Statistical theory forms a foundation for how epidemiologists learn about populations in public health and medical studies and is fundamental for the understanding of more advanced epidemiological methods (e.g., in causal inference and machine learning). Textbooks provide in-depth coverage of probability and statistical theory, but with such comprehensive coverage that it can be easy to miss the forest for the trees. Here, we provide a summary of fundamental concepts from large-sample statistical theory to allow for more focused understanding tailored to epidemiologists and health science researchers. This primer aims to promote appropriate understanding and application of statistical methods in epidemiologic research. We clarify several often-confused statistical topics and provide a motivation for the application of large-sample inferential methods to data from population health and medical studies. Assumptions underlying commonly used statistical methods that must be considered for valid inference are also discussed. These ideas are contextualized with an example from the Women's Interagency HIV Study.

Abstract Objectives Highly flexible nonparametric estimators have gained popularity in causal inference and epidemiology. Popular examples of such estimators include targeted maximum likelihood estimators (TMLE) and double machine learning (DML). TMLE is often argued or suggested to be better than DML estimators and several other estimators in small to moderate samples – even if they share the same large-sample properties – because TMLE is a plug-in estimator and respects the known bounds on the parameter, while other estimators might fall outside the known bounds and yield absurd estimates. However, this argument is not a rigorously proven result and may fail in certain cases. Methods In a carefully chosen simulation setting, I compare the performance of several versions of TMLE and DML estimators of the average treatment effect among treated in small to moderate samples. Results In this simulation setting, DML estimators outperforms some versions of TMLE in small samples. TMLE fluctuations are unstable, and hence empirically checking the magnitude of the TMLE fluctuation might alert cases where TMLE might perform poorly. Conclusions As a plug-in estimator, TMLE is not guaranteed to outperform non-plug-in counterparts such as DML estimators in small samples. Checking the fluctuation magnitude might be a useful diagnosis for TMLE. More rigorous theoretical justification is needed to understand and compare the finite-sample performance of these highly flexible estimators in general.

Abstract Objectives Determining the causal relationship between exposure and outcome is the goal of many observational studies. However, the selection of subjects into the study population, either voluntary or involuntary, may result in estimates that suffer from selection bias. To assess the robustness of the estimates as well as the magnitude of the bias, bounds for the bias can be calculated. Previous bounds for selection bias often require the specification of unknown relative risks, which might be difficult to provide. Here, alternative bounds based on observed data and unknown outcome probabilities are proposed. These unknown probabilities may be easier to specify than unknown relative risks. Methods I derive alternative bounds from the definitions of the causal estimands using the potential outcomes framework, under specific assumptions. The bounds are expressed using observed data and unobserved outcome probabilities. The bounds are compared to previously reported bounds in a simulation study. Furthermore, a study of perinatal risk factors for type 1 diabetes is provided as a motivating example. Results I show that the proposed bounds are often informative when the exposure and outcome are sufficiently common, especially for the risk difference in the total population. It is also noted that the proposed bounds can be uninformative when the exposure and outcome are rare. Furthermore, it is noted that previously proposed assumption-free bounds are special cases of the new bounds when the sensitivity parameters are set to their most conservative values. Conclusions Depending on the data generating process and causal estimand of interest, the proposed bounds can be tighter or wider than the reference bounds. Importantly, in cases with sufficiently common outcome and exposure, the proposed bounds are often informative, especially for the risk difference in the total population. It is also noted that, in some cases, the new bounds can be wider than the reference bounds. However, the proposed bounds based on unobserved probabilities may in some cases be easier to specify than the reference bounds based on unknown relative risks.