A random effect model is a model all of whose factors represent random effects. (See Random Effects.) Such models are also called variance component models. Random effect models are often hierarchical models. A model that contains both fixed and random effects is called a mixed model. Repeated measures and split-plot models are special cases of mixed models. Often the term mixed model is used to subsume random effect models.
A random effect is a factor whose levels are considered a random sample from some population. Often, the precise levels of the random effect are not of interest, rather it is the variation reflected by the levels that is of interest (the variance components). However, there are also situations where you want to predict the response for a given level of the random effect. Technically, a random effect is considered to have a normal distribution with mean zero and nonzero variance.
 • Y is an n x 1 vector of responses
 • X is the n x p design matrix for the fixed effects
 • β is a p x 1 vector of unknown fixed effects with design matrix X
 • Z is the n x s design matrix for the random effects
 • γ is an s x 1 vector of unknown random effects with design matrix Z
 • ε is an n x 1 vector of unknown random errors
 • G is an s x s diagonal matrix with identical entries for each for each level of the random effect
 • In is an n x n identity matrix
 • γ and ε are independent
The diagonal elements of G, as well as σ2, are called variance components. These variance components, together with the vector of fixed effects β and the vector of random effects γ, are the model parameters that must be estimated.
The covariance structure for this model is sometimes called the variance component structure (SAS Institute Inc. 2017, ch. 79). This covariance structure is the only one available in the Standard Least Squares personality.
 • REML, which stands for restricted maximum likelihood (always the recommended method)
 • EMS, which stands for expected mean squares (use only for teaching from old textbooks)
The EMS method, also called the method of moments, was developed before the availability of powerful computers. Researchers restricted themselves to balanced situations and used the EMS methodology, which provided computational shortcuts to compute estimates for random effect and mixed models. Because many textbooks still in use today use the EMS method to introduce models containing random effects, JMP provides an option for EMS. (See, for example, McCulloch et al., 2008; Poduri, 1997; Searle et al., 1992.)
Models with random effects are specified in the Fit Model launch window. To specify a random effect, highlight it in the Construct Model Effects list and select Attributes > Random Effect. This appends &Random to the effect name in the model effect list. (For a definition of random effects, see Random Effects.) Random effects can also be specified in a separate effects tab. (See Construct Model Effects Tabs in Model Specification.)
There are two different approaches to parameterizing the variance components: the unrestricted and the restricted approaches. The issue arises when there are mixed effects in the model, such as the interaction of a fixed effect with a random effect. Such an interaction term is considered to be a random effect.
You should leave the Unbounded Variance Components option selected if you are interested in fixed effects. Constraining the variance estimates to be nonnegative leads to bias in the tests for the fixed effects.

Help created on 7/12/2018