A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful in settings where repeated measurements are made on the same statistical units (see also longitudinal study), or where measurements are made on clusters of related statistical units. Mixed models are often preferred over traditional analysis of variance regression models because they don't rely on the independent observations assumption. Further, they have their flexibility in dealing with missing values and uneven spacing of repeated measurements. The Mixed model analysis allows measurements to be explicitly modeled in a wider variety of correlation and variance-covariance avoiding biased estimations structures. This page will discuss mainly linear mixed-effects models rather than generalized linear mixed models or nonlinear mixed-effects models.
Qualitative Description Linear mixed models (LMMs) are statistical models that incorporate fixed and random effects to accurately represent non-independent data structures. LMM is an alternative to analysis of variance (ANOVA). Often, ANOVA assumes the statistical independence of observations within each group, however, this assumption may not hold in non-independent data, such as multilevel/hierarchical, longitudinal, or correlated datasets. Non-independent sets are ones in which the variability between outcomes is due to correlations within groups or between groups. Mixed models properly account for nested structures/hierarchical data structures where observations are influenced by their nested associations. For example, when studying education methods involving multiple schools, there are multiple levels of variables to consider. The individual level/lower level comprises individual students or teachers within the school. The observations obtained from this student/teacher is nested within their school. For example, Student A is a unit within the School A. The next higher level is the school. At the higher level, the school contains multiple individual students and teachers. The school level influences the observations obtained from the students and teachers. For Example, School A and School B are the higher levels each with its set of Student A and Student B respectively. This represents a hierarchical data scheme. A solution to modeling hierarchical data is using linear mixed models.
LMMs allow us to understand the important effects between and within levels while incorporating the corrections for standard errors for non-independence embedded in the data structure. In experimental fields such as social psychology, psycholinguistics, cognitive psychology (and neuroscience), where studies often involve multiple grouping variables, failing to account for random effects can lead to inflated Type I error rates and unreliable conclusions. For instance, when analyzing data from experiments that involve both samples of participants and samples of stimuli (e.g., images, scenarios, etc.), ignoring variation in either of these grouping variables (e.g., by averaging over stimuli) can result in misleading conclusions. In such cases, researchers can instead treat both participant and stimulus as random effects with LMMs, and in doing so, can correctly account for the variation in their data across multiple grouping variables. Similarly, when analyzing data from comparative longitudinal surveys, failing to include random effects at all relevant levels—such as country and country-year—can significantly distort the results.
The Fixed Effect Fixed effects encapsulate the tendencies/trends that are consistent at the levels of primary interest. These effects are considered fixed because they are non-random and assumed to be constant for the population being studied. For example, when studying education a fixed effect could represent overall school level effects that are consistent across all schools. While the hierarchy of the data set is typically obvious, the specific fixed effects that affect the average responses for all subjects must be specified. Some fixed effect coefficients are sufficient without corresponding random effects where as other fixed coefficients only represent an average where the individual units are random. These may be determined by incorporating random intercepts and slopes. In most situations, several related models are considered and the model that best represents a universal model is adopted.
The Random Effect, ε A key component of the mixed model is the incorporation of random effects with the fixed effect. Fixed effects are often fitted to represent the underlying model. In Linear mixed models, the true regression of the population is linear, β. The fixed data is fitted at the highest level. Random effects introduce statistical variability at different levels of the data hierarchy. These account for the unmeasured sources of variance that affect certain groups in the data. For example, the differences between student 1 and student 2 in the same class, or the differences between class 1 and class 2 in the same school.
History and current status
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