Fixed effects, random effects, and mixed models

 While classical statistical methods are based on an assumption of independent observations, many currently used statistical models include observations that are related instead of independent, such as repeated measurements from the same patient and patients randomised at the same centre. Fixed effects models, random effects models, and mixed models provide different ways to deal with independent and related, and a mix of independent and related observations (1).

Fixed effects estimate a population-average association or a specific comparison that is central to the research question. For example, in a clinical study, treatment group, follow-up time, baseline age, and sex may be modelled as fixed effects. The estimated treatment effect then represents the expected difference between treatment groups, conditional on the other predictors.

Random effects represent the variation among units drawn from a wider population. Instead of estimating a separate, unrelated parameter for each hospital or participant, the model treats their deviations from the overall average as draws from a common distribution. For example, a random-intercept model allows each patient to have their own baseline outcome level. A random-slope model additionally allows the association with time or treatment to differ from patient to patient.

Random effects are especially useful for analysing repeated measures and hierarchical data as they account for correlation between observations from the same unit. Moreover, random effects values shrink towards their expected value, often the overall population mean, reducing the influence of extreme outcomes. By this correction for extreme values, the shrunken estimates are generally more accurate and reliable than their uncorrected counterparts, which means that the overall model tends to better reflect the underlying data structure.

Mixed models (also known as multilevel or hierarchical models), include both fixed and random effects. They are commonly used in the analysis of randomised trials because of their ability to handle both repeated measurements and including data for patients with partially missing observations (2). This explains, for example, why MMRM ANOVA (mixed model for repeated measures analysis of variance) has become the standard method for analysing continuous endpoints in confirmatory trials.

For binary, counts, ordinal, or otherwise non-normal outcomes, the corresponding approach is a generalized linear mixed model (GLMM), which combines random effects with an appropriate distribution and link function. For example, logistic mixed models for binary outcomes and Poisson mixed models for counts.

References

1. Silveira LTYD, Ferreira JC, Patino CM. Mixed-effects model: a useful statistical tool for longitudinal and cluster studies. J Bras Pneumol. 2023 May 15;49(2):e20230137. doi: 10.36416/1806-3756/e20230137. PMID: 37194822; PMCID: PMC10171296.

2. Ranstam J, Turkiewicz A, Boonen S, Van Meirhaeghe J, Bastian L, Wardlaw D. Alternative analyses for handling incomplete follow-up in the intention-to-treat analysis: the randomized controlled trial of balloon kyphoplasty versus non-surgical care for vertebral compression fracture (FREE). BMC Med Res Methodol. 2012 Mar 24;12:35. doi: 10.1186/1471-2288-12-35. PMID: 22443312; PMCID: PMC3323461.

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