R help - Feb 2022 - Question About lm()

The models are NOT equivalent. Why would you?ll think they were?

? 
David

Sent from my iPhone
> On Feb 9, 2022, at 11:10 PM, Bromaghin, Jeffrey F via R-help <r-help at
r-project.org> wrote:
> 
> ?Hello,
> 
> I was constructing a simple linear model with one categorical (3-levels)
and one quantitative predictor variable for a colleague. I estimated model
parameters with and without an intercept, sometimes called reference cell coding
and cell means coding.
> 
> Model 1: yResp ~ -1 + xCat + xCont
> Model 2: yResp ~ xCat + xCont
> 
> These models are equivalent and the estimated coefficients come out fine,
but the R-squared and F statistics returned by summary() differ markedly. I
spent some time looking at the code for both lm() and summary.lm() but did not
find the source of the difference. aov() and anova() results also differ, so I
suspect the issue involves how the sums of squares are being computed. I've
also spent some time trying to search online for information on this, without
success. I haven't used lm() for quite a while, but my memory is that these
differences didn't occur in the distant past when I was teaching.
> 
> Thanks in advance for any insights you might have,
> Jeff
> 
> Jeffrey F. Bromaghin
> Research Statistician
> USGS Alaska Science Center
> 907-786-7086
> Jeffrey Bromaghin, Ph.D. | U.S. Geological Survey
(usgs.gov)<https://www.usgs.gov/staff-profiles/jeffrey-bromaghin>
> Ecosystems Analytics | U.S. Geological Survey
(usgs.gov)<https://www.usgs.gov/centers/alaska-science-center/science/ecosystems-analytics>
> 
> 
>    [[alternative HTML version deleted]]
> 
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On Wed, 9 Feb 2022 23:16:13 -0800
David Winsemius <dwinsemius at comcast.net> wrote:
> The models are NOT equivalent. Why would you?ll think they were?
'Scuse me, David, but they *are* "equivalent".  They are simply
different parametrisations of the same model.  *However* the different
parametrisations imply, according to the conventions of lm(), different
*null* models.

For yResp ~ xCat + xCont, the null model is yResp ~ 1.

For yResp ~ -1 + xCat + xCont, the null model is yResp ~ 0.

For the first null model, the residual sum of squares is
ssr1 = sum((yResp-mean(yResp))^2).

For the second null model, the residual sum of squares is
ssr2 = sum(yResp^2).

Thus for the first parametrisation one gets

RSquared = 1 - ssr1/ssr

and for the second parametrisation one gets

RSquared = 1 - ssr2/ssr

in both cases "ssr" is the sum of squares of the residuals from the
full model (which is of course the same in both cases).

I hope that this clarifies things a bit for the OP.

One further comment:  RSquared is a bit of a dubious concept;  it is
particularly dubious for models with no intercept.  Caveat utilitor.

cheers,

Rolf

-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276