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Question:

I'm using a poisson fit; is the residual deviance = $ \chi^{2}$, and residual deviance / residual degrees of freedom = $\chi^{2}_{reduced}$?

Does this method provide a valid tool comparable to a reduced $\chi^{2}$ for goodness of fit?

Context:

This is in relation to a radioactive decay experiment. The data is made up of 50 points, taken at 18 seconds intervals, and are discrete counts. I am looking for a way to test the goodness of fit. I have used Mathematica to plot and fit with Generalised Linear Model with exponential family set to Poisson, fitting $y=Ae^{-\lambda t}$. The graph produced:

enter image description here

The 'Reduced $\chi^{2}$' on this graph is actually 'residual deviance / dof' I just haven't changed the label yet.

However there's another way to process the data also, which involves plotting a histogram of the response locations on the detector, fitting a gaussian on them, then using the area under the peak as the points for the data set, then plot and use NonLinearModel to fit $y=Ae^{-\lambda t}$, and the goodness of fit being $\chi^{2}$ / dof. enter image description here

I was wondering how comparable these two methods are for goodness of fit. Can i reasonably compare one to the other directly?

Also, the Mathematica Stack Exchange answer that explained the fit that should be used can be found here

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  • $\begingroup$ Are you saying you are trying to perform a $\chi^2$ type fit on data using a Poisson model? $\endgroup$
    – Mister Mak
    Apr 8 '21 at 16:05
  • $\begingroup$ Also note reduced $\chi^2$ is not sufficient for goodness of fit, you need both the $\chi^2$ and the number of degrees of freedom. $\endgroup$
    – Mister Mak
    Apr 8 '21 at 16:06
  • $\begingroup$ More that I want something that is a parameter for goodness of fit for a poisson model, somewhat akin to a chi^2 or chi^2 itself. I just wasn't sure if residual deviance would be that thing and if it was identical to chi^2 or similar or not $\endgroup$
    – Epideme
    Apr 8 '21 at 16:07
  • $\begingroup$ True... Is Chi^2 / dof = Reduced Chi^2? That is what I had understood that to be $\endgroup$
    – Epideme
    Apr 8 '21 at 16:08
  • $\begingroup$ Can you explain your fit a bit more clearly? $\endgroup$
    – Mister Mak
    Apr 8 '21 at 16:09
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If the fluctuations are of Poisson type, and too small to be Gaussian, then the standard weighted-least-squares estimator $\sum \frac{(data - model)^2}{(model fluctuations)^2}$ is not guaranteed to follow a $\chi^2$ distribution.

The good news is that if the fluctuations are of Poisson type, you may be able to use a likelihood method with a saturated data set to recover a $\chi^2$: https://www.sciencedirect.com/science/article/abs/pii/0167508784900164

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