Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?

Abstract: We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates $X$ and an outcome $Y$ under different experimental conditions (environments) but do not observe them jointly; we either observe $X$ or $Y$. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument-covariate sample and prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via $\ell_1$-regularized estimation and post-selection refitting.

• The paper introduces a novel GMM-type estimator that remains consistent across many experimental conditions despite limited paired observations.
• It leverages instrumental variable regression with ℓ1-regularization to effectively manage high-dimensional data and sparse causal effects.
• The framework mitigates hidden confounding and measurement error biases, making it valuable for biomedical applications with unpaired data.

"Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?" (2601.15254)

Introduction

The paper addresses the challenge of estimating causal effects in an environment where data are available under multiple experimental conditions, but observations are unpaired. Specifically, covariates $X$ and outcomes $Y$ are not jointly observed; rather, one might observe $X$ or $Y$, but not both simultaneously. This setting poses significant challenges due to hidden confounding and limited paired data, a problem often encountered in biological experiments where multiple experiments are conducted but repetitions per experiment are sparse.

Methodology

The authors leverage instrumental variable (IV) regression methods by treating the experimental environment as an instrument. In scenarios with many environments and few observations per environment, classical two-sample IV estimators may fail to be consistent due to the lack of paired data. The paper introduces a novel GMM-type estimator that remains consistent as the number of experimental conditions grows, despite a constant number of samples per condition. Sparse causal effects are targeted by extending the approach through $\ell_1$-regularized estimation and refitting post-selection to enhance efficiency.

Identifiability and Estimation

Identifiability

The paper emphasizes the concept of identifiability within high-dimensional and sparse settings. Under the assumption that the rank of the covariance between the instrument and covariates allows identification, the causal effect is deemed identifiable. Two primary scenarios are discussed:

• Finite-Dimensional Instruments: When the rank of the instrument's covariance suffices, identifiability follows traditional models.
• High-Dimensional Instruments: In more complex scenarios, identifiability hinges on conditions like the limit of scaled covariance matrices.

Estimation Techniques

The proposed GMM-type estimator is designed to function optimally with cross-validation techniques, such as sample splitting to address measurement error biases inherent in high-dimensional settings. The challenges associated with estimation error, especially with high-dimensional data, are managed through a combination of biased correction and sparsity assumptions, allowing for effective estimation even when data are unpaired.

Figure 1: Setting 1 (finite-dimensional instruments; sparse $\beta^*$). We compare TS-IV, (with $\ell_1$ regularization), and (with $\ell_1$ regularization) on data with finite-dimensional instruments and a sparse $\beta^*$. Both and are consistent, whereas the estimation error of TS-IV does not vanish even at large sample sizes.

Applications and Implications

The methodology is particularly well-suited for applications in biomedical domains where large-scale molecular data are collected under various causal regimes. Such data collection is often unpaired due to experimental limitations, such as the destruction of samples during data acquisition. The paper suggests its approach is relevant for applications involving the estimation of intervention effects on observable phenotypes under different experimental conditions. It holds potential for enhancing causal inference in high-throughput biological experiments, such as those involving gene expression or chemical compound screenings.

Figure 2: Setting 2 (high-dimensional instruments; dense $\beta^*$). In the high-dimensional instrument regime, the naive plug-in denominator induces a persistent measurement-error bias in two-sample IV, so both TS-IV and remain asymptotically biased.

Conclusion

The research presented introduces a robust framework for causal inference under the constraints of unpaired data and sparse observational frequencies. The proposed methodologies address limitations of traditional IV methods while extending applicability in high-dimensional settings through sparsity-induced assumptions. This work expands the boundaries of causal inference methodologies, making it feasible to draw consistent inferences under challenging data collection scenarios commonly found in modern experimental and observational sciences. In the future, this framework could be adapted to even broader contexts of causal analysis, potentially encompassing more complex networks of interrelated variables or extending into fully unobserved confounding structures.