Generalized TVEM Mathematical Model

Time-varying effect models (TVEMs) are a natural extension of linear regression models. The fundamental difference is that, in linear regression models a single estimate of each covariate’s effect is provided, but in TVEM the coefficients can vary over time. Intensive longitudinal data are generally collected to capture temporal changes in a process, so not only might the outcome change over time, but the relationships between the covariates and the outcome might also change over time. TVEM is designed to evaluate whether and how the effects of covariates change over time. The following example is specific to normal distributions.

Suppose we observe intensive longitudinal data {(xij, yij, tij), for individuals i = 1,2…n, over observation times j = 1,2,…,mi} , where yij is the response variable for the i-th individual measured at time tij, and xij = (xij1, xij2, …, xijp,)’ is a corresponding p-dimensional covariate vector. The traditional linear model is specified as

yij = β1xij1 +…+ βpxijp + εij (1)

where we set xij1= 1 to represent an intercept, and ε is a random, normally distributed error term.

The model can be extended to include time-varying effects. A TVEM is defined as

yij = β1(tij)x +…+ βp(tij)xijp + εij (2)

where β1(t),…,βp(t) are unknown coefficient functions that are assumed to be smooth functions of time.

The %TVEM macros were developed to estimate the coefficient functions in model (2). The %TVEM macro uses all available data for every individual over time. Time-specific observations with missing values either for response or for any covariates are automatically excluded. Note that the %TVEM macros can also include time-invariant covariates, such as gender, and the effects of time-invariant covariates may vary over time or be static. Details about the TVEM mathematical model can be found in Tan et al. (2012).


Tan, X., Shiyko, M., Li, R., Li, Y., & Dierker, L. (2012). A time-varying effect model for intensive longitudinal data. Psychological Methods, 17(1), 61-77. doi: 10.1037/a0025814 PMCID: PMC3288551