Ashley Isaac Naimi

The g-formula is a foundational tool for identifying causal effects in observational data. This tool is based on the law of iterated expectation, a key mathematical identity in statistics. However, the notation with which the law of iterated expectation and the g-formula is expressed can be opaque to those with little background in statistics. We provide a primer introducing the law of iterated expectation, the integration notation used to express it, and its role for causal effect identification via the g-formula. Combined with the assumptions of causal consistency, positivity, and conditional exchangeability, the law of iterated expectation yields a causal standardization formula (the g-formula) in two nonparametrically equivalent forms: a non-iterative conditional expectation (NICE) form involving a single weighted average of conditional outcome means, and an iterative conditional expectation (ICE) form involving nested expectations. We illustrate both forms using three progressively complex numerical examples: a time-fixed example with a single binary confounder, a time-fixed example with discrete and continuous confounders, and a time-varying example with two timepoints. We provide clarity on what the law of iterated expectation is, how it is related to the g-formula, and how to gain intuition of its mathematical formulations in actual data examples that can be generalized to a range of settings.

Background: We aimed to examine whether fruit and vegetable intake densities modify the association between a periconceptional high-fat, sugar, and sodium diet and adverse pregnancy outcomes. Methods: We analyzed data from 8,259 participants in the Nulliparous Pregnancy Outcomes Study: Monitoring Mothers-to-Be (2010–2013). Periconceptional diet was assessed via food frequency questionnaire. The outcome was a composite of preeclampsia, gestational diabetes, preterm birth, or small-for-gestational-age birth; individual outcomes were also examined. Effect modification by vegetable and fruit intake densities was evaluated using a doubly robust learner with ensemble machine learning. Results: Approximately 23% of participants experienced an adverse outcome, and 29% consumed a high-fat, sugar, and sodium diet. This dietary pattern was associated with 1.0–2.9 excess cases per 100 pregnancies for the composite outcome and most individual outcomes (except preterm birth), compared with other diets. For the composite outcome, associations varied across levels of vegetable but not fruit intake density, although estimates were imprecise. For example, risk differences decreased from 3.7 (95% CI, −0.7 to 8.0) to 2.6 (95% CI, −1.4 to 6.6) excess cases per 100 pregnancies at 0.5 versus 1.25 cups of vegetables per 1,000 kcal. Attenuation of risk at higher vegetable intake was slightly greater when combined with higher fruit intake density, but estimates remained imprecise, limiting interpretation of joint effect modification. Patterns of associations differed across individual outcomes. Conclusions: We did not find evidence that the risk of adverse pregnancy outcomes associated with a high-fat, sugar, and sodium dietary pattern was meaningfully different across vegetable and/or fruit intake densities.

Many research findings are based on a nested subset of an original cohort followed over time, but results are rarely generalized back to the original population. Here, we conduct a “proof of concept” analysis demonstrating the application of simple methods to generalize the association between prepregnancy obesity and preeclampsia in women from the Nulliparous Pregnancy Outcomes Study: Monitoring Mothers to Be (nuMoM2b, target sample), using a subset of women whose follow-up was extended to 3 years postpartum in the Heart Health Study (nuMoM2b-HHS, source sample). We constructed inverse probability of selection weights and estimated three risk ratios for the association between obesity and preeclampsia: in the target nuMoM2b sample (N = 9,920), in the source nuMoM2b-HHS sample (N = 4,486), and in the weighted source nuMoM2b-HHS sample generalized back to the target sample using inverse probability of selection weights (pseudo N = 4,468). In the target, source, and weighted samples, the estimated risk ratios (95% confidence intervals) were 2.0 (1.7, 2.3), 2.2 (1.8, 2.6), and 2.0 (1.6, 2.4), respectively. We discuss the assumptions involved in generalizing study findings and methods available for doing so. When relevant, researchers should deploy methods for generalizing study nested cohort findings back to a target sample.

This study evaluated high-dimensional mediation analysis methods (HIMA by Zheng et al. and HDMA by Gao et al.) and the "Meet-in-the-Middle" (MITM) approach using simulated metabolomics data. Simulations varied in sample size, mediator set size, correlation structure, proportion of true mediators, and mediation effect size (beta). We assessed each method's ability to estimate the total indirect effect (TIE), component indirect effects (CIEs), sensitivity, and specificity. In scenarios with independent metabolites, HIMA and HDMA reliably estimated CIEs, while HDMA provided the most accurate estimate of the TIE. MITM generally underestimated the TIE, and HIMA showed improved TIE estimates with higher mediator effect sizes. In correlated settings, CIE estimation was not feasible due to the lack of identifiable causal contrasts, and all methods underestimated the TIE. Sensitivity declined in low beta, small sample size, and high-dimensional scenarios, though specificity remained high (>90%) across all methods. Findings suggest that HIMA offers the most accurate mediation results but may exclude meaningful features through dimensionality reduction. Therefore, applying parallel mediation approaches, such as MITM and HIMA, and focusing on the overlapping findings would be recommended. These results underscore the need for the development of robust, scalable mediation methods tailored to untargeted metabolomics data.

