Concept Architecture
Concept
Theoretically, Mixed Model Repeated Measures (MMRM) is a likelihood-based statistical method for analysing longitudinal continuous outcomes measured repeatedly on the same individuals. It is a specialised application of the linear mixed model that incorporates fixed effects for treatment, time and other covariates while modelling the covariance structure of repeated observations without explicitly including subject-specific random effects in the standard formulation. MMRM exists to estimate treatment effects over time while accounting for within-subject correlation and incomplete follow-up under the Missing at Random assumption.
Mathematically, MMRM models the expected outcome as a linear function of explanatory variables and specifies a covariance matrix describing the correlation between repeated measurements within each participant. Parameters are estimated using Restricted Maximum Likelihood (REML) or Maximum Likelihood (ML), and treatment comparisons are obtained from the estimated fixed effects. The covariance structure may be unstructured, autoregressive, compound symmetric or another recognised form selected according to model fit.
In practice, MMRM is widely used in randomised clinical trials to analyse repeated measurements of quality of life, healthcare costs, clinical outcomes and patient-reported outcomes. In health economics, it is frequently applied to longitudinal utility scores, repeated cost measurements and treatment-effect estimation because it uses all available observations without requiring imputation of missing values when the Missing at Random assumption is considered reasonable.
Purpose
Used to analyse repeated continuous outcomes, estimate longitudinal treatment effects, account for within-subject correlation and provide valid inference from incomplete longitudinal data under the Missing at Random assumption.
Mathematical Formulae
Primary Formula
Y = X? + �
where:
- Y = vector of repeated outcomes
- X = fixed-effects design matrix
- ? = vector of fixed-effect parameters
- � ~ N(0, V)
with:
V = R
where R is the within-subject covariance matrix.
Supporting Formulae
Marginal distribution:
Y ~ N(X?, V)
Restricted Maximum Likelihood estimator:
?? = (X?V??X)??X?V??Y
Log-likelihood:
?(?, V) = ?(1/2)[ln|V| + (Y ? X?)?V??(Y ? X?) + nln(2�)]
Related Mathematical Methods
- Linear Mixed Models
- Maximum Likelihood Estimation
- Restricted Maximum Likelihood
- Repeated Measures Analysis
- Generalised Estimating Equations
- Longitudinal Data Analysis
- Covariance Structure Modelling
Example
A health economist analyses EQ-5D utility scores collected at baseline, 3 months, 6 months and 12 months from 850 patients participating in a randomised trial.
An MMRM is fitted with fixed effects for treatment group, visit, treatment-by-visit interaction and baseline utility, using an unstructured covariance matrix.
The estimated treatment difference at 12 months is:
?? = 0.043
95% CI = 0.021 to 0.065
The analysis includes participants with incomplete follow-up, provided at least one post-baseline observation is available, and estimates are obtained using Restricted Maximum Likelihood under the Missing at Random assumption.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B851,C2:F851,TRUE,TRUE) | Estimate simplified fixed effects for exploratory analyses. |
| MMULT | =MMULT(A2:C851,D2:F4) | Perform matrix calculations used in likelihood estimation. |
| MINVERSE | =MINVERSE(MMULT(TRANSPOSE(A2:C851),A2:C851)) | Calculate matrix inverses required for estimation. |
| Solver | Maximise the restricted likelihood by varying covariance parameters. | Illustrate estimation of MMRM parameters. |
| AVERAGEIFS | =AVERAGEIFS(B:B,A:A,Visit) | Summarise repeated outcomes by visit during exploratory analysis. |
VBA (Optional)
A VBA routine can organise repeated-measures datasets, prepare covariance structures and generate longitudinal summaries for subsequent Mixed Model Repeated Measures analyses.
Sources
- Mallinckrodt CH, Clark WS, David SR. Accounting for Dropout Bias Using Mixed-Effects Models. Journal of Biopharmaceutical Statistics. 2001.
- Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis. Wiley.
- Verbeke G, Molenberghs G. Linear Mixed Models for Longitudinal Data. Springer.
- Mallinckrodt CH, Lane PW, Schnell D, Peng Y, Mancuso JP. Recommendations for the Primary Analysis of Continuous Endpoints in Longitudinal Clinical Trials. Drug Information Journal. 2008.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a mixed model for repeated measures?
A longitudinal trial analysis method using a mixed effects framework under the missing at random assumption, an alternative to last observation carried forward.
Source: Mallinckrodt et al. 2001
How does a mixed model for repeated measures use a patient's available data?
A mixed model for repeated measures analyses longitudinal trial data with a mixed effects framework, drawing on every measurement a patient did provide rather than discarding those with some missing visits. Under the assumption that data are missing at random, it estimates the treatment effect over time using the observed measurements and the correlation between a patient's own readings, so partial records still contribute. This makes it a principled alternative to last observation carried forward, which crudely freezes the final value. Using all observed measurements is its strength. Little and Rubin (2002) discuss such approaches.
Source: Little & Rubin 2002
How does a mixed model for repeated measures work?
A mixed model for repeated measures works by modelling the repeated outcomes over time with fixed effects for treatment, time, and their interaction, and a structure for the within-participant correlation, using all observed measurements and estimating treatment effects at the time points of interest. It does not require imputing missing values, using the available data directly. So a mixed model for repeated measures works by fitting a mixed effects model to the longitudinal outcomes, accounting for the correlation and using each participant's available data, which yields valid estimates of the treatment effect under the missing at random assumption without the need to carry forward or impute missing observations explicitly.
Source: Mallinckrodt et al. 2001
Why is a mixed model for repeated measures preferred over last observation carried forward?
A mixed model for repeated measures is preferred over last observation carried forward because it makes valid use of the available data under the plausible missing at random assumption and properly reflects uncertainty, whereas last observation carried forward assumes outcomes stay unchanged after dropout, which is unrealistic and can bias results and understate uncertainty. So a mixed model for repeated measures is preferred for giving more valid and honest analyses of longitudinal trial data with dropout, since it avoids the implausible constancy assumption of last observation carried forward and its potential for bias, which is why methodological and regulatory guidance has favoured it for the primary analysis of such trials.
Source: Mallinckrodt et al. 2001
What assumption does a mixed model for repeated measures rely on?
A mixed model for repeated measures relies chiefly on the missing at random assumption, that the probability of missing outcomes depends only on observed data, not on the unobserved values, under which it gives valid estimates using the available data. It also assumes the model, including the correlation structure, is reasonably specified. So a mixed model for repeated measures rests on missing at random and correct model specification, and its validity depends on these, which is why the plausibility of missing at random is considered and sensitivity analyses are used to examine robustness to departures, since if missingness is not at random the estimates could still be biased despite the method's advantages over last observation carried forward.
Source: Mallinckrodt et al. 2001
When is a mixed model for repeated measures used?
A mixed model for repeated measures is used for the analysis of longitudinal clinical trials with repeated outcome measurements and dropout, where a valid method under missing at random is wanted for estimating treatment effects over time. So a mixed model for repeated measures is used as a primary analysis approach for longitudinal trial data, particularly where missing data from dropout must be handled validly, which is why it has become a standard method in clinical trials, replacing older approaches such as last observation carried forward and analysis of variance that make less reasonable assumptions about the missing data and the correlation structure.
Source: Mallinckrodt et al. 2001
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 18 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-126
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