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My aim is to find out if infection state (pos / neg) is associated with the group of patients (A, B or C). For this analysis I have followed suggestion from How to analyze $ 2\times n$ contingency table? and ran propensity test and pairwise propensity test with Bonferroni correction:

Inf_mat <- matrix(c(15,35, 12,29, 18,15), 2, 3,
             dimnames = list(infection = c("pos", "neg"),
                             group = c("A", "B", "C")))

prop.test(t(Inf_mat))

    3-sample test for equality of proportions without continuity correction

data:  t(Inf_mat)
X-squared = 6.4866, df = 2, p-value = 0.03903
alternative hypothesis: two.sided
sample estimates:
   prop 1    prop 2    prop 3
0.3000000 0.2926829 0.5454545

pairwise.prop.test(t(Inf_mat), p.adjust.method ="bonferroni")

    Pairwise comparisons using Pairwise comparison of proportions

data:  t(Inf_mat)

  A    B  
B 1.00 -  
C 0.13 0.15

P value adjustment method: bonferroni

As results of these tests do differ I am not sure how to interpret it. From prop.test it seems that $H0:pA=pB=pC$ could be rejected as probability of false discovery if there is no association is ~4%. However, as pairwise.prop.test shows there is no pair of groups where we could reject $H0:pX=pY$ as Bonferroni corrected p-values are rather high.

zubenel
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  • Bonferroni sucks...away your power. You have three comparisons, meaning that, if you want to reject at 0.05, you need a p-value less than 0.016. Your 0.039 is more than double that. What happens if you try Bonferroni-Holm or some other correction as your p.adjust.method? – Dave Jan 28 '20 at 12:03
  • @Dave With Holm method the results are similar. The only change is that adjusted p-value between group B and C proportions changes from `0.15` to `0.13`. – zubenel Jan 28 '20 at 12:31
  • The intuition behind this to me is that you have sufficient evidence to reject (at $\alpha=0.05$) the null hypothesis that all three proportions are equal. However, since the p-value is rather close to $\alpha$, you don’t have enough firepower to pin down where the differences are. – Dave Jan 28 '20 at 12:35
  • This is an example of (and, mathematically, is exactly the same thing as) a set of phenomena where a multiple regression is significant overall but none of the individual variables is. See https://stats.stackexchange.com/questions/9751 and https://stats.stackexchange.com/questions/310628 *inter alia.* – whuber Jan 28 '20 at 14:20

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