# comparing proportions before and after two different interventions

I am trying to compare the proportions before and after TWO interventions in a small sample.

Basically, 10 people had intervention A, and I want to compare proportions of their success rate before and after to the 10 people who had intervention B.

My null hypothesis is that intervention A is equivalent to intervention B with respect to how they change the success rate.

Should I just be doing a Chi-Squared to just compare the Delta Success rates for the two interventions?

Also, Is McNemar something I should be considering?

The McNemar test is for pair data, and I presume that the subjects in intervention A are different from the ones in intervention B.

Although you could use a chi square test of homogeneity to test if the proportion of responders is the same under both interventions, the small counts may make the Fisher exact test more suitable.

Here is a simulation in R:

> Interv <- matrix(c(8, 10 - 8, 4, 10 - 4), nrow = 2)
> dimnames(Interv) = list(Response = c("Success", "Failure"),
+                         Intervention = c("A", "B"))
Intervention
Response   A  B Sum
Success  8  4  12
Failure  2  6   8
Sum     10 10  20
>
> chisq.test(Interv)

Pearson's Chi-squared test with Yates' continuity correction

data:  Interv
X-squared = 1.875, df = 1, p-value = 0.1709

Warning message:
In chisq.test(Interv) : Chi-squared approximation may be incorrect


You can see that the expected frequencies in two cells are below $5$:

> (addmargins(expect <- chisq.test(Interv)\$expected))
Intervention
Response   A  B Sum
Success  6  6  12
Failure  4  4   8
Sum     10 10  20
Warning message:
In chisq.test(Interv) : Chi-squared approximation may be incorrect


Here is the Fisher test:

> fisher.test(Interv)

Fisher's Exact Test for Count Data

data:  Interv
p-value = 0.1698
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.6026805 79.8309210
sample estimates:
odds ratio
5.430473