Polynomial contrasts for regression I cannot understand the usage of polynomial contrasts in regression fitting. In particular, I am referring to an encoding used by R in order to express an interval variable (ordinal variable with equally spaced levels), described at this page.
In the example of that page, if I understood correctly, R fits a model for an interval variable, returning some coefficients which weights its linear, quadratic, or cubic trend. Hence, the fitted model should be:
$${\rm write} = 52.7870 + 14.2587X - 0.9680X^2 - 0.1554X^3,$$
where $X$ should take values $1$, $2$, $3$, or $4$ according to the different level of the interval variable.
Is this correct? And, if so, what was the purpose of polynomial contrasts?
 A: I will use your example to explain how it works. Using polynomial contrasts with four groups yields following.
\begin{align}
E\,write_1 &= \mu -0.67L  + 0.5Q -0.22C\\
E\,write_2 &= \mu -0.22L -0.5Q + 0.67C\\
E\,write_3 &= \mu + 0.22L -0.5Q -0.67C\\
E\,write_4 &= \mu + 0.67L + 0.5Q + 0.22C
\end{align}
Where first equation works for the group of lowest reading scores and the fourth one for the group of best reading scores. we can compare these equations to the one given using normal linear regression (supposing $read_i$ is continous)
$$E\,write_i=\mu+read_iL + read_i^2Q+read_i^3C$$
Usually instead of $L,Q,C$ you would have $\beta_1, \beta_2, \beta_3$ and written at first position. But this writing resembles the one with polynomial contrasts. So numbers in front of $L, Q, C$ are actually instead of $read_i, read_i^2, read_i^3$. You can see that coefficients before $L$ have linear trend, before $Q$ quadratic and before $C$ cubic.
Then R estimates parameters $\mu, L,Q,C$ and gives you
$$
\widehat{\mu}=52.79, \widehat{L}=14.26, \widehat{Q}=−0.97, \widehat{C}=−0.16
$$
Where $\widehat{\mu}=\frac{1}{4}\sum_{i=1}^4E\,write_i$ and estimated coefficients $\widehat{\mu}, \widehat{L}, \widehat{Q}, \widehat{C}$ are something like estimates at normal linear regression. So from the output you can see if estimated coefficients are significantly different from zero, so you could anticipate some kind of linear, quadratic or cubic trend. 
In that example is significantly non-zero only $\widehat{L}$. So your conclusion could be: We see that the better scoring in writing depends linearly on reading score, but there is no significant quadratic or cubic effect.
