Detecting significant predictors out of many independent variables In a dataset of two non-overlapping populations (patients & healthy, total $n=60$) I would like to find (out of $300$ independent variables) significant predictors for a continuous dependent variable. Correlation between predictors is present. I am interested in finding out if any of the predictors are related to the dependent variable "in reality" (rather than predicting the dependent variable as exactly as possible). As I got overwhelmed with the numerous possible approaches, I would like to ask for which approach is most recommended. 


*

*From my understanding stepwise inclusion or exclusion of predictors is not recommended

*E.g. run a linear regression separately for every predictor and correct p-values for multiple comparison using FDR (probably very conservative?)

*Principal-component regression: difficult to interpret as I won't be able to tell about the predictive power of individual predictors but only about the components. 

*any other suggestions?
 A: I would recommend trying a glm with lasso regularization. This adds a penalty to the model for number of variables, and as you increase the penalty, the number of variables in the model will decrease.
You should use cross-validation to select the value of the penalty parameter.  If you have R, I suggest using the glmnet package.  Use alpha=1 for lasso regression, and alpha=0 for ridge regression.  Setting a value between 0 and 1 will use a combination of lasso and ridge penalties, also know as the elastic net.
A: To expand on Zach's answer (+1), if you use the LASSO method in linear regression, you are trying to minimize the sum a quadratic function and an absolute value function, ie:
$$\min_{\beta} \; \; (Y-X\beta)^{T}(Y-X\beta) + \sum_i |\beta_i| $$
The first part is quadratic in $\beta$ (gold below), and the second is a square shaped curve (green below). The black line is the line of intersection.

The minimum lies on the curve of intersection, plotted here with the contour curves of the quadratic and square-shaped curve:
 
You can see the minimum is on one of the axes, hence it has eliminated that variable from the regression.
You can check out my blog post on using $L1$ penalties for regression and variable selection (otherwise known as Lasso regularization).  
A: What is your prior belief on how many predictors are likely to be important? Is it likely that most of them have an exactly zero effect, or that everything affects the outcome, some variables only less than others?
And how is the health status related to the predictive task?
If you believe that only few variables are important, you may try the spike and slab prior (in the R's spikeSlabGAM package, for example), or L1. If you think all predictors affect the outcome, you may be out of luck. 
And in general, all caveats related to causal inference from observational data apply. 
A: Whatever you do, it is worthwhile getting bootstrap confidence intervals on the ranks of importance of the predictors to show that you can really do this with your dataset.  I am doubtful that any of the methods can reliably find the "true" predictors.
A: I remember Lasso regression doesn't perform very well when $n \leq p$, but I'm not sure. I think in this case Elastic Net is more appropriate for variable selection.
