[R] lm -- significance of x coefficient when I(x^2) is used

[Martyn Plummer]

On 26-Sep-00 Michael Pronath wrote:
> 
>  In "Modern Applied Statistics with S-Plus" 3rd ed., footnote on page 153
>  regarding a model lm(Gas~Insul/(Temp+I(Temp^2))-1,whiteside), I read
> 
>      "Notice that when the quadratic terms are present, first degree
>       coefficients mean 'the slope of the curve at temperature zero', so a
>       non-significant value does not mean that the linear term is not
>       needed. Removing the non-significant linear term for the 'after'
>       group, for example, would be unjustified."
> 
>  AFAIK, t-test for significance of a coefficient is not based on the
>  assumption that the variables of the linear model are "independent".  What
>  if I only got the model matrix X and I don't know, that one column is
>  simply the square of another: Do I have to examine the model matrix for
>  polynomial dependencies between its columns, to know if t-test significance
>  is "significant"?

I would hope that you never conduct an analysis without knowing what
the variables mean! Seriously, though, it is not a question that can
be answered by the formal representation of the model, but by its
interpretation. If `the slope of the curve at x=0' has some physical
interpretation, you _might_ be justified in eliminating the linear term
while keeping the quadratic one.  But temperature is a good example
of where this is not the case.  If you eliminate the linear term,
then change your temperature scale (from Centigrade to Fahrenheit),
the linear term will pop back out again.  In effect you are testing
a "hypothesis" that has no meaning. Don't do this.

Martyn
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