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What is an acceptable phrase for reporting a Fisher's exact test where the number of rows and the number of columns are both $>2$. An example in R:

> x <- as.matrix(data.frame("A"=c(10,4,6),"B"=c(45,2,6),"C"=c(23,4,20),
         row.names=c("df1","df2","df3")))
> x
    A  B  C
df1 10 45 23
df2  4  2  4
df3  6  6 20
> fisher.test(x,, simulate.p.value=TRUE, B=1e5)

Fisher's Exact Test for Count Data with simulated p-value 
(based on 1e+05 replicates)

data:  x 
p-value = 0.00026
alternative hypothesis: two.sided 

I'm looking for a phrase / sentence that would encapsulate the above.

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  • 1
    $\begingroup$ Kind of the same as with 2x2 tables. Under the assumption of independence, these data are highly unlikely. If you want to report on the magnitude of the deviation from independence, you could take a look at pearson residuals. $\endgroup$ – miura Oct 17 '12 at 13:49
  • $\begingroup$ @miura Could you put an example phrase in an answer (so I can accept)? Also any elaboration on pearson residuals in the above example? Thanks. $\endgroup$ – oisyutat Oct 17 '12 at 13:58
  • $\begingroup$ You could write some phrase relating to association/non-independence of the two sets of grouping variables. $\endgroup$ – Glen_b Jul 23 '13 at 0:20
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Fisher's exact test tells you only about the probability of your or "more extreme" data given that the null hypothesis holds. It does not tell you about where in the table your data deviate from independence.

Pearson residuals are one way of expressing that. Try chisq.test(x)$residuals to obtain them. Larger absolute values correspond to greater deviance from what would be expected under the null hypothesis, positive values indicate "more than expected", negative values "less than expected".

They can be visualized by mosaicplot(x, shade=T)

If you square them and sum them up, you get Pearson's Chi-squared statistic. They partition this statistic, making visible how much each table entry contributes to it.

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