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Principal Stratification with Bayesian Additive Regression Trees for Count-Valued Intermediate Variables: Estimating the Effect of Fertility on Women's Employment

This paper introduces a novel Bayesian Additive Regression Trees (BART) framework for principal stratification that addresses the challenges of estimating causal effects with count-valued intermediate variables, such as fertility, by flexibly modeling covariate-dependent instrument validity and effect heterogeneity, ultimately revealing nuanced negative impacts of fertility on women's employment in specific African contexts that traditional methods fail to detect.

Original authors: Lucas Godoy Garraza, Leontine Alkema

Published 2026-08-12

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CC BY 4.0

Original authors: Lucas Godoy Garraza, Leontine Alkema

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to figure out why a plant stops growing. Is it because the plant is just naturally small, or is it because you stopped watering it? In the world of science, this is called finding a "causal effect." But life is messy. Often, the thing we think is causing a change (like having a baby) is tangled up with other things we can't see (like a mother's desire to work or her family's support). It's like trying to taste the salt in a soup while someone else is simultaneously adding pepper and sugar.

To solve this, scientists use a clever trick called an "instrument." Think of an instrument like a lucky coin flip that happens to change one thing (like how many kids a woman has) without directly changing the other thing (like whether she goes to work). If the coin flip is truly random, any difference in work habits between the "lucky" and "unlucky" groups must be because of the number of kids, not because of other hidden reasons. However, finding a perfect coin flip in real life is incredibly hard. Sometimes the "coin" isn't random at all; it depends on the person's age or background. When that happens, the old math tools scientists usually use break down, giving them blurry or wrong answers. This is the puzzle this paper tries to solve: how to measure the true cost of having a child on a woman's job, even when the "lucky coin" isn't perfectly random.

The authors of this paper, Lucas Godoy Garraza and Leontine Alkema, decided to build a new, super-flexible tool to solve this puzzle. They took a method called "Principal Stratification"—which is like sorting people into secret groups based on how they would react to the lucky coin—and combined it with a powerful machine-learning technique called "Bayesian Additive Regression Trees" (or BART). If standard math tools are like a rigid ruler that can only measure straight lines, BART is like a flexible, stretchy tape measure that can twist and turn to fit any shape. They also tweaked this tool to handle "count-valued" variables. Since you can't have half a baby, the number of children is a whole number (1, 2, 3...), not a smooth slide. Their new tool, called PrinceBART, is designed specifically to handle these whole-number counts while still being flexible enough to untangle the messy relationships between age, education, and work.

To test if their new tool actually works, the authors ran thousands of computer simulations. They created fake worlds where the "lucky coin" (infecundity, or the inability to conceive) was tricky and depended on other factors. In these simulations, the old, rigid tools often got the answer wrong or were very unstable. In contrast, PrinceBART consistently found the correct answer, even when the data was messy and the effects were different for different people. It was like showing that while a rigid ruler might snap when trying to measure a winding river, their stretchy tape measure followed the curves perfectly.

When they applied this new tool to real data from Nigeria, Senegal, and Kenya, the results were fascinating and showed that the "cost" of having a child isn't the same for everyone. In Nigeria, the study suggests that having one more child lowers a woman's chance of being employed by about 6.3 percentage points. However, in Senegal and Kenya, the data didn't show a clear average drop in employment for the whole country. But here is the most important part: the average hides a huge story. The authors found that the penalty for having a child is heavily concentrated among specific groups. In all three countries, the women who suffer the most are the youngest and those with the least education. For example, in Nigeria, a woman under 20 with no education faces a massive drop in employment chances, while an older, educated woman might see no drop at all.

The paper suggests that these younger, less-educated women likely have fewer job protections or less family support to help with childcare, so an extra child forces them to stop working. Older or more educated women seem better able to handle the extra responsibility without losing their jobs. The authors are careful to say that while their new method is much better at finding these hidden patterns than the old tools, the results are still based on survey data and rely on certain assumptions. They didn't prove these findings with a magic wand, but their simulations show their method is robust, and their real-world application suggests that the "cost" of fertility is not a single number for everyone, but a story that changes depending on who you are and where you live.

Technical Summary: Principal Stratification with Bayesian Additive Regression Trees for Count-Valued Intermediate Variables

Problem Statement
Estimating the causal effect of fertility on women's employment is complicated by the joint determination of fertility and labor-market decisions, leading to confounding by unobserved factors (e.g., preferences, household bargaining power). While instrumental variable (IV) strategies are standard, they face significant limitations in this context:

Covariate-Dependent Validity: Instruments like infecundity (the inability to conceive) are strongly associated with age and other covariates. Consequently, the assumption of instrument independence often holds only after conditioning on observed covariates, not marginally.
Inflexibility of Standard IVs: Two-stage least squares (2SLS), the dominant IV implementation, typically relies on additive covariate adjustments. This approach fails to flexibly accommodate nonlinearities and may fail to identify a clearly interpretable complier-specific estimand when instrument validity is covariate-dependent.
Treatment Structure: Fertility is a count-valued intermediate variable (number of children). Standard binary IV methods (e.g., standard principal stratification) do not naturally extend to multivalued ordered treatments, and continuous approximations may not respect the discrete nature of parity.
Heterogeneity: A single average treatment effect often obscures substantial heterogeneity in how additional children affect employment across different demographic groups (age, education, etc.).

