1 Backgroud- univariable MR

MR employs genetic variants as instrumental variables (Z) to estimate the causal effect of an exposure (X) on an outcome (Y) using observational data, even in the presence of unmeasured confounding (C).

For univariable MR to provide valid causal estimates of the effect of the exposure on the outcome, there are three assumptions that must be satisfied:


When these assumptions are satisfied, univariable MR can test for a causal effect of an exposure on an outcome without bias from unobserved confounding. With an additional assumption, univariable MR will give an estimate of the size of the causal effect of the exposure on the outcome.


The “exclusion restriction” assumption requires no pleiotropy, i.e., no other pathways from Z to Y except through X.


In practice, however, many variants are pleiotropic. For example, SNPs for lipid fractions.


2 Multivariable MR

Multivariable MR is an extension of univariableMR that allows us to estimate the causal effects of multiple exposures on an outcome. Multivariable MR estimates the “direct” causal effects of each exposure included in the estimation on the outcome, conditional on the other exposures.

There are also three assumptions:


3 Simulation

3.1 Data generating rules:

  • Genetic risk scores:

\[ \begin{bmatrix} Z1 \\ Z2 \end{bmatrix} \sim N (\begin{bmatrix} 0 \\ 0 \end{bmatrix},\,\begin{bmatrix} 1 ,0.5\\ 0.5 ,1 \end{bmatrix})\, \]

  • Confounder C:

\[ C \sim N (0,1) \]

  • Exposures X1 & X2:

\[ X1 = 0.4\times Z_1 + 0.3\times Z_1 \times Z_2 + 0.1\times Z_2 + 0.5 \times C + \epsilon_1 \\ X2 = 0.1 \times Z_1 + 0.5\times Z_2 + 0.3\times Z_2^2 + 0.5 \times C + \epsilon_2 \\ \epsilon_1 {\perp \!\!\! \perp} \epsilon_2 \sim N(0,1) \]

  • Outcome Y:

\[ Y = -0.3\times X1 + 0.5\times X2 - 0.8 \times C + \epsilon, \,\,\, \epsilon \sim N(0,1) \]

  • Other parameters: sample size n=2000, simulation iteration b=500


3.2 Models:

  • No adjustment for C:

\[ Y\sim X_1+X_2 \]

  • Adjustment for C:

\[ Y\sim X_1+X_2 + C \]

  • Multivariable MR:

\[ X_1 \sim Z_1 + Z_2 \\ X_2 \sim Z_1 + Z_2 \\ Y \sim {\hat{X}}_1 + {\hat X}_2 \]

3.3 Results:

  • No adjustment for C:

  • Adjustment for C:

  • Multivariable MR:

4 Discussion

Limitations:


Several previous studies using multivariable MR to disentangle lipids effects or effects on lipids: