# Instrument variables in Mendelian Randomization (MR)

As I know, in Mendelian Randomization (MR) analysis, many researchers will select SNPs which are correlated with exposure X (or Risk Factor) as Instrument Variables (IVs), but not all of these SNPs are correlated with outcome Y (Sometimes, all SNPs are not correlated with outcome).

However, it doesn't make sense to me. My opinion is that we should select SNPs which are correlated with X and Y (which implies that in univariate regression, the beat coefficient is not zero) as IVs, and then we start to analyze whether there is a causal effect of X on Y. If all IVs are not correlated with outcome, then "I think" we will only get the inference like: there is no causal effect of X on Y. (Or maybe I am wrong?)

Where do I misunderstanding?

BTW, I learn the Logic of Instrumental Variables form this video. (https://www.youtube.com/watch?v=4xF_DMbL14w&app=desktop)

## 1 Answer

No, you should select SNPs that are correlated with $$X$$ (strictly speaking, those that are correlated with $$X$$ because they are in linkage disequilibrium with variants that affect $$X$$: correlation is not enough), and that are correlated with $$Y$$ only because they are correlated with $$X$$ (the so-called exclusion restriction)

It's easier to think about if we first think about one SNP at a time and pretend the SNP genotype is binary rather than binomial, and pretend that the SNP is actually the functional variant rather than just near it.

Let $$\lambda_1$$ be the mean of $$X$$ for people with the variant allele and $$\lambda_0$$ be the mean of $$X$$ for people with the reference allele. The difference $$\lambda_1-\lambda_0$$ is the effect of the SNP on $$X$$. It's the effect, because (by assumption) the only way $$X$$ is correlated with genotype is through the effect of genotype on $$X$$.

Let $$\mu_1$$ be the mean of $$Y$$ for people with the variant allele and $$\mu_0$$ be the mean of $$Y$$ for people with the reference allele. The difference $$\mu_1-\mu_0$$ is the effect of the SNP on $$Y$$. It's the effect, because (by assumption) the only way $$Y$$ is correlated with genotype is through the effect of genotype on $$X$$ and the effect of $$X$$ on $$Y$$. The assumption that the genotype has no effect on $$Y$$ except through $$X$$ is a strong one and is critical: violations of it matter.

Under these assumptions, if the effect of $$X$$ on $$Y$$ is linear, we can estimate it by $$(\mu_1-\mu_0)/(\lambda_1-\lambda_0)$$.

Now, going back to the oversimplifications. It doesn't matter if the SNP we used is just a marker rather than the functional variant, because the impact of this on the numerator and denominator cancel.

It doesn't matter that genotype has three levels (for bi-allelic diploid SNPs) as long as the effect of SNP on $$X$$ is linear (an additive genetic model), and these only needs to be approximately true

It does potentially matter that the effect of $$X$$ on $$Y$$ is linear, and this is typically not true, but the genetic effects being studied by Mendelian randomisation are usually weak enough that linearity is a reasonable approximation (and the test for non-zero effect still makes sense).

It does seriously matter that the correlation of genotype with $$X$$ is causal. For example, there is a well-studied polymorphism in the alcohol dehydrogenase 1B gene that is correlated with using chopsticks to eat (that is: it's more common in East Asia than in other parts of the world), but the correlation is not causal and a Mendelian randomisation analysis would be invalid.

And, again, the assumption that all the effect of genotype on $$Y$$ goes through $$X$$ is important. For example, a polymorphism affecting bitter taste perception might decrease $$X$$ defined as broccoli consumption and increase $$Y$$ defined as weight, but the effect might be due to, say, increased consumption of sweet foods, and a Mendelian randomisation analysis would be bogus.

• Thank you for your reply, and it helps me a lot! but I still have a question. I agree that I should select SNPs that are correlated with X, and that are correlated with Y only because they are correlated with X. But I notice that some researchers will only select the SNPs that are correlated with X, and these SNPs are not correlated with Y ( lm(Y~SNP) ). In that case, can I say these SNPs are not instrument variables, and the results of MR analysis are not appropriate? – Amy Chang Sep 28 '20 at 6:56
• No, it could be that $X$ actually has no effect on $Y$, which is a very important possibility. Or it could be that the effects of genotype on $X$ or $X$ on $Y$ are small enough that you can't tell -- and these SNPs would still be useful as a contribution to the analysis. – Thomas Lumley Sep 28 '20 at 6:59
• Ah, that makes sense! So, in the MR analysis, it is fine that SNPs are not correlated with Y. But when we observe that SNPs are correlated with Y, we need to check that whether SNPs are not correlated with Y given X. Am I right? – Amy Chang Sep 28 '20 at 7:17
• You need to do more than check: testing isn't powerful enough. You need to have good reasons why the SNPs can't affect $Y$ except through $X$. – Thomas Lumley Sep 28 '20 at 7:27
• I see! It seems that comprehensive knowledge are required to support the assumptions in Mendelian Randomization (MR) analysis. Oh!! I appreciate your help very much!! – Amy Chang Sep 28 '20 at 7:39