Wald ratio MR estimate from pooled profile likelihood

Takes the pooled profile log-likelihood from poolLikelihoodProfiles and the instrument table to compute the MR causal estimate via the Wald ratio. The SNP-outcome association (beta_ZY) is estimated from the profile likelihood peak, and the causal estimate is beta_MR = beta_ZY / beta_ZX. For multi-SNP analyses, beta_ZX is the association of the exact weighted allele score used at each site, not a simple mean of the component SNP effects. Uncertainty is propagated using the delta method.

Confidence intervals are computed both from the profile likelihood (likelihood ratio method) and via the delta method (normal approximation). The likelihood- based CI is preferred as it does not assume normality.

Usage

computeMREstimate(combinedProfile, instrumentTable, ciLevel = 0.95)

Arguments

combinedProfile: Output of poolLikelihoodProfiles.
instrumentTable: Data frame. Output of getMRInstruments. Used as a fallback for beta_ZX and se_ZX values when the pooled profile does not carry a stored score definition.
ciLevel: Numeric. Confidence interval level. Default is 0.95.

Value

A list with class "medusaMREstimate" containing:

betaZY: Point estimate of SNP-outcome association (profile MLE).
seZY: Standard error of betaZY from profile curvature.
betaMR: Causal effect estimate: betaZY / betaZX.
seMR: Standard error of betaMR via delta method.
ciLower: Lower confidence limit for betaMR.
ciUpper: Upper confidence limit for betaMR.
pValue: Two-sided p-value for betaMR.
oddsRatio: Exp(betaMR) — causal odds ratio for binary outcomes.
orCiLower: Lower CI for odds ratio.
orCiUpper: Upper CI for odds ratio.
ciLevel: The confidence level used.
betaZX: The exposure effect of the fitted allele score.
seZX: The SE of betaZX used.
nInstruments: Number of instruments used.
combinedProfile: The full combined profile for plotting.

Details

Compute Mendelian Randomization Causal Estimate

Wald ratio: For a single instrument, beta_MR = beta_ZY / beta_ZX. When multiple instruments are pooled into a single allele score, the denominator must be the exposure effect of that same score: $$\beta_{ZX, score} = \sum_j w_j \gamma_j$$ where $w_j$ are the fixed score weights used at the sites and $\gamma_j$ are the SNP-exposure coefficients.

Delta method SE: se_MR = sqrt( (se_ZY/beta_ZX)^2 + (beta_ZY * se_ZX / beta_ZX^2)^2 ).

Likelihood-based CI: The region of betaZY values where the log-likelihood does not drop below peak - qchisq(ciLevel, df=1)/2. This CI is then transformed to the MR scale via division by beta_ZX.

References

Burgess, S., Butterworth, A., & Thompson, S. G. (2013). Mendelian randomization analysis with multiple genetic variants using summarized data. Genetic Epidemiology, 37(7), 658-665. doi:10.1002/gepi.21758. Open access: https://pmc.ncbi.nlm.nih.gov/articles/PMC4377079/

Examples

profiles <- simulateSiteProfiles(nSites = 3, trueBeta = 0.5)
combined <- poolLikelihoodProfiles(profiles)
#> Pooling profile likelihoods from 3 site(s)...
#> Pooling complete: 3 sites, 402 total cases, 5598 total controls.
instruments <- simulateInstrumentTable(nSnps = 5)
estimate <- computeMREstimate(combined, instruments)
#> MR estimate: beta = 1.3561 (95% CI: 0.7195, 1.9926), p = 6.42e-05
#> Odds ratio: 3.881 (95% CI: 2.054, 7.335)
print(estimate$betaMR)
#> [1] 1.356072