When do I need something "fancier" than multiple regression? Let me describe my data, without getting TOO specific.
I have ~1000 words. They have all been rated on some dimension (i.e., "DV"). There are 30 properties that can either be present or absent in the words (i.e., "X1":"X30").
I want to know if any properties are more common in words that are higher (or lower) on the DV.
I've taken a few swings at answering this. I have run an adaptive LASSO. I've also run a random forests model with boruta feature selection.
I was told that these two approaches may be good for prediction but not so good for explanation. My main objective isn't to predict new data, but to understand which of my 30 properties are related to the DV.
So I just ran a plain old multiple regression. The residuals are all distributed nicely. The VIF for each predictor is also < 3.
So, is this enough? For some reason, I've gotten it in my head that it isn't, but now that I think about it, I can't explain why. I think my worry was that it is too "liberal" and that with a large enough sample some predictors are bound to be significant.
So my questions are:

*

*Is there any reason that this multiple regression isn't enough?


*Are there ways to "augment" it if not? I've thought about stepwise regression, but in all honesty I'm not sure why I would need that if there is no multicolinearity.
Edit: I will also add the detail that 69/703 pairwise correlations among my predictors are significant at p < .01.

For my money, if your goal is to understand the relationship between your predictors and the outcome, multiple regression is absolutely fine here, BUT you need to worry a bit about multiple comparisons.
You have lots of predictors. Even if none of your predictors are really related to the outcome, just by chance you would expect ~5% of them to come out as significantly associated with it in your data, using multiple regression or any maximum-likelihood method.
LASSO and related methods deal with this by finding a set of regression weights that fit your data reasonably well, while trying to keep the weights small, and having as many weights of $0$ as possible. This is great for prediction, but, for reasons I won't cover here, you can't really interpret the weights estimated using LASSO, particularly when your predictors are correlated.
An alternative way to deal with this is just to use multiple regression, identify all of the significant predictors ($p < .05$), and then, to control your false discovery rate, use something like the Benjamni-Hochberg procedure to throw out the predictors whose effects are too weak.
PS: It's worth mentioning that depending on how many observations and how many features you have, there comes a point where multiple regression no longer works, even with correction for multiple comparisons. I think $\frac{1000}{30} \approx 30$ observations per predictor is probably still fine.
