cluster_fit)^2

icc <- clinic_variance /
  (clinic_variance + residual_variance)

design_effect <- 1 + (18 - 1) * icc

c(
  clinic_variance = clinic_variance,
  residual_variance = residual_variance,
  icc = icc,
  design_effect = design_effect
)

Selected analysis output

Fixed effect                     Estimate   SE       95% CI
Clinic intervention               -5.316    0.916    -7.111 to -3.521
Age                                0.009    0.013    -0.017 to 0.035
Male versus female                 1.249    0.313     0.636 to 1.862

Clinic variance = 4.445
Residual variance = 10.564
ICC = 0.296
Approximate design effect = 6.03

Visualising clinic variation

The figure should not be interpreted as a league table. Clinic means differ in case mix and sampling uncertainty.

Individual-level and cluster-level predictors

A participant-level age effect compares participants within and across clinics under the model.

A clinic-level intervention effect compares clinics.

For a participant variable such as socioeconomic status, separate:

• within-clinic individual effect;
• between-clinic contextual effect.

Group-mean centring can distinguish these components.

Ecological and atomistic fallacies

An association between clinic-level smoking prevalence and clinic mean outcome does not prove the same individual-level association.

Conversely, an individual association does not necessarily describe differences between clinics.

The level of the exposure and effect must match the interpretation.

Unequal and informative cluster size

Larger clinics often contribute more participants.

If cluster size is related to outcome after conditioning on the model, standard approaches may target a participant-average effect weighted towards large clusters.

The desired estimand may instead weight clinics equally. Informative cluster size should be considered explicitly.

Few clusters

With few clusters:

• sandwich SEs can be biased;
• variance components are imprecise;
• treatment degrees of freedom are limited;
• chance imbalance between arms can be substantial;
• complex covariate adjustment can be unstable.

Small-sample corrections or cluster-level analysis may be needed.

Interpreting the output

Presenting the results

Suggested reporting

Report number of clusters per arm, participant counts, cluster-size range, allocation level, analysis level, ICC and precision, variance method, small-sample correction and cluster loss [2], [3], [4].

Common mistakes

Knowledge check

What is the independent treatment unit?

Answer and reasoning

The clinic, because intervention is allocated at clinic level.

What does ICC 0.296 mean?

Answer and reasoning

About 29.6% of residual variance under the model is between clinics.

Why does a small ICC still matter with large clusters?

Answer and reasoning

The design effect increases with both ICC and cluster size.

What is a contextual effect?

Answer and reasoning

A between-cluster association that differs from the corresponding within-cluster individual association.

Why are few clusters challenging?

Answer and reasoning

Treatment information and asymptotic variance estimation depend on the number of independent clusters, not merely participant count.

Key take-home messages

References

1. Campbell MK, Fayers PM, Grimshaw JM. Determinants of the intracluster correlation coefficient in cluster randomized trials: the case of implementation research. Clin Trials. 2005;2(2):99–107. https://doi.org/10.1191/1740774505cn071oa
2. Donner A, Klar N. Design and analysis of cluster randomization trials in health research. London: Arnold; 2000.
3. Eldridge SM, Kerry S. A practical guide to cluster randomised trials in health services research. Chichester: Wiley; 2012. https://doi.org/10.1002/9781119966241
4. Campbell MK, Piaggio G, Elbourne DR, Altman DG, CONSORT Group. CONSORT 2010 statement: extension to cluster randomised trials. BMJ. 2012;345:e5661. https://doi.org/10.1136/bmj.e5661