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

Martyn Plummer plummer at iarc.fr
Tue Sep 26 11:00:45 CEST 2000

```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,
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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