# π§¬ nonlinear-causal ο

nonlinear-causal is a Python module for nonlinear causal inference, including hypothesis testing and confidence interval for causal effect, built on top of instrument variables and Two-Stage least squares (2SLS).

## Modelsο

nonlinear-causal considers two instrument variable causal models:

Illustrated by the above image example, letβs denote $$\mathbf{z}$$ as the valid/invalid instrument variables (such as SNPs), $$x$$ as the exposure (such as gene expression), and $$y$$ as the outcome (such as AD).

### Two-Stage least squares (2SLS)ο

$x = \mathbf{z}^\prime \mathbf{\theta} + w, \quad y = \beta x + \mathbf{z}^\prime \mathbf{\alpha} + \epsilon,$

where $$(w,\varepsilon)$$ are the error terms independent of the instruments $$\mathbf{z}$$, however, $$w$$ and $$\varepsilon$$ may be correlated due to underlying confounders, and $$\beta\in\mathbb{R}$$, $$\mathbf{\alpha}\in\mathbb{R}^p$$, $$\mathbf{\theta}\in\mathbb{R}^p$$ are unknown parameters.

### Two-Stage Sliced Inverse Regression (2SIR)ο

$\phi(x) = \mathbf{z}^\prime \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^\prime \mathbf{\alpha} + \epsilon,$

where $$(w,\varepsilon)$$ are the error terms independent of the instruments $$\mathbf{z}$$, however, $$w$$ and $$\varepsilon$$ may be correlated due to underlying confounders, and $$\beta\in\mathbb{R}$$, $$\mathbf{\alpha}\in\mathbb{R}^p$$, $$\mathbf{\theta}\in\mathbb{R}^p$$ are unknown parameters.

Remarks

• 2SLS / 2SIR. $$\mathbf{\alpha} \neq \mathbf{0}$$ indicates the violation of the second and/or third IV assumptions. The models may not be identifiable with the presence of invalid IVs. In the literature, additional structural constraints are imposed to avoid this issue, such as $$\|\mathbf{\alpha}\|_0 < p/2$$.

• 2SIR. $$\beta$$ and $$\phi$$ are identifiable by fixing $$\|\mathbf{\theta}\|_2 = 1$$ and $$\beta \geq 0$$.

Strengths of 2SIR

• Model assumptions of 2SIR are weaker than the classical 2SLS: the model admits an arbitrary nonlinear transformation $$\phi(\cdot)$$ across $$\mathbf{z}$$, $$x$$ and $$y$$, relaxing the linearity assumption in the standard TWAS/2SLS.

• 2SIR includes 2SLS and Yeo-Johnson power transformation 2SLS (PT-2SLS) as special cases. It is worth mentioning that the proposed method remains competitive against 2SLS/PT-2SLS even if the linear assumption holds.

• The implicit linear structure in both 2SLS and 2SIR allows the use of GWAS summary data of our method, in contrast to requiring individual-level data by the other (non-linear) models.

## What We Can Do:ο

2SLS

• Estimate $$\beta$$: marginal causal effect from $$X \to Y$$

• Hypothesis testing (HT) and confidence interval (CI) for marginal causal effect $$\beta$$.

2SIR

• Estimate $$\beta$$: marginal causal effect from $$X \to Y$$

• Hypothesis testing (HT) and confidence interval (CI) for marginal causal effect $$\beta$$.

• Estimate nonlinear causal link $$\phi(\cdot)$$.

For implementation usage of nonlinear_causal, kindly refer to the provided examples and notebooks.

## Installationο

# Install the latest version nonlinear-causal in Github:
pip install git+https://github.com/nl-causal/nonlinear-causal
# or Install nonlinear-causal lib from pypi
pip install nonlinear-causal


## Simulation Performanceο

• We examine four cases: (i) $$\beta = 0$$, (ii) $$\beta = .05$$, (iii) $$\beta = .10$$, (iv) $$\beta = .15$$. Note that case (i) is for Type I error analysis, while $$\beta > 0$$ in (ii) - (iv), suggests power analysis.

• Six transformations are considered: (1) linear: $$\phi(x) = x$$; (2) logarithm: $$\phi(x) = \log(x)$$; (3) cube root: $$\phi(x) = x^{1/3}$$; (4) inverse: $$\phi(x) = 1/x$$; (5) piecewise linear: $$\phi(x) = xI(x\leq 0) + 0.5 x I(x > 0)$$; (6) quadratic: $$\phi(x) = x^2$$.

For more information, please check our paper (Section 3) or the Jupyer Notebook for the simulation examples.

## Referenceο

If you use this code please star π the repository and cite the following paper:

• Dai, B., Li, C., Xue, H., Pan, W., & Shen, X. (2024). Inference of nonlinear causal effects with GWAS summary data. In Conference on Causal Learning and Reasoning. PMLR.

@inproceedings{dai2022inference,
title={Inference of nonlinear causal effects with GWAS summary data},
author={Dai, Ben and Li, Chunlin and Xue, Haoran and Pan, Wei and Shen, Xiaotong},
booktitle={Conference on Causal Learning and Reasoning},
pages={},
year={2024},
rganization={PMLR}
}