Package 'MRMix'

Two-sample Mendelian randomization analysis using mixture models

Description

This function conducts Mendelian randomization analysis using an underlying mixture model incorporating a fraction of the genetic instruments to have direct effect on the outcome (horizontal pleiotropy). MRMix takes GWAS summary statistics as inputs to estimate causal effects of one trait on another. For stability of the method, we recommend using summary statistics in the standardized scale: 1) For both binary and continuous traits, summary-statistics should be standardized by genotypic variance; 2) In addition, for continuous phenotype, summary-statistics should be standardized by phenotypic variance. If the data are not in the standardized scale, users may use the standardize function to standardize their data. See Details and Examples for more information.

Usage

MRMix(betahat_x, betahat_y, sx, sy, theta_temp_vec = seq(-1, 1, by =
  0.01), pi_init = 0.6, sigma_init = 1e-05, profile = FALSE)

Arguments

betahat_x	GWAS effect estimates of the exposure, recommended to be in standardized scale. Vector of length K, where K is the number of instruments (SNPs).
betahat_y	GWAS effect estimates of the outcome, recommended to be in standardized scale. Vector of length K.
sx	Standard error of betahat_x, recommended to be in standardized scale. Vector of length K.
sy	Standard error of betahat_y, recommended to be in standardized scale. Vector of length K.
theta_temp_vec	A vector of the grid search values for the causal effect theta. Default to be seq(-1,1,by=0.01). Users may adjust the grid if larger effects are possible.
pi_init	Initial value of the probability mass at the null component of the mixture model corresponding to underlying valid instruments. Default to be 0.6. See Details.
sigma_init	Initial value of the variance of the non-null component of the mixture model which corresponds to underlying invalid instruments with pleiotropic effect. Default to be 1e-5. See Details.
profile	Whether to include the profile matrix. Default to be FALSE. If TRUE, include the profile matrix in the output. See Value profile_mat for details.

Details

The algorithm searches over a grid of possible values of the causal effect theta. For each fixed theta, it fits mixture model pi0*N(0,sy^2+theta^2*sx^2)+(1-pi0)*N(0,sigma2) on the residual betahat_y-theta*betahat_x. It then chooses the value of theta that leads to the maximum pi0 as the estimate of causal effect. Summary statistics can be standardized using the standardize() function if they are estimates from linear or logistic regression. Do not use Standardize() for other models.

Value

A list that contains

theta	Estimate of causal effect. Assuming the summary statistics are standardized, theta represents increase in mean value of Y in s.d. unit of Y (for continuous outcomes) or log-OR of Y (for binary outcomes) associated with per s.d. unit increase in values of X (for continuous exposures) or values of X changing from 0 to 1 (for binary exposures).
pi0	The probability mass of the null component corresponding to the estimated theta.
sigma2	The variance of the non-null component corresponding to the estimated theta.
SE_theta	Standard error of causal effect estimate.
zstat_theta	Z-statistic for test of the causal effect estimate.
pvalue_theta	P-value from the z test for the causal effect.
profile_mat	A matrix of 3 columns containing details of the grid search. The first column is theta_temp_vec. The second and third columns are the corresponding pi0 and sigma2 values. Only returned if profile=TRUE.

References

1. Qi, Guanghao, and Nilanjan Chatterjee. "Mendelian randomization analysis using mixture models for robust and efficient estimation of causal effects." Nature Communications 10.1 (2019): 1941.

2. Qi, Guanghao, and Nilanjan Chatterjee. "A Comprehensive Evaluation of Methods for Mendelian Randomization Using Realistic Simulations of Genome-wide Association Studies." bioRxiv (2019): 702787.

Examples

data("sumstats", package = "MRMix")
# Convert summary statistics to standardized scale
data_std = standardize(sumstats$betahat_x, sumstats$betahat_y, sumstats$sx, sumstats$sy, xtype = "continuous", ytype = "continuous", sumstats$nx, sumstats$ny, MAF = NULL)
# MRMix analysis
est = MRMix(data_std$betahat_x_std, data_std$betahat_y_std, data_std$sx_std, data_std$sy_std)
str(est) # True causal effect is 0.2.
# Include profile matrix
est = MRMix(data_std$betahat_x_std, data_std$betahat_y_std, data_std$sx_std, data_std$sy_std, profile = TRUE)
str(est)

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Standard error of the MRMix estimator

Description

This function calculates the standard error of the MRMix estimator using asymptotic theory.

