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Summary - inference of causal metabolite networks

13 Aug 2023

A summary of: Chen, S., Lin, Z., Shen, X., Li, L., & Pan, W. (2023). Inference of causal metabolite networks in the presence of invalid instrumental variables with GWAS summary data. Genetic Epidemiology, 1–15, https://doi.org/10.1002/gepi.22535, (Chen et al., n.d.).

Briefly, this paper is about using instrumental variables (SNPs) in causal inference with applications to genome-wide association studies (GWAS).

  • Exposure:
    • Metabolites: They serve as intermediate phenotypes connecting genetic variants to clinical outcomes and play a crucial role in biological processes.
  • Instrumental Variable (IV):
    • SNPs (Single Nucleotide Polymorphisms): These are used as instrumental variables to isolate the variability in metabolites that is independent of confounders.
  • Outcome:
    • Clinical outcomes: These are affected by metabolites, which in turn are influenced by SNPs. The causal effect of metabolites on clinical outcomes is what the study aims to determine.

For a more gradual background jump down to the “In context” section.

Abstract summary

Structural equation models (SEMs) for inferring causal networks in metabolites and other complex traits. The method:

  1. Performs causal analysis to discover relationships among multiple traits.
  2. Accounts for potential invalid IVs.
  3. Allows for data analysis using only GWAS summary statistics.
  4. Considers bidirectional relationships between traits.

The approach uses a stepwise selection to identify invalid IVs, and demonstrates its superior performance using both real and simulated GWAS data.

Key methods summary

2.1 One-sample data

For one-sample GWAS individual-level data:

  • For each of \(n\) individuals:
    • We have \(p\) SNPs as IVs and \(M\) traits.
    • Traits are denoted by an \(n x M\) matrix \(Y = [y_1, y_2, ... , y_M]\).
      • Where \(y_i = (y_1i, y_2i, ..., y_ni)^T\) is the vector of \(n\) observations for trait \(i = 1, 2, ..., M\).
    • IVs are denoted as \(Z = [z_1, ... , z_p]\).
      • Where \(z_j = (z_1j, ..., z_nj)^T\) is the vector of the \(n\) observations for IV \(j = 1, 2, ..., p\).

2.2 SEM with individual-level data

  • SEM is used as a tool for multivariate causal inference in this framework.
  • The analysis is based on one-sample GWAS individual-level data.
  • The system considers linear equations for the ‘n’ individuals in the sample.
  • Notations follow Wang et al. 2016.
  • The model has random errors denoted by a ‘n x M’ matrix.
  • The matrix ‘E’ consists of vectors for the ‘n’ random errors for each trait from 1 to M.
  • The assumption is made that the expected value of error ‘e_i’ is 0.
  • For each trait, the error follows a normal distribution with mean 0 and variance ‘σ_i^2’.
  • The matrix ‘M x M’ denotes the coefficients for the traits, represented by ‘Γ’.
  • The coefficients for each trait ‘i’ range from 1 to M.
  • The linear model for the ‘i-th’ trait is represented by an equation where traits are connected linearly and the sum is 0.

2.3 SEM with some invalid IVs

  • Discusses the scenario when some IVs may be invalid.
  • Violation of one or more of the three valid IV Assumptions (A1)–(A3).
  • Proposes a method to account for invalid IVs in SEM.
  • For trait i, the first \({p}_{0i}\) IVs are considered invalid IVs.
  • Represents the matrix of invalid IVs with \({Z}_{ {\mathscr{A}}_{i}}=[{z}_{1},{\rm{\ldots }},{z}_{p_{0i}}]\).
  • \({z}_{j}={({z}_{1j},{\rm{\ldots }},{z}_{nj})}^{T}\) is the vector for the n observations of IV \(j=1,2,{\rm{\ldots }},{p}_{0i}\).
  • \(M\) vectors are present for the coefficients of invalid IVs: \({B}_{1},{\rm{\ldots }},{B}_{M}\).
  • \({B}_{i}={({\beta }_{1i},{\beta }_{2i},{\rm{\ldots }},{\beta }_{p_{0i}i})}^{T}\) represents the direct or (horizontal) pleiotropic effects of the invalid IVs on trait \(i=1,2,{\rm{\ldots }},M\).
  • The linear SEM for trait i is represented as: \({y}_{1}{\gamma }_{1i}+\cdots +{y}_{i}{\gamma }_{ii}+\cdots +{y}_{M}{\gamma }_{Mi}+{z}_{1}{\beta }_{1i}+\cdots +{z}_{p_{0}i}{\beta }_{p_{0i}i}+{e}_{i}=0,\).
  • \(\gamma\)’s and \(\beta\)’s are unknown parameters in the model.

