Generalized Linear Mixed Models (GLMMs) are a statistical framework for analyzing complex datasets. This approach is well-suited for situations where the outcome variable does not follow a normal distribution and when observations are related or grouped. GLMMs offer a flexible way to model these intricate relationships, providing insights into phenomena across various scientific disciplines. They account for structures in data, such as repeated measurements on the same subject or observations nested within different clusters.
Deconstructing the Generalized Linear Part
A basic linear model assumes the outcome variable is continuous and follows a normal distribution, like an individual’s height. Generalized Linear Models (GLMs) extend this by accommodating response variables that do not adhere to a normal distribution. For instance, GLMs can analyze binary outcomes, such as whether a patient recovers, or count data, like the number of events occurring within a timeframe. This flexibility is achieved through error distributions and link functions.
The error distribution describes the probability distribution of the response variable. Common choices include the binomial distribution for binary or proportion data, modeling the probability of success or failure. For count data, such as insect bites or hospital admissions, the Poisson distribution is frequently used, as it is appropriate for discrete, non-negative integer outcomes. These distributions allow the model to correctly characterize variability in non-normal data.
The link function transforms the mean of the response variable to a linear scale. This transformation enables the linear combination of predictor variables to relate to the transformed mean. For binary data, the logit link function is common, mapping probabilities (between 0 and 1) to a continuous scale. For Poisson count data, the logarithmic link function is often applied, ensuring predicted counts remain non-negative and aligning with the multiplicative nature of count processes.
Understanding the Mixed Component
The “Mixed” aspect of GLMMs refers to the inclusion of both fixed and random effects. Fixed effects represent predictor variables whose specific influence is directly quantifiable and of primary interest. These effects are constant across all observations and are used to estimate the average influence of a treatment or characteristic across the study population. For example, in a medical study, the effect of a drug dosage would be a fixed effect, aiming to determine its consistent impact.
Random effects account for non-independence or grouping within the data, representing sources of variability not the main focus of the study. They capture variability among different groups or clusters of observations, where the overall variance across those groups is modeled. For instance, if data are collected from multiple patients, a random effect for patient ID accounts for repeated measurements from the same patient being more similar to each other. This approach allows for partial pooling of information, meaning estimates for groups with fewer data points can borrow strength from groups with more data.
These random effects allow the model to acknowledge that observations from the same cluster, such as students within a classroom or plants within an experimental plot, might exhibit more similarity. By incorporating random effects, GLMMs correctly model the correlation structure within grouped data, preventing inflated statistical significance and inaccurate conclusions. The goal is to understand population-level effects while accounting for varying conditions or characteristics of sampled units.
When to Use a Generalized Linear Mixed Model
A GLMM is a statistical tool when a dataset exhibits two characteristics: a response variable that is not normally distributed and observations that are grouped or clustered. This dual requirement highlights the capability of GLMMs to handle complex data structures that simpler models cannot adequately address. The “generalized linear” part addresses the non-normal outcome, while the “mixed” part addresses the grouped nature of the data.
Consider an ecological study investigating the presence or absence of a plant species across multiple forest plots over several years. The outcome, presence or absence, is binary and non-normally distributed, requiring a generalized linear approach. Since multiple measurements are taken from the same plots over time, or multiple plots exist within larger forest regions, observations are grouped, necessitating random effects to account for plot-specific or year-specific variations. Ignoring either the binary nature of the data or the grouping structure would lead to an incomplete or incorrect analysis.
Another example is in medicine, when studying the number of seizures experienced by patients over several weeks or months. The number of seizures is count data, which typically follows a Poisson distribution and is not normally distributed. Since each patient provides multiple observations over time, these measurements are grouped within each patient. A GLMM can model the expected number of seizures while accounting for variability among different patients, allowing for more accurate assessment of treatment effects or other covariates.
Contrasting GLMM with Simpler Models
To understand the utility of GLMMs, it is helpful to contrast them with simpler statistical models. One comparison is with a Generalized Linear Model (GLM), which handles non-normal response variables by employing appropriate error distributions and link functions. However, a standard GLM assumes all observations are independent. If data are grouped, such as repeated measurements on the same individual, using a GLM violates this assumption. This could lead to biased parameter estimates, underestimated standard errors, and incorrect statistical inferences.
Another comparison is with a Linear Mixed Model (LMM). An LMM accounts for grouped or hierarchical data structures by incorporating random effects, similar to a GLMM. However, a limitation of LMMs is their assumption that the response variable is normally distributed and continuous, like blood pressure readings. Therefore, an LMM cannot be used when the outcome is binary, a proportion, or count data, as these variables do not follow a normal distribution.
The GLMM combines the strengths of both GLMs and LMMs. It provides the framework to analyze data where the response is non-normally distributed, as handled by a GLM, and where observations are grouped or correlated, as handled by an LMM. This means a GLMM is the appropriate choice when both conditions are present: for example, analyzing binary outcomes from repeatedly measured subjects, or count data collected from individuals nested within different experimental blocks. It resolves the problem of analyzing non-normal, dependent data, which neither a GLM nor an LMM can address on its own.