Abstract
In this chapter, we guide the reader through model specification and interpretation in time-varying effect modeling (TVEM) with a continuous dependent variable. We begin by discussing selecting data that meet the requirements for TVEM and structuring the dataset. Next, we discuss technical details, including two variants of spline regression to estimate the coefficient functions, weighted estimation, nested structures in repeated-measures data, model selection, and significance testing. We then walk readers through a progression of four hypothetical models: an intercept-only TVEM, a model with a single independent variable (i.e., main effect model with a single covariate), an extended model that incorporates a statistical control variable, and a model to examine time-varying moderation. For each, we describe the mathematical model, the variables included in the analysis, specification of the model in SAS, and interpretation of the resultant coefficient functions. Finally, we present an empirical example examining depressive symptoms across age using data from a longitudinal panel study to facilitate a concrete understanding of the concepts presented earlier in this chapter. In this section, we walk the reader through the various decisions an analyst must make when using TVEM. A sample results section based on this empirical example, similar to what one might include in an applied paper, is included.
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Notes
- 1.
The reader may wonder why we explain the reason for the name “P-spline” but not the reason for the name “B-spline.” When they were proposed in the 1940s, B-splines meant “basic splines,” but this did NOT mean that they were simpler or easier than other kinds of splines—basic referred to the fact that they used different functions of time, which formed a geometric basis for constructing the spline. (Basis here means a set of functions or dimensions that you take a linear combination of, in order to make objects in a space of possible objects.) This does not provide any intuition however because any kind of spline has some kind of basis functions. Thus, it is a historical artifact. When we use the word P-splines in the TVEM macro, we use it in a slightly nonstandard way to mean penalized truncated powers of time. Others (e.g., Eilers and Marx (1996)) have used it to refer to penalized B-spline bases.
- 2.
This is in contrast to simultaneous confidence bands, which are constructed such that the probability that the confidence intervals at every point across t contain their true values simultaneously is 0.95. In practice, simultaneous bands with the same coverage probability as pointwise confidence bands would be much wider.
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Lanza, S.T., Linden-Carmichael, A.N. (2021). Specifying, Estimating, and Interpreting Time-Varying Effect Models. In: Time-Varying Effect Modeling for the Behavioral, Social, and Health Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-70944-0_2
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