Linear model: comparing predictive power of two different measurement methods I'm interested in predicting Y and am studying different two measurement techniques X1 and X2. It could be for instance that I want to predict the tastiness of a banana, either by measuring how long it has been lying on the table, or by measuring the number of brown spots on the banana.
I want to know which one of the measuring techniques is better, should I choose to perform only one.
I can create a linear model in R:
m1 = lm(Y ~ X1)
m2 = lm(Y ~ X2)

Now let's say X1 is a superior predictor of banana tastiness than X2. When calculating the $R^2$ of the two models, the $R^2$ of model m1 is clearly higher than model m2. Before writing a paper on how method X1 is better than X2, I want to have some sort of indication that the difference is not by chance, possibly in the form of a p-value.
How would one go about this? How to do it when I'm using different brands of bananas and move to a Linear Mixed Effect model that incoporates banana brand as a random effect?
 A: There's a good 19th century answer in risk of neglect here. To compare two different straight-line fits, plot the data and the fitted lines and think about what you see. It's quite likely that one model will be clearly better, and that need not mean higher $R^2$. For example, it is possible that a straight-line model is qualitatively wrong in one or other case. Even better, the data and fit may suggest a better model. If the two models appear about equally good or poor, that's another answer. 
The banana example is presumably facetious here, but I would not expect straight-line fits to work well at all.... 
The inferential machinery wheeled out by others in answer is a thing of intellectual beauty, but sometimes you don't need a state-of-the-art sledgehammer to crack a nut. Sometimes it seems that anyone publishing that night is darker than day would always have some one asking "Have you tested that formally? What is your P-value?". 
A: Do a Cox test for non-nested models.
y <- rnorm( 10 )
x1 <- y + rnorm( 10 ) / 2
x2 <- y + rnorm( 10 )

lm1 <- lm( y ~ x1 )
lm2 <- lm( y ~ x2 )

library( lmtest )

coxtest( lm1, lm2 )
?coxtest

(you will find references to other test).
See also this comment and this question. In especially, consider using AIC/BIC.
