Multilevel Models with Crossed Random Effects

Most researchers know by now to use Mixed Models when observations are clustered. Examples include studies in which patients share the same doctor, plants grow in the same field, or multiple responses are observed from the same study participant. The observations at Level 1 (patient, plant, response) are clustered at Level 2 (doctor, field, or participant), generally making them correlated.

In these models, the Level 2 cluster is often not of interest. Even so, its effects need to be controlled for. If the researcher would like to generalize the results to all doctors, fields, or participants, these clustering variables are random effects.

The observations of the dependent variable are always measured at Level 1 (the patient, plant, or time point). Predictor variables (fixed effects) can be measured at either Level 1 or Level 2. For example, number of years of experience of a doctor would be at Level 2, but patient age would be measured at Level 1. The observations within cluster are assumed to be correlated, but the observations between clusters are assumed to be independent.

In one kind of 2 level model, there is not one random effect at Level 2, but two crossed effects. Each observation at Level 1 is nested in the combination of these two random effects. These models need special consideration in order to capture both random effects at Level 2.

Here are the same examples with crossed random effects:

Example 1: Every patient (Level 1) sees their Doctor (Random Effect at Level 2) at one of four Hospitals (Random Effect at Level 2) for a study comparing a new drug treatment for diabetes to an old one. Each doctor sees patients at each of the four hospitals. Patient responses vary across doctors and hospitals. Because each Patient sees a single doctor at a single hospital, patients are nested in the combination of Doctor and Hospital. The response is measured at Level 1--the patient. Predictors can occur at Level 1 (age, diet) or either Level 2 effect (years of practice by doctor, size of hospital).

Example 2: An agricultural study is studying plants in 6 fields. While there are many species of plants in each field, the researcher randomly chooses 5 species to study. Each individual plant (Level 1) lies within one combination of species and field. But since every species is in every field, Species and Field are crossed at Level 2. The response is measured at Level 1--the plant, and predictors can occur at either Level 1 (height of plant) or either Level 2 effect (fertilizers applied to the field, whether the species is native or introduced).

Example 3: In a psychological experiment, subjects are asked to rate statements that describe behaviors done by a fictional person, Bob. On each trial, subjects rate whether or not Bob was friendly and the response time of the rating is recorded. Each subject sees the same 10 friendly and 10 unfriendly behaviors. The behaviors are not in themselves of interest to the experimenter, but are representative of all friendly and unfriendly behaviors that Bob could perform. Because responses to the same behavior tend to be similar, it is necessary to control for their effects. Each trial of the experiment (Level 1) is nested both withi
Subject and Behavior, which are both random effects at Level 2. Subject and Behavior are crossed at Level 2 since every Subject rates every Behavior. The response is measured at Level 1--the trial, and predictors can occur at either Level 1 (a distractor occurs on some trials) or ei ther Level 2 effect (Behavior is friendly or not, Subject is put into positive, neutral, or negative mood).

Luckily, specifying a crossed random effects model can be easily done in standard mixed modeling procedures, such as SAS Proc Mixed or SPSS Mixed. It should be done with care, however, because like most mixed models, specifying a Crossed Random Effects model correctly can be tricky.