# Is there a relationship between R squared and linear regression coefficients variances?

I performed a multivariate meta-analysis for a set of regression lines, that each of them represent the relationship between soybean-yield a and a fungal disease in a field experiment.

The meta-analysis uses the regression coefficient variances to weight each trial pair of coefficients.

The estimated overall regression line and the trials EBLUPs are plotted in the figure below.

The reason I am asking this question is that I want to plot the original regression lines with a color intensity according to each weight, or reliability of the trials.

So, a first shot was to use the R² o each regressions. However it would interesting to use a metric that account for both coefficient variances.

In simple linear regression, we have $y = \beta_0 + \beta_1 x + u$, where $u \sim iid\;\mathcal N(0,\sigma^2)$.

The variance of $\hat\beta_1$ and $\hat\beta_0$: $$\text{Var}(\hat{\beta_1}) = \frac{\sigma^2(1 - \frac{1}{n})}{\sum_i (x_i - \bar{x})^2}\ .$$

$$\text{Var}(\hat{\beta}_0) = {\rm Var} (\bar{Y} - \hat{\beta}_1 \bar{x}) \\$$

And

$$R^2 = 1 - \frac{SS(Res)} {SS(Tot)}$$

Is there a relationship between $\text{Var}(\hat{\beta_1}); \text{Var}(\hat{\beta_1})$ and $R^2$?

I have particular interest on giving a weight to a group of linear regressions regarding both coefficients variances: the higher the variances, the lower the weight. I did something like that idea with considering the R²s.

However if there is not any relationship between R squared and linear regression coefficients variances this plot is useless...

• Have you noticed that all three variances are expressed in different units? That suggests there is no general relationship. You might instead look into the possibility of a relationship when all variables have been standardized (which means $\hat\beta_0$ has zero variance and so can be ignored). Note, too, that as $n$ increases, $R^2$ should approach the population $R^2$ while the variance of the coefficient estimate should become vanishingly small (assuming observations are random samples of the bivariate random variable $(x,y)$), implying any such relationship must depend on $n$.
– whuber
Commented Aug 9, 2017 at 20:23
• In your formula for $R^2$ what does "Res" stand for? It seems like you mean residual, when the fitted Sum of Squares is what belongs in the numerator. Commented Aug 9, 2017 at 20:23
• Thanks @whuber and doubletrouble (I edited the typo). So, if there is no difference between them, is there a statistic that can summarize both variances? Commented Aug 9, 2017 at 21:23
• I mean, I´m trying to give a weight for several linear regressions (by groups) so those cases where the coefficients variances are higher, would have a lower weight... Commented Aug 9, 2017 at 21:46
• You should not give a different/smaller weight to certain lines just because they have a smaller estimated linear $\beta_1$ coefficient. A regression line with $\beta_1 \sim 1$ is still a line and may be a relevant line. The variance in $\beta_1$ coefficients could be relevant and should not be averaged out. I will think about you question and maybe edit it. But in the meantime, please provide more information about your meta-analysis, goals and questions. There are lot's of different angles, which I can't foresee and tackle all. Do you expect homogeneity? Then make a funnel plot. Commented Aug 10, 2017 at 15:58