A three-level mixed-effects location scale model with an application to ecological momentary assessment data.

In studies using ecological momentary assessment (EMA), or other intensive longitudinal data collection methods, interest frequently centers on changes in the variances, both within-subjects and between-subjects. For this, Hedeker et al. (Biometrics 2008; 64: 627-634) developed an extended two-level mixed-effects model that treats observations as being nested within subjects and allows covariates to influence both the within-subjects and between-subjects variance, beyond their influence on means. However, in EMA studies, subjects often provide many responses within and across days. To account for the possible systematic day-to-day variation, we developed a more flexible three-level mixed-effects location scale model that treats observations within days within subjects, and allows covariates to influence the variance at the subject, day, and observation level (over and above their usual effects on means) using a log-linear representation throughout. We provide details of a maximum likelihood solution and demonstrate how SAS PROC NLMIXED can be used to achieve maximum likelihood estimates in an alternative parameterization of our proposed three-level model. The accuracy of this approach using NLMIXED was verified by a series of simulation studies. Data from an adolescent mood study using EMA were analyzed to demonstrate this approach. The analyses clearly show the benefit of the proposed three-level model over the existing two-level approach. The proposed model has useful applications in many studies with three-level structures where interest centers on the joint modeling of the mean and variance structure.