TVEM was introduced in the statistical literature nearly 20 years ago (Hastie and Tibshirani 1993
; Hoover et al. 1998
). In the area of psychology, Li et al. (2006)
demonstrated a variation of the model with applications to smoking. However, despite the abundance of ILD being collected in social and behavioral sciences, models with time-varying effects are not used in practice. This is likely due to the fact that user-friendly software was not available until now, and that the literature has lacked demonstrations of how TVEM could be applied to behavioral data.
To introduce the model conceptually, we provide a hypothetical example in , in which ILD measures of negative affect and smoking urges for a single individual are summarized in the first two panels. To explore variation in the relation between negative affect and smoking urges over time, we split the time scale into three equally spaced intervals. An association between negative affect and smoking urges is positive in the beginning (r=.9), negative in the middle (r=−.8), and absent at the end (r=.05; the bottom panel of ). Correlating data from the entire observational period (a similar approach to MLM) yielded a nearly null association between the variables (r=.13). This simplified example demonstrates the importance of considering time when studying the relation between a TVC and a time-varying outcome. Moreover, it makes intuitive sense that the dynamic phenomena studied using ILD might have a time-varying association with correlates, because many ILD studies are designed to elicit or capture changes in behavior.
With TVEM, there is no need to divide time into arbitrary intervals or to assume a linear change pattern. Instead, directionality and strength of association are evaluated along the continuum of a time scale by using multiple individuals in ILD studies. Thus, time is measured on a continuous scale and values of parameter estimates are allowed to change with time.
Considering a simple case of one TVC from the current empirical example on smoking data, TVEM can be formulated in the following way:
(smoking urges) and NAij
(negative affect) are intensively measured longitudinal variables for a subject i
measured at time t
. Assessments j
can be sampled at different time points across individuals, a feature that differentiates TVEM from traditional functional data analysis methods. The outcome SUij
is assumed to be normally distributed. β0
are intercept and slope parameters, respectively. Making use of intensively sampled phenomenon under investigation, TVEM assumes that any relations described by intercept β0
) and slope β1
) follow a smooth curve. As a result, both model parameters are functions
that summarize relations in forms of curves, with values changing along the time continuum. Thus, β0
) corresponds to an intercept function, varying over time. In this example, β0
) represents the course of smoking urge intensity over time for a centered value of negative affect (the exact interpretation depends on how negative affect is centered). Similarly, β1
) is a slope function describing the progressive time-varying association between negative affect and urges. The shape and magnitude of each function are specified separately, allowing both intercept and slope to flexibly and independently change over time. Because the parameter estimates vary with time, it is helpful to summarize estimates graphically by plotting their values and corresponding confidence intervals. Hypothesis testing can be done in relation to an estimated confidence interval of a function.
TVEM is a nonparametric model, requiring no constraints on shapes of intercept and slope functions. Instead, shapes are estimated from the available data; the only assumption is that temporal progression happens gradually, in a smooth way (with no sudden peaks or jumps). For model-fitting purposes, the P-spline method is used to estimate shapes of parameter functions. With this flexible approach, a complex function is split into several (usually equally spaced) intervals, and each portion of a function is estimated with a polynomial (in our case, cubic) model. With this method, any complex function can be successfully approximated if a sufficient number of intervals is specified. The splitting points between intervals are referred to as knots
. Thus, in the process of model selection, models with a different number of knots (or intervals) and of different complexity are compared. More technical details about model fitting and estimation can be found in Tan et al. (2010)