Changing polynomial degree leads to changing p-values in OLS regression I have a question about interpreting coefficient $p$-values when fitting a polynomial function with ordinary least squares.
When I sequentially fit a linear, then quadratic, then cubic etc. polynomial to some data, the $p$-values change.
For instance, I compute the $p$-values for coefficients $a$ and $b$ of a linear fit (with null hypothesis that a = b = 0):
$$
y = a1 + bx
$$
and I happen to get a $p$-value for b of 0.002.  Then I try:
$$
y = a1 + bx + cx^2
$$
and the $p$-value of $b$ now has changed 0.014.


*

*Why is this?  I tried using Chebyshev polynomials of $x$, but find the same thing happening.  

*How can I do polynomial regression in this fashion where the $p$-values won't change in this way? 

*If I can't, how do I interpret the changing $p$-values?
 A: You can't keep the p-value the same, since adding terms will change the standard error of residuals, which necessarily alters the p-value.
To keep the coefficients the same, you need orthogonal polynomial regression, which is set up to do exactly that.
This is usually already implemented in stats packages. In R, you do it like so (output edited down):
> summary(lm(dist~poly(speed,1),cars))

Call:
lm(formula = dist ~ poly(speed, 1), data = cars)

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)      42.980      2.175  19.761  < 2e-16 ***
poly(speed, 1)  145.552     15.380   9.464 1.49e-12 ***

Residual standard error: 15.38 on 48 degrees of freedom
Multiple R-squared:  0.6511,    Adjusted R-squared:  0.6438 
F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

> summary(lm(dist~poly(speed,2),cars))

Call:
lm(formula = dist ~ poly(speed, 2), data = cars)

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)       42.980      2.146  20.026  < 2e-16 ***
poly(speed, 2)1  145.552     15.176   9.591 1.21e-12 ***
poly(speed, 2)2   22.996     15.176   1.515    0.136    

Residual standard error: 15.18 on 47 degrees of freedom
Multiple R-squared:  0.6673,    Adjusted R-squared:  0.6532 
F-statistic: 47.14 on 2 and 47 DF,  p-value: 5.852e-12

Where the additional term doesn't add a lot, the earlier p-values probably won't change much.
