Causal diagram for IV

Goal. Infer the causal effect from exposure to outcome

Issues. Simple regression?

• yields biased estimator of β (unobserved confounders)
• The promise of instrumental variables (IVs):

  • unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.

Causal diagram for IV

Goal. Infer the causal effect from exposure to outcome

Issues. Simple regression?

• yields biased estimator of β (unobserved confounders)
• The promise of instrumental variables (IVs):

  • unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.

Source: Howell et al. (2018)

Causal diagram for IV

Goal. Infer the causal effect from exposure to outcome

Issues. Simple regression?

• yields biased estimator of β (unobserved confounders)
• The promise of instrumental variables (IVs):

  • unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.

• Random allocation alleles suggests SNPs are IVs for gene testing

TWAS data types

• Controlled access to indiv. level data, e.g., GTEx dataset

• The sample size is much smaller than GWAS

TWAS accepts various forms of input data types:

     Individual-level gene expression data  + GWAS

SNPs -> Gene expression

SNPs -> Outcome

• GWAS boasts a large sample size: ukb-b (~400K)

Recall: 2SLS (with invalid IVs)

$x = \mathbf{z}^T \mathbf{\theta} + w, \qquad  y = \beta x + \mathbf{z}^T \mathbf{\alpha}  + \varepsilon.$      (1)

• $\beta\in\mathbb{R}$, $\pmb\alpha\in\mathbb{R}^p$, $\pmb\theta\in\mathbb{R}^p$ are unknown parameters

• $(w, \varepsilon)$ are correlated (confounder), and $(w, \varepsilon) \perp \mathbf{z}$ (IVs)

• $\bm\alpha\neq \mathbf{0}$ indicates the violation of IV assumptions

Goal: estimation and statistical inference on $\beta$

• application: potential causal genes for AD

Recall: 2SLS

$x = \mathbf{z}^T \mathbf{\theta} + w, \qquad  y = \beta x + \mathbf{z}^T \mathbf{\alpha}  + \varepsilon.$        (1)

By plugging the Stage 1 into the Stage 2, we obtain

$y = \mathbf{z}^T \mathbf{\theta}\beta + \mathbf{z}^T\mathbf{\alpha} + e, \quad e = w\beta + \varepsilon,\ E(e) = 0, \ E (e^2) = \sigma_e^2.$

(now, $\mathbf{z}$ is uncorrelated with $e$)

Obs

Recall: 2SLS

$x = \mathbf{z}^T \mathbf{\theta} + w, \qquad  y = \beta x + \mathbf{z}^T \mathbf{\alpha}  + \varepsilon.$        (1)

solves $\pmb{\theta}$ and $(\beta, \pmb{\alpha})$ separately based on two independent data.

$D_1 = (\mathbf{Z}_1, \mathbf{x}_1)$ with $n_1$; and $D_2 = (\mathbf{Z}^T_2 \mathbf{Z}_2, \mathbf{Z}^T_2 \mathbf{y}_2)$ with $n_2$

$\hat{\pmb \theta} = (\mathbf{Z}^T_1 \mathbf{Z}_1)^{-1} \mathbf{Z}_1^T\mathbf{x}_1$, and impute $\hat{\mathbf{x}} = \mathbf{z}^T \hat{\pmb{\theta}}$

S1

By plugging the Stage 1 into the Stage 2, we obtain

$y = \mathbf{z}^T \mathbf{\theta}\beta + \mathbf{z}^T\mathbf{\alpha} + e, \quad e = w\beta + \varepsilon,\ E(e) = 0, \ E (e^2) = \sigma_e^2.$

(now, $\mathbf{z}$ is uncorrelated with $e$)

2SLS

Obs

$\min_{\beta, \pmb{\alpha}} (\hat{\mathbf{\theta}}\beta + \mathbf{\alpha})^T \mathbf{Z}_2^T \mathbf{Z}_2 (\widehat{\mathbf{\theta}}\beta + \mathbf{\alpha}) - 2\mathbf{y}_2^T\mathbf{Z}_2 (\hat{\mathbf{\theta}}\beta + \mathbf{\alpha}), \quad  \|\pmb{\alpha}\|_0 \leq K$

S2

• $\|\cdot\|_0$ penalty can be replaced by SCAD and MCP
• 2SLS only requires summary statistics:
  • ($\mathbf{Z}^T_1 \mathbf{Z}_1, \mathbf{Z}_1^T \mathbf{x}_1$) and $(\mathbf{Z}^T_2 \mathbf{Z}_2, \mathbf{Z}^T_2 \mathbf{y}_2)$
• Identifiability conditions: majority / plurality rules

Haavelmo (1943), Theil (1953), Kang et al (2016) and Guo et al. (2018)

Ref

Nonlinear effect?

