# [R] Query on R-squared correlation coefficient for linear regression through origin

peter dalgaard pd@|gd @end|ng |rom gm@||@com
Thu Sep 27 16:22:49 CEST 2018

```This is an old discussion. The thing that R is doing is to compare the model to the model without any regressors, which in the no-intercept case is the constant zero. Otherwise, you would be comparing non-nested models and the R^2 would not satisfy the property of being between 0 and 1.

A similar issue affects anova tables, where the regression sum of squares is sum(yhat^2) rather than sum((yhat - ybar)^2).

-pd

> On 27 Sep 2018, at 12:56 , Patrick Barrie <pjb10 using cam.ac.uk> wrote:
>
> I have a query on the R-squared correlation coefficient for linear
> regression through the origin.
>
> The general expression for R-squared in regression (whether linear or
> non-linear) is
> R-squared = 1 - sum(y-ypredicted)^2 / sum(y-ybar)^2
>
> However, the lm function within R does not seem to use this expression
> when the intercept is constrained to be zero. It gives results different
> to Excel and other data analysis packages.
>
> As an example (using built-in cars dataframe):
>> cars.lm=lm(dist ~ 0+speed, data=cars)     # linear regression through
> origin
>> summary(cars.lm)\$r.squared # report R-squared  0.8962893 >
> 1-deviance(cars.lm)/sum((cars\$dist-mean(cars\$dist))^2)     # calculates
> R-squared directly  0.6018997 > # The latter corresponds to the value
> reported by Excel (and other data analysis packages) > > # Note that we
> expect R-squared to be smaller for linear regression through the origin
>> # than for linear regression without a constraint (which is 0.6511 in
> this example)
>
> Does anyone know what R is doing in this case? Is there an option to get
> R to return what I termed the "general" expression for R-squared? The
> adjusted R-squared value is also affected. [Other parameters all seem
> correct.]
>
> Thanks for any help on this issue,
>
> Patrick
>
> P.S. I believe old versions of Excel (before 2003) also had this issue.
>
> --
> Dr Patrick J. Barrie
> Department of Chemical Engineering and Biotechnology
> University of Cambridge
> Philippa Fawcett Drive, Cambridge CB3 0AS
> 01223 331864
> pjb10 using cam.ac.uk
>
>
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>
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--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: pd.mes using cbs.dk  Priv: PDalgd using gmail.com

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