Proper Statistical Test for Binary Data

Question

I looking for the best statistical test to apply in a particular situation and I hope I can find here the answer(s) I'