Assessing the Total Effect of Time-Varying Predictors in Prevention Research

Observational data are often used to address prevention questions such as, “If alcohol initiation could be delayed, would that in turn cause a delay in marijuana initiation?” This question is concerned with the total causal effect of the timing of alcohol initiation on the timing of marijuana initiation. Unfortunately, when observational data are used to address a question such as the above, alternative explanations for the observed relationship between the predictor, here timing of alcohol initiation, and the response abound. These alternative explanations are due to the presence of confounders. Adjusting for confounders when using observational data is a particularly challenging problem when the predictor and confounders are time-varying. When time-varying confounders are present, the standard method of adjusting for confounders may fail to reduce bias and indeed can increase bias. In this paper, an intuitive and accessible graphical approach is used to illustrate how the standard method of controlling for confounders may result in biased total causal effect estimates. The graphical approach also provides an intuitive justification for an alternate method proposed by James Robins [Robins, J. M. (1998). 1997 Proceedings of the American Statistical Association, section on Bayesian statistical science (pp. 1–10). Retrieved from http://www.biostat.harvard.edu/robins/research.html; Robins, J. M., Hernán, M., & Brumback, B. (2000). Epidemiology, 11(5), 550–560]. The above two methods are illustrated by addressing the motivating question. Implications for prevention researchers who wish to estimate total causal effects using longitudinal observational data are discussed.

APPENDIX: WEIGHT COMPUTATION

Complete details and explanation of generic SAS programming code that can be used for the weight creation, naïve method, standard method, and weighted method are available at http://methodology.psu.edu/publications/tvpappen.html. Also provided are two simulated datasets that allow for practice and analysis in conjunction with a review of the SAS code. At each measured time point, $t$, where a participant is at risk for response initiation (e.g., marijuana initiation) a weight component is created. Each weight component is the ratio of two predicted probabilities. The numerator is the predicted probability of a participant's observed predictor initiation or noninitiation in period $t$, given past predictor initiation status (e.g., alcohol initiation or noninitiation) and baseline variables (e.g., sex and race), for those still at risk of response initiation. The denominator is the predicted probability of a participant's observed predictor initiation or noninitiation in period $t$, given past predictor initiation status, baseline variables, and confounders, for those still at risk of response initiation. Thus, the numerator and denominator models only differ in that the confounders are present in the denominator predicted probability. Note that if alcohol initiation (i.e., predictor initiation) occurs prior to time $t$, but before marijuana initiation (i.e., response initiation), the ratio at time $t$ is 1 because both the numerator and denominator predicted probabilities are 1 (i.e., the predicted probability of initiating alcohol use after the initiation of alcohol use is 1 for any values of the confounders and baseline variables). Hence, the weight component does not need to be computed after predictor or response initiation (whichever occurs first). Because the numerator and denominator probabilities are computed only for those still at risk of predictor and response initiation, conditioning on $\overline{{\rm Alc}}_{i-1}$ and $\overline{{\rm Mj}}_{i-1}$ (complete past predictor and response patterns, respectively), as shown in Eq. (5), is not necessary. Note that conditioning on $\overline{{\rm Mj}}_{i-1}$ in Eq. (5) is implicit because this is a survival analysis setting in which a participant is no longer in the dataset if the participant has initiated marijuana use; hence $\overline{{\rm Mj}}_{i-1}=\mathbf{0}$ for all participants in the dataset at time $i$. Thus, the equations below do not include past Alc or Mj. The model for the numerator regression model is

$$\log\left(\frac{{\rm numpr}_{ti}}{1-{\rm numpr}_{ti}}\right)=\alpha\,{\rm Schyr} + \beta_{1}\,{\rm Sex}_i + \beta_{2}\,{\rm Race}_i,$$

(A.1)

whereas the model for the denominator regression model is

$$\log\left(\frac{{\rm denpr}_{ti}}{1-{\rm denpr}_{ti}}\right) = \alpha\,{\rm Schyr} + \beta_{1}\,{\rm Sex}_i+ \beta_{2}\,{\rm Race}_i\\ + \theta\,{\rm Conf}_{ti}.$$

(A.2)

Thus, the weight component for participant $i$ before alcoholinitiation is

$$\frac{1-{\rm numpr}_{ti}}{1-{\rm denpr}_{ti}},$$

(A.3)

and the weight component for participant $i$ at alcohol initiation is

$$\frac{{\rm numpr}_{ti}}{{\rm denpr}_{ti}}.$$

(A.4)

The weight at time $t$, $W_t$, is the product of these weight components up to time $t$. If at time $t-1$ participant $i$ has yet to initiate alcohol use then the weight at time $t-1$ is

$$W_{t-1}=\left(\frac{1-{\rm numpr}_{ti}}{1-{\rm denpr}_{ti}}\right)_{t-1} \cdots\left(\frac{1-{\rm numpr}_{ti}}{1-{\rm denpr}_{ti}}\right)_1\kern-3pt.\qquad$$

(A.5)

If at time $t$ participant $i$ initiates alcohol use then the form of the weight at time $t$ is

$$W_t = \left(\frac{{\rm numpr}_{ti}}{{\rm denpr}_{ti}}\right)_{t} \left(\frac{1-{\rm numpr}_{ti}}{1-{\rm denpr}_{ti}}\right)_{t-1}\\ \cdots\left(\frac{1-{\rm numpr}_{ti}}{1-{\rm denpr}_{ti}}\right)_1.$$

(A.6)

The weight $W_s$ for all times, $s$ larger than $t$, remains equal to $W_t$ as each weight component is now equal to 1. Each participant has a weight for each time point until either the participant initiates the response or the study ends.

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