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Mixed Models: An introduction in R

9 Summary

Here is a suggested workflow:

  1. Start with a model containing only fixed effects: Begin by fitting a model with the fixed effects that are relevant to your research question. Include the main predictors and any potential interactions you hypothesize might exist. For example, if you have predictors A and B, you might start with a model like response ~ A + B + A:B.

  2. Assess fixed effects and interactions: Evaluate the significance, direction, and magnitude of the fixed effects coefficients. Look for interactions that show significant effects and consider their interpretation in the context of your research question. This step allows you to identify the key variables and interactions that are important in explaining the variation in the response variable.

  3. Model evaluation and refinement: Assess the goodness of fit of the fixed effects model using appropriate measures like AIC, BIC, or model deviance. Consider conducting model comparison to evaluate different models with alternative fixed effects structures. This process helps you refine the model and select the most appropriate combination of variables and interactions. However - this should be no substitution for carefully considered hypotheses and experimental design.

  4. Incorporate random effects: Once you have identified the significant fixed effects and relevant interactions, you can then consider the inclusion of random effects. Random effects capture the variation at different levels and can account for individual differences or clustering within groups. Evaluate the need for random intercepts, random slopes, or crossed random effects based on your research design and the nature of the data.

  5. Assess and compare models with random effects: Fit models with random effects and compare their fit to the fixed effects model. Consider appropriate measures such as likelihood ratio tests, AIC, or BIC for model comparison. Evaluate the contribution of the random effects to the model.However - this should be no substitution for carefully considered hypotheses and experimental design. And we have seen examples where we leave random effects in place despite LRT tests.

  6. Validate and interpret the final model: Validate the final model by assessing assumptions, checking for influential observations, and performing sensitivity analysis. Interpret the estimated coefficients, including fixed effects and random effects, in the context of your research question. Report the results with figures, summary tables and carefully considered text summarising the analysis.

By initially focusing on the fixed effects, you can establish the foundation of your model and identify the significant predictors and interactions. This step allows you to better understand the relationships in your data and guide the subsequent inclusion of random effects if appropriate.

9.1 Mixed Model extensions

9.2 Practical problems

Below are two common issues or warnings you will likely encounter when fitting linear models:

  • boundary (singular) fit: see help('isSingular'): Your model did fit, but it generated that warning because some of your random effects are very small, common with complex mixed-effect models. You can read more about this in help page

  • Convergence warnings: Values for the mixed-effects models are determined using optimisation algorithms. Sometimes these algorithms fail to converge on a best parameter estimate, which will produce an error. There are several possible solutions:

    • Normalise and rescale: Rescaling the variables can mitigate issues caused by the differences in scales or magnitudes of the predictors, which can affect the optimization process. You can rescale the fixed effects predictors by subtracting the mean and dividing by the standard deviation. This centers the variables around zero and scales them to have a standard deviation of 1. This can be done using the scale() function in R.
    • Try alternative optimisation algorithms: e.g. lmer3 <- lmer(y ~ x + (x | group), data = data, control = lmerControl(optimizer ="Nelder_Mead"))
    • Finally, although it pains me to admit it, you should try running the model using a different package - each has their own unique and slightly different optimisation protocols.

9.4 References

Bates, D., Maechler, M., Bolker, B., & Walker, S. (2023). lme4: Linear mixed-effects models using eigen and S4. https://github.com/lme4/lme4/
Brooks, M., Bolker, B., Kristensen, K., Maechler, M., Magnusson, A., Skaug, H., Nielsen, A., Berg, C., & van Bentham, K. (2023). glmmTMB: Generalized linear mixed models using template model builder. https://github.com/glmmTMB/glmmTMB
Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press. https://doi.org/10.1017/CBO9780511790942
Harrison, X. A., Donaldson, L., Correa-Cano, M. E., Evans, J., Fisher, D. N., Goodwin, C. E. D., Robinson, B. S., Hodgson, D. J., & Inger, R. (2018). A brief introduction to mixed effects modelling and multi-model inference in ecology. PeerJ, 6, e4794. https://doi.org/10.7717/peerj.4794
Hartig, F. (2022). DHARMa: Residual diagnostics for hierarchical (multi-level / mixed) regression models. http://florianhartig.github.io/DHARMa/
Krzywinski, M., Altman, N., & Blainey, P. (2014). Nested designs. Nature Methods. https://doi.org/10.1038/nmeth.3137
Kuznetsova, A., Bruun Brockhoff, P., & Haubo Bojesen Christensen, R. (2020). lmerTest: Tests in linear mixed effects models. https://github.com/runehaubo/lmerTestR
Lüdecke, D. (2023a). Ggeffects: Create tidy data frames of marginal effects for ggplot from model outputs. https://strengejacke.github.io/ggeffects/
Lüdecke, D. (2023b). sjPlot: Data visualization for statistics in social science. https://strengejacke.github.io/sjPlot/
Lüdecke, D., Makowski, D., Ben-Shachar, M. S., Patil, I., Waggoner, P., & Wiernik, B. M. (2023). Performance: Assessment of regression models performance. https://easystats.github.io/performance/
Makowski, D., Lüdecke, D., Patil, I., Thériault, R., Ben-Shachar, M. S., & Wiernik, B. M. (2023). Report: Automated reporting of results and statistical models. https://easystats.github.io/report/
Pinheiro, J., Bates, D., & R Core Team. (2023). Nlme: Linear and nonlinear mixed effects models. https://svn.r-project.org/R-packages/trunk/nlme/
Silk, M. J., Harrison, X. A., & Hodgson, D. J. (2020). Perils and pitfalls of mixed-effects regression models in biology. PeerJ, 8, e9522. https://doi.org/10.7717/peerj.9522
Zuur, A. F., Ieno, E. N., Walker, N., Saveliev, A. A., & Smith, G. M. (2009). Mixed effects models and extensions in ecology with r. Springer.