Code

onesample mvstepIV

The original source code is here: https://github.com/chen-siyi7/one-sample-stepwise-IV-selection/blob/main/one-sample%20stepwise%20IV%20code.R

The function onesample_mvstepIV conducts one-sample stepwise IV.

Input Parameters:

  • p: Total number of predictors.
  • R: Correlation matrix.
  • betaZX: Regression coefficients for predictors.
  • betaZY: Regression coefficients for outcomes.
  • se_betaZY: Standard error of betaZY.
  • n: Sample size.
  • gamma_hat: Gamma hat values (prior information).

Main Computations:

  1. Initialize ZTZ using R as: \(ZTZ = R\)
  2. Compute ZTY as the element-wise product of ZTZ diagonal and betaZY: \(ZTY = \text{diag}(ZTZ) \times \beta{ZY}\)
  3. Calculate the median of YTY for each predictor SNP as: \(YTY[SNP] = (n-1) \times ZTZ[SNP,SNP] \times (se_\beta{ZY}^2)[SNP] + ZTY[SNP] \times \beta{ZY}[SNP]\), excluding NA values.
  4. Compute Bayesian Information Criterion (BIC) for each predictor. For each predictor i:
    • a. Create a matrix test11 with diagonal element i set to 1.
    • b. Create matrix W1 by combining test11 and gamma_hat.
    • c. Solve for W1 using: \(\text{solve.W1} = W1^T \times ZTZ \times W1\)
    • d. Compute beta estimates beta1 as: \(\beta1 = (solve.W1^{-1} \times W1^T \times ZTY)\)
    • e. Calculate BIC as: \(testbic[i] = n \times \log(YTY - \beta1^T \times W1^T \times ZTY) + \log(n) \times \sum_{i} \text{diag}(test11)\)
  5. Determine the optimal instrument variables (IVs) based on BIC:
    • a. For each iteration j, select the predictor i with the smallest BIC.
    • b. Repeat the process by adding one predictor at a time.
    • c. Stop if the current and previous IV are the same.
  6. Compute the final beta estimates using the invalid IVs:
    • a. Extract the invalid IVs from whichIV and set their diagonal elements in test11 to 1.
    • b. Compute beta estimates beta1 as: \(\beta1 = (solve.W1^{-1} \times W1^T \times ZTY)\)
    • c. Calculate variance of beta Varbeta as: \(\text{Var\beta} = \text{diag}(solve.W1 \times n) \times \sigma_u2\) where \(\sigma_u2 = YTY - \beta1^T \times W1^T \times ZTY\).

Output:

  • invalidIV: Indices of invalid IVs.
  • beta_est: Estimated beta values.
  • beta_se: Standard error of beta estimates.
  • K: Number of invalid IVs.

onesample mvstepIV ind

The original source code is here: https://github.com/chen-siyi7/one-sample-stepwise-IV-selection/blob/main/onesample_mvstepIV_ind%20code.R

The function onesample_mvstepIV_ind performs one-sample stepwise IV for independent SNPs.

Input Parameters:

  • Y: Response variable.
  • Z: Predictor matrix.
  • n: Sample size.
  • gamma_hat: Gamma hat values (prior information).

Main Computations:

  1. Initialize testbic for Bayesian Information Criterion.
  2. For each predictor i:
    • a. Initialize a zero vector l with length dim(Z)[2] and set the ith element to 1.
    • b. Modify matrix Z22 such that for each row j, Z22[j,] is Z[j,]*l.
    • c. Perform a linear regression (lm_stage2) of Y on Z22 and Z*gamma_hat.
    • d. Calculate BIC for this predictor using: \(testbic[i] = n \times \log\left(\frac{\sum(lm\_stage2\text{residuals}^2)}{n}\right) + \log(n) \times \sum(l)\)
  3. Determine the optimal instrument variables (IVs) based on BIC:
    • a. For each iteration j, select the predictor i with the smallest BIC.
    • b. Modify matrix Z22 for the selected predictors and add one predictor at a time.
    • c. Repeat the linear regression and calculate BIC as in step 2.
    • d. Stop if the current and previous IV are the same.
  4. Extract the invalid IVs, which.invalid, from whichIV and sort them to obtain K.
  5. Compute the final beta estimates using the invalid IVs:
    • a. Extract columns K from Z to form Z22.
    • b. Perform linear regression (lm_stage2) of Y on Z22 and Z*gamma_hat.
    • c. Calculate beta estimates betaest as: \(\beta{est} = \text{summary}(lm\_stage2)\text{coef[,1]}\)
    • d. Calculate variance of residuals sigma_u2 as: \(\sigma_u2 = \frac{\sum(lm\_stage2\text{residuals}^2)}{n}\)
    • e. Calculate variance of beta Varbeta using: \(\text{Varbeta} = \text{diag}(ginv(X^TX)) \times \sigma_u2\) where \(X = \text{cbind}(Z22, Dhat)\) and \(Dhat = Z*gamma\_hat\).
    • f. Compute the standard error betase as: \(\beta{se} = \sqrt{\text{Varbeta}}\)

