In a sense, every model has at least one random effect, which is the effect that makes up the residual error. The individual observations are assumed to be randomly selected from a much larger population, and the error term is assumed to have a mean of zero and variance σ2.
The most common random effects model is the repeated measures or split plot model. Types of Effects in a Split Plot Model lists the types of effects in a split plot model. In these models, the experiment has two layers. Some effects are applied on the whole plots or subjects of the experiment. Then these plots are divided or the subjects are measured at different times and other effects are applied within those subunits. The effects describing the whole plots or subjects are whole plot effects, and the subplots or repeated measures are subplot effects. Usually the subunit effect is omitted from the model and absorbed as residual error.
Form an F-ratio by dividing the whole plot mean squares by the whole plot ID mean squares.
Consider the data in the sample data table (the data are fictional). The study collected information about differences in the seasonal hunting habits of foxes and coyotes. Each season for one year, three foxes and three coyotes were marked and observed periodically. The average number of miles that they wandered from their dens during different seasons of the year was recorded (rounded to the nearest mile). The model is defined by the following aspects:
The species effect with values fox or coyote
The season effect with values fall, winter, spring, and summer
An animal identification code called subject, with nominal values 1, 2, and 3 for both foxes and coyotes
The top layer is the between-subject layer, in which the effect of being a fox or coyote (species effect) is tested with respect to the variation from subject to subject.
The bottom layer is the within-subject layer, in which the repeated-measures factor for the four seasons (season effect) is tested with respect to the variation from season to season within a subject. The within-subject variability is reflected in the residual error.
The season effect can use the residual error for the denominator of its F-statistics. However, the between-subject variability is not measured by residual error and must be captured with the subject within species (subject[species]) effect in the model. The F-statistic for the between-subject effect species uses this nested effect instead of residual error for its F-ratio denominator.
Select Help > Sample Data Library and open
Select Analyze > Fit Model.
Select miles and click Y.
Select species and subject and click Add.
Click Nest.
This adds the subject within species (subject[species]) effect to the model.
Select the nested effect subject[species].
Select Attributes > Random Effect.
This nested effect is now identified as an error term for the species effect and appears as subject[species]&Random.
When you define an effect as random using the Attributes menu, the Method options (REML and EMS) appear at the top right of the dialog, with REML selected as the default. The populated launch window is shown in Fit Model Dialog.
Fit Model Dialog
Click Run.
The report is shown in Partial Report of REML Analysis. Both fixed effects, species and season, are significant. The REML Variance Component Estimates report gives estimates of the subject within species and residual variances.
Partial Report of REML Analysis