# 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 Aug 9 '17 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. – doubletrouble Aug 9 '17 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? – Juanchi Aug 9 '17 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... – Juanchi Aug 9 '17 at 21:46
• Your idea about $R^2$ and relation with estimated variances of model parameters $\beta_i$ is failing, but your problem about the soybean yields and spot severity is interesting. Could you rephrase your question focusing on that particular interest as the question ('how to solve ...')? Also provide some more information about that problem. By very rough eyeballing I'd say that your relation is mostly negative, but some of your lines are relatively horizontal (and gray) because they may be infected by noise or non-linearity. Such issues might also explain why you think $R^2$ and var are related – Sextus Empiricus Aug 10 '17 at 0:22