# Is using "Normal Approximation to binomial distribution" to test mutation enrichment in genomic region correct?

We are analyzing cancer patient mutation data. We defined set of region on the human genome as binding events, (for the ones who is interested in to the subject, it is a transcription factor binding regions obtained by ChIP-seq) and would like to prove that some of the region have significantly higher number of mutations than others. To prove this, we decided to set a mutation number threshold to say, there is at least X number of mutations are required to say this region is a hotspot.

Parameters:

1. We assume that mutation probability is constant within these regions.
2. We have total number of 196 patients in this project.
3. We have 4500 binding events (interested regions)
4. Have total 960 mutation found in the proximity of the regions.
5. 750bp is the median of the all binding events ( for our discussion lets assume they are all 750 bp)
6. We have total number of 196 patients in this project

Attempt: To solve this problem, I thought implementing "Normal Approximation to binomial distribution" could be useful.

$$\mu = np$$

where $n$ will be the binding region width 750, and $p$ will be the mutation probability per base pair.

$$\begin{array}{rcl} \sigma & = & \sqrt{np(1-p)} \\ Z & = & \displaystyle\frac{X-np}{\sqrt{np(1-p)}} = \frac{X-np}{\sigma} \end{array}$$

Questions:

1. I will test each binding region mutation number, X, with Uo. As a final procedure, I will do multiple hypothesis testing with respect to number of total binding regions. Is this correct?

Referring to this question, the OP asked a very similar question. But the OP is mainly focused on patient wise which I don't think it is not relevant in my case. Therefore:

2. Could implementing a Poisson distribution be more accurate?

I am very confused so any guidance will be very helpful.

• I don't have the answers to your questions but I do have a suggestion regarding your assumptions and investigation into normality wrt binomials. A.W.F. Edwards 1992 book Likelihood has an extensive discussion of these very topics in the context of evaluating genomic data and models. It's worth checking out, not to mention that it's a well written and elegantly done book. Commented Mar 24, 2017 at 12:53
• Thank you for your comment. I will definitely take a look at that book. Also one question quick was my title clear? Commented Mar 24, 2017 at 13:58
• Not being an expert in your topic area, I'm not the person to evaluate it. Commented Mar 24, 2017 at 14:31
• There seems (naively to me, since I'm not an expert) like at least two issues, which you could answer better than I could: (1) Does a normal distribution accurately model the data? (2) Are the assumptions for normality approximately satisfied in your situation? en.wikipedia.org/wiki/Normal_distribution#Approximate_normality It doesn't really matter so much what the underlying distribution is, as much whether the assumptions of the Central Limit Theorem are (approximately) satisfied. Commented Mar 24, 2017 at 17:35
• Thank you for spending time on question and commenting @Will . I don't think my data can be accurately explained by the normal distribution because the histogram of it shows that number of mutations are aggregated on (0,1,2,3,4,5) mutations and there is certainly no bell curve. (If this is how i am supposed to check it). docdro.id/esCUngV Commented Mar 25, 2017 at 8:27