Objective: To evaluate whether the well-established age-related reduction in antral follicle counts (AFC) is greater among women with higher concentrations of endocrine-disrupting chemicals. Design: Prospective cohort study using the doubly robust (DR) learner, a flexible machine learning method that models the relationship between an exposure and outcome of interest and identifies potential modifiers of this relation. Subjects: Seven hundred seventy-five women aged 21-46 years enrolled in The Environment and Reproductive Health (EARTH) Study at the Massachusetts General Hospital Fertility Center, Boston, USA, between 2004 and 2019. Exposure: Age at AFC assessment calculated as the date of antral follicle scan minus the patient’s birthdate, categorized into <35 years or ≥35 years. We assessed urinary concentrations of 16 endocrine-disrupting chemicals, including 11 urinary phthalate metabolites, 3 urinary parabens, urinary bisphenol A, and hair mercury, as potential modifiers of the age- AFC association. Main Outcome Measures: AFC assessed by transvaginal ultrasonography. Results: In this cohort of 775 women, 43% were less than 35 years of age. Women greater than or equal to aged 35 years had lower AFC (mean [SD]: 11.63 [6.24] vs. 16.89 [8.64]) Three endocrine-disrupting chemicals significantly modified the age-AFC association. Women with higher urinary mono-isobutyl phthalate (MiBP) and monocarboxyisooctyl phthalate (MCOP) concentrations experienced stronger negative associations between age and AFC. Threshold analyses revealed that the age-AFC association strengthened, declining at a faster rate when MiBP, MCOP, or MEHP concentrations were higher. Women with higher hair Hg concentrations experienced weaker negative associations between age and AFC. The remaining 13 endocrine- disrupting chemicals did not modify the age-AFC relationship. Conclusion: Select phthalate metabolites, specifically MiBP, MCOP, and MEHP, were associated with a modified age-related decline in ovarian reserve in a curvilinear dose- dependent manner. These findings add to the growing evidence that exposure to certain phthalates may accelerate reproductive aging in women. CAPSULE: Women with higher urinary concentrations of select phthalate metabolites also had accelerated age-related declines in ovarian reserve, with evidence of exposure thresholds for MiBP and MCOP.

Context: Disordered eating behaviors are linked to reproductive health and are more common in women with polycystic ovary syndrome (PCOS). However, it remains unclear whether they are risk factors for PCOS. Objective: To investigate the associations of disordered eating behaviors during adolescence and PCOS in adulthood. Design: Prospective, longitudinal cohort (1996-2021). Setting: United States. Patients or Other Participants: Female participants (n = 6208) from the Growing Up Today Study. Disordered eating behaviors were self-reported on repeated questionnaires during follow-up. Main Outcome Measures: Physician-diagnosed PCOS was reported on each questionnaire starting in 2010. Multivariable logistic regression with generalized estimating equations to calculate the adjusted odds ratios (aOR) and 95% confidence intervals (CIs) for the association between past-year frequency of binge eating, laxative use, and self-induced vomiting and PCOS. Results: We identified 410 cases (7%) of PCOS. Binge eating was the most common behavior, with 2331 (38%) girls binge eating at least once during follow-up. Additionally, 1103 (18%) girls reported self-induced vomiting and 701 (11%) reported using laxatives for weight control at least once. Compared to girls who never binge ate, cumulative binge eating more than monthly was associated with higher odds of PCOS (aOR: 1.59, 95% CI: 1.09, 2.33). Cumulative laxative use less than monthly was associated with lower odds of PCOS (aOR: 0.45, 95% CI: 0.29, 0.72) while at least monthly laxative use was associated with higher odds of PCOS (aOR: 1.67, 95% CI: 0.89, 3.14). Self-induced vomiting was not associated with PCOS. Conclusion: Girls who binge eat during adolescence may have increased odds of developing PCOS.

Background: The Super Learner is an ensemble learning method that has been widely used with doubly robust causal effect estimators. It is recommended to deploy the Super Learner with a diverse library of algorithms. To our knowledge, however, the magnitude of the improvements gained by including many algorithms has not yet been systematically evaluated in common epidemiologic research settings. Methods: We applied Super Learning with two doubly robust estimators, augmented inverse probability weighting (AIPW) and targeted minimum loss-based estimation (TMLE), to estimate the average treatment effect (ATE) of high periconceptional dietary fruit and vegetable density on the risk of preeclampsia among 7,923 women from the nuMoM2b study. Using a reference ensemble with a diverse library of algorithms, we compared estimates under different sets of algorithms included in the Super Learner to evaluate whether ATE estimates were sensitive to library choices. Results: The doubly robust estimators fitted with the reference Super Learner ensemble suggested ≥2.5 cups/1,000 kcal of total fruit and vegetable density was associated with a lower risk of preeclampsia. ATE estimated on the risk difference scale by AIPW was −0.019 (95% confidence interval = −0.036, −0.003) and by TMLE was −0.023 (95% confidence interval = −0.039, −0.007). Excluding any individual algorithm from the reference ensemble had little impact on estimates from either AIPW or TMLE. However, relying on a single algorithm (e.g., extreme gradient boosting) yielded results that were much more variable. Conclusion: Our empirical findings support recommendations to build ensemble learners for doubly robust estimators using a diverse array of flexible machine learning algorithms.

Simulation studies are used to evaluate and compare the properties of statistical methods in controlled experimental settings. In most cases, performing a simulation study requires knowledge of the true value of the parameter, or estimand, of interest. However, in many simulation designs, the true value of the estimand is difficult to compute analytically. Here, we illustrate the use of Monte Carlo integration to compute true estimand values in simple and more complex simulation designs. We provide general pseudocode that can be replicated in any software program of choice to demonstrate key principles in using Monte Carlo integration in two scenarios: a simple three-variable simulation where interest lies in the marginally adjusted odds ratio and a more complex causal mediation analysis where interest lies in the controlled direct effect in the presence of mediator-outcome confounders affected by the exposure. We discuss general strategies that can be used to minimize Monte Carlo error and to serve as checks on the simulation program to avoid coding errors. R programming code is provided illustrating the application of our pseudocode in these settings.