Linear Mixed-Effects Models

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

$$y=\underset{fixed}{\underbrace{X\beta }}+\underset{random}{\underbrace{Zb}}+\underset{error}{\underbrace{\epsilon }},$$

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:

  $$\begin{array}{l}b~N\left(0,{\sigma }^{2}D\left(\theta \right)\right),\\ \epsilon ~N\left(0,\sigma {}^{2}I\right),\end{array}$$

  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

$$\begin{array}{l}{y}_{im}={\beta }_{0m}+{\beta }_1{x}_{im}+{\epsilon }_{im},i=1,2,\mathrm{..},n,m=1,2,\mathrm{...},M,{\epsilon }_{im}~N\left(0,{\sigma }^2\right),\\ {\beta }_{0m}={\beta }_{00}+b_{0m},b_{0m}~N\left(0,{\sigma }_0^2\right),\end{array}$$

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

$$y_{im}=\underset{fixedeffects}{\underbrace{{\beta }_{00}+{\beta }_1{x}_{im}}}+\underset{randomeffects}{\underbrace{b_{0m}}}+{\epsilon }_{im}.$$

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

$$\begin{array}{l}{y}_{im}={\beta }_{0m}+{\beta }_{1m}{x}_{im}+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M,{\epsilon }_{im}~N\left(0,{\sigma }^2\right),\\ {\beta }_{0m}={\beta }_{00}+b_{0m},b_{0m}~N\left(0,{\sigma }_0^2\right),\\ {\beta }_{1m}={\beta }_{10}+b_{1m},b_{1m}~N\left(0,{\sigma }_1^2\right),\end{array}$$

or

$$b_m=\left(\begin{array}{l}{b}_{0m}\\ b_{1m}\end{array}\right)~N\left(0,\left(\begin{array}{cc}{\sigma }_0^2& 0\\ 0& {\sigma }_1^2\end{array}\right)\right).$$

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

$$b_m=\left(\begin{array}{l}{b}_{0m}\\ b_{1m}\end{array}\right)~N\left(0,\sigma {}^{2}D\left(\theta \right)\right),$$

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

$$y_{im}=\underset{fixedeffects}{\underbrace{{\beta }_{00}+{\beta }_{10}{x}_{im}}}+\underset{randomeffects}{\underbrace{b_{0m}+b_{1m}{x}_{im}}}+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M.$$

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

$$y_{im}=\underset{fixedeffects}{\underbrace{{\beta }_{00}+{\beta }_{10}{x}_{im}}}+\underset{randomeffects}{\underbrace{b_{0m}+b_{1m}{z}_{im}}}+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M.$$

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

$$\begin{array}{l}{y}_{im}={\beta }_{0im}+{\beta }_{1im}{x}_{im}+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M,{\epsilon }_{im}~N\left(0,{\sigma }^2\right),\\ {\beta }_{0im}={\beta }_{00}+{\beta }_{01}{v}_{im}+b_{0m},b_{0m}~N\left(0,{\sigma }_0^2\right),\\ {\beta }_{1im}={\beta }_{10}+{\beta }_{11}{v}_{im}+b_{1m},b_{1m}~N\left(0,{\sigma }_1^2\right).\end{array}$$

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

$$\begin{array}{l}{y}_{im}={\beta }_{00}+{\beta }_{01}{v}_{im}+b_{0m}+\left({\beta }_{10}+{\beta }_{11}{v}_{im}+b_{1m}\right)x_{im}+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M,\\ =\underset{fixedeffects}{\underbrace{{\beta }_{00}+{\beta }_{10}{x}_{im}+{\beta }_{01}{v}_{im}+{\beta }_{11}{v}_{im}{x}_{im}}}+\underset{randomeffects}{\underbrace{b_{0m}+b_{1m}{x}_{im}}}+{\epsilon }_{im}.\end{array}$$

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

$$\begin{array}{l}{y}_{im}=\left[\begin{array}{cccc}1& x_1{}_{im}& v_{im}& v_{im}{x}_{1im}\end{array}\right]\left[\begin{array}{c}{\beta }_{00}\\ {\beta }_{10}\\ {\beta }_{01}\\ {\beta }_{11}\end{array}\right]+\left[\begin{array}{cc}1& x_{1im}\end{array}\right]\left[\begin{array}{c}{b}_{0m}\\ b_{1m}\end{array}\right]+{\epsilon }_{im},i=1,2,\mathrm{...},n,m=1,2,\mathrm{...},M,\\ \end{array}$$

which corresponds to the standard form given earlier,

$$y=X\beta +Zb+\epsilon .$$

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

$$y_i=x_i^T\beta +{\displaystyle \sum _{r=1}^R{z}_{ir}{b}_{m(r,i)}^{(r)}}+{\epsilon }_i,i=1,2,\mathrm{...},n,$$

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.

References

[1] Pinherio, J. C., and D. M. Bates. Mixed-Effects Models in S and S-PLUS. Statistics and Computing Series, Springer, 2004.

[2] Hariharan, S. and J. H. Rogers. “Estimation Procedures for Hierarchical Linear Models.” Multilevel Modeling of Educational Data (A. A. Connell and D. B. McCoach, eds.). Charlotte, NC: Information Age Publishing, Inc., 2008.

[3] Hox, J. Multilevel Analysis, Techniques and Applications. Lawrence Erlbaum Associates, Inc., 2002

[4] Snidjers, T. and R. Bosker. Multilevel Analysis. Thousand Oaks, CA: Sage Publications, 1999.

[5] Gelman, A. and J. Hill. Data Analysis Using Regression and Multilevel/Hierarchical Models. New York, NY: Cambridge University Press, 2007.