Goal. Infer the causal effect from exposure to outcome
Issues. Simple regression?
unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.
Goal. Infer the causal effect from exposure to outcome
Issues. Simple regression?
unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.
Source: Howell et al. (2018)
Goal. Infer the causal effect from exposure to outcome
Issues. Simple regression?
unbiased estimation of the causal effect is possible without explicitly enumerating all confounders.
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)
\( x = \mathbf{z}^T \mathbf{\theta} + w, \qquad y = \beta x + \mathbf{z}^T \mathbf{\alpha} + \varepsilon. \) (1)
\((w, \varepsilon)\) 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 \(\beta\)
\( x = \mathbf{z}^T \mathbf{\theta} + w, \qquad y = \beta x + \mathbf{z}^T \mathbf{\alpha} + \varepsilon. \) (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
Kang et al (2016a, 2016b) and Guo at al. (2018)
Ref
\( x = \mathbf{z}^T \mathbf{\theta} + w, \qquad y = \beta x + \mathbf{z}^T \mathbf{\alpha} + \varepsilon. \) (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
Haavelmo (1943), Theil (1953), Kang et al (2016) and Guo et al. (2018)
Ref
Common Lab Tests Normal Ranges. (Source: Healthline)
ongevityalcohol consumption -> CAD
exercise -> immune responsive disease resistance
Component | Normal range |
---|---|
White blood cells | 3,500 to 10,500 cells/mcL |
Platelets glucose CO2 Ca+ |
150,000 to 450,000/mcL 70-99 mg/dL 23-29 mEq/L 8.6-10.2 mg/dL |
Common Lab Tests Normal Ranges. (Source: Healthline)
ongevityalcohol consumption -> CAD
exercise -> immune responsive disease resistance
Component | Normal range |
---|---|
White blood cells | 3,500 to 10,500 cells/mcL |
Platelets glucose CO2 Ca+ |
150,000 to 450,000/mcL 70-99 mg/dL 23-29 mEq/L 8.6-10.2 mg/dL |
Difficulty
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\(\phi(\cdot)\) is an arbitrary nonlinear transformation
Incorporates the classical 2SLS and PT-2SLS
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\( \beta \) is called the marginal causal effect
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\( \phi(\cdot) \) is called the nonlinear causal transformation
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\( \beta \phi(\cdot) \) is called the nonlinear causal effect
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\(\phi(\cdot)\) can also be estimated
If the model is well-specified, \(\beta \phi(\cdot) \to\) ATE
Difficulty
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
Observation
WITHOUT estimating \( \phi(\cdot) \) !!!
Once \( \hat{\mathbf{\theta}} \) is obtained ...
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
Observation
WITHOUT estimating \( \phi(\cdot) \) !!!
Once \( \hat{\mathbf{\theta}} \) is obtained ...
Suppose \( (\mathbf{z}, x, y) \) satisfy a nonlinear causal model:
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
Observation
WITHOUT estimating \( \phi(\cdot) \) !!!
Once \( \hat{\mathbf{\theta}} \) is obtained ...
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 \( \phi(\cdot) \)
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.
\( \phi(x) = \mathbf{z}^T \mathbf{\theta} + w, \quad y = \beta \phi(x) + \mathbf{z}^T \pmb{\alpha} + \varepsilon.\)
\(\phi\) can be estimated by a two-stage procedure.
Estimate \(\mathbb{E}(\mathbf{z}^T \mathbf{\theta}\mid x)\) by a non-parametric regression
\(\widehat\rho\) is est via the uncorrelatedness between \(\mathbf{z}^T \mathbf{\theta}\) and \(w\)
Six transformations are considered in the simulation:
linear: \( \phi(x) = x\);
logarithm function: \( \phi(x) = \log(x)\);
inverse function: \(\phi(x) = 1/x\)
piecewise linear function: \(\phi(x) = xI(x\leq 0) + 0.5 x I(x > 0)\)
cube root function: \(\phi(x) = x^{1/3}\);
quadratic function: \(\phi(x) = x^2\)
Empirical Type I error (\(\beta_0 = 0\)) and power (\(\beta_0 = 0.05, 0.10, 0.15\)) of the proposed nonlinear causal test for the simulated example (marginal effect inference).
Empirical Type I error (\(\beta_0 = 0\)) and power (\(\beta_0 = 0.05, 0.10, 0.15\)) of the proposed nonlinear causal test for the simulated example (marginal effect inference).
Empirical Type I error (\(\beta_0 = 0\)) and power (\(\beta_0 = 0.05, 0.10, 0.15\)) of the proposed nonlinear causal test for the simulated example (marginal effect inference).
The bar-plot of \(p\)-values of significant genes for AD by at least one method, where the \(y\)-axis represents \(-\log_{10}(p)\). The results are based on ADNI + IGAP GWAS datasets.
7 genes, including TOMM40, are only identified by Comb-2SIR. We searched these genes in GWAS results and found ALL of them have been reported to be significantly associated with AD.
APOC1: a significant gene over all methods.
APOC1: a significant gene over all methods.
BCL3: a significant gene only identified by 2SIR/Comb-2SIR.
APOC1: a significant gene over all methods.
Negative control derived from ADNI, where outcomes are permuted.
APOC1: a significant gene over all methods.
Negative control derived from ADNI, where outcomes are permuted.
More simulated examples based on different sample sizes and dimensions with:
Strength
2SIR relaxes the linear assumption underlying the relationships between \((\mathbf{z}, x, y)\).
Compatibility: The method exhibits minimal power loss when the underlying true model is linear and the same datasets are used.
Easy to use, well-documented software, more power
Weakness
Additional assumptions on instrumental variables (IVs): zz should follow an elliptical symmetric distribution; however, this issue appears to be relatively minor in TWAS, see Example 3.
Cannot use summary statistics data in Stage 1: \( \mathbf{Z}^T \mathbf{x} \)
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