Lecture 42—Monday, April 3, 2006

What was covered?

• Heterogeneity versus correlation
• Basic multilevel models
• Pseudo-R² measures

Terminology defined

• pseudo-R² measure
• random intercepts model
• random slopes and intercepts model
• unconditional growth model
• unconditional means model

Heterogeneity versus correlation

Correlation viewpoint

• Last time we demonstrated that starting with a very simple multilevel model (a level-1 model with only an intercept and a level-2 in which this intercept is allowed to be random) a correlation structure is induced in the response. The correlation pattern that is obtained is very simple though. For observations arising from the same level-2 unit, say the i^th unit, we showed that the covariance matrix is the following.

A covariance matrix that exhibits this structure is said to show compound symmetry.

• The entire collection of response variables can then be assembled into the following block-diagonal matrix.

Here we see the basic structure. Observations coming from the same level-2 unit share the correlation structure shown above, while observations coming from different level-2 units are uncorrelated.

Heterogeneity viewpoint

• A composite model that yields the compound symmetry covariance structure for observations coming from the same level-2 unit is the following.

• The way to think about the term in this equation is that it includes all those details about level-2 unit i that make it different from the other units. If there were level-2 predictors in the model, then would include all those details about the level-2 unit i that are not accounted for by the predictors.
• So by including a term such as we end up making the observations on level-2 unit i systematically different from the observations made on other units. By making them different in the same way from other observations we of course necessarily make them appear more similar to each other. Thus heterogeneity and correlation are flip sides of the same coin.

Basic multilevel models

• When building a multilevel model there should be a natural progression in the models that one builds. In this progression the two most important models are the ones occupying the endpoints. These are the model with minimal structure, the unconditional means model, and the model with maximal structure, the random slopes and intercepts model. This last model is also called the unconditional growth model when the primary level-1 predictor is time-varying.
• I consider next four standard preliminary multilevel models that one should always fit and I describe the comparisons one can make using them. These four models are the unstructured ordinary least squares (OLS) model, the unconditional means model, the random intercepts model, and the random slopes and intercepts model. In all the models that follow I assume there is single level-1 predictor of interest, x, with a response labeled y. There are no level-2 predictors. The issue of level-2 predictors will be covered in a subsequent lecture.

Basic OLS model

• This is not a multilevel model at all. It is just an ordinary regression model in which the basic structure in the data set is ignored. Thus we fit the model

Even though I'm still using the double subscript notation it has no meaning here. The fact that observation (1,1) and observation (1,2) come from the same level-2 unit, unit 1, but that observation (2,1) comes from a different level-2 unit, unit 2, is completely ignored in this model. We treat all observations as if they are completely equivalent, ignoring the heterogeneity that is inherent in the data set. The result is that the apparent sample size is inflated and the standard errors of parameter estimates are underestimated typically leading us to find significant differences where none exist.

• The primary reason for fitting this model is that it allows us to perform a formal test of the structure. This model is directly comparable to the random intercepts model to be discussed below and differs from it only in the absence of any attempt to account for observational heterogeneity. A test of this model against the random intercepts model is a test of the need for including random effects.

The unconditional means model

• The unconditional means model is substantively uninteresting because it contains no predictors. It's value statistically is that it's the simplest multilevel model and it allows us to estimate variance components. In particular it allows us to estimate how much variance lies within subjects versus how much variance occurs between subjects. The basic model is the following.

where and where and are independent for all i and j.

• The primary reason for fitting this model is to obtain estimates of and . The relative sizes of these quantities can be used as a guide for determining where the subsequent modeling effort should be invested. In multilevel modeling inclusion of level-1 predictors will act primarily to reduce , while adding level-2 predictors will serve primarily to reduce . Thus if one of these quantities is much larger than the other, it makes sense to focus more attention on that level.
• Recall for the unconditional means model that the correlation for observations from the same level-2 unit is given by . Thus the unconditional means model allows us to assess the importance of observational heterogeneity because it estimates the quantities that appear in the expression for the correlation. If the correlation is large that argues strongly in favor of fitting a multilevel model.
• The last benefit in fitting this model is that it provides a benchmark for assessing the importance of any level-1 predictors that are added at a subsequent stage (random intercepts model).

The random intercepts model

• The random intercepts model is typically the model in which the maximal level-1 model has been developed. This model differs from the unconditional means model in that one or more level-1 predictors are added. The basic form of the random intercepts model with one level-1 predictor is the following.

where again we have and where and are independent for all i and j.

• This model has two primary uses.
  1. We can compare this model to the basic OLS model. These two models differ only in the inclusion of the random effects terms . A likelihood ratio test comparing these two models tests whether it is necessary to take into account the data structure, the fact that level-1 observations are nested within level-2 units.
  2. We can also compare this model to the unconditional means model. These two models differ only in the inclusion of the level-1 predictor . Likelihood ratio tests can be used to test the importance of the level-1 predictors. In addition we can compare the magnitude of in the two models and use the relative difference as a kind of pseudo-R² statistic as follows.

• From the form of the pseudo-R² statistic we see it can be interpreted as the fraction of the original level-1 variance that has been explained by the addition of the level-1 predictor . It's called a pseudo-R² statistic because the different variance terms that appear in the multilevel models are not entirely independent of each other. A predictor added at one level can also have ramifications at other levels making interpretation of these statistics difficult.

The random intercepts and slopes model

• The random intercepts and slopes model (also called the unconditional growth model) is a natural extension of the random intercepts model. It differs from the random intercepts model in that it allows both the slopes and intercepts to be random. Thus there are two equations at level 2.

where in addition

• At this point we have three variance components, , , and . These aren't additive or directly comparable because is measured on a different scale from the rest. Its random effect multiplies and hence needs to be scaled accordingly . They also don't partition the variability in any sensible way because the covariance term is typically not zero.
• Still this model can be used as a baseline model. We can add level-2 predictors to either or both the slope or the intercept equation of the unconditional growth model depending on how we suspect the predictor should affect the level-1 model.
• Furthermore the unconditional growth model allows us to obtain two additional pseudo-R² statistics of the form

• If the predictor is added to intercept equation then the that appears in this equation should be . If on the other hand the predictor is added to the level-2 slope equation, should be .
• We'll consider the basic problem of adding level-2 predictors to multilevel models in Friday's lecture.

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