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ABSTRACT How do we use observational data to address questions such as “If we could delay the initiation of conduct disorder, would that in turn lead to a delay in marijuana initiation?” In observational studies, confounders often provide alternate explanations for the observed relationship between predictor and response. Thus, we must control for confounding in order to address the above question. When the values of the predictor and confounders vary over time (time-varying), traditional adjustments for confounders do not eliminate bias and can cause further bias (Robins & Greenland, 1994; Barber et al, 2002). In this poster we illustrate Hernán, Brumback, and Robins’ (2000) method of using sample weights to adjust for confounding due to time-varying confounders when using survival analysis. The current study examined 210 adolescents from a subset of the Lexington Longitudinal Study. The results of this study suggest that biases result when confounding is controlled through the standard method of including confounders as covariates in the response regression model. This project emphasizes the importance of appropriately including confounders in order to prevent misleading answers to prevention questions. • When compositional differences in confounders are not controlled, the coefficient of the predictor is a biased estimate of the total effect. • The estimated effect of conduct disorder initiation on marijuana initiation reflects the compositional differences between the predictor groups in addition to the causal effect. • In experimental settings, compositional differences are minimized by randomization. In observational studies, statistical methods and scientific assumptions are required. Coefficient Estimates • Confounders are usually controlled with the “standard” response regression model. • The standard model includes confounders as covariates in the response regression model. • Thus, peer pressure resistance would be included as a covariate in the response regression model of marijuana initiation on conduct disorder initiation. • When confounders are affected by the predictor, controlling for confounders with the standard model creates a spurious correlation between the predictor and response. • Consider a simplified model of conduct disorder initiation, marijuana initiation, and peer pressure resistance: • Peer pressure resistance is affected by past conduct disorder initiation, via path “b.” • If peer pressure resistance is included as a covariate in the response regression model, a spurious correlation is created between conduct disorder initiation at time 1 and marijuana initiation at time 3, via the red paths “b” and “c.” THE STANDARD MODEL Usual Solution • “Does delaying conduct disorder initiation lead to a delay in marijuana initiation?” • Estimate the total effect of conduct disorder initiation on marijuana initiation. • Confounders: common correlates of both the predictor (conduct disorder initiation) and the response (marijuana initiation). • Confounders provide alternate explanations of the observed relationship between the predictor and response. • In this example, peer pressure resistance is correlated with both conduct disorder initiation and marijuana initiation. Hence, peer pressure resistance is a confounder of the observed relationship between conduct disorder and marijuana. • To estimate the total effect of conduct disorder initiation on marijuana initiation, confounders must be controlled in the final response regression analysis. • Compositional differences: unequal distribution of confounders among levels of the predictor. • In this example, peer pressure resistance is distributed differently among initiators and non-initiators of conduct disorder. THE PROBLEM Our Question Goal Complication Confounders Compositional Differences Notes: Many arrows that would naturally be in this figure are omitted for simplicity. This is not an SEM diagram. Spurious Correlation This research was supported by the National Institute on Drug Abuse Grant # P50 DA10075 and Award # 1 K02 DA15674-01.
Spurious correlations can cause the estimate of the total effect to be biased. • Hence, including confounders as covariates in the standard response regression model results in biased estimates of the total effect of a time-varying predictor on a response. • Hernán, Brumback, and Robins (2000) use sample weights to statistically control for time-varying confounders. This method can produce unbiased estimates of the total effect. • Sample weights equalize predictor patterns – simulating the predictor pattern as if individuals were randomly assigned to levels of the predictor. • Weighting controls for confounders by equalizing the compositional differences in the confounders between levels of the predictor. Further, weighting eliminates the path of the spurious correlation by not conditioning on the confounder. • Once weights equalize the compositional differences in peer pressure resistance, the correlation between peer pressure resistance and conduct disorder initiation no longer exists. Consider our model with this path eliminated: • With this path eliminated we have this new model: • Each individual has a weight for each time period, t, in which he or she is still at risk for marijuana initiation. • Weights are created from ratios of two predicted probabilities at each time period, t. • The denominator is the predicted probability of an individual’s observed predictor status from a regression of the predictor on all confounders and baseline variables. • The numerator is the same as the denominator except the predictor is regressed on only the baseline variables (non-time-varying moderators). • The weight at time period t is the product of these ratios up to time t. • These weights are used in the weighted response regression model of marijuana initiation on conduct disorder initiation. Weight Creation THE SOLUTION Weighting • A convenience sample from the Lexington Longitudinal Study was used to estimate the total effect of alcohol initiation on marijuana initiation. • Three models were used to compare results: 1. Naïve Model: confounders are omitted 2. Standard Model: confounders are included as covariates 3. Weighted Model: response regression model using weights created as above DATA EXAMPLE NOTES: Coefficients for intercepts and baseline variables are omitted. Naïve and Standard models do not include confounders by definition. One tailed tests: *p<0.05 **p<0.01 ***p<0.001 • When randomization is not possible, confounders are alternate explanations for our findings. The common method for controlling for confounders is the standard model, which produces a biased estimate of the total effect because of spurious correlations. The weighting method is one way to get at this problem. The weighting method eliminates spurious correlations yet controls for confounders, allowing for an unbiased estimate of the total effect. SUMMARY • Barber, J. S., Murphy, S. A., & Verbitsky, N. (2002). Adjusting for time-varying confounding in survival analysis. Manuscript submitted for publication. • Hernán, M., Brumback, B., & Robins, J. M. (2000). Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology, 11, 5, 561-570. • Robins, J. M. & Greenland, S. (1994). Adjusting for differential rates of PCP prophylaxis in high- versus low-dose AZT treatment arms in an AIDS randomized trial. Journal of the American Statistical Association, 89, 737-749. SELECTED REFERENCES