# [R] Adjusted R-squared formula in lm()

Ista Zahn istazahn at gmail.com
Mon Jan 28 14:34:00 CET 2013

```Hi Nicole,

One nice thing about R is that it is often easy to see the code for
many functions. For summary.lm just type the name at the command
prompt (no brackets) to see the function definition. There you will
find

ans\$adj.r.squared <- 1 - (1 - ans\$r.squared) * ((n -
df.int)/rdf)

Best,
Ista

On Mon, Jan 28, 2013 at 6:03 AM, Nicole Janz <nicolejanz at gmail.com> wrote:
> What is the exact formula used in R lm() for the Adjusted R-squared? How can I interpret it?
>
> There seem to exist several formula's to calculate Adjusted R-squared.
>
> Wherry’s formula [1-(1-R2)·(n-1)/(n-v)]
>
> McNemar’s formula [1-(1-R2)·(n-1)/(n-v-1)]
>
> Lord’s formula [1-(1-R2)(n+v-1)/(n-v-1)]
>
> Stein 1-(n-1/n-k-1)(n-2)/n-k-2) (n+1/n)
>
> Theil's formula (found here: http://en.wikipedia.org/wiki/Coefficient_of_determination)
>
> According to the textbook Field, Discovering Statistics Using R (2012, p. 273) R uses Wherry's equation which "tells us how much variance in Y would be accounted for if the model had been derived from th. population from which the sample was taken". He does not give the formula for Wherry. He recommends using Stein's formula (by hand) to check how well the model cross-validates.
> Kleiber/Zeileis, Applied Econometrics with R (2008,p. 59) claim it's "Theil's adjusted R-squared" and don't say exactly how its interpretation varies from the multiple R-squared.
>
> Dalgaard, Introductory Statistics with R (2008, p.113) writes that "if you multiply [adjusted R-squared] by 100%, it can be interpreted as '% variance reduction'. He does not say to which formula this corresponds.
>
> I had previously thought, and read widely, that R-squared penalizes for adding additional variables to the model. Now the use of these different formulas seems to call for different interpretations?
>
> My two questions in short: Which formula is used by R lm()? How can I interpret it?
>
> Thank you!
>
>
>
>
> Nicole Janz, PhD Cand.
> Lecturer at Social Sciences Research Methods Centre 2012/13
> University of Cambridge
> Department of Politics and International Studies
> www.nicolejanz.de | nj248 at cam.ac.uk | Mobile: +44 (0) 7905 70 1 69 4
> Skype: nicole.janz
>
>
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