# 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.