Can I use linear model on each variable to determine which variables are important? Suppose we have a n*p matrix X and a n*1 matrix Y, where n is the number of samples and p is the number of variables. p>>n. Also suppose this data is from a biology field experiment. My goal is to select the potential biomarkers (variables). 
I know there are a bunch of variable selection tools, such as stepwise procedure, criterion based methods and other methods that are nested with the algorithms (random forest, PLS, SVM etc.). But I think the aim of these methods is to build a model to make a good prediction accuracy, FOLLOWED by telling people in this model, which variables are relatively important.
In my opinion, because my goal is not to predict but to select the possible biomarkers which will be confirmed in further experiments, can I simply do linear regression between each variable and Y and see which are significant and also have a high R^2 score? 
Thank you!
I understand this is not a simple answer question. Please provide a relatively thorough description of your idea. 
 A: If your predictors (biomarkers) are colinear, univariate regressions may grossly over / underestimate effect sizes, depending on sign of the colinearity and the sign of the product of their effect sizes. This is known as Simpson's paradox, or in general as omitted-variable bias, as mentioned above. I would therefore not recommend this approach. 
I am not aware of a perfect solution for the p>>n case, and neither do I think that one exists. Yet, if the goal is to prioritize predictors for later testing, and you think effects can be well expressed by linear relationships, I would go for a regularization method such as ridge regression and lasso, and simply take the variables that come out with the strongest effects - the advantage over AIC-based model selection is less sensitivity to colinearity in the predictors (because predictors are not removed).
A: Aristotle said that, “The whole is greater than the sum of its parts.” Each simple linear regression is merely testing  a part. However, I imagine that many diseases are associated with combinations of markers (the whole). What you really care about are the combination of markers.  As a result, your algorithm may not work well because you are not testing the combination.  
