I measured a binary variable from two different populations, and now I'm trying to find out whether the different populations differ with regards to this variable. I could use a Chi-Square test, but that would necessitate that both populations have the same length. Is there an appropriate test for these circumstances? Thank you.
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asked Mar 26, 2012 at 16:30
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$\begingroup$ Are you sure the Chi² requires the same size ? (I do not clearly remember the Chi²). But anyway, you can use Fisher's exact test. $\endgroup$– Stéphane LaurentMar 26, 2012 at 18:49
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$\begingroup$ If you have a decent sample size you can use the normal approximation to the binomial: en.wikipedia.org/wiki/… and use a $z$-test $\endgroup$– MacroApr 5, 2012 at 13:47
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4 Answers
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Chi Square doesn't require equal size groups. In R you can use either prop.test() or chisq.test().
I do this often with A/B direct mail tests with unequal size groups. For example, 100K donors are split 90% and 10%: the 90% are sent an email appeal, and 10% are sent nothing. The binary outcome is whether they donated to the appeal.
The nice thing about prop.test vs chisq.test is that prop.test will both calculate the p-value of the hypothesis that the groups are equal and calculate the confidence interval for the difference
This page gives an example of prop.test() with two groups: http://cran.r-project.org/doc/contrib/Lemon-kickstart/kr_prop.html
sexsmoke<-matrix(c(70,120,65,140),ncol=2,byrow=T)
rownames(sexsmoke)<-c("male","female")
colnames(sexsmoke)<-c("smoke","nosmoke")
prop.test(sexsmoke)
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You can do a two sample t-test, perhaps after transforming the proportions using e.g. the arcsine transformation.
answered Mar 26, 2012 at 17:03
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2$\begingroup$ ... "perhaps after transforming" because the $\arcsin$ transformation is a variance stabilizing transform enhancing precision, as the variance depends on the only parameter of the Bernoulli distribution. $\endgroup$ May 23, 2015 at 10:41
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You can actually use logistic regression / glm with the outcome as dependent variable and group belonging as explanatory factor variable.
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Weighting by sample size is built into how the expected vaues are computed. The only thing to worry about are the rules about how small an expected value can be.
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1$\begingroup$ Welcome to the site,@JamalMunshi. This is a little more of a comment than an answer, would you mind expanding it a bit? $\endgroup$ Nov 14, 2014 at 9:34
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$\begingroup$ maybe i did not understand the question very well as i am unsure what is meant by the length of a population. $\endgroup$ Nov 14, 2014 at 12:17