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.