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I started reading some papers on X-wide association studies, where X can be metabolome, epigenome, etc... The authors usually describe which are the dependent and which are the independent variables of their models. Taking as an example a metabolome-wide association study (MWAS), the authors use as model the following: metabolite ~ condition. That is, each metabolite is considered the dependent variable and the condition (e.g., smoker vs. non-smoker) is considered the independent variable. I was a little bit confused, so I looked into an example study using GWAS: in this case the model can be written as condition ~ SNPs, which is what I would expect (knowing nothing about GWAS but something about Data Science) and quite the opposite from the MWAS described above.

Does the choice of the dependent variable depend on the questions asked in the study, or does it exist some sort of agreement? Moreover, regarding the GWAS model described above, I read that usually researchers fit a model for each SNP: what's wrong with fitting a model which consists of several independent variables (i.e., y ~ SNP_1 + SNP_2 + ... + SNP_N)?

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To answer the first part of your question, the dependent and independent variable of X-WAS is kind of arbitrary and dependent on the question you asked. But it gradually becomes a convention in the field after the initial name and concept are accepted by the community. For example, GWAS from the very beginning is written as condition ~ SNP, and there is no doubt about it. Later, people did something like condition ~ gene expression and called it TWAS, which is also widely accepted now. In your case, MWAS is indeed metabolite ~ condition, and the only important thing is to know the convention in the research community so people are on the same page when they talk to each other.

As for why running GWAS using single SNP. In a GWAS, it is typical to use 500K to 7M SNPs. So this is a large number. It is obvious that we cannot include all the SNP in one single model, then the question becomes how many SNPs to include in each model, and how to choose those SNPs. This is very hard to decide. Also, due to link disequilibrium, many SNPs are correlated, which means co-linearity, a thing you want to avoid in linear models. It is not clear how to correctly interpret the effect sizes in a model with many SNPs as well. Therefore, due to feasibility, simplicity, and interpretability, it might be best to perform regression on every single SNP.

Now since we are running GWAS in an SNP-by-SNP fashion, there are a lot of work that needs to be done after GWAS in order to find the true causal SNP. Many fine-mapping methods have been developed to tackle this problem.

Interestingly, there are some multi-variate GWASs methods, but they do not come in the flavor of simple linear models. Instead, sometimes complex Bayesian statistics are used, for instance.

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