# Chapter 11 Linear Mixed-Effects Models

- Mixed-effects model
\(y = X\beta + Zu + e\)

- the elements of \(\beta\) are considered to be non-random and are called
*fixed effects*. - the elements of \(u\) are random variables and are called
*random effects*. - We assume that \(E(e) = 0, Var(e) = R, E(u) = 0, Var(u) = G, Cov(u, e) = 0\)
- \(E(y) = X\beta\), \(Var(y) = ZGZ' + R \equiv \Sigma\), \(E(y\mid u) = X\beta + Zu\)

When there are \(m\) random factors, we can partition \(Z\) and \(u\) as \(Z = [Z_1, \ldots, Z_m]\) and \(u = [u_1', \ldots, u_m']'\) so that \(Zu = \sum_{j = 1}^m Z_ju_j\).

- We often assume that all random effects are mutually independent and random effects associated with \(j\)th random factor have variance \(\sigma_j^2\). Then, \(Var(y) = ZGZ' + R = \sum_{j=1}^R \sigma_j^2 Z_jZ_j' + \sigma_e^2I\).
- The unknown variance parameters \(\sigma_j^2, \sigma_e^2\) are called
*variance components*.

Experimental Design Terminology

**Experiment**: An investigation in which the investigator applies some treatments to experimental units and then observes the effect of the treatments on the experimental units by measuring one or more response variables.**Treatment**: a condition or set of conditions applied to experimental units in an experiment.**Experimental Unit**: the physical entity to which a treatment is randomly assigned and independently applied.**Response Variable**: a characteristic of an experimental unit that is measured after treatment and analyzed to assess the effects of treatments on experimental units.**Observational Unit**: the unit on which a response variable is measured.**Completely Randomized Design (CRD)**– experimental design in which, for given number of experiment units per treatment, all possible assignments of treatments to experimental units are equally likely.**Block**– a group of experimental units that,**prior to treatment**, are expected to be more like one another (with respect to one or more response variables) than experimental units in general.**Randomized Complete Block Design (RCBD)**– experimental design in which separate and completely randomized treatment assignments are made for each of multiple blocks in such a way that all treatments have at least one experimental unit in each block.

Whenever an experiment involves multiple observations per experimental unit, it is important to include a random effect for each experimental unit.

Without a random effect for each experimental unit, a one-to-one correspondence between observations and experimental units is assumed.