# [R] lm without intercept

peter dalgaard pdalgd at gmail.com
Sun Jul 29 09:28:45 CEST 2012

```On Feb 18, 2011, at 14:20 , Jan wrote:

> Hello Achim,
>
>> Not quite. Consult your statistics textbook for the correct interpretation
>> of p-values. Under the null hypothesis of a true intercept of zero, it is
>> very likely to observe an intercept as large as 13.52 or larger.
> thank you for that help. I suppose the net doesn't have a detailed
> explanation of the output of summary.lm for someone with very little
> knowledge about statistics? I worked through J. Verzani "simple R" but
> it does assume some pre-knowledge.
>
>>> So I repeat the regression forcing the intercept to zero:
>>
>> Do you have a good interpretation for that?
> In this case, my knowledge of the physical reality behind the numbers
> tells me that the intercept should be zero.
>
>> The model without intercept needs to be interpreted differently. The
>> p-value pertains to a regression with intercept zero and slope 0.292
>> against a model with both intercept zero and slope zero.
> In other words, of course the slope of 0.292 is almost infinitely better
> than a zero slope? But the same would be true for most slopes >0, I
> suppose.
> So what is the correct way to compare the quality of the regression with
> and without intercept? Assuming that I don't know from the physical
> reality that the intercept should be zero, what can I say to support one
> model against the other?

R^2 is overused as a quality measure anyway, the Residual Standard Error is often more to the point. In your case, it is essentially the same in the two models, as would be expected when the test for the intercept is not significant.

(Notwithstanding the no-intercept case, R^2 is popular because it sort of lets you know what the scatterplot looks like without actually drawing it. E.g. if you are predicting weight by age based on a group of 25-75 year olds, you'll get a larger R^2 that if you base it on 30-40 year olds, but it isn't going to predict the value for a 35-year old any better.)

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

```