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 fixed 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/STAT 9.2 User’s Guide, 2008, p. 3955). 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, Searle, and Neuhaus, 2008, Poduri, 1997, and Searle, Casella, and McCulloch, 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 Introduction to Fit Model.)
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.
Fit Least Squares Report for REML Method shows the report obtained for a fit to the Investment Castings.jmp sample data using the REML method. Run the script Model - REML, and then fit the model. Note that Casting is a random effect and is nested within Temperature.
Fit Least Squares Report for REML Method
For each term in the model, this report gives an empirical estimate of its best linear unbiased predictor (BLUP) and a test for whether the corresponding coefficient is zero.
Gives an empirical estimate of the best linear unbiased predictor (BLUP) for each random effect. (See Best Linear Unbiased Predictors.)
Gives the t ratio for testing that the effect is zero. The t ratio is obtained by dividing the BLUP by its standard error.
Gives the p-value for the test.
The term best linear unbiased predictor (BLUP) refers to an estimator of a random effect. Specifically, it is an estimator that, among all unbiased estimators, minimizes mean square prediction error. The Random Effect Predictions report gives estimates of the BLUPs, or empirical BLUPs. These are empirical because the BLUPs depend on the values of the variance components, which are unknown. The estimated values of the variance components are substituted into the formulas for the BLUPs, resulting in the estimates shown in the report.
The estimates of σ2 and the variance components in G are obtained by maximizing a residual log-likelihood function that depends on only these parameters. An iterative procedure attempts to maximize the residual log-likelihood function, or equivalently, to minimize –2 times the residual log likelihood (–2LogLike). The Iterations report provides details about this procedure.
The convergence criterion is based on the gradient, with a default tolerance of 10-8. You can change the criterion in the Fit Model launch window by selecting the option Convergence Settings > Convergence Limit and specifying the desired tolerance.
Gives the p-value for the effect test.
When you use the REML method, five additional options appear in the Save Columns menu. These option names start with the adjective Conditional. This prefix indicates that the calculations for these columns use the predicted values for the terms associated with the random effects, rather than their expected values of zero.
Expected Mean Squares Report shows the Expected Mean Squares report for the Investment Castings.jmp sample data table. Run the Model - EMS script and then run the model.
Expected Mean Squares Report
where is the sum of the squares of the effects for Treatment divided by the number of levels of Treatment minus one.
Gives the F ratio for the test. It is the ratio of the numerator mean square to the denominator mean square, which can be obtained from the Test Denominator Synthesis report.
Gives the p-value for the effect test.
Caution: Standard errors for least squares means and denominators for contrast F tests use the synthesized denominator. In certain situations, such as tests involving crossed effects compared at common levels, these tests might not be appropriate. Custom tests are conducted using residual error, and leverage plots are constructed using the residual error, so these also might not be appropriate.