# Test of Significance of Different Surgeries

I have a dataset about nose surgeries done in several years. Some surgeries were combined, e.g. multiple procedures were done during one surgery. After some surgeries, some complications appeared. My task is to determine if there is a statistically significant difference in the frequency of surgeries between combined and independent surgeries. It is assumed that the combined surgeries should have a higher percentage of complications, yet it has to be prooved.

|                        | Independent | Combined |
|------------------------|-------------|----------|
| Without a complication | 31          | 123      |
| With a complication    | 10          | 65       |


Please note there were complications in 24 % of independent surgeries and 35 % of combined surgeries, which is quite a big difference in my opinion.

I used the Chi-Square test of independence, which is as-far-as-I-know the most typical test for this type of task, yet the result is not statistically significant ($$\alpha = 0.05$$), I had the same result with the Fisher's Exact test. Is there some other test I can try? I found the usage of t-test in some research papers for similar tasks, yet in my opinion, it is not possible to use it here. Or I am missing something?

You make the following hypothesis : "there is no difference IN THE FREQUENCY OF COMPLICATION following simple VS combined surgery.

And you want to refute that hypothesis using statistics ? Fine.

This should be translated into "the difference between frequency of complication after simple surgery and the frequency of complication after combined surgery is not statistically different from zero"

Then your first step is to calculate that difference. That, you can do only once because you only have one estimate of frequency of complication for each group.

However you actually need a SAMPLE of INDEPENDENT such estimates, for using statistics. For example, one estimate of each frequency for N hospitals.

In your case, you have only one estimate for each group. Then you can calculate only one estimate for the difference between those two groups.

The logical conclusion is that what you have AT BEST is a point-estimate of that difference.

You CAN NOT estimate significance using such point-estimate.

I am really sorry for you but at best you can argue that the difference is not zero IN YOUR STUDY and that it could mean more INDEPENDENT indentical studies should be made, maybe in different hospitals, for strengthening your conclusion WHICH IS NOT BACKED BY STATISTICS for the time being.

Tests you used are fine, you should just accept your results. The point of statistics is not to keep trying tests until you get results you like.

HOWEVER, just because the difference is not statistically significant, that does not mean that the rate of complications is the same. That's the old absence of evidence is not evidence of absence. What you can do is to look at confidence intervals around the difference between the surgeries and say that 0 difference is compatible with the observed data, but huge difference is also compatible with the data. So if you think that there is a difference, than this trial would not convince you otherwise, but also not really prove it either.