It seems that if I have a regression model such as $y_i \sim \beta_0 + \beta_1 x_i+\beta_2 x_i^2 +\beta_3 x_i^3$ I can either fit a raw polynomial and get unreliable results or fit an orthogonal polynomial and get coefficients that don't have a direct physical interpretation (e.g. I cannot use them to find the locations of the extrema on the original scale). Seems like I should be able to have the best of both worlds and be able to transform the fitted orthogonal coefficients and their variances back to the raw scale. I've taken a graduate course in applied linear regression (using Kutner, 5ed) and I looked through the polynomial regression chapter in Draper (3ed, referred to by Kutner) but found no discussion of how to do this. The help text for the
poly() function in R does not. Nor have I found anything in my web searching, including here. Is reconstructing raw coefficients (and obtaining their variances) from coefficients fitted to an orthogonal polynomial...
- impossible to do and I'm wasting my time.
- maybe possible but not known how in the general case.
- possible but not discussed because "who would want to?"
- possible but not discussed because "it's obvious".
If the answer is 3 or 4, I would be very grateful if someone would have the patience to explain how to do this or point to a source that does so. If it's 1 or 2, I'd still be curious to know what the obstacle is. Thank you very much for reading this, and I apologize in advance if I'm overlooking something obvious.