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I want to know what test to use to compare 2 percentages from the same sample.

My study is as follows: I conducted a survey on 50 people regarding consent for 2 tests (X and Y). 48 people consented to test X, 40 people consented to test Y. So the consent rate for test X would be 48/50 (96%) and the consent rate for test Y would be 40/50 (80%).

How to compare these two percentages? Can I say the consent rate for test X is better than the consent rate for test Y? How to know if this is statistically significant? I tried Fisher's exact test and I tried to generate a contingency table, but I don’t know whether I can apply Fisher's exact test for non mutually exclusive data.

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I guess you would be having the data on how many subjects consented to both tests. Draw a simple Venn diagram with the information you have. Find out how many are there who consented only for test X and only for test Y and those who consented to both. Now, that makes the categories mutually exclusive. Consider those who provide consent to both the tests as a separate category "agrees to both tests X and Y". This creates three categories and depending on the size of the cells, you can use Chi-square test or Fisher's exact. Try this. May help.

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  • $\begingroup$ I think the chi square test is only for two samples, no? Does it mean I should ignore the third group that consents to both? Thanks. $\endgroup$ Aug 10 '19 at 19:31
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You can try the McNemar chi squared test and Z test for paired samples.

Quoted from the book, the type of situation typically involves measurements made on a variable before and after some sort of intervention.

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If each person answers about both questions, McNemar's test is best.

Imagine a 2x2 table where the rows are the people who answer Yes/No to X and the columns are the people who answer Yes/No to Y. The table is then...

$$ \begin{bmatrix} e & f \\ g & h \end{bmatrix}$$

Here, $e$ people answer yes to both, $h$ people answer no to both, $g$ answer No to X but Yes to Y, and $f$ answer Yes to X but no to Y.

McNemar's test is

$$\dfrac{(f-g)^2}{f+g}$$

and is chi-square distributed on one degree of freedom.

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