On page 661 of the book "Numerical Recipes in C:The art of scientific computing", we fit a set of N data points $(x_{i}, y_i)$ to a straight line model $$y(x)=a+bx.$$ We do this by minimizing the chi-square merit function with fitting parameters a,b, as in eq. (15.2.2) of the book $$\chi²(a,b)=\sum_{i=1}^N\big(\frac{y_i-a-bx_i}{\sigma_i}\big)^2.$$ Here $\sigma$ is the uncertainity for each $y_i$ measurement. The goodness-of-fit of the data is then calculated using the incomplete gamma function as in equation eq. (6.2.3) $$Q=\text{gammaq}\left(\frac{(N-2)}{2}, \chi^2/2\right)= \\ \frac{1}{\Gamma((N-2)/2)}\int_{\chi^2/2}^{\infty}e^{-t} t^{(N-2)/2-1}dt,$$ where $\Gamma$ is the complete gamma function. Since this is way more abstract to me than other fitting routines, I want to ask if someone has a deeper understanding of why we use the incomplete gamma function gammaq as the goodness-of-fit estimation. What connects the $\chi^2$ fitting with the gammaq function, e.g. how do we know that the incomplete gamma function returns us a gof estimation for chi-square fitting?
1 Answer
$\chi^2$ distribution is a special case of the Gamma distribution : a $\chi^2$ distribution with $n$ degrees of freedom is the same as a Gamma distribution with shape parameter $n/2$ and scale parameter $2$ : $\chi^2_n \sim \text{Gamma}(n/2,2)$, or equivalently $\chi^2_n/2 \sim \text{Gamma}(n/2,1)$ by the scaling property of the Gamma distribution.
So the quantity $Q$ is just the tail probability of a $\chi^2$ distribution with $N-2$ degrees of freedom: $Q = P(X > \chi^2_{obs})$ where $\chi^2_{obs}$ is the observed value and $X \sim \chi^2_{N-2}$. It tells you what is the probability to observe a similar or larger value than $\chi^2_{obs}$ under the hypothesis that the model is correct. (you have $N-2$ degrees of freedom because of the two free parameters in the fit)
