The Phylogenetic Ornstein-Uhlenbeck Mixed Model

The Phylogenetic Ornstein-Uhlenbeck Mixed Model (POUMM) allows to estimate the phylogenetic heritability of continuous traits, to test hypotheses of neutral evolution versus stabilizing selection, to quantify the strength of stabilizing selection, to estimate measurement error and to make predictions about the evolution of a phenotype and phenotypic variation in a population. The package implements combined maximum likelihood and Bayesian inference of the univariate Phylogenetic Ornstein-Uhlenbeck Mixed Model, fast parallel likelihood calculation, maximum likelihood inference of the genotypic values at the tips, functions for summarizing and plotting traces and posterior samples, functions for simulation of a univariate continuous trait evolution model along a phylogenetic tree. So far, the package has been used for estimating the heritability of quantitative traits in macroevolutionary and epidemiological studies, see e.g. Bertels et al. (2017) and Mitov and Stadler (2018) . The algorithm for parallel POUMM likelihood calculation has been published in Mitov and Stadler (2019) .

Generalised Linear Mixed Model Selection

Provides tools for fitting sparse generalised linear mixed models with l0 regularisation. Selects fixed and random effects under the hierarchy constraint that fixed effects must precede random effects. Uses coordinate descent and local search algorithms to rapidly deliver near-optimal estimates. Gaussian and binomial response families are currently supported. For more details see Thompson, Wand, and Wang (2025) .

Multilevel/Mixed Model Helper Functions

A collection of miscellaneous helper function for running multilevel/mixed models in 'lme4'. This package aims to provide functions to compute common tasks when estimating multilevel models such as computing the intraclass correlation and design effect, centering variables, estimating the proportion of variance explained at each level, pseudo-R squared, random intercept and slope reliabilities, tests for homogeneity of variance at level-1, and cluster robust and bootstrap standard errors. The tests and statistics reported in the package are from Raudenbush & Bryk (2002, ISBN:9780761919049), Hox et al. (2018, ISBN:9781138121362), and Snijders & Bosker (2012, ISBN:9781849202015).

Bias Diagnostic for Linear Mixed Models

Provides a function to perform bias diagnostics on linear mixed models fitted with lmer() from the 'lme4' package. Compares an internal random-effects alignment diagnostic with a fitted permutation reference distribution, following Karl and Zimmerman (2021) . Conditional response-refit simulations separately record random-effects alignment and residual-weight contributions to estimation error. Karl and Zimmerman (2020) provide R code for implementing the test using 'mvglmmRank' output. Development and review were assisted by 'GPT o1-preview', 'GPT 5 Pro', and 'GPT6 Astra'; the author is responsible for the code and its interpretation.

Solving Mixed Model Equations in R

Fits generalized linear and non linear mixed models with flexible covariance structures for multiple random effects. Supports structured random-effect and Gaussian residual covariances, including unstructured, autoregressive, spatial, factor-analytic and reduced-rank models. These structures can be combined using Kronecker products and user-defined covariance functions. Known relationships among random-effect levels, including pedigree and genomic relationship matrices, can be incorporated. Gaussian models are fitted by maximum likelihood or restricted maximum likelihood using complementary coefficient-space Henderson and observation-space direct-inversion engines, with average-information optimization and expectation-maximization recovery. Numerical methods include sparse matrix factorization, preconditioned iterative solvers and stochastic trace and log-determinant approximations for eligible large-scale problems. Generalized models are fitted by penalized quasi-likelihood or native Laplace-approximated maximum likelihood, including joint models of responses from different families linked through shared random effects. Provides prediction, uncertainty quantification and inference tools for applications including quantitative genetics, genomic prediction, multi-environment trials and spatial analysis (Covarrubias-Pazaran, 2016 ; Maier et al., 2015 ; Jensen et al., 1997).

Heritability Estimation from Mixed Models

Reporting heritability estimates is an important to quantitative genetics studies and breeding experiments. Here we provide functions to calculate various broad-sense heritabilities from 'asreml' and 'lme4' model objects. All methods we have implemented in this package have extensively discussed in the article by Schmidt et al. (2019) .

Analysis of Factorial Experiments

Convenience functions for analyzing factorial experiments using ANOVA or mixed models. aov_ez(), aov_car(), and aov_4() allow specification of between, within (i.e., repeated-measures), or mixed (i.e., split-plot) ANOVAs for data in long format (i.e., one observation per row), automatically aggregating multiple observations per individual and cell of the design. mixed() fits mixed models using lme4::lmer() and computes p-values for all fixed effects using either Kenward-Roger or Satterthwaite approximation for degrees of freedom (LMM only), parametric bootstrap (LMMs and GLMMs), or likelihood ratio tests (LMMs and GLMMs). afex_plot() provides a high-level interface for interaction or one-way plots using ggplot2, combining raw data and model estimates. afex uses type 3 sums of squares as default (imitating commercial statistical software).

Likelihood-Based Boosting for Generalized Mixed Models

Likelihood-based boosting approaches for generalized mixed models are provided.

Mixed Models for Biological, Clustered and Longitudinal Data

Fits and interprets mixed-effects models for clustered, longitudinal and heterogeneous biological data. Provides variance partitioning, intraclass correlation, penalized likelihood summaries, a heterogeneous-data information criterion, model comparison, diagnostics, and ensemble-style summaries for multilevel data. The package is designed as a complementary, interpretable workflow around established mixed-model methods. Methods for intraclass correlation and variance partitioning are informed by Nakagawa and Schielzeth (2010) and Nakagawa et al. (2017) . Mixed-effects modeling approaches are described by Zuur et al. (2009) .

Longitudinal Drift-Diffusion Mixed Models (LDDMM)

Implementation of the drift-diffusion mixed model for category learning as described in Paulon et al. (2021) .