# Simple Linear Regression - High $R^2$, High P-Value

I recently ran a series of 18 simple linear regressions. Some gave me results that are easy to interpret. For example, one has an $$R^2$$ of 0.24 ($$R_{adj}^2$$ = 0.2) and a p-value of 0.025.

But others gave me results that seem to me strange and difficult to interpret. For example, one yielded an $$R^2$$ of 0.7 ($$R_{adj}^2$$ = 0.55) and a p-value of 0.17. The scatter on this regression looks perfectly fine to me, with a clear positive relationship between the varaible.

How should I interpret this?

• I wonder if this question helps you? stats.stackexchange.com/questions/50425/… Dec 21, 2021 at 15:10
• Do you have a tiny sample size? I have trouble believing that the overall F-test will give that high of a p-value for that high of an $R^2_{adj}$ unless your sample size is small.
– Dave
Dec 21, 2021 at 15:12
• @ jcken, I may be wrong, but I believe that this only applies to multivariate regressions?
– Phil
Dec 21, 2021 at 15:16
• This seems even more closely related: stats.stackexchange.com/questions/257603/… I.e., in a simple linear regression the only way to have high $R^2$ and small $t$ values (and hence large $p$-values) is small $n$. Dec 21, 2021 at 17:02
• Hint: from $R^2$ and $p$ you can compute $n.$
– whuber
Dec 21, 2021 at 19:38