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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?

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  • $\begingroup$ How many observations do you have? $\endgroup$
    – Dave
    Jun 15, 2020 at 10:56
  • $\begingroup$ thanks, Dave, 1500 $\endgroup$
    – user330
    Jun 15, 2020 at 10:58

2 Answers 2

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user330
    Jun 15, 2020 at 15:36
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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 $R^2$.

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