# Statistical limitations of conducting nonlinear regression of $x$ vs.$f(y)$ as opposed to $y$ vs. $f(x)$

I derived a solution to a physics problem using a method of analysis that gives me an inversed relation, the independent variable $t$ expressed explicitly in terms of a nontrivial algebraic set of functions with the dependent variable $v$, namely $t= f(v, a,b,c,d)$,

where a,b,c,d, are parameters. From a physics standpoint it is variable $t$ that determines $v$, but obtaining such a solution of $v(t)$ it very complicated. My equation can be inverted using series functions but one sacrifices accuracy and some level of functionality. I need to determine the parameters a, b, c and d. For that purpose, I can use nonlinear regression vs. a reliable data set of ${(v_i,t_i)}$ values. The main sources of errors in the data are associated with variable $t$.

My question is if performing such a nonlinear regression for parameter estimation is a statistically valid approach. That is, the parameters and their confidence intervals could be considered accurate; having obtained very acceptable levels of significance in the statistical inferences of t statistics values, probability $p$, intrinsic curvature and parameter effect values, and well behaved standardized residuals

• Just to be clear, it's only nonlinear regression if the coefficients you'r trying to find show up in nonlinear ways. So, for example, if you're trying to fit a quadratic $y=ax^2+bx+c$ to some data, it's linear regression because it's linear in the coefficients $a, b,$ and $c$. The fact that this relationship is nonlinear in $x$ is irrelevant to the terminology. Second, such a regression is definitely a statistically valid approach. You always want to look at your coefficient of determination, $R^2,$ to see how much of the variation in $t$ your fit explains. – Adrian Keister Jun 26 '18 at 13:38
• Thank you for your remarks. I failed to mention that my solution is indeed nonlinear with respect to the parameters. It seems that the $R^2$ coefficient is not an applicable goodness-of-fit measure for nonlinear regression calculations, because it violates the basic premise about errors: SS_{ Regression} + SS_{ Error} = SS _{Total}.. – chavavic Jun 27 '18 at 15:00

It is a legitimate method. There is no close form solution for the nls, hence it is hard to tell something concrete without knowing nothing about $f(\cdot)$. For some types of non-linear models (e.g., the logistic model) - there are well established theoretical (inferential) results. For a general nls, assuming that the LS function is well behaving w.r.t the parameters and your initial values of them leads to the global minima - for large enough sample size you should be on the safe side.