How to analyze a small dataset? I have this dataset and I'm not sure how to analyze it. I threw the classical regression methods such as OLS at it and haven't achieved much success.

My response variables are Tensile.Strength, Elongation, and Compression.Set and the predictor variables are the other columns. So far, I've tried linear regression and random forest regression. Is there anything else I can try here besides polynomial fitting? I want to examine the relationships between our response variables and these predictors, but none of the methods seem to do well.
In general, I'm just confused on how to analyse this when the standard regression methods fail. Thoughts?
 A: Your problem is that most of your values are zero.  Even if this were a massive data set, you probably wouldn't find a standard relationship.  Additionally, polymers three and four only have two variables due to ties.  When you have ties like that, it functionally turns polymers three and four into categorical variables.  The size doesn't matter so much as the absence of variation.
Nonetheless, I do have a suggestion.  There is a one-way measure of association called Somers' D.  It isn't used very often, but it would probably work here.
Normal correlation tests the bidirectional association of two variables.  Somers' D only tests the association in a single direction.  It was designed to distinguish between statements like, "if it is raining, then it is cloudy," versus "if it is cloudy, then it is raining."
For the purpose of Somers' D, the association of rains and clouds should be perfect in that clouds are necessary for rain, but the converse is not true.  
You would use it here by treating the independent variables as the clouds and the dependent variables as the rain.   Do your independent variables cause the dependent ones?  It is a bivariate only tool though, so you will have to go through it pairwise.  If memory serves me, its sampling distribution quickly converges to the normal distribution, so I believe it is easy to test.  There are quite a few software packages that have it buried as a standard statistical test, but no one ever turns the feature on.
The only purpose of this suggestion is exploratory.  Even if it were reported out as significant, the most it means is that there may be some type of monotonically increasing or decreasing relationship between the variables, but with an implicitly causal relationship.  It isn't a true causal relationship but is enough to be thought of as Granger causation.
It isn't the size of the set, it is the zeros and the ties that are the problem.  You will almost definitely have no relationship at all between polymer 3, 4 and filler 3 and any dependent variable.  I could be wrong though and be surprised.
Unless there is a theoretical relationship present, you should avoid polynomial regression as your set is so small that you are guaranteed a significant model, but with no meaning.  The only reason to use a polynomial model is if there should be a polynomial relationship according to theory.
Finally, some of your columns seem to add to 100, which implies your regression matrix will not be of full rank.  You may also have a similar problem with other variables, but I didn't go in and take a close look because I am not really sure what I am looking at.  It isn't my subject area and it is past midnight.  You will need to reduce variables though.
