# Likelihood ratio test to compare two predictions

I have two predictions from two different types of methods. "predictedHousePrices1" is a continuous variable and the output of a prediction from a RandomForest model, "predictedHousePrices2" is the output of the predict() function a RidgeRegression model. I'd like to compare which one better explains the variability in the real data. I'm wondering if a likelihood ratio test is the best way to do this, for example:

m1 <- lm(realHousePrices~predictedHousePrices1) # R^2 = 0.25
m2 <- lm(realHousePrices~predictedHousePrices2) # R^2 = 0.30


Is it correct to use a likelihood ratio test to check if an $R^2$ of .25 is "significantly" more than 0.30, for example:

m3 <- lm(realHousePrices~predictedHousePrices1+predictedHousePrices2)
library("epicalc")
lrtest(m3, m1)


Or is there a better way to do this?

Whether R² = .3 is better then .25 also depends largely on your field, substantive theory, and the variables in the models. If m1 has only one parameter but m2 has 20, the first may be preferable. If m1 misses a theoretically very relevant predictor, you may choose m2 although an increase of .05 may sound not that much.
Just to make sure: Your predictedValues is just a dummy for one or more predictors, i.e., variables, right? Because otherwise m3 does not make much sense, to me at least.
• Just to make sure: If you want to use the LR test (how and when, see above), your model statements need to include the actual variables, e.g., lm(Y~X1+X2+X3) and not the predictedValues you get from m1\$fitted.values. Because otherwise, the LR test can not correctly count the number of parameters, which you need for the dfs. – hplieninger Jul 24 '13 at 8:20