# Interpretation R-squared and F statistics

Hope I get some help

I have two predictor variables and one outcome. I have used multiple regression to get estimates. I get a value of 0.005 for adjusted Squared R. I got significant results for the model. The p-value of the F-statistic is 0.003, meaning that at least, one of the predictor variables is significantly related to the outcome variable. The coefficients table shows both predictors are significant.

I do not understand what happens given a very low value of Squared R. Which one I need to consider Squared R or the coefficient table for each predictor? Can anyone explain it?

• How many observations do you have?
– Dave
Jun 15, 2020 at 10:56
• thanks, Dave, 1500 Jun 15, 2020 at 10:58

R Squared (adjusted or unadjusted) can be low even with a low F statistic p-value. Consider a simple linear regression (one regressor), which has the property that the f statistic p-value equals the t statistic p-value and, providing an intercept is included, the R squared value equals the (Pearson) correlation between the dependent variable and the regressor.

If there is a lot of unexplained variation in the regression, then a plot of the independent variable against the regressor would show wide variation of points about the line. The R squared value would be low since this is the proportion of the dependent variable that is “explained”, statistically at least, by the regressor.

R squared is a goodness of fit statistic, it is used to give an overall idea of the amount the independent variable is “explained” by the regressors.

• Thanks, Single, I know that ( sorry). But I think the issue is related to sample size. when the sample size is very large, any small difference would be significant. Jun 15, 2020 at 15:36

The F stat is:

$$F = \frac{(RSS_0 - RSS)/p}{RSS/(n - p -1)}$$

Where $$RSS_0$$ is the residual sum of squares from the intercept only model and $$RSS$$ is the residual sum of squares from the full model. $$n$$ is the sample size and $$p$$ is the number of regressors.

Asymptotically, $$RSS_0$$ and $$RSS$$ will both grow at rate $$n$$, these will cancel out. This leaves the $$n$$ in the denominator of the denominator, meaning that $$F$$ will grow at rate $$n$$.

$$R^2$$ does not grow with $$n$$, so if the sample size is large you can get an big F-stat even with very small $$R^2$$.