how to get perfect fit of lm if response is constant

Thanks for the answer.
The situation is that I don't know anything of y a priori. Of course I then would not do a regression on constant y's, but isn't it a problem of stability of the algorithm, if I get an adj RSquare of 0.6788 for
a least square fit on this type of data? I think lm should give me a correct result even in case of y is perfectly fittable, because the situation is that I never know if my data could become so. If I have to offset y in this case, then my question becomes how noisy do my y's have to be, so that I can rely on the lm result, if I specify the formula y~x without offset. What if my y's become nearly linear (or nearly perfect fittable with another linear model). I think my question now becomes 'how to rely on lm's result if the formula is specified the way y~x without offset? or 'How do I prevent my result to become numerically incorrect if I may get nearly perfect fittable y's'.

Greetings

Henrik


-----Urspr?ngliche Nachricht-----
Von: Peter Ehlers [mailto:ehlers at ucalgary.ca]
Gesendet: Freitag, 8. Januar 2010 19:44
An: Jan-Henrik P?tter
Cc: r-help at r-project.org
Betreff: Re: [R] how to get perfect fit of lm if response is constant

You need to review the assumptions of linear models:
y is assumed to be the realization of a random variable,
not a constant (or, more precisely: there are assumed to
be deviations that are N(0, sigma^2).

If you 'know' that y is a constant, then you have
two options:

1. don't do the regression because it makes no sense;
2. if you want to test lm()'s handling of the data:

fm <- lm(y ~ x, data = df, offset = rep(1, nrow(df)))

(or use: offset = y)

?-Peter Ehlers

Jan-Henrik P?tter wrote:

Hello.

Consider the response-variable of data.frame df is constant, so analytically
perfect fit of a linear model is expected. Fitting a regression line using
lm result in residuals, slope and std.errors not exactly zero, which is
acceptable in some way, but errorneous. But if you use summary.lm it shows
inacceptable error propagation in the calculation of the t value and the
corresponding p-value for the slope, as well R-Square ? just consider the
adj R-Square of 0.6788! This result is independent of which mode used for
the input vectors. Is there any way to get the perfect fitted regression
curve using lm and prevent this error propagation? I consider rounding all
values of the lm-object afterwards to somewhat precision as a bad idea.
Unfortunately there is no option in lm for calculation precision.

df<-data.frame(x=1:10,y=1)

myl<-lm(y~x,data=df)

myl

Call:

lm(formula = y ~ x, data = df)



Coefficients:

(Intercept) ? ? ? ? ? ?x

? 1.000e+00 ? ?9.463e-18

summary(myl)

Call:

lm(formula = y ~ x, data = df)



Residuals:

? ? ? ?Min ? ? ? ? 1Q ? ? Median ? ? ? ? 3Q ? ? ? ?Max

-1.136e-16 -1.341e-17 ?7.886e-18 ?2.918e-17 ?5.047e-17



Coefficients:

? ? ? ? ? ? ?Estimate Std. Error ? t value Pr(>|t|)

(Intercept) 1.000e+00 ?3.390e-17 2.950e+16 ? <2e-16 ***

x ? ? ? ? ? 9.463e-18 ?5.463e-18 1.732e+00 ? ?0.122

---

Signif. codes: ?0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1



Residual standard error: 4.962e-17 on 8 degrees of freedom

Multiple R-squared: 0.7145, ? ? Adjusted R-squared: 0.6788

F-statistic: 20.02 on 1 and 8 DF, ?p-value: 0.002071




? ? ? [[alternative HTML version deleted]]



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--
Peter Ehlers
University of Calgary
403.202.3921