Generalized linear mixed-effects models are a family of statistical methods to analyze a variety of nonindependent (correlated) outcome data.
In this issue of Anesthesia & Analgesia, Kimura Kuroiwa et al1 report results of a randomized trial in which they studied the effects of phrenic nerve block on ipsilateral shoulder pain (ISP) following video-assisted thoracic surgery. While their primary outcome—incidence of ISP—was compared between the groups with Pearson’s χ2 test,2 these authors used a linear mixed-effects model to analyze and to compare pain scores repeatedly measured over time.
Most of the commonly used statistical techniques assume that the data being analyzed are independent of each other. Repeatedly measured outcomes in the same subjects (longitudinal data) violate this assumption because measurements obtained in the same subject are likely more similar to each other than observations from different subjects.3,4 For example, a patient with a lower pain score at one time point is more likely to have lower pain scores at other time points and vice versa, and therefore, repeatedly measured outcomes in the same subject tend to be correlated.
While this Statistical Minute focuses on the use of linear mixed-effects models for longitudinal data, correlated data do not exclusively occur with repeated measurements in the same subjects. Patients who belong to a certain group (cluster) in which they share certain characteristics—for example, patients from the same family or neighborhood, or subjects treated in the same hospital in a multicenter study—also tend to have more similar outcomes than patients from another cluster. For such correlated data, linear mixed-effects models can also be appropriately used.
Longitudinal data have traditionally been analyzed using techniques like the paired t test or repeated measures analysis of variance (RM-ANOVA). In recent years, linear mixed-effects models—also referred to as multilevel models or hierarchical models—are becoming increasingly popular because they are much more flexible and overcome many of the limitations of more traditional methods.4 For example, traditional methods do not allow for multiple levels of nonindependence, such as repeated measurements in patients who are also clustered by study site; however, this situation can conveniently be modeled with linear mixed-effects models. Unlike RM-ANOVA, these models can also accommodate multiple independent variables—either continuous or categorical variables—for example, to control for confounding in observational research.5
Linear mixed-effects models can model time as a categorical or continuous variable depending on the aim of the analysis. Unlike RM-ANOVA, these models do not exclude a patient from the analysis when outcome data are partially missing in a series of repeated measurements. Moreover, traditional methods make assumptions that often are not met in practice. For example, RM-ANOVA assumes that the correlation between repeated data is the same irrespective how far apart in time the measurements occur. In reality, the correlation is usually highest between adjacent time points and tends to decrease with the increasing time intervals. Such more complex correlation structures can be appropriately modeled with linear mixed-effects models.
Linear mixed-effects models are actually an extension of linear regression (Figure).6 Linear regression only includes so-called fixed effects, whereas linear mixed models are termed “mixed” because they additionally include “random” effects. Fixed effects refer to effects of covariates that are of specific scientific interest and on which researchers wish to make inferences. In longitudinal study designs, these typically include treatment (or exposure) group, time, the interaction between group and time, as well as other factors of interest (like patient age or sex). Random effects refer to the cluster in which the correlated measurements are observed—in other words, the individual patient in whom the same outcome is repeatedly measured over time and/or other clustering variables like the study site.
Without going into technical details, several options are available to model the nonindependence of correlated observations within patients or clusters. These include random intercepts and/or random slopes, in which each patient or cluster is basically allowed to have its own regression line, and/or direct specification of the assumed correlation structure. A more detailed discussion can be found in a previous statistical tutorial on this topic published in Anesthesia & Analgesia.4
Linear regression is used for continuous outcomes, and a family of extensions of this technique—generalized linear models—is available to accommodate a variety of other outcome types (Figure). These models make use of the so-called link function to connect a linear combination of independent (predictor) variables to the dependent (outcome) variable. Examples of generalized linear models include logistic regression for binary data and Poisson regression for count data. Likewise, generalized linear mixed-effects models can be used for a variety of nonindependent (correlated) outcome measures and provide a flexible framework to address a broad spectrum of research questions.