Effect of Correlations Between Model Parameters and Nuisance Parameters When Model Parameters are Fit to Data

The effect of correlations between model parameters and nuisance parameters is discussed, in the context of fitting model parameters to data. Modifications to the usual $\chi^2$ method are required. Fake data studies, as used at present, will not be optimum. Problems will occur for applications of the Maltoni-Schwetz \cite{ms} theorem. Neutrino oscillations are used as examples, but the problems discussed here are general ones, which are often not addressed.