Usage

MRMix_se(betahat_x, betahat_y, sx, sy, theta, pi0, sigma2)

Arguments

betahat_x	GWAS effect estimates of the exposure, recommended to be in standardized scale. Vector of length K, where K is the number of instruments.
betahat_y	GWAS effect estimates of the outcome, recommended to be in standardized scale. Vector of length K.
sx	Standard error of betahat_x, recommended to be in standardized scale. Vector of length K.
sy	Standard error of betahat_y, recommended to be in standardized scale. Vector of length K.
theta	Estimate of causal effect.
pi0	The probability mass of the null component corresponding to the estimated theta.
sigma2	The variance of the non-null component corresponding to the estimated theta.

Value

The standard error of MRMix estimator.

References

Qi, Guanghao, and Nilanjan Chatterjee. "Mendelian randomization analysis using mixture models for robust and efficient estimation of causal effects." Nature Communications 10.1 (2019): 1941.

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Standardize summary statistics for MRMix analysis

Description

1) For both binary and continuous traits, this function standardizes GWAS summary statistics by genotypic variance; 2) In addition, for continuous phenotype, this function standardizes summary statistics by phenotypic variance. This function is designed for GWAS estimates from linear or logistic regression. Do not use for other models.

Usage

standardize(betahat_x, betahat_y, sx, sy, xtype, ytype, nx, ny, MAF)

Arguments

betahat_x	GWAS effect estimates of the exposure. Vector of length K, where K is the number of instruments (SNPs).
betahat_y	GWAS effect estimates of the outcome. Vector of length K.
sx	Standard error of betahat_x. Vector of length K.
sy	Standard error of betahat_y. Vector of length K.
xtype	Is the exposure a continuous or binary trait? Set to xtype="continuous" or xtype="binary". Or set to xtype="n" if exposure summary statistics do not need to be standardized.
ytype	Is the outcome a continuous or binary trait? Set to ytype="continuous" or ytype="binary". Or set to ytype="n" if outcome summary statistics do not need to be standardized.
nx	SNP-specific sample size (recommended) or total sample size of the study associated with the exposure. Vector of length K or a single number. Set to NULL if trait is binary. Summary statistics for binary traits are standardized by the genotypic variance which can be calculated using the minor allele frequency (MAF) under Hardy-Weinberg equilibrium. Hence sample size is not needed for binary traits.
ny	SNP-specific sample size (recommended) or total sample size of the study associated with the outcome. Vector of length K or a single number. Set to NULL if trait is binary for the same reason as for nx.
MAF	Minor allele frequency. Vector of length K. Set to NULL if both traits are continuous. Summary statistics for continuous traits are standardized as z statistics rescaled by sample size, hence MAF is not needed.

Details

Using the exposure X as an example: 1) For continuous phenotypes analyzed with linear regression, data are standardized by betahat_x_std=betahat_x/(sx*sqrt(nx)); sx_std=1/sqrt(nx). Note that this standardization assumes that GWAS was conducted without covariate adjustment or the covariates do not have strong effects on Y. If the covariates have strong effects on Y, set nx equal to the effective sample size, which can be approximated by N/(1-R2), where N is the sample size associated with the study for X and R2 is the R-squared for the covariates. 2) For binary phenotypes analyzed with logistic regression, data are standardized by betahat_x_std=betahat_x*sqrt(2*MAF*(1-MAF)); sx_std=sx*sqrt(2*MAF*(1-MAF)). Same formulas apply to the outcome Y.

Value

A list that contains

betahat_x_std	Standardized betahat for the exposure.
betahat_y_std	Standardized betahat for the outcome.
sx_std	Standard error of betahat_x_std.
sy_std	Standard error of betahat_y_std.

References

1. Qi, Guanghao, and Nilanjan Chatterjee. "Mendelian randomization analysis using mixture models for robust and efficient estimation of causal effects." Nature Communications 10.1 (2019): 1941.

2. Qi, Guanghao, and Nilanjan Chatterjee. "A Comprehensive Evaluation of Methods for Mendelian Randomization Using Realistic Simulations of Genome-wide Association Studies." bioRxiv (2019): 702787.

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