I'm conducting an analysis where the primary goal is to understand the data. The dataset is large enough for cross-validation (10k), and predictors include both continuous and dummy variables, and the outcome is continuous. Main goal was to see if it makes sense to kick out some predictors, in order to make the model easier to interpret.
My question is "which vars explain the outcome and are a 'strong enough' part of that explanation". But to select the lambda parameter for lasso, you use cross-validation, i.e, predictive validity as the criterion. When doing inference, is predictive validity a good enough proxy for the general question I am asking?
Say LASSO kept only 3 out of 8 predictors. And now I ask myself: "what effect do these have on the outcome". For example, I found a gender difference. After the lasso shrinkage, the coefficient suggests that women score 1 point higher than men. But without the shrinkage (i.e., on the actual dataset), they score 2.5 points higher.
- Which one would I take as my "real" gender effect? Going only by predictive validity, it would be the shrunk coefficient.
- Or in a context, say that I'm writing a report for people not well versed in statistics. Which coefficient would I report to them?