Volume 1, Issue 2 -

Estimation of the causal dose–response curve is an old problem in statistics. In a non-parametric model, if the treatment is continuous, the dose–response curve is not a pathwise differentiable parameter, and no -consistent estimator is available. However, the risk of a candidate algorithm for estimation of the dose–response curve is a pathwise differentiable parameter, whose consistent and efficient estimation is possible. In this work, we review the cross-validated augmented inverse probability of treatment weighted estimator (CV A-IPTW) of the risk and present a cross-validated targeted minimum loss–based estimator (CV-TMLE) counterpart. These estimators are proven consistent and efficient under certain consistency and regularity conditions on the initial estimators of the outcome and treatment mechanism. We also present a methodology that uses these estimated risks to select among a library of candidate algorithms. These selectors are proven optimal in the sense that they are asymptotically equivalent to the oracle selector under certain consistency conditions on the estimators of the treatment and outcome mechanisms. Because the CV-TMLE is a substitution estimator, it is more robust than the CV-AIPTW against empirical violations of the positivity assumption. This and other small sample size differences between the CV-TMLE and the CV-A-IPTW are explored in a simulation study.

Large randomized controlled clinical trials are the gold standard to evaluate and compare the effects of treatments. It is common practice for investigators to explore and even attempt to compare treatments, beyond the first round of primary analyses, for various subsets of the study populations based on scientific or clinical interests to take advantage of the potentially rich information contained in the clinical database. Although subjects are randomized to treatment groups in clinical trials, this does not imply the same degree of randomization among sub-populations of the original trials. Therefore, comparisons of treatments in sub-populations may not produce fair and unbiased results without properly addressing this issue. Covariate adjustments in regression analysis and propensity score matching are commonly used to address the non-randomized nature of the sub-populations issue with various degrees of success. However, further improvements to these methods are still possible. In this article, we propose an analysis strategy that shows improvement to conventional methods. Treatment effects and their differences are estimated after adjustment for background imbalances. Treatment groups are then compared using confidence intervals whose limits are determined using the Robbins–Monro stochastic approximation. Data from a recent clinical trial are used to illustrate the methodology.

The before-and-after study with multiple unaffected control groups is widely applied to study treatment effects. The current methods usually assume that the control groups’ differences between the before and after periods, i.e. the group time effects, follow a normal distribution. However, there is usually no strong a priori evidence for the normality assumption, and there are not enough control groups to check the assumption. We propose to use a flexible skew- t distribution family to model group time effects, and consider a range of plausible skew- t distributions. Based on the skew- t distribution assumption, we propose a robust- t method to guarantee nominal significance level under a wide range of skew- t distributions, and hence make the inference robust to misspecification of the distribution of group time effects. We also propose a two-stage approach, which has lower power compared to the robust- t method, but provides an opportunity to conduct sensitivity analysis. Hence, the overall method of analysis is to use the robust- t method to test for the overall hypothesized range of shapes of group variation; if the test fails to reject, use the two-stage method to conduct a sensitivity analysis to see if there is a subset of group variation parameters for which we can be confident that there is a treatment effect. We apply the proposed methods to two datasets. One dataset is from the Current Population Survey (CPS) to study the impact of the Mariel Boatlift on Miami unemployment rates between 1979 and 1982.The other dataset contains the student enrollment and grade repeating data in West Germany in the 1960s with which we study the impact of the short school year in 1966–1967 on grade repeating rates.

Causal effects in right-censored survival data can be formally defined as the difference in the marginal cumulative event probabilities under particular interventions. Conventional estimators, such as the Kaplan-Meier (KM), fail to consistently estimate these marginal parameters under dependent treatment assignment or dependent censoring. Several modern estimators have been developed that reduce bias under both dependent treatment assignment and dependent censoring by incorporating information from baseline and time-dependent covariates. In the present article we describe a recently developed targeted minimum loss-based estimation (TMLE) algorithm for general longitudinal data structures and present in detail its application in right-censored survival data with time-dependent covariates. The treatment-specific marginal cumulative event probability is defined via a series of iterated conditional expectations in a time-dependent counting process framework. The TMLE involves an initial estimator of each conditional expectation and sequentially updates these such that the resulting estimator solves the efficient influence curve estimating equation in the nonparametric statistical model. We describe the assumptions required for consistent estimation of statistical parameters and additional assumptions required for consistent estimation of the causal effect parameter. Using simulated right-censored survival data, the mean squared error, bias, and 95% confidence interval coverage probability of the TMLE is compared with those of the conventional KM and the inverse probability of censoring weight estimating equation, conventional maximum likelihood substitution estimator, and the double robustaugmented inverse probability of censoring weighted estimating equation. We conclude the article with estimation of the causal effect of warfarin medical therapy on the probability of “stroke or death” within a 1-year time frame using data from the ATRIA-1 observational cohort of persons with atrial fibrillation. Our results suggest that a fixed policy of warfarin treatment for all patients would result in 2% fewer deaths or strokes within 1-year as compared with a policy of withholding warfarin from all patients.

The paradox described below aims to clarify the principles by which empirical data are harnessed to guide decision making. It is motivated by the practical question of whether empirical assessments of the effect of treatment on the treated (ETT) can be useful for either policy evaluation or personal decisions.