First, to introduce you to my situation, I have a dataset containing n = 16 observations and p = 17 variables. My variable set contains 16 independent variables (14 variables I'm interested in and two serving as control variables) and one outcome variable. I want to perform a regression analysis to see which of the 14 independent variables is statistically significant in regards to the outcome variable.
The known problem hereby is that p > n, which makes OLS regression unusable. During research I went into the topic of regularization, in particular Shrinkage methods (e.g. Ridge/Lasso regression) or Dimension reduction, as these alternatives tackle the problem of having too little observations for too many variables. But these alternatives usually are used to model fitting or predictions, which I'm not quite focused on in my scenario.
In James et al. (2013) book An Introdcution to Stastitical Learning, they mention in a p > n setting (High-Dimensional) the problem with multicollinearity is severe, as it increases variances of coefficients, thus making usual statistical inference (tests, R² statistics, etc.) not applicable, even when applying Ridge, Lasso or other similar methods in high-dimensional setting.
Researching here on old posts, some posts (e.g. Ridge regression in R with p values and goodness of fit) show that there are methods to calculate p-values from Ridge or Lasso for individual significance of the variables, but other posts https://stats.stackexchange.com/q/276266 are neglecting the idea.
I'm confused now whether or not I can use Ridge or Lasso for my problem. Furthermore if not, are there other methods to get results for individual variable significance in a highdimensional setting, which I haven't thought of? I'm thankful for any advice I can get.