I have a (mixed) model in which one of my predictors should a priori only be quadratically related to the predictor (due to the experimental manipulation). Hence, I would like to add only the quadratic term to the model. Two things keep me from doing so:


I think I read somehwere that you should always include the lower order polynomial when fitting higher order polynomials. I forgot where I found it and in the literature I looked at (e.g., Faraway, 2002; Fox, 2002) I cannot find a good explanation.
When I add both, the linear and quadratic term, both are significant. When I add only one of them, they are not significant. However, a linear relation of predictor and data is not interpretable.


The context of my question is specifically a mixed-model using lme4, but I would like to get answers that could explain why it is or why it is not okay to inlcude a higher order polynomial and not the lower order polynomial.

If necessary I can provide the data.

1. Why include the linear term?
It is illuminating to notice that a quadratic relationship can be written in two ways:
$$y = a_0 + a_1 x + a_2 x^2 = a_2(x - b)^2 + c$$
(where, equating coefficients, we find $-2a_2 b = a_1$ and $a_2 b^2 + c = a_0$).  The value $x=b$ corresponds to a global extremum of the relationship (geometrically, it locates the vertex of a parabola).
If you do not include the linear term $a_1 x$, the possibilities are reduced to
$$y = a_0 + a_2 x^2 = a_2(x - 0)^2 + c$$
(where now, obviously, $c = a_0$ and it is assumed the model contains a constant term $a_0$).  That is, you force $b=0$.
In light of this, question #1 comes down to whether you are certain that the global extremum must occur at $x=0$.  If you are, then you may safely omit the linear term $a_1 x$.  Otherwise, you must include it.
2. How to understand changes in significance as terms are included or excluded?
This is discussed in great detail in a related thread at https://stats.stackexchange.com/a/28493.
In the present case, the significance of $a_2$ indicates there is curvature in the relationship and the significance of $a_1$ indicates that $b$ is nonzero: it sounds like you need to include both terms (as well as the constant, of course).
