mr_cML function

Constrained maximum likelihood (cML) method

Constrained maximum likelihood (cML) based Mendelian Randomization method robust to both correlated and uncorrelated pleiotropy. methods

mr_cML(
  object,
  MA = TRUE,
  DP = TRUE,
  K_vec = 0:(length(object@betaX) - 2),
  random_start = 0,
  num_pert = 200,
  random_start_pert = 0,
  maxit = 100,
  random_seed = 314,
  n,
  Alpha = 0.05
)

## S4 method for signature 'MRInput'
mr_cML(
  object,
  MA = TRUE,
  DP = TRUE,
  K_vec = 0:(length(object@betaX) - 2),
  random_start = 0,
  num_pert = 200,
  random_start_pert = 0,
  maxit = 100,
  random_seed = 314,
  n,
  Alpha = 0.05
)

Arguments

• object: An MRInput object.
• MA: Whether model average is applied or not. Default is TRUE.
• DP: Whether data perturbation is applied or not. Default is TRUE.
• K_vec: Set of candidate K's, the constraint parameter representing number of invalid IVs. Default is from 0 to (#IV - 2).
• random_start: Number of random starting points for cML, default is 0.
• num_pert: Number of perturbation when DP is TRUE, default is 200.
• random_start_pert: Number of random start points for cML with data perturbation, default is 0.
• maxit: Maximum number of iterations for each optimization. Default is 100.
• random_seed: Random seed, default is 314. When random_seed=NULL, no random seed will be used and the results may not be reproducible.
• n: Sample size. When sample sizes of GWAS for exposure and outcome are different, and/or when sample sizes of different SNPs are different, the smallest sample size is recommended to get conservative result and avoid type-I error. See reference for more discussions.
• Alpha: Significance level for the confidence interval for estimate, default is 0.05.

Returns

The output from the function is an MRcML object containing:

• Exposure: A character string giving the name given to the exposure.

• Outcome: A character string giving the name given to the outcome.

• Estimate: Estimate of theta.

• StdError: Standard error of estimate.

• Pvalue: p-value of estimate.

• BIC_invalid: Set of selected invalid IVs if cML-BIC is performed, i.e. without MA or DP.

• GOF1_p: p-value of the first goodness-of-fit test.

• GOF2_p: p-value of the second goodness-of-fit test.

• SNPs: The number of SNPs that were used in the calculation.

• Alpha: Significance level for the confidence interval for estimate, default is 0.05.

• CILower: Lower bound of the confidence interval for estimate.

• CIUpper: Upper bound of the confidence interval for estimate.

• MA: Indicator of whether model average is applied.

• DP: Indicator of whether data perturbation is applied.

Details

The MRcML method selects invalid IVs with correlated and/or uncorrelated peliotropic effects using constrained maximum likelihood. cML-BIC gives results of the selected model with original data, while cML-MA-BIC averages over all candidate models. cML-BIC-DP and cML-MA-BIC-DP are the versions with data-perturbation to account for selection uncertainty when many invalid IVs have weak pleiotropic effects.

When DP is performed, two goodness-of-fit (GOF) tests are developed to check whether the model-based and DP- based variance estimates converge to the same estimate. Small p-values of GOF tests indicate selection uncertainty is not ignorable, and results from DP is more reliable. See reference for more details.

As the constrained maximum likelihood function is non-convex, multiple random starting points could be used to find a global minimum. For some starting points the algorithm may not converge and a warning message will be prompted, typically this will not affect the results.

Examples

# Perform cML-MA-BIC-DP:
mr_cML(mr_input(bx = ldlc, bxse = ldlcse, by = chdlodds,
byse = chdloddsse), num_pert=5, MA = TRUE, DP = TRUE, n = 17723)
# num_pert is set to 5 to reduce computational time
# the default value of 200 is recommended in practice

# Perform cML-BIC-DP:
mr_cML(mr_input(bx = ldlc, bxse = ldlcse, by = chdlodds,
byse = chdloddsse), MA = TRUE, DP = FALSE,, n = 17723)

References

Xue, H., Shen, X., & Pan, W. (2021). Constrained maximum likelihood-based Mendelian randomization robust to both correlated and uncorrelated pleiotropic effects. The American Journal of Human Genetics, 108(7), 1251-1269.