How do you choose the best random effects structure for your GLM?

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Generalized linear models (GLMs) are a powerful tool for analyzing data with different distributions and link functions. However, when you have repeated or clustered observations, you need to account for the non-independence of the data by adding random effects to your GLM. But how do you choose the best random effects structure for your GLM? In this article, you will learn some tips and tricks to make this decision easier and more informed.

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1 What are random effects?

Random effects are variables that capture the variability in the data that is not explained by the fixed effects or the error term. They are usually factors that have multiple levels, such as subjects, groups, or locations. Random effects can have different components, such as intercepts or slopes, that vary across the levels of the factor. For example, if you have a GLM that predicts the weight of a dog based on its breed and age, you can include a random intercept for each dog to account for the individual differences in weight.

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Carlos Miranda Durand

Artificial Intelligence & Business Intelligence Consulting | Innovation | Data Science & Technology
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More clear if both types defined: - Fixed effects are factors whose levels include all possible levels or a subset picked intentionally following a non-random process. - Random effects are factors whose levels are a random sample of the population of all possible levels. - Fixed effects allow inferences for each selected level but random effects allow only broad inferences about the factor (not about specific levels.) - Example, longitudinal study with 2 factors: drug and patient. The drug intentionally selected for this study is a fixed effect. If patients are randomly selected, we don't need individual-patient effects, but need to control for patient correlations (repeated observations), then patient can be modeled as a random effect.

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Yogesh Gudipati

Data Scientist @Microsoft || Ex.Amazon || Ms Statistics
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Random effects capture the unobserved variability in the data that is not explained by fixed effects or the error term. They are factors with multiple levels, like subjects or locations, and can have different components that vary across the levels of the factor. Random effects are particularly useful for hierarchical or nested data structures where observations are not independent but grouped by some higher-level factor. They help us account for similarities within groups, improve the accuracy of estimates, and avoid overfitting. By including random effects in statistical models, we can better understand and model complex data structures. Random effects are a powerful and flexible tool that should be considered in any statistical analysis.

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Gerard Kinneen

CoFounder and CXO of iSort AB | Blending Innovation in Robotics with Practical Solutions for Heavy Load Management | Passionate about Autonomous Vehicles for Construction Sites.
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I select the best random effects structure for my GLM by considering the hierarchical nature of the data and assessing whether certain groups or clusters significantly impact the response variable. I use methods like likelihood ratio tests and model comparison to determine the most appropriate random effects structure that accounts for the data's underlying structure and improves model fit.

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2 Why are random effects important?

Random effects are important because they allow you to model the heterogeneity and correlation in the data, which can improve the accuracy and generalizability of your GLM. By including random effects, you can avoid biased estimates of the fixed effects, inflated standard errors, and incorrect inference. Random effects can also help you to answer more complex research questions, such as how the effect of a predictor varies across different levels of a factor, or how the variability in the outcome depends on the factor.

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Dr. Muhammad Mustapha Ishaq (CSAA, FIFP)

Financial Econometrics | Islamic Finance | Digital Innovation
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Random effects add an extra layer to the model, allowing for different intercepts (and sometimes slopes) for each cross-section or group. This prevents averaging over hidden patterns within groups or cross-section.

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3 How to specify random effects?

To specify random effects, you need to use a special notation that indicates which factor is the random effect, and which component is varying across its levels. The notation can vary depending on the software or package you use, but a common one is the lme4 syntax in R. In this syntax, you use parentheses to enclose the random effects, and separate them by a vertical bar from the factor. For example, (1 | dog) means a random intercept for each dog, and (age | dog) means a random intercept and a random slope for age for each dog.

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4 How to test random effects?

To test random effects, you need to compare different models with different random effects structures, and see which one fits the data better. There are different ways to do this, but one common method is to use likelihood ratio tests (LRTs). LRTs compare the log-likelihoods of two nested models, and calculate a chi-square statistic that indicates how much the more complex model improves the fit. The more complex model is preferred if the chi-square statistic is significant at a certain level, such as 0.05.

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5 How to simplify random effects?

Sometimes, you may end up with a very complex random effects structure that includes many components and interactions. This can make your model difficult to interpret and estimate, and may cause convergence problems or overfitting. To avoid these issues, you may want to simplify your random effects structure by removing some components or interactions that are not necessary or supported by the data. One way to do this is to use backward elimination, which means starting with the most complex model and testing whether removing a component or interaction improves or worsens the fit.

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6 How to report random effects?

To report random effects, you need to provide some information about the model and the results. This may include the software or package you used, the notation you used to specify the random effects, the criteria you used to select the best random effects structure, and the statistics you used to test the random effects. You may also want to report the estimates and confidence intervals of the random effects components, such as the standard deviations of the random intercepts and slopes, and the correlations between them. You can use tables or graphs to present this information in a clear and concise way.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you choose the best random effects structure for your GLM?

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1 What are random effects?

2 Why are random effects important?

3 How to specify random effects?

4 How to test random effects?

5 How to simplify random effects?

6 How to report random effects?

7 Here’s what else to consider

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