# Statistical significance test with 1x2 data and faked 2x2 contingency table

Medical scientists love p-values and in a publication, I found the following patient statistics table. I stumbled over the Number of patients line.

They divided their subject group into two subgroups TAV and BAV and I asked how they calculated the p-value, because you only have one characteristic BAV/TAV. If you look for instance at the Male row, you see that you can use a 2x2 contingency table consisting of 2 characteristics BAV/TAV and Male/Female and make e.g. a Fisher test.

What was done in the Number of patients row was to fake a 2x2 contingency table repeating and reversing the numbers:

$$\begin{array}{c|c} 347&32\\\hline 32&347 \end{array}$$

From the viewpoint that they want to express how similar the number of patients are in the BAV and TAV group, this kind of makes sense because it gives a p-value of 1 (two tailed Fisher) when BAV and TAV have the same number of patients. However, I have never seen this and I cannot find any reference to such an approach.

Question: Can someone tell me whether it is correct what they did?

• Do you mean 2x2 table??? – SmallChess Apr 15 '16 at 11:32
• In the title of the question, I really meant 1x2 because basically we have only 1x2 data. But I see your point that it might be confusing. Maybe we find a different title for the question that describes the problem better. Any suggestions? – halirutan Apr 15 '16 at 11:33
• No need. If you mean 1x2, let it be. – SmallChess Apr 15 '16 at 11:34

The only "right" way I can think of to analyze that type of patient row is to test whether the proportions are equal. This can be done with a one way chi-square test, which is, indeed, a 1x2 table.

I've never seen the sort of procedure you mention - did they say in the article that that is what they did, or are you guessing/figuring out that that is what they did?

The next question is whether the two methods give the same results. The table you show doesn't give much, so let's test:

type <- c(TAV = 347, BAV = 32)
chisq.test(type)


and

x <- matrix(c(347,32,32,347),ncol = 2)
chisq.test(x)


give identical p values. The first is what I suggested, the second is what you said.

• Thanks for pointing this out. I really know what they did because I asked. +1 – halirutan Apr 15 '16 at 12:09
• Note, these are giving different p-values, it's just hard to tell since the p-value is so small. If you change 347 to 50, you'll see different p-values. I think this should be a one proportion z-test, please see my response – Peter Calhoun Apr 21 '16 at 7:02

The p-value calculations are a little confusing. The p-values are trying to test whether the percentages are similar for the two subgroups. For the first row "Number of patients" the % in TAV has to be 1-% in BAV. Therefore, one should test whether p=0.5 for one proportion:

prop.test(221, 244, p=0.5, alternative="two.sided")


For the "Male Gender" row, the % of males in TAV does not have to be 1-% of males in BAV, so we have a different test. You can actually fill out the table:

       Male   Female  Total
TAV    221     126     347
BAV    23      9       32
Total  244     135


This can be tested using Fisher's exact test (although Barnard's exact test is superior) giving the p-value reported.

> data<-matrix(c(221,126,23,9),ncol=2,byrow=TRUE)
> fisher.test(data,alternative="two.sided")

Fisher's Exact Test for Count Data

data:  data
p-value = 0.4418
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.2710563 1.5995457
sample estimates:
odds ratio
0.686984