Output:

  • beta_est: Estimated beta values.
  • beta_se: Standard error of beta estimates.
  • invalid IVs: Indices of invalid IVs.
  • no. of invalid IV: Number of invalid IVs.

In context

Recap:

  • Exposure:
    • Metabolites: They serve as intermediate phenotypes connecting genetic variants to clinical outcomes and play a crucial role in biological processes.
  • Instrumental Variable (IV):
    • SNPs (Single Nucleotide Polymorphisms): These are used as instrumental variables to isolate the variability in metabolites that is independent of confounders.
  • Outcome:
    • Clinical outcomes: These are affected by metabolites, which in turn are influenced by SNPs. The causal effect of metabolites on clinical outcomes is what the study aims to determine.

Background

In GWAS, associations are generally sought between single nucleotide polymorphisms (SNPs) and a single trait. But GWAS data can also be used to analyze multiple related traits, leading to improved power and new biological insights. Specifically, network analysis of multiple traits is gaining interest, especially when it comes to causal network analysis. This is pivotal for elucidating relationships among multiple traits, such as in gene network and protein network analyses. Metabolite network analysis, the focal point of this research, posits that metabolites are integral parts of many biological processes, often interacting with each other in regulatory networks. By inferring these networks, we can gain insight into relationships among metabolites in biological processes.

In causal networks, traits, including metabolites, proteins, and genes, serve as the nodes. Their causal relationships are represented by directed edges connecting them. SNPs are utilized as instrumental variables (IVs). To model these intricate biological networks, structural equation models (SEMs) have been adopted.

What is an Instrumental Variable (IV)?

An instrumental variable is associated with the exposure but does not have a direct association with the outcome, except through its relationship with the exposure. Its role is to isolate the variability in the exposure that is independent of the confounders.

Key Assumptions

  • Relevance: The IV is correlated with the exposure.
  • Exclusion: The IV only affects the outcome through its effect on the exposure.
  • Exchangeability: The IV is not associated with unobserved confounders.

How Does It Work?

IV analysis uses the variation in the exposure explained by the instrument to estimate the causal effect of the exposure on the outcome.

Why is it Necessary for Causal Inference?

  • Control for Unmeasured Confounding: IVs can provide unbiased estimates of causal effects when unmeasured or unobserved confounding is present.
  • Endogeneity: IVs can solve the problem of endogeneity.
  • Natural Experiments: IVs can be employed in “natural experiments” where random assignment of treatments is not feasible.

Usage

In the context of GWAS and metabolite network analysis, IV methods are crucial. They help determine causal relationships in complex biological processes, especially when metabolites, which do not function in isolation, interact within metabolite regulatory networks.

Limitations

  • Weak Instruments: Weak correlation between the IV and exposure can lead to biased IV estimates.
  • Violations of Assumptions: IV estimates can be biased if any core assumptions are violated.
  • Interpretability: The causal effect estimated through IV is often specific to a particular population, reducing generalizability.

Summary

Instrumental Variables are a pivotal tool in causal inference, especially in genome-wide association studies (GWAS). When utilized properly, they can provide valuable insights into causal relationships in settings laden with confounding and endogeneity. However, they come with their own assumptions and potential limitations.

References

  1. Chen, S., Lin, Z., Shen, X., Li, L., & Pan, W. Inference of causal metabolite networks in the presence of invalid instrumental variables with GWAS summary data. Genetic Epidemiology, n/a(n/a). https://doi.org/https://doi.org/10.1002/gepi.22535
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