THE adjusted R^2 is [1-(1-R2)·(n-1)/(n-v-1)], which you call McNemar’s 
formula.  It was actually proposed first by Fisher in 1924.  Theil's 
formula is equal to Fisher's.

Wherry's formula, as you give it,  is correct but was proposed to estimate 
the cross-validated R2, which is different from R2.  Neither Lord nor 
Stein actually proposed their respective formulas.  They were instead 
proposed by Darlington to estimate the CVR2 but are based on a mistaken 
assumption. Neither Wherry's, Lord's, or Stein's formulas estimates what 
they had hoped to estimate, and most likely are not appropriate to your 
problem.  Browne found the correct estimator of CVR2.

R actually uses Fisher's formula but misattributes it to Wherry .  The 
adjusted-R2 is a better estimator of the population coefficient of 
determination than is R2 itself.  It has much less bias and, unlike R2, 
its expectation is not a function of v, the number of variables.   In 
particular, if the population coefficient of determination is truly zero, 
R2 can be expected to give the value v/(n-1), whereas the adjR2 will have 
an expected value of 0.





Ista Zahn <istazahn@gmail.com> 
Sent by: r-help-bounces@r-project.org
01/28/2013 08:34 AM

To
Nicole Janz <nicolejanz@gmail.com>, 
cc
r-help@r-project.org
Subject
Re: [R] Adjusted R-squared formula in lm()






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@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@cam.ac.uk | Mobile: +44 (0) 7905 70 1 69 4
> Skype: nicole.janz
>
>
>
>
>
>
>
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>
>
> ______________________________________________
> R-help@r-project.org mailing list
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> PLEASE do read the posting guide 
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

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