R help - Nov 2022 - Is this *always* the intended R^2 value for no intercept in lm?

I am puzzled by the computation of R^2 with intercept omitted that is
already illustrated by the following example taken from help("lm")

## Annette Dobson (1990) "An Introduction to Generalized Linear
Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
weight <- c(ctl, trt)
lm.D9 <- lm(weight ~ group)
lm.D90 <- lm(weight ~ group - 1) # omitting intercept

The calculations for the R^2 for both models  are consistent with the
help("summary.lm") description:
"y* is the mean of y[i] if there is an intercept and zero otherwise."
Which causes a dramatic difference in the resulting R^2 values.

r2.D9 <- summary(lm.D9)$r.squared
r2.D90 <- summary(lm.D90)$r.squared

all.equal(r2.D9, 0.0730775989903856) #TRUE
all.equal(r2.D90, 0.981783272435264) #TRUE

This is counter-intuitive to say the least since the two models have
identical predictions and both models could be described more
accurately as two intercepts rather than zero. I see three
possibilities:

1. This is the intended result, in which case no fix is required, but
I?d be curious to understand the argument better.
2. This is an unfortunate outcome but not worth fixing as the user can
easily compute the correct R^2. In this case, I'd suggest that this
unintuitive behavior should be explicitly called out in the
documentation.
3. This is a bug worth fixing.

I look forward to hearing the community?s opinion on this.
Thanks in advance!

FAQ 7.41
and
https://stackoverflow.com/questions/57415793/r-squared-in-lm-for-zero-intercept-model

(among numerous others that could no doubt be found with a bit of
searching).

In short, the "null models" against which you are comparing the fitted
model are different with and without an intercept.

--Bert



On Sat, Nov 5, 2022 at 11:52 AM Thierry Zell <thierry.zell at gmail.com>
wrote:
> I am puzzled by the computation of R^2 with intercept omitted that is
> already illustrated by the following example taken from
help("lm")
>
> ## Annette Dobson (1990) "An Introduction to Generalized Linear
Models".
> ## Page 9: Plant Weight Data.
> ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
> trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
> group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
> weight <- c(ctl, trt)
> lm.D9 <- lm(weight ~ group)
> lm.D90 <- lm(weight ~ group - 1) # omitting intercept
>
> The calculations for the R^2 for both models  are consistent with the
> help("summary.lm") description:
> "y* is the mean of y[i] if there is an intercept and zero
otherwise."
> Which causes a dramatic difference in the resulting R^2 values.
>
> r2.D9 <- summary(lm.D9)$r.squared
> r2.D90 <- summary(lm.D90)$r.squared
>
> all.equal(r2.D9, 0.0730775989903856) #TRUE
> all.equal(r2.D90, 0.981783272435264) #TRUE
>
> This is counter-intuitive to say the least since the two models have
> identical predictions and both models could be described more
> accurately as two intercepts rather than zero. I see three
> possibilities:
>
> 1. This is the intended result, in which case no fix is required, but
> I?d be curious to understand the argument better.
> 2. This is an unfortunate outcome but not worth fixing as the user can
> easily compute the correct R^2. In this case, I'd suggest that this
> unintuitive behavior should be explicitly called out in the
> documentation.
> 3. This is a bug worth fixing.
>
> I look forward to hearing the community?s opinion on this.
> Thanks in advance!
>
> ______________________________________________
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