Common Lab Tests Normal Ranges. (Source: Healthline)

• U-shaped (nonlinear) causal effect
• Other examples:
  • body weight / cholesterol levels -> longevity

  • ongevityalcohol consumption -> CAD

  • exercise -> immune responsive disease resistance

Nonlinear effect?

Difficulty

• The sample size of individual data is relatively small
• GWAS can not be used to learn nonlinear pattern but you still to use it
  • It may be too "expensive" to accurately estimate the non-parametric nonlinear causal effect

Nonlinear causal model

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

• $\beta$ and $\phi$ are only identifiable up to a multiplicative scalar. Thus, we fix $\|\bm\theta\|_2 = 1$ and $\beta \geq 0$

• $\phi(\cdot)$ is an arbitrary nonlinear transformation

• Incorporates the classical 2SLS and PT-2SLS

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Interpretation

$\beta$ is called the marginal causal effect

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Interpretation

$\phi(\cdot)$ is called the nonlinear causal transformation

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Interpretation

$\beta \phi(\cdot)$ is called the nonlinear causal effect

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Interpretation

• $\beta > 0$ indicates the presence of the causal relation, and its hypothesis testing and CI are developed

• $\phi(\cdot)$ can also be estimated

• If the model is well-specified, $\beta \phi(\cdot) \to$ ATE

Difficulty

• The sample size of individual data is relatively small
• GWAS can not be used to learn nonlinear pattern but you still to use it
  • It may be too "expensive" to accurately estimate the non-parametric nonlinear causal effect

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Method

Observation

• Very similar to the single index model: $x \perp \mathbf{z} \mid \mathbf{z}^T\mathbf{\theta}$
• $\pmb{\theta}$ can be estimated via sliced inverse regression (SIR; Li (1991)) or sufficient dimension reduction (SDR; Cook (2009))

• WITHOUT estimating $\phi(\cdot)$ !!!

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Method

Observation

• Very similar to the single index model: $x \perp \mathbf{z} \mid \mathbf{z}^T\mathbf{\theta}$
• $\pmb{\theta}$ can be estimated via sliced inverse regression (SIR; Li (1991)) or sufficient dimension reduction (SDR; Cook (2009))

• WITHOUT estimating $\phi(\cdot)$ !!!

Suppose $(\mathbf{z}, x, y)$ satisfy a nonlinear causal model:

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

Method

Observation

• Very similar to the single index model: $x \perp \mathbf{z} \mid \mathbf{z}^T\mathbf{\theta}$
• $\pmb{\theta}$ can be estimated via sliced inverse regression (SIR; Li (1991)) or sufficient dimension reduction (SDR; Cook (2009))

• WITHOUT estimating $\phi(\cdot)$ !!!

Once $\hat{\mathbf{\theta}}$ is obtained ...

• Impute $\phi(x)$ as $\mathbf{z}^T \hat{\pmb{\theta}}$
• Plugging into Stage 2, solve $\beta$ via a sparse reg as in 2SLS

Inference

Consider the hypotheses:

$H_0: \beta = 0, \qquad H_1: \beta > 0$

where rejecting the null hypothesis $H_0$ indicates an evidence for causal influence of the exposure $x$ on the outcome $y$.

Define the pivotal test statistic:

$\widehat{T} = \frac{n_2^{1/2}\widehat\beta}{\widehat\sigma_e (\widehat{\mathbf \theta}^T\widehat{\mathbf \Sigma}\widehat{\mathbf \theta} - \widehat{\mathbf \theta}^T \widehat{\mathbf \Sigma}_{*A}(\widehat{\mathbf\Sigma}_{AA})^{-1}\widehat{\mathbf \Sigma}_{A*}\widehat{\mathbf{\theta}} )^{1/2}}$

where $A = \{ j : \alpha_j \neq 0 \}$, $\mathbf \Sigma_{*A},\mathbf{\Sigma_{A*}}$ denote the columns and rows of $\pmb{\Sigma}$ indexed by $A$, respectively.

Does NOT require an estimation of $\phi(\cdot)$

Misspecified nonlinearity

It is possible that the nonlinear transformation $\phi(\cdot)$ could be misspecified in practice, especially when two structural equations do not share the same transformation for the exposure:

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

where $\phi\neq \psi$ are two different nonlinear functions, hypothesis testing remains valid

Corollary 1. In the above model, with the same conditions and the same test, then the Type-I error is controlled by $\alpha$ under the null hypothesis.

$\phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.$

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