R help - Jan 2013 - Adjusted R-squared formula in lm()

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







	[[alternative HTML version deleted]]

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
>
>
>
>
>
>
>
>         [[alternative HTML version deleted]]
>
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

Nicole Janz <nicolejanz <at> gmail.com> writes:
> 
> What is the exact formula used in R lm() for the Adjusted R-squared? How
can I
interpret it?
>From the code of summary.lm():
   ans$r.squared <- mss/(mss + rss)
        ans$adj.r.squared <- 1 - (1 - ans$r.squared) * ((n - 
            df.int)/rdf)


  Does that answer your question?

  Ben Bolker