Statistical Inference in Mixed Models

Mixed models combine fixed effects, which represent population-level influences, with random effects that capture variation between clusters or subjects. They are widely used in fields ranging from ecology and medicine to social sciences, enabling analysts to account for hierarchical or longitudinal data structures. Statistical inference in mixed models seeks to estimate the key parameters—particularly the variance components that quantify random effects—and to test hypotheses about both fixed and random contributions to the data. Rigorous approaches include maximum likelihood and restricted maximum likelihood estimation, non-parametric bootstrap, permutation tests and empirical Bayes methods. Recent developments have focused on improving the accuracy of variance component testing near model boundaries and singularities, on incorporating robust estimation for non-Gaussian data and outliers, and on unifying frequentist and Bayesian workflows. Practical applications include optimising the specification of random effects structures to balance model complexity and power, developing software tools for variance component testing, and refining hypothesis tests that remain valid under irregular conditions. These advances ensure that mixed models continue to offer reliable inference in increasingly complex data settings.

Research from Nature Portfolio

A new framework integrates empirical Bayes concepts within a frequentist goodness-of-fit paradigm, offering a unified view that combines the strengths of both inferential approaches. This methodology estimates priors directly from data while preserving the objectivity of frequentist tests, thereby improving parameter estimation in mixed models with limited or noisy observations. Examples demonstrate its utility across clinical trials, ecological studies and metrology, where it delivers more stable variance component estimates and enhances the reproducibility of inferences.

Research from all publishers

The varTestnlme software package provides an asymptotic likelihood-ratio testing procedure for selecting among variance components in linear, generalised linear and non-linear mixed models. It enables simultaneous testing of fixed effects and random variance terms, filling a gap in existing tools for random-effect selection and facilitating more parsimonious model building. Numerical illustrations highlight its performance on diverse real datasets.

A Monte Carlo permutation method for generalised linear mixed models employs a test statistic analogous to ANOVA but relies on fitting only the null model. Simulations show that this approach controls type I error effectively and achieves superior power compared with bootstrap-based score tests. Its computational efficiency and straightforward implementation make it attractive for applications where rapid inference on variance components is required.

Comparative simulations of strategies for specifying optimal random-effects structures in linear mixed models reveal that search-based algorithms, including forward and backward selection, often outperform both maximal and minimal approaches. When predictors occur at the cluster level, simpler random-intercept models suffice, whereas within-cluster variables demand random slopes to preserve statistical power. These findings guide researchers in balancing model complexity against inferential precision, especially in small-sample or high-variance scenarios.

Statistical Inference in Mixed Models publication trend

The graph below shows the total number of articles in statistical inference in mixed models across all publications each year (not limited to Nature Index journals).

Technical terms

Mixed model: A statistical model containing both fixed effects (parameters associated with an entire population) and random effects (parameters associated with individual experimental units).

Fixed effect: A model term that represents systematic, reproducible influences believed to be constant across clusters or subjects.

Random effect: A model term that captures variation across clusters or subjects, typically assumed to follow a probability distribution.

Variance component: A parameter quantifying the variability attributable to a random effect within a mixed model.

Likelihood ratio test: A hypothesis test comparing nested models by evaluating the ratio of their maximised likelihoods, often used to test for zero variance components.

Empirical Bayes: An approach that estimates prior distributions from the data itself, providing a bridge between frequentist and Bayesian inference.

References

1. varTestnlme: An R Package for Variance Components Testing in Linear and Nonlinear Mixed-Effects Models. Journal of Statistical Software (2023).
2. Hypothesis testing near singularities and boundaries. Electronic Journal of Statistics (2019).
3. Generalized Empirical Bayes Modeling via Frequentist Goodness of Fit. Scientific Reports (2018).
4. A Monte Carlo permutation procedure for testing variance components in generalized linear regression models. Computational Statistics (2023).
5. A Comparison of Methods for Specifying Optimal Random Effects Structures. Methodology (2023).