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 Tammy Kochel posted on Monday, February 11, 2013 - 2:31 pm
In a two-level model predicting level 1 outcome y, the standardized output returns an r-squared within and r-squared between.

My question is whether the r-squared between tells me the proportion of variance on y that is explained by the level 2 predictor and thus whether the r-squared for the complete multi-level model is the sum of the two r-squared outputs (total explained variance of y is r-squared from the within variables plus the r-squared from the between variables)?

Thank you.
 Bengt O. Muthen posted on Monday, February 11, 2013 - 5:46 pm
The sum of R-squares is not what you want because

R-square(W) = within-explained/(within-total),

R-sqauare(B) = between-explained/(between-total),

where for each level

total = explained plus residual.

If you want the total r-square for y, I think you want the numerator to be

within-explained + between-explained

and for the denominator you add the two residual terms. You can get all these quantities from the output.
 Tammy Kochel posted on Tuesday, February 12, 2013 - 1:20 pm
Thank you for this explanation. If I understand correctly, you are suggesting that I would add the estimates for the between and within R-squares for the numerator, and add together the between and within standardized estimates for the residuals?

My within R-square is .39 and my between is 0.148 (sum .538).

The standardized residual estimate (STDYX) for within = .609 and between=.852. I estimate a total R-square for the y = (.538/1.461) .368. Correct?

If that is correct, why would my total explained variance be less than the model r-square when I only have level 1 predictors? If that is not correct, what should I have done instead?

Thanks.
 Bengt O. Muthen posted on Tuesday, February 12, 2013 - 3:57 pm
No, that's not what I suggested.

Your within R-square is for instance

R_W = a/(a+b),

and your between is

R_B = c/(c+d).

Then your total R-square would be

R_T = (a+c)/(a+c+b+d).
 Tammy Kochel posted on Wednesday, February 13, 2013 - 6:43 am
Thanks, so do I used the standardized or unstandardized residual estimate for d and b?
 Bengt O. Muthen posted on Wednesday, February 13, 2013 - 2:32 pm
Unstandardized everywhere.
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