Fixed, random and mixed effects models, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Linear mixed-effects models are extensions of linear regression models for data that are collected and summarized in groups. These models describe the relationship between a response variable and independent variables, with coefficients that can vary with respect to one or more grouping variables. A mixed-effects model consists of two parts, fixed effects and random effects. Fixed-effects terms are usually the conventional linear regression part, and the random effects are associated with individual experimental units drawn at random from a population. The random effects have prior distributions whereas fixed effects do not. Mixed-effects models can represent the covariance structure related to the grouping of data by associating the common random effects to observations that have the same level of a grouping variable. The standard form of a linear mixed-effects model is

where

• y is the n-by-1 response vector, and n is the number of observations.

• X is an n-by-p fixed-effects design matrix.

• β is a p-by-1 fixed-effects vector.

• Z is an n-by-q random-effects design matrix.

• b is a q-by-1 random-effects vector.

• ε is the n-by-1 observation error vector.

The assumptions for the linear mixed-effects model are:

• Random-effects vector, b, and the error vector, ε, have the following prior distributions:

where D is a symmetric and positive semidefinite matrix, parameterized by a variance component vector θ, I is an n-by-n identity matrix, and σ² is the error variance.

• Random-effects vector, b, and the error vector, ε, are independent from each other.

Mixed-effects models are also called multilevel models or hierarchical models depending on the context. Mixed-effects models is a more general term than the latter two. Mixed-effects models might include factors that are not necessarily multilevel or hierarchical, for example crossed factors. That is why mixed-effects is the terminology preferred here. Sometimes mixed-effects models are expressed as multilevel regression models (first level and grouping level models) that are fit simultaneously. For example, a varying or random intercept model, with one continuous predictor variable x and one grouping variable with M levels, can be expressed as

where y_im corresponds to data for observation i and group m, n is the total number of observations, and b_0m and ε_im are independent of each other. After substituting the group-level parameters in the first-level model, the model for the response vector becomes

A random intercept and slope model with one continuous predictor variable x, where both the intercept and slope vary independently by a grouping variable with M levels is

or

You might also have correlated random effects. In general, for a model with a random intercept and slope, the distribution of the random effects is

where D is a 2-by-2 symmetric and positive semidefinite matrix, parameterized by a variance component vector θ.

After substituting the group-level parameters in the first-level model, the model for the response vector is

If you express the group-level variable, x_im, in the random-effects term by z_im, this model is

In this case, the same terms appear in both the fixed-effects design matrix and random-effects design matrix. Each z_im and x_im correspond to the level m of the grouping variable.

It is also possible to explain more of the group-level variations by adding more group-level predictor variables. A random-intercept and random-slope model with one continuous predictor variable x, where both the intercept and slope vary independently by a grouping variable with M levels, and one group-level predictor variable v_m is

This model results in main effects of the group-level predictor and an interaction term between the first-level and group-level predictor variables in the model for the response variable as

The term β₁₁v_mx_im is often called a cross-level interaction in many textbooks on multilevel models. The model for the response variable y can be expressed as

which corresponds to the standard form given earlier,

 y=Xβ+Zb+ε.

In general, if there are R grouping variables, and m(r,i) shows the level of grouping variable r, for observation i, then the model for the response variable for observation i is

where β is a p-by-1 fixed-effects vector, b^(r)_m(r,i) is a q(r)-by-1 random-effects vector for the rth grouping variable and level m(r,i), and ε_i is a 1-by-1 error term for observation i.