Methodology
The authors extend the PrinceBART framework (Principal Stratification with Bayesian Additive Regression Trees), originally designed for binary intermediate variables, to settings with count-valued intermediate variables.

Causal Framework: The study employs principal stratification to define causal effects within latent strata determined by how a woman's potential parity ( $W(0)$ if fecund, $W(1)$ if infecund) responds to the instrument (infecundity). The target estimand is the Conditional Average Treatment Effect (CATE) among women "affected" by the instrument (those whose parity would decrease by one child due to infecundity).
Statistical Model:
- Mixture Model: The likelihood is formulated as a mixture model where the latent stratum membership $G^* = \{W(0), W(1)\}$ is unobserved for any individual.
- BART for Outcomes and Strata: Bayesian Additive Regression Trees (BART) are used for two components:
  1. Outcome Model: Modeling the probability of employment ( $Y$ ) as a function of covariates ( $X$ ), the instrument ( $Z$ ), and potential parities.
  2. Stratum Model: Modeling the joint distribution of potential parities. To handle the count nature of $W$ , the authors introduce a latent continuous variable $W^*(z)$ modeled via BART, which is then mapped to discrete counts via a rounding mechanism ( $W(z) = \max(0, \lfloor W^*(z) \rfloor)$ ).
- Propensity Score Adjustment: To mitigate regularization-induced confounding, the instrument propensity score $e(X)$ is estimated via a separate BART model and included as a covariate in the main models.
- Identification Assumptions: The approach relies on standard IV assumptions (conditional unconfoundedness, overlap, monotonicity, exclusion restriction) and an additional assumption (conditional independence of potential parities given covariates) to identify the joint distribution of $\{W(0), W(1)\}$ . External validity (generalizing to a target population) is achieved via a scaled Bayesian bootstrap to re-weight results to the target population's covariate distribution.
Heterogeneity Summarization: To interpret complex CATEs, the authors fit a "surrogate shallow tree" to the posterior mean of individual CATEs, identifying subgroups with homogeneous effects.

Key Contributions

Methodological Extension: The paper presents the first adaptation of principal stratification with BART to multivalued nonnegative integer (count) intermediate variables, bridging the gap between binary and continuous implementations.
Flexible IV Estimation: The method accommodates covariate-dependent instrument validity and nonlinear treatment effects without relying on restrictive parametric forms (e.g., linearity) required by 2SLS.
Software Implementation: The approach is implemented in the R package PrinceBART.

Simulation Results
The authors conducted simulations comparing PrinceBART against conventional and flexible IV estimators (2SLS additive/interacted/binned, DML-PLIV, $\kappa$ -weighting) under data-generating processes featuring nonlinear confounding, heterogeneous treatment effects, and weak overlap.

Performance: PrinceBART demonstrated substantially lower Mean Squared Error (MSE) than all competitors, particularly in settings with small sample sizes, covariate-dependent compliance, and heterogeneous effects.
Stability: While competitors like 2SLS-binned and DML-PLIV showed improvements over additive 2SLS in specific designs, they lacked the stability of PrinceBART across all scenarios.
Coverage: PrinceBART provided conservative and valid uncertainty quantification, with coverage rates consistently at or near 1.00, whereas other methods occasionally fell below nominal levels under weak overlap.

Empirical Application
The method was applied to Demographic and Health Survey (DHS) data from Nigeria, Senegal, and Kenya to estimate the effect of an additional child on women's employment using infecundity as an instrument.

Average Effects:
- Nigeria: A significant negative average effect was found; an additional child reduces the probability of employment by approximately 6.3 percentage points among affected women.
- Senegal and Kenya: No clear average effect was detected; estimates were statistically indistinguishable from zero.
- These findings were consistent with a covariate-adjusted 2SLS benchmark, though the simulations suggest 2SLS may not be robust in all contexts.
Heterogeneity: Strong evidence of effect heterogeneity was found across all three countries. The employment penalty of an additional child is concentrated among younger and less-educated women.
- In Nigeria, the sharpest penalty was among non-Muslim women under 20.
- In Kenya and Senegal, the most pronounced declines were among women under 18 or those with no education.
- Older or more-educated women often showed no negative impact or even slight positive effects.
Generalization: Re-weighting to the full population (PATE) yielded results similar to the affected population (MATE), suggesting the covariate distribution differences between the instrument-affected group and the general population did not materially alter the conclusions.

Significance and Claims
The paper claims that by combining principal stratification with Bayesian nonparametric regression, researchers can estimate the causal effects of count-valued treatments (like fertility) while flexibly handling covariate-dependent instrument validity and uncovering complex effect heterogeneity.

The authors caution that while 2SLS provided consistent results in this specific application, the simulation evidence suggests that relying on standard linear IV specifications can be risky in settings with nonlinear confounding and heterogeneous effects.
The study highlights that the "cost" of an additional child is not uniform; it is heavily concentrated among vulnerable subgroups (younger, less-educated women), a nuance that standard aggregate approaches would obscure.
The authors acknowledge limitations, specifically the reliance on cross-sectional data and the assumption of conditional unconfoundedness, noting that unobserved confounders (e.g., household wealth) could still bias results. They suggest that longitudinal data would strengthen future evaluations.

Technical Summary: Principal Stratification with Bayesian Additive Regression Trees for Count-Valued Intermediate Variables

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