Collaborative targeted maximum likelihood for time to event data.

Current methods used to analyze time to event data either rely on highly parametric assumptions which result in biased estimates of parameters which are purely chosen out of convenience, or are highly unstable because they ignore the global constraints of the true model. By using Targeted Maximum Likelihood Estimation (TMLE) one may consistently estimate parameters which directly answer the statistical question of interest. Targeted Maximum Likelihood Estimators are substitution estimators, which rely on estimating the underlying distribution. However, unlike other substitution estimators, the underlying distribution is estimated specifically to reduce bias in the estimate of the parameter of interest. We will present here an extension of TMLE for observational time to event data, the Collaborative Targeted Maximum Likelihood Estimator (C-TMLE) for the treatment specific survival curve. Through the use of a simulation study we will show that this method improves on commonly used methods in both robustness and efficiency. In fact, we will show that in certain situations the C-TMLE produces estimates whose mean square error is lower than the semi-parametric efficiency bound. We will also demonstrate that a semi-parametric efficient substitution estimator (TMLE) outperforms a semi-parametric efficient non-substitution estimator (the Augmented Inverse Probability Weighted estimator) in sparse data situations. Lastly, we will show that the bootstrap is able to produce valid 95 percent confidence intervals in sparse data situations, while influence curve based inference breaks down.