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I am trying to report the r-squared value for the results of a linear regression in R using the lm package. I noticed that the summary function provides two r-squared values for my equation: adjusted r-squard and multiple r-squared. In most of the analyses I have run the two values are identical, but in a small minority of results they give different values. Given this, I am trying to figure out in what situations I should report adjusted r-squared and which should I report multiple r-squared. It was my impression that one reports multiple r-squared if they have a multivariate regression, but I was not sure.

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The short answer is that you should almost always report adjusted R-squared in favor of R-squared.

The long answer is that it depends and you have many more choices available than only R-squared and adjusted R-squared. I wrote a whole paper on this topic, see https://doi.org/10.1525/collabra.343. This answer provides a nice summary: https://stats.stackexchange.com/a/451772/30495

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