# Finding the true "causative" covariate - in joint versus separate modeling

This is regarding some genetic assignment. Assume we have two random covariates (SNPs) $$X1,X2$$, and a random response $$Y$$ (disease). I believe that only one of $$X1,X2$$ is “causative” for $$Y$$ , but do not know which one, and the goal is to find out that it’s $$X1$$, based on a random sample of $$n$$ examples of $$X1$$,$$X2$$,$$Y$$.

For this we have two possible strategies:

(I) Two univariate models : $$Y\sim X_{1},Y\sim X_{2}$$, and choose between them (using a $$p_{val}$$ comparison or something).
(II) Single regression model $$Y\sim X_{1}+X_{2}$$ (and choose the one that has a lower $$p$$-value or higher).

I need to justify/not justify using each strategy when:

(a) $$X_1,X_2$$ are independent (e.g., two SNPs on separate chromosomes).
(b) $$X_1,X_2$$ are positively correlated (e.g., two neighboring SNPs).

My attempt was going with the basics of linear regression, looking at

$$I:\hat{\beta}=X^{T}Y\ \ \ \ \ II:\hat{\beta}=\left(X^{T}X\right)^{-1}X^{T}Y$$

and from that I think that in case (a) there is not much of a difference and in case (b) we must use II because approach (I) doesn't takes into account the correlation between the covariates. But this is in "predicting" wise, not in "causative" wise. Thus I have trouble "formalizing" this.

## EDIT

Thank you all for the answers! but I think I'v made myself not clear.

1. The question is not a qusetion is casual inference, but in regression and modeling. I think my lecturer means more in the "associative" than "causative" in this matter.
2. We as modeleres KNOW that one of them is associative. Now assume that this is $$X_1$$. The question is what strategy will be more effienct of finding it (error and power wise).
3. This is suppose to be sound theoretical question, not practical one. An idea for simulation is also acceptable. Thank you so much!

Neither strategy in my opinion is an option. This one doesn't work:

(I) Two univariate models : $$Y∼X1$$, $$Y∼X2$$, and choose between them (using a $$p$$ value comparison or something).

Simply because a $$p$$ value alone cannot ascertain causation at all. You can get many spurious effects with $$p$$ values that have nothing to do with causation. Consider Simpson's paradox for example, where one can find a statistically significant and strong association that overlooks between clusters that has an essentially non-existent association. With respect to model comparison here, you can also have the case where one predictor is an effective control only when it is entered into the model, where these two simple regressions don't account for that. The other has some seemingly opposite problems:

(II) Single regression model $$Y∼X1+X2$$ (and choose the one that has a lower $$p$$-value or higher).

For this, you can get a low $$p$$ value from anything, and controlling for another predictor doesn't alone help. This is especially problematic if you include a collider, which can bias the model rather than account for causality. So for both of these methods, not carefully ascertaining causality (through something like a DAG) won't really alone solve the issue.

Some ways you could instead get around this are through methods that use randomization more effectively...randomized control trials (RCTs), propensity score matching, etc. These would at least attempt to model the problem explicitly rather than guesswork via $$p$$ values.

Strategy I and II work in fairly limited and unrealistic circumstances.

• Strategy I works if and only if X1 is the only cause of Y (meaning for example, there are no other unmeasured confounders).

• Strategy II only works if X1 is the only cause of Y, or X2 is a confounder for the relationship of X1 and Y and there are no other unmeasured confounders.

Furthermore, I would refrain from making any decisions on if something is a cause based on the p value. The p value is of limited use in causal inference, aside from the statistical purpose for which it was designed. To put it bluntly, statistical significance tells you very little in causal inference, and you need a lot more to claim something is a cause than just a small p value.

• Hi, see me edit. We can assume only one of them is associative to Y. In what metrics will you use besides p value (with only regression as a tool). I thought akso about taking the bigger coeeffients as metric but i think this is not a robust method... Commented Aug 5 at 14:58

RCT, as proposed in the other comments, are of course the golden standard. However, I have doubts it is a practical thing to require in your setting. Instead, given your domain knowledge, a causal graph of the DGP looks like

$$Y \leftarrow X_1 \leftarrow U \rightarrow X_2$$ or $$Y \leftarrow X_1 \rightarrow X_2$$. Alternatively, $$X_1, X_2$$ change places.

An approach would then be to test which of the 2 conditional independencies hold: either $$Y \perp X_2 | X_1$$ (thus $$X_1$$ is the cause) or $$Y \perp X_1 | X_2$$.