With numbers so small (40 or so total cases), its tough to identify cause and effect or analyze statistical deviations. It may simply be (for example) that als is found sooner in military people because they are more physically active and get more physical exams than the rest of the population.
The numbers actually aren't small at all. They say 40 of the 696,000 Gulf War troops now have ALS, compared to 67 of nearly 1.8 million who stayed home. The number of positive cases is small, yes, but the sample sizes themselves are very large.
We have pALS|Gulf War = 40/696,000 = .0000575, compared with pALS|Stayed Home = 67/1,800,000 = .0000372. The p-value for a double tailed hypothesis test with H0: pALS|Gulf War = pALS|Stayed Home and HA: pALS|Gulf War != pALS|Stayed Home is 0.02852-- in other words, if there is no real difference between the odds of Gulf War troops contracting ALS vs troops who stayed home, there is only a 2.852% chance that we would see the sample statistics that have been compiled. I haven't done a power test since the equation is long and nasty, but with sample sizes that big it's almost certainly a safe bet that we can rule out a Type II error as well as Type I. In other words, the numbers are statistically significant-- something strange went on in the Gulf War, making American troops who fought in it more susceptible to ALS than American troops who did not.
Its the number of cases, not the sample size that matters because the comparison is of the rates themselves. If for example their number is off by 1 for some reason (maybe a false positive or by chance someone got hit by a truck before being diagnosed) thats a 2.5% change in the incidence rate. Thats huge.
Any random factors affecting the number of ALS cases recorded in one group should theoretically apply equally well to the other, especially with such large sample sizes-- so they should more or less cancel out. Even if they don't, from the numbers posted it's clear that the number of ALS cases on either side could be off by a bit and still yield us statistically significant results. This is especially true since the quick statistical analysis above is based on a conservative double tailed test. If I had tested a one tailed alternative hypothesis (which is really what is in question here-- are the Gulf War troops more likely to contract ALS?) HA: pALS|Gulf War > pALS|Stayed Home, the results would be even more statistically significant and thus even more resistant to small errors in measurements.
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