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I want to show that if you don't take enough biopsies from the oesophagus for certain patients then you are more likely to not detect an illness . The counts in the table are the number of investigations where a sufficient number of biopsies were taken (based on a threshold number) versus whether that investigation resulted in a diagnosis of Illness.

                                Illness No Illness
Insufficient number of biopsies  18        52
Sufficient number of biopsies    11        18

It seems obvious to me when looking at the table that you are much more likely to not pick up the disease with an insufficient number of biopsies but Chi-squared doesnt show this.

Illness<-c(18,11)
No_Illness<-c(52,18)
data.frame(Illness,No_Illness)
chisq.test(data.frame(EoE,No_EoE))

Am I missing something here?

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You are looking at the difference between ~26% and ~38% (insufficient and sufficient, respectively) which, given the number of counts, is not that much to make a convincing case.

prop.test(c(18, 11), c(70, 29), alternative= 'less')

    2-sample test for equality of proportions with continuity correction

data:  c(18, 11) out of c(70, 29)
X-squared = 0.94659, df = 1, p-value = 0.1653
alternative hypothesis: less
95 percent confidence interval:
 -1.00000000  0.07352868
sample estimates:
   prop 1    prop 2 
0.2571429 0.3793103 
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