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## Appendix B: Statistics in XSPEC

The version 5 documentation indicates the goodness-of-fit chi-squared statistic reported with the results of Poisson regression is a test of the null hypothesis that the dependent variable is Poisson distributed. My question is why this statistic and perhaps the resulting inference regarding the appropriateness of Poisson regression varies with the composition of the right-hand-side variables. The goodness-of-fit chi-squared statistic in the poisson command is a simple Pearson's chi-squared statistic:. If you split up or group the counts and exposures differently, you get different cells for the Pearson's chi-squared and thus a different statistic.

Now we will group the data by the unique covariate patterns of the model.

### State the Hypotheses

In this case that simply amounts to grouping by XYZowned and summing counts injuries and exposure n within this grouping:. Note that the IRR and std error are the same, but the goodness-of-fit test is different. From the standpoint of the Poisson regression, both the original and collapsed datasets are equivalent, but the first dataset has more information about the Poisson-ness of the data since you can examine the counts for small portions of exposure. When the portions of exposure get too small, one gets the well-known problem of the expected counts for the Pearson chi-squared becoming small.

Perhaps Stata should automatically group by covariate pattern before doing the Pearson's chi-squared as lfit does after logistic.

### Interpret the results

Based on the chi-squared distribution with 14 degrees of freedom, the p-value of the test statistic is 0. Thus, there is insufficient evidence to suggest that the Poisson distribution is a bad fit.

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Flipping that double negative, the Poisson distribution seems like a good fit. This is confirmed by the scatter plot of the observed counts as proportions of the total number of counts; it is close to the Poisson PMF plotted with dpois in R with rate parameter 8.

## Poisson Distribution - Chi Square Test for Goodness of Fit

The full R code of my analysis is here:. Rice, John A. Mathematical statistics and data analysis. Duxbury press, See Chapter 8. Filed under Applied Statistics , Chemistry , Data Visualization , Nuclear Chemistry , Physical Chemistry , R programming , Radiochemistry , Statistical Computing , Statistics , Tutorials Tagged with alpha decay , alpha particle , Am-o , americium , americum , applied statistics , chemistry , chi-square , chi-squared , count , counts , dpois , expression , goodness of fit , helium , helium-4 , neptunium , neptunium , neutron , neutrons , Np , nuclear chemistry , nucleus , Pearson's chi-square test , Pearson's chi-squared test , plot , plots , plotting , plutonium , plutonium , Poisson , Poisson distribution , Poisson model , proton , protons , Pu , R , R programming , Radiochemistry , smke detectors , smoke detector , statistics.

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