Using GCV to tune ridge parameter: alternatives for small number of data points

Question

I have previously used ridge regression, with suitable ridge parameter $\lambda$ tuned by generalized cross validation, to fit a dataset of around 100 points to around 10 predictor variables (with strong correlations between predictors). It worked well.

I am currently trying the same thing on a smaller data set, around 30 points and 6 predictors. The out-of-set prediction from GCV tuned ridge regression is always poor (r2<0.05). I have tried cutting the model down to 2 not-strongly-correlated predictors, still a poor result.

For the bivariate model (one predictor) out of set prediction is reasonable (r2=0.5). This leads me to think that there are enough data points to use GCV, but not enough data points to tune a ridge parameter using GCV.

Is there any way to estimate the minimum number of data points I need?

Also if I can't get any more data what alternatives are open to me?

Answer 1

To follow up on this, it turned out cv.glmnet in R was failing to automatically find the optimal value of lambda. If I manually set the range of lambda I could find it, and the model fitted just fine with cross-validated $r^2 \approx 0.5$.

So, there were enough data points after all.