mixed model

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There are two ways to run a repeated measures analysis.The traditional way is to treat it as a multivariate test–each response is considered a separate variable.The other way is to it as a mixed model.While the multivariate approach is easy to run and quite intuitive, there are a number of advantages to running a repeated measures analysis as a mixed model.

First I will explain the difference between the approaches, then briefly describe some of the advantages of using the mixed models approach. (more…)

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It seems very many researchers are needing to learn multilevel and mixed models, and I have to say, it’s not so easy on your own.

I too went to graduate school before it was taught in classes–we did learn mixed models as in Split Plot designs, but things have progressed a bit since then.  So I too have had to learn them without benefit of a class, or teacher.  So I feel your pain.  But I’ve struggled through and learned a (more…)

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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.

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A great article for specifying Mixed Models in SAS:

Mixed up Mixed Models
by Robert Harner & P.M. Simpson

Anyone doing mixed modeling in SAS should read this paper, originally presented at SUGI: SAS Users Group International conference. It compares the output from Proc Mixed and Proc GLM when specified different ways. There are some subtle distinctions in the meaning of the defaults in the Repeated and Random statements, and this paper does an excellent job of clarifying them.

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