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The Relative Performance of Targeted Maximum Likelihood Estimators Under Violations of the Positivity Assumption


Observational studies often present the challenge of data sparsity due to violations of the positivity assumption. Such violations occur when some subgroups never or rarely receive a particular treatment or never or rarely are uncensored. Bias due to actual or practical positivity violations often goes undiagnosed, and such bias can threaten valid inference for estimation of the target parameter. It is important to recognize that different estimators perform differently under a lack of positivity - in terms of both bias and variance. Common estimators across many fields often perform poorly in this setting.

Alternatively, targeted maximum likelihood estimators (TMLE's) tend to be relatively robust under a lack of positivity. This dissertation compares the performance of TMLE's to many common estimators under violations of the positivity assumption for three different target parameters: (1) a causal effect focused on the difference in mean outcomes for two treatments, (2) a mean outcome that is subject to missingness but for which all possible covariates for predicting missingness are measured, and (3) conditional relative risk in a semi-parametric multiplicative regression model.

For each of these parameters, the parameter-specific positivity assumption is formally presented and discussed. Also for each parameter, the theoretical properties of existing methods are compared to the those of TMLE's. The theoretical properties indicate how we expect different estimators to behave under positivity violations. Also, using a variety of simulations with various degrees of and reasons for positivity violations, the performance of TMLE's, relative to other estimators, is demonstrated. This dissertation also discusses how to diagnose bias due to positivity violations and how to respond to resulting bias.

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