Finding the Optimal Number of Splits and Repetitions in Double Cross-Fitting Targeted Maximum Likelihood Estimators

Flexible machine learning algorithms are increasingly utilized in real-world data analyses. When integrated within double robust methods, such as the Targeted Maximum Likelihood Estimator (TMLE), complex estimators can result in significant undercoverage-an issue that is even more pronounced in singly robust methods. The Double Cross-Fitting (DCF) procedure complements these methods by enabling the use of diverse machine learning estimators, yet optimal guidelines for the number of data splits and repetitions remain unclear. This study aims to explore the effects of varying the number of splits and repetitions in DCF on TMLE estimators through statistical simulations and a data analysis. We discuss two generalizations of DCF beyond the conventional three splits and apply a range of splits to fit the TMLE estimator, incorporating a super learner without transforming covariates. The statistical properties of these configurations are compared across two sample sizes (3000 and 5000) and two DCF generalizations (equal splits and full data use). Additionally, we conduct a real-world analysis using data from the National Health and Nutrition Examination Survey (NHANES) 2017-18 cycle to illustrate the practical implications of varying DCF splits, focusing on the association between obesity and the risk of developing diabetes. Our simulation study reveals that five splits in DCF yield satisfactory bias, variance, and coverage across scenarios. In the real-world application, the DCF TMLE method showed consistent risk difference estimates over a range of splits, though standard errors increased with more splits in one generalization, suggesting potential drawbacks to excessive splitting. This research underscores the importance of judicious selection of the number of splits and repetitions in DCF TMLE methods to achieve a balance between computational efficiency and accurate statistical inference. Optimal performance seems attainable with three to five splits. Among the generalizations considered, using full data for nuisance estimation offered more consistent variance estimation and is preferable for applied use. Additionally, increasing the repetitions beyond 25 did not enhance performance, providing crucial guidance for researchers employing complex machine learning algorithms in causal studies and advocating for cautious split management in DCF procedures.