I have a situation where the skeletons from one of three archaeological sites do not display a bone condition called periostitis that is found at the other two.
The relevant statistics are:
Site 1: freq=.167, n=48 Site 2: freq=.000, n=5 Site 3: freq=.250, n=36.
Since it is unlikely that individuals from Site 2 would not have periostitis (for anthropological reasons), it seems reasonable that the small sample size (n=5) is to blame for the freq=0.
As a means of testing independence between the sites and frequency of periostitis, I performed a Fisher-Freeman-Halton test, which yielded p=.495; but, in accordance with the ASA statement on p-values, I'm trying to get away from null hypothesis significance testing (NHST) as the only support for my findings.
With other skeletal traits, I've showed that the overlap between the 95% CIs for various frequencies support the findings of the NHST, but the freq=0 reduces the value of this approach in this case. However, if I assume that Site 2 has a freq=.200 (which is the average frequency for all three sites), there is considerable overlap between the 95% CIs for the three sites, thereby supporting the NHST result.
This approach is reasonable since the 3 sites are close to each other geographically, and periostitis is a very common condition in prehistoric populations. Also, the binomial distribution shows a prob=.328 that samples of 5 would have no periostitis if the actual frequency is 1 in 5 (freq=.200).
Is this a legitimate analysis? Is projecting a frequency, based on logic but not supported by data, acceptable? Please give me your opinions.