Mixed Effects Model

Mixed Effects Model: Concept Architecture

Orientation and learning roadmap

This page moves from clustered health observations to a model with population-level coefficients and group-specific deviations. It then works through repeated utility measurements, explains estimation and model checks, and shows what the results can contribute to an economic evaluation.

Why clustered observations change the analysis

Repeated measurements from one patient tend to be more alike than measurements from unrelated patients, and people treated within one clinic may share local influences. An analysis that treats all those observations as independent can misrepresent uncertainty and obscure meaningful variation between patients, clinics, or studies.

The analyst should identify the grouping unit and the target effect before choosing a random intercept, random slope, or another covariance structure. A patient may have multiple visits, patients may be nested within clinics, and a multi-study analysis may have further levels; these are different designs rather than interchangeable labels. The number of distinct groups, not just the number of observations, matters for estimating between-group variation.

Fixed coefficients and random deviations

Fixed effects describe coefficients that apply across the analysed population, conditional on the model's covariates. Random effects represent deviations for sampled groups from a specified population distribution, with an estimated variance or covariance rather than a separate unrelated coefficient for every group.

For a continuous outcome $Y_{it}$ for patient $i$ at time $t$, a simple random-intercept model with treatment indicator $T_i$ is

$$Y_{it}=\beta_0+\beta_1t+\beta_2T_i+\beta_3tT_i+b_{0i}+\varepsilon_{it}, \qquad E(b_{0i})=0.$$

Here $\beta_0$ is the modelled comparator mean at $t=0$, $\beta_1$ its time trend, $\beta_2$ the treatment-group difference at $t=0$, and $\beta_3$ the difference in time trends. The patient-specific $b_{0i}$ shifts the intercept, while $\varepsilon_{it}$ captures remaining observation-level error under a specified residual structure. In the simplest Gaussian version, random effects and residual errors have mean zero, modelled variances, and assumptions about their independence from each other and relevant covariates.

Worked example: repeated health utility

Suppose a fitted linear model for one-year follow-up has $\beta_0=0.70$, $\beta_1=-0.04$ utility units per year, $\beta_2=0$, and $\beta_3=0.06$ utility units per year. These invented values give equal modelled baseline means across strategies and an average difference in change from baseline of $0.06$ utility units after one year.

Each row has a treatment-minus-comparator contrast of $0.06$ at year 1 because this illustration has no random treatment slope. The patient-specific rows show conditional predictions for hypothetical intercept deviations, not observations that the same person can receive both treatments. A one-year endpoint difference of $0.06$ is not automatically a $0.06$-QALY gain over that year; area under the utility trajectories and survival must be considered.

Correlation and alternative structures

A random intercept induces correlation among repeated observations within a patient. Under the simplest homoscedastic random-intercept model, the intraclass correlation coefficient (ICC) is $\sigma_b^2/(\sigma_b^2+\sigma_\varepsilon^2)$; for illustrative variances $0.01$ and $0.04$, it is $0.20$.

If patients change at different rates, a random time slope can allow both starting levels and trends to vary, with an estimated intercept–slope covariance. Clinic-level effects can be added when patients share a provider or facility, while crossed effects may be needed when classifications overlap instead of nesting cleanly. A random-effects distribution cannot compensate for a wrongly specified mean trend, unmeasured confounding, or an unsupported assumption that group effects are independent of the included predictors.

Fitting, assumptions, and diagnostics

Linear mixed models are commonly estimated using maximum likelihood (ML) or restricted maximum likelihood (REML), with the choice depending on whether fixed-effect structures are being compared and what inference is planned. For binary outcomes, counts, or other non-Gaussian responses, a generalised linear mixed model uses an appropriate link and outcome distribution, so its conditional coefficients are not interpreted on the same scale as a linear utility model.

Check the observed trajectories against fitted values, residual patterns, estimated variance components, and plausible correlation structure. Sparse groups or near-zero random-effect variances can make complex models unstable; convergence warnings and boundary estimates require investigation. Missing follow-up data need a stated missingness assumption and sensitivity analysis, since fitting a mixed model does not by itself cure informative dropout.

Interpretation for health economics

Mixed effects models can analyse repeated utilities, costs, service use, and outcomes in cluster-randomised or multi-centre studies, with the likelihood and scale matched to each outcome. In individual participant data meta-analysis, study-level random effects may represent between-study variation while within-study treatment contrasts remain the relevant evidence for a causal treatment comparison.

For an economic model, report the effect scale, time point, target population, uncertainty, and whether predictions are conditional on a particular group or averaged across groups. Predictions for a new clinic or patient must account for the uncertainty in its unobserved random effect rather than borrowing a known fitted deviation from an existing group. A statistically significant coefficient does not on its own establish clinical importance or cost effectiveness.

Common distinctions and source basis

The word random refers to a probability model for group-specific effects, not to random assignment of treatment. A random-effects meta-analysis of published study estimates is related but uses a different data structure and estimand from an individual-level mixed model; the grouping level and outcome must always be named.

• Laird and Ware (1982), “Random-Effects Models for Longitudinal Data”, Biometrics, 38:963–974. This foundational paper develops mixed models for repeated observations.
• Bates, Mächler, Bolker, and Walker (2015), “Fitting Linear Mixed-Effects Models Using lme4”, Journal of Statistical Software, 67(1). This provides model formulation and estimation detail.
• Cochrane, Handbook for Systematic Reviews of Interventions, Chapter 26: Individual participant data. This describes one-stage multilevel approaches in evidence synthesis.