# Linear regression: slope (b) confidence interval and R^2

Possibly quite a basic question: I am unsure of the relationship between the confidence interval for a regression coefficient (b) and R^2. That is, is the width of the confidence interval directly related to R^2?

Regards

• This is an interesting question. Do you have any qualifiers? E.g. is the null hypothesis true? How much noise is in the data? etc etc – Demetri Pananos Feb 6 '19 at 4:33
• @DemetriPananos it looks as if the (admittedly fascinating) question is a more general one. It does seem plausible that there is some formal relationship between these two things. – James Phillips Feb 6 '19 at 12:42
• Let's take the classic linear regression model $y_i = b x_i + \epsilon_i$ with spherical errors so that $\operatorname{Var}(\epsilon) = \sigma^2 I$. Taking $X$ as given, a higher variance $\sigma^2$ of the error term will lower $R^2$ and increase your standard errors. – Matthew Gunn Feb 6 '19 at 16:55
• Considering that the regression coefficient is unitless and the units of the coefficient are (units of y) divided by (units of x), there cannot possibly be any universal relationship. At a minimum you will need to know something about the dispersion of the $x$ data. When there is more than one explanatory variable (including a constant!), all bets are off because its relationship with $x$ can play an important role too. – whuber Feb 6 '19 at 21:45
• Hi folks, thanks all for your responses. I was initially thinking with respect to simple linear regression (a single predictor). The question could be extended to ask how confidence intervals for each b map to partial eta-squared though for multiple predictors. My simple intuition was that there should be some degree of correspondence between the two. That is, narrower confidence intervals corresponding to higher R^2. Again, just my intuition though, I've not sat down to work through any of the maths. How might qualifies like those you've listed affect the answer, @Demetri Pananos? – shrub Feb 7 '19 at 5:19