I have 5 zones(categories) in which a certain percentage of total sinkholes exist. I have 5 different maps that I am testing to see which one provides me with the best fit to my expected percentages per category.

My hope is that 0% of the sinkholes will be in category 1, 10% in category 2, 20% in category 3, 30% in category 4, and 40% in category 5 (summing up to 100% total).

I am trying to compare my observed set of percentages (for each map) with my desired set of percentages.

I thought using a Chi-Squared Goodness of Fit test would be the best option, but when I attempt to calculate the Test Statistic, I run into problems, obviously, since one of my desired values is 0%, and 0 can't be in the denominator.

Any ideas as to how I can approach this statistical analysis without using the Chi-Square Test?

Thank you!


Actually, I think you can take the limit just fine.

Consider $\lim_{E_i\to 0} (O_i-E_i)^2/E_i$ under two cases:

Case 1: $O_i > 0$. In this case the term goes off to infinity in the limit, and the overall chi-square statistic goes with it.

Case 2: $O_i = 0$. In this case the term equals $E_i^2/E_i = E_i$, which goes to $0$ in the limit. So if $O_i=0$, the term adds nothing to the chi-square statistic.

Your statistic doesn't have a chi-squared distribution under the null, but it's perfectly possible to simulate it under the null (whence $O_i$ must be $0$), as long as you replace the contribution $(O_i-E_i)^2/E_i$ for the cell with $E_i=0$ with the limiting value $0$.

Then if $O_i$ for that cell is 0, you can compute the overall chi-square and compare with the simulated distribution. If $O_i$ is anything but 0, you can reject the null immediately; it's not possible to observe that if the null were true.

You can't simply run it through a canned routine as-is, but with a little bit of effort you can still do a test as described.

An alternative might be to use something from the power-divergence statistics


where $\lambda$ is chosen so the statistic will always exist.

I believe an appropriate reference for these is Cressie and Read (1984)$^{[1]}$.

e.g. something like the Freeman-Tukey $F^2 =4\sum_i (\sqrt{O_i}-\sqrt{E_i})^2$

However, instead of auto-rejecting, an $O_i$ of 1 would only contribute 4 to the statistic. You would likely need to simulate the null distribution of the statistic here; the asymptotic chi-square approximation may not be so good.

A bigger issue, perhaps, is to note that your categories are ordered. I don't think it would make sense to use a chi-square-like statistic in any case, since it throws away a lot of power in the rest of the table. You might use something like an Anderson-Darling statistic, but (again) with simulated distribution under the null (you have a similar issue to the above in applying this test, but it should have better power if you use an analogous approach to solving it).

[I wouldn't use Kolmogorov-Smirnov because it won't automatically reject the impossible case. At least I wouldn't use it as-is, but a test of that form could be adapted to behave as you'd hope.]

[1]: Cressie, N. A. C. and Read, T. R. C., (1984),
"Multinomial goodness-of-fit tests,"
J. Roy. Statist. Soc. Ser. B, 46, 440-464

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.