Penalized Inverse-Variance Weighted (pIVW) Method for Mendelian Randomization

The penalized inverse-variance weighted (pIVW) estimator is a Mendelian randomization method for estimating the causal effect of an exposure variable on an outcome of interest based on summary-level GWAS data. The pIVW estimator accounts for weak instruments and balanced horizontal pleiotropy simultaneously.

Setup

Use the following command in R to install the package (The latest version v0.1.3 updated on April 26, 2024):

library(devtools)
install_github("siqixu/mr.pivw",ref="main") 

Usage

mr_pivw(Bx, Bxse, By, Byse, lambda = 1, over.dispersion = TRUE, delta = 0, sel.pval = NULL, Boot.Fieller = NULL, n.boot = 1000, alpha = 0.05)

Bx: A numeric vector of beta-coefficient values for genetic associations with the exposure variable.

Bxse: The standard errors associated with the beta-coefficients Bx.

By: A numeric vector of beta-coefficient values for genetic associations with the outcome variable.

Byse: The standard errors associated with the beta-coefficients By.

lambda: The penalty parameter in the pIVW estimator. It plays a role in the bias-variance trade-off of the estimator. It is recommended to choose lambda=1 to achieve the smallest bias and valid inference. By default, lambda=1.

over.dispersion: Should the method consider overdispersion (balanced horizontal pleiotropy)? Default is TRUE.

delta: The z-score threshold for IV selection. delta should be greater than or equal to zero. By default, delta=0 (i.e., no IV selection will be conducted).

sel.pval: A numeric vector containing the P-values of the SNP effects on the exposure, which will be used for the IV selection. sel.pval should be provided when delta is not zero.

Boot.Fieller: If Boot.Fieller=TRUE, then the P-value and the confidence interval of the causal effect will be calculated based on the bootstrapping Fieller method. Otherwise, the P-value and the confidence interval of the causal effect will be calculated from the normal distribution. By default, Boot.Fieller=TRUE when Condition is smaller than 10 (see 'Details' in R Help Documentation), and Boot.Fieller=FALSE otherwise.

n.boot: The number of bootstrap samples used in the bootstrapping Fieller method. It will be used only when Boot.Fieller=TRUE. By default, n.boot=1000. A larger value of n.boot should be provided when a more precise P-value is needed.

alpha: The significance level used to calculate the confidence intervals. The default value is 0.05.

Value

Over.dispersion: TRUE if the method has considered balanced horizontal pleiotropy, FALSE otherwise.

Boot.Fieller: TRUE if the bootstrapping Fieller method is used to calculate the P-value and the confidence interval of the causal effect, FALSE otherwise.

N.boot: The number of bootstrap samples used in the bootstrapping Fieller method.

Lambda: The penalty parameter in the pIVW estimator.

Delta: The z-score threshold for IV selection.

Estimate: The causal point estimate from the pIVW estimator.

StdError: The standard error associated with Estimate.

CILower: The lower bound of the confidence interval for Estimate.

CIUpper: The upper bound of the confidence interval for Estimate.

Pvalue: P-value associated with Estimate.

Tau2: The variance of the balanced horizontal pleiotropy. Tau2 is calculated by using all IVs in the data before conducting the IV selection.

SNPs: The number of SNPs after IV selection.

Condition: The estimated effective sample size. It is recommended to be greater than 5 for the pIVW estimator to achieve reliable asymptotic properties.

Example

library(mr.pivw)  # load the mr.pivw package
mr_pivw(Bx = Bx_exp, Bxse = Bxse_exp, By = By_exp, Byse = Byse_exp)  # analyze the example data with the pIVW method. 

# results 
Penalized inverse-variance weighted method

Account for over-dispersion: TRUE 
CI and P-value: Normal approximation 
Penalty parameter (lambda): 1 
IV selection threshold (delta): 0 
Number of variants: 1000 

-----------------------------------------------------------
Method Estimate Std Error  95% CI       p-value Condition
  pIVW    0.560     0.291 -0.011, 1.131  0.0545    10.071
-----------------------------------------------------------  

Reference

Xu S., Wang P., Fung W.K. and Liu Z. (2022). A Novel Penalized Inverse-Variance Weighted Estimator for Mendelian Randomization with Applications to COVID-19 Outcomes. Biometrics. doi:10.1111/biom.13732