# 2x2 contigency table non-independent but not paired data

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#### stweb

##### Guest
I am looking at male-biased sexual size dimorphism in a bird species (males average larger than females). I am interested in the proportion of males larger than females in breeding pairs.

I have one proportion result "A" from observed breeding pairs. I then did random bootstrap resampling for a comparison result "B". I want to show that results "B" agrees with result "A", meaning that the observed proportion is no different than what would could happen from random pairing.

I have been using 2 x 2 contingency tables and chi-squared tests, but these proportions "A" and "B" are correlated: they are derived from the same size measures and from the same subjects (plus a few extras in the bootstrap sample). These data are not paired but obviously correlated.

How might I test this problem? I am aware of Pearson's chi-squared, Fisher's exact test, McNemar's test and sign test. None of these are suitable. I was thinking perhaps to use a binomial test, with one of the proportions as reference. I am using R for statistical computing.

Cheers.

#### Disvengeance

##### New Member
If you want to test whether the proportions are significantly different you can perform a binomial test. If you want to determine the amount of agreement between the two then you could use one or more of the agreement statistics like kappa, rand index, etc.

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#### stweb

##### Guest
Actually, I soon realised that the binomial test should NOT be used since both results, observed and simulated, have associated inaccuracy. I don't think a parametric test exists for this evaluation.

What I have done is a bootstrap where BOTH the observed and simulated pairs are resampled 1000 times. The difference between proportions is calculated for each bootstrap resample. Then I calculated a simple percentile confidence interval (95%) from the 1000 difference in proportion results.

I have read that bootstrapped proportions are a bit dodgy, particulary for very high or very low proportions. Better bootstrap c.i. estimators are more difficult to compute and some can also have problems with proportions data. So the best I have now is a crude estimate of the confidence interval. No surprises, there are no "significant" differences detected.

I welcome any suggestions on how I might improve this analysis or any alternatives.

#### Karabiner

##### TS Contributor
I have one proportion result "A" from observed breeding pairs. I then did random bootstrap resampling for a comparison result "B". I want to show that results "B" agrees with result "A", meaning that the observed proportion is no different than what would could happen from random pairing.
If I understand it correctely, you used your sample
data and randomly paired female with male subjets,
a 1000 times, in order to estimate the theoretically
expected value (=chance value) within the given
sample
, which needn't be generalized to a population.

Assuming that 1000 replications give a reasonable
estimate, close to the true chance value (don't know
whether this is true), I'd just then use the binomial test
in order to compare the sample value with the theoretical
(chance) value.

With kind regards

K.