Crossed random effects models are a little trickier than most mixed models, but they are quite common in many fields. Recognizing when you have one and knowing how to analyze the data when you do are important statistical skills.
The Nested Multilevel Design
The most straightforward use of Mixed Models is when observations are clustered or nested in some higher group.
It’s also so common that it often has its own name: multilevel model.
Examples include studies where patients share the same doctor, plants grow in the same field, or participants respond to multiple experimental conditions.
The observations at Level 1 (patient, plant, response) are clustered at Level 2 (doctor, field, or participant). This makes the responses from the same cluster correlated.
In these models, the Level 2 cluster is often not of interest. It’s what we call a “blocking factor.” Even so, we need to control for its effects.
If the researcher would like to generalize the results to all doctors, fields, or participants, these clustering variables are random factors.
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.
We assume the observations within cluster are are correlated, but the observations between clusters are independent.
A third level is possible as well. This would happen if each doctor sees all their patients at one of four hospitals or each field has only one of 5 species.
The Crossed Multilevel Design
In one kind of 2-level model, there is not one random factor at Level 2, but two crossed factors.
Each observation at Level 1 is nested in the combination of these two random factors. These models need to be specified correctly to capture the effects of both random factors 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 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 factor (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. Predictors can occur at Level 1 (height of plant) or either Level 2 factor (fertilizers applied to the field, whether the species is native or introduced).
Example 3:
In a psychological experiment, subjects rate statements that describe behaviors done by a fictional person, Bob.
On each trial, subjects rate whether or not they find Bob friendly. 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 within 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. Predictors can occur at Level 1 (a distractor occurs on some trials) or either Level 2 factor (Behavior is friendly or not, Subject is put into positive, neutral, or negative mood).
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Luckily, standard mixed modeling procedures such as SAS Proc Mixed, SPSS Mixed, Stat’s xtmixed, or R’s lmer can all easily run a crossed random effects model. (R’s lme can’t do it).
Use care, however, because like most mixed models, specifying a crossed random effects model correctly can be tricky.