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With Multiple Regression, the R-Squared gives the researcher an estimate of the explanatory power of the regression equation.

What is the equivalent for Decision Trees?

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Decision tree usually "overfit" the data (in the sense that every point is assigned to a specific class) if you don't provide them early stopping criterion when you grow the tree. However, you can use out-of-bags estimate to get a pseudo R squared.

Per example, for a regression random forest, R offers an implementation of the pseudo R squared.

Value
An object of class randomForest, which is a list with the following components:
[...]
rsq (regression only) “pseudo R-squared”: 1 - mse / Var(y)
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  • $\begingroup$ Let's say I'm trying to find the best tree... so I'm trying different stopping criteria and evaluating the resulting trees... how would I compare these various trees to select the best one? What's the standard heuristic used in research papers? $\endgroup$ – thanks_in_advance Sep 11 '15 at 23:13
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The measure of model accuracy I see most is the "confusion matrix". It's a table that shows which observations were fitted correctly and which were misclassified. Here's an example with R code. From a table of Actual vs. Predicted, you could calculate many measures like

  1. What percentage of observations did my model predict correctly?
  2. What percentage is my type I/II error?
  3. etc.

In general, however, R-sq can be calculated for most models - you can square the correlation of the observed values with the fitted values from a (tree) model. I haven't seen R-sq used with decision trees, but I've calculated it in the past for superiors who really like seeing R-sq values for any and all modeling analyses.

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