| Three-level data occur frequently in behavior and medical sciences. For example, in a multi-center trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level3). Mixed-effects models have been developed to analyze such three-level data when the response is binary. The model, however, cannot be applied when the response is ordinal. The three-level mixed-effects model for binary data also assumes that the variances at the second and/or the third level of data are the same. Unfortunately, this assumption does not hold in several situations, but especially in twin studies. For example, in the case of the twin data mentioned above, the model assumes that the association within the MZ (monozygotic) and DZ (dizygotic) twins at the third level is the same, that is, the groups have the same intratwin correlation. However, MZ and DZ twins are very different genetically and assuming constant variance is therefore implausible and could lead to misleading results.; An extension of the three-level mixed-effects model for binary data is proposed which is appropriate for three-level ordinal response data, and which allows for heterogeneous variances between groups at either higher level of data (that is, levels 2 or 3). Parameters are estimated using a maximum marginal likelihood (MML) solution using Gauss-Hermite quadrature to numerically integrate over the distribution of the random effects.; Simulation studies show that the fit of the heterogeneous model increases as the magnitude of the difference in variation between the groups increase. The method is illustrated using data from the Chronic Fatigue Syndrome Twin Registry. One part of this data set is the completion of a sleep diary for 14 days. Characteristics of this data set include questions with an ordinal response, three-level data structure, and MZ and DZ twins. |
