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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?

Pitouille
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Phil
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  • I wonder if this question helps you? https://stats.stackexchange.com/questions/50425/what-is-the-relationship-between-r-squared-and-p-value-in-a-regression – jcken Dec 21 '21 at 15:10
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    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 '21 at 15:12
  • @ jcken, I may be wrong, but I believe that this only applies to multivariate regressions? – Phil Dec 21 '21 at 15:16
  • @Dave, yes, sample size is small. Perhaps it is too small - although what constitutes 'too small' is an interminable debate. But I have run regressions on small samples before and not got such a high p-value with such a high R^2. Hence my confusion. – Phil Dec 21 '21 at 15:17
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    This seems even more closely related: https://stats.stackexchange.com/questions/257603/high-r2-squared-and-high-p-value-for-simple-linear-regression/257634#257634 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 $n$. – Christoph Hanck Dec 21 '21 at 17:02
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    Hint: from $R^2$ and $p$ you can *compute* $n.$ – whuber Dec 21 '21 at 19:38

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