When to transform predictors in regression when response may be quadratic? I am analyzing data from an experiment in which treatment levels increase quadratically, e.g. the treatment levels are $0, 1, 4, 9$.
When analyzing the response using regression, would it make sense to use the square root of the treatment level as a predictor? 
If so, how would this affect interpretation?
 A: When you don't know the functional form ahead of time (which is a common setting) and you have no reason to assume it's linear, it's best to be flexible.  If there were more levels of treatment you could fit a quadratic or restricted cubic spline shape, for example.  For only 4 levels it may be best to assign 3 degrees of freedom to treatment using 3 dummy variables.
A: Why not look at a bivariate X-Y scatterplot in advance of running a regression.  That'll show you the shape of the line or curve, especially if you have software that can give a lowess/loess fit (locally weighted smoothed fit).
As to interpretation, it'll no doubt be easier for you than for your audience, but if you do have a quadratic fit, then for each increment of one on the sq. rt. of X, Y will change by b, your coefficient.
If you really only have 4 levels of X, I agree with @Frank's point and would add that you might make your job easier by running an ANOVA instead of regression.  Or, some software makes it easy to combine continuous and categorical predictors, fusing regression and anova into a general linear model without the need for dummy variables (if you use SPSS, search 'Unianova').
