## Panel Modeling: Random Coefficients and Covariance Structures

The multilevel regression model for longitudinal data is a straightforward application of the standard multilevel regression mode, with measurement occasions within subjects replacing the subjects within groups structure. At the lowest, the repeated measures level, we have:

Yti = π0i + π1iTti + π2iXti + eti. (5.1)

In repeated measures applications, the coefficients at the lowest level are often indicated by the Greek letter π. This has the advantage that the subject level coefficients, which are in repeated measures modeling at the second level, can be represented by the usual Greek letter β, and so on. In Equation

5.1, Yti is the response variable of individual i measured at time point t, T is the time variable that indicates the time point, and Xti is a time-varying covariate. Subject characteristics, such as gender, are time invariant covariates, which enter the equation at the second level:

π0i = β00 + β01Zi + u0i, (5.2)

π1i = β10 + β11Zi + u1i, (5.3)

π2i = β20 + β21Zi + u2i. (5.4)

By substitution, we get the single equation model:

Yti = β00 + β10Tti + β20Xti + β01Zi + β11ZiTti + β21ZiXti + u1iTti (5.5) + u2iXti + u0i + eti

In multilevel models for subjects within groups, there is an assumed dependency between the subjects who are in the same group. Most often, there is no need to assume a specific structure for this dependency. Subjects within the same group are assumed exchangeable, and the intraclass correlation refers to the average correlation between two randomly chosen subjects from the same group. In multilevel models for occasions within subjects, measurement occasions are not freely exchangeable, because they are ordered in time. In such models, it often does make sense to assume a structure for the relationships between measurements across time. For example, an intuitively attractive assumption is that correlations between measures taken at different measurement occasions are higher when these occasions are close to each other in time.