Flexible Treatment of Time-Varying Covariates with Time Unstructured Data

Abstract

Time-varying covariates (TVCs) are a common component of growth models. Though mixed effect models (MEMs) and latent curve models (LCMs) are often seen as interchangeable, LCMs are generally more flexible for accommodating TVCs. Specifically, the standard MEM constrains the effect of TVCs across time-points whereas the typical LCM specification can estimate time-specific TVC effects, can include lagged TVC effects, or constrain some TVC effects based on theoretically appropriate phases. However, when data are time-unstructured, LCMs can have difficulty providing TVC effects whose interpretation aligns with typical research questions. This paper shows how MEMs can be adapted to yield TVC effects that mirror the flexibility of LCMs such that the model likelihoods are identical in ideal circumstances. We then extend this adaptation to the context of time-unstructured data where MEMs tend to be more flexible than LCMs. Examples and software code are provided to facilitate implementation of these methods.

Keywords:

• Growth model
• time-varying covariate
• time-unstructured data
• unbalanced data
• multilevel model
• random effects model

Notes

¹ Note that a single variable could be differentially treated as a TIC or TVC depending on how and for whom the data are collected. For instance, if Depression was collected only at baseline, it would be treated as a TIC because the value of this variable is constant within individuals. However, if Depression was assessed at each time-point, then it would be treated as a TVC because its values change within each person.

² The MEMs discussed are not limited to SAS, but can be estimated in any MEM software. Mplus is used because, to the authors’ knowledge, it is currently the only LCM software that permits the definition variable method discussed in one of our subsequent examples.

³ It is important to note that missing values are a prevalent concern for growth modeling in general, and if a TVC value is missing because the data were not provided rather than because of the structure of the data, then the concerns of Grimm, Ram et al. (2016) are warranted because MEMs will end up listwise deleting in such contexts. Alternatively, a multiple imputation approach could be employed.