Mixed model

Statistical model containing both fixed effects and random effects

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 (longitudinal study), or where measurements are made on clusters of related statistical units. The Mixed model analysis allows measurements to be explicitly modeled in a wider variety of correlation and variance covariance structures. This page will discuss mainly linear mixed effects models rather than generalized linear mixed models or nonlinear mixed effects models.

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. Often, ANOVA assumes the 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 nest 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.

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. In most situations, several related models are considered and the model that best represents a universal model is adopted.

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. In the 1950s, Charles Roy Henderson
provided best linear unbiased estimates of fixed effects and best linear unbiased predictions of random effects. Subsequently, mixed modeling has become a major area of statistical research, including work on computation of maximum likelihood estimates, non linear mixed effects models, missing data in mixed effects models, and Bayesian estimation of mixed effects models. Mixed models are applied in many disciplines where multiple correlated measurements are made on each unit of interest. They are prominently used in research involving human and animal subjects in fields ranging from genetics to marketing, and have also been used in baseball and industrial statistics. The mixed linear model association has improved the prevention of false positive associations. Populations are deeply interconnected and the relatedness structure of population dynamics is extremely difficult to model without the use of mixed models. Linear mixed models may not, however, be the only solution. LMM's have a constant residual variance assumption that is sometimes violated when accounting or deeply associated continuous and binary traits.