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?

  • $\begingroup$ I wonder if this question helps you? stats.stackexchange.com/questions/50425/… $\endgroup$
    – jcken
    Dec 21, 2021 at 15:10
  • 1
    $\begingroup$ 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. $\endgroup$
    – Dave
    Dec 21, 2021 at 15:12
  • $\begingroup$ @ jcken, I may be wrong, but I believe that this only applies to multivariate regressions? $\endgroup$
    – Phil
    Dec 21, 2021 at 15:16
  • 3
    $\begingroup$ 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 $n$. $\endgroup$ Dec 21, 2021 at 17:02
  • 1
    $\begingroup$ Hint: from $R^2$ and $p$ you can compute $n.$ $\endgroup$
    – whuber
    Dec 21, 2021 at 19:38


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