# Odds Ratio and gene mutations association

I have a simple (yet to me still trivial) problem to submit to you.

I have a dataset of a group of patients affected by a disease, for which the presence of several genes mutations was inferred. Each gene is a variable with either 0 for negative and 1 for positive. I need to assess the presence of associations between these genes, to establish whether some tend to be co-mutated while some other tend to be mutually exclusive. For doing this, I first analyzed all the genes possible combinations in 2x2 contingency tables such as:

in this case for example, the p value is very significant, so I thought it could be useful to compute the OR to establish a relationship. Here, for example, the OR obtained from the formula (OR=A x D/ C x B) is 0.53, hence it should mean that the two genes tend to be more in opposite directions (0-1 or 1-0) compared to same directions (0-0 or 1-1). However, my concern is that in this way it is not clear whether the two genes have a positive or negative correlation. Should I just compare the double positive (1-1) against the total of discordant cases (0-1 and 1-0)? In this case it would be 69/(131+428)=69/559=0,12. Is it useful?

However, each gene has a different % of mutation within the population, so for example gene 1 here has a .18 probability of being mutated whilst gene 2 has a .46 probability. Should I take this into account? I played around and tried to see how these 4 combinations would look like if they were only due to the each of the two genes expected mutation frequencies, so something like that came out: final numbers are the same, but if you look at it, numbers are ridistributed according to the expected frequencies (ie: total no of mut gene 1 cases is 195/1059=0.18 which is the expected mut frequency of the gene). I then computed another OR for these numbers (12.76) and compared it with the previous one using Tarone´s test of homogeneity between the two tables (in this case, p-value is significant). From the simple division of each category from the "real life" table / the "expected frequencies" table I obtained a ratio (ie: 0/0 ratio=431/542=0.79, there are less double negative than expected). Do you think this is a correct reasoning? If so, should i use the 1/1 ratio to know if the relation is positive or negative (in this case 69/172=0,4, there are less double positive than expected so the genes are inversely correlated)?

I thank you in advance and look forward for your help! Best, Luca

• It's not completely clear from your question whether you care about the co-mutation/mutual exclusivity relations simply within your disease group, or whether you want to know whether those relations differ from those in the population as a whole. There may be co-mutation/mutual exclusivity relations in the population as a whole that your second table (which seems to assume independence between the "mutations") doesn't take into account. Also, are these actually "mutations" from a normal genome (like somatic mutations in cancer cells) or are they normal variants? – EdM Jun 5 '18 at 17:07
• Hi EdM, thanks for the answer. I´m sorry I did not specify, but this is a population of patients affected by AML (Acute Myeloid Leukemia), which I should study for certain genes mutations (somatic mutations in AML cancer cells). What I would need to do is compare several genes sequencing results to check whether there is an association between them (positive when they are usually mutated together or negative when they tend to exclude each other). So i do not need to compare these numbers with the general population. – luke_hats Jun 6 '18 at 7:37
• The second table is just an attempt of seeing how the mutation of the two same genes would look like if they were randomly distributed and not associated to each other. I made it based on the actual mut frequencies of the single genes so that it respect the proportion of mutated cases among them. – luke_hats Jun 6 '18 at 7:39

The observed contingency table (your first table) with a statistical test (Fisher's exact test or chi-square) and the associated odds ratio is the way to go. The chi-square and Fisher's tests compare the observed contingency table against what would have been observed if the 2 events were independent. With multiple pairs of genes being compared, however, you will have to correct for the multiple testing (e.g., with the Benjamini-Hochberg procedure).

A significant odds ratio less than 1 (as for your first table) is equivalent to what you seem to mean by "negative correlation." If the occurrence of the 2 mutations has a "positive correlation" then there will be an odds ratio greater than 1.

There seems to be a problem with the way you calculated your second table, as it does not represent your intended independence between the 2 gene mutations (as its odds ratio of 12.76 implies; your second table shows a very high frequency of gene 2 mutations in the cases with gene 1 mutations, inconsistent with independence). The following table represents independence between the 2 mutations:

             gene 1 Total
0   1
gene 2  0   469 103  572
1   399  88  487
Total      868 192 1059


with an odds ratio of 1.

This NEJM paper by Papaemmanuil et al illustrates how to evaluate co-occurrent and mutually exclusive mutations in Acute Myeloid Leukemia (AML), your disease of interest. Its supplemental Figure S3 is a very useful display, showing both co-occurrence numbers and odds ratios in a single display, with annotations of statistical significance. The authors used the patterns of co-occurrence to develop genomic classifications of AML, which were then related to patient outcomes.

In response to comment:

It's easy to get confused here; odds are already ratios of probabilities to start with. So the odds ratio is the ratio of two ratios, which can be difficult to grasp until you play with some examples. In a 2-way table, a single odds ratio symmetrically represents the association of either property with the other (gene 1 or gene 2 mutation, in your case). The extensive Wikipedia page on Odds ratio shows a few ways to think about this, documents this symmetry property, and shows many further properties and examples.

The wide range of values of odds ratios in the Papemmanuil et al paper simply represents the wide range of associations among the pairs of mutations. Remember that an odds ratio closer to 0 represent a pair of mutations that are nearly mutually exclusive, while high odds ratios much greater than 1 represent more frequently co-occurring mutations.

The p.adjust() function in R allows you to compare several different simple approaches to the multiple comparisons problem. The multcomp package in R provides more extensive methods, as does the multtest Bioconductor package.

The simplest multiple-comparison corrections assume independence among the observations, unlikely with gene mutation or expression data. This superb answer has useful discussion and extensive references on ways to deal with this issue.

• Thanks EdM!!!You got the exact point of my question. the paper I was trying to refer to is exactly the Papaemmanuil et al one. I was trying to build your table 2, but I was wrongly not assuming that the OR should be 1. I will then use the OR of the first table. – luke_hats Jun 7 '18 at 7:35
• The question now is: how to interpret the OR for a clinical purpose? I mean: if the OR in this example is 0.53, can it be read as: gene 2 is 0.53 times less mutated when gene 1 is mutated ? And also gene 1 is 0.53 times more mutated when gene 2 is negative? Also, since you got the exact figure I was trying to arrive, why do OR values range so widely from 0.001 to 1000? Do you have a suggestion on how to perform nicely a Benjamini-Hochberg procedure in R and everything around FDR and FWER (some good guides)? – luke_hats Jun 7 '18 at 7:35