# [R] Statistically significant in linear and non-linear model

(Ted Harding) Ted.Harding at manchester.ac.uk
Tue Oct 7 20:16:07 CEST 2008

```On 07-Oct-08 17:46:52, Hsiao-nan Cheung wrote:
> Hi,
> I have a question to ask. if in a linear regression model, the
> independent variables are not statistically significant, is it
> necessary to test these variables in a non-linear model?
> Since most of non-linear form of a variable can be represented
> to a linear combination using Taylor's theorem,

That depends on the coefficients in the Taylor's series expansion.
It is quite possible to have the linear coefficient zero, and the
quadratic coefficient non-zero.

> so I wonder whether the non-linear form is also not statistically
> significant in such a situation.
>
> Best Regards
> Hsiao-nan Cheung
> 2008/10/08

Example:

X <-  0.2*((-10):10)
Y <-  0.5*(X^2) + 0.2*rnorm(21)
X2 <- X^2

[A] Linear regression, Y on X:
summary(lm(Y ~ X))\$coef
#               Estimate Std. Error   t value     Pr(>|t|)
# (Intercept) 0.72840442  0.1554215 4.6866382 0.0001606966
# X           0.06570652  0.1283351 0.5119919 0.6145564688

So the coefficient of X is not significant.

[B] Quadratic regression, Y on X and X^2:
summary(lm(Y ~ X + X2))\$coef
#                Estimate Std. Error     t value     Pr(>|t|)
# (Intercept) 0.003425041 0.07203265  0.04754846 9.625997e-01
# X           0.065706524 0.03957727  1.66020864 1.141924e-01
# X2          0.494304121 0.03666239 13.48259513 7.570563e-11

So the coefficient of X is still not significant (P = 0.14),
but the coefficient of X^2 is *highly* significant!

So it all depends ... of course the original coefficients
(Taylor) could be anything.
Ted.

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Date: 07-Oct-08                                       Time: 19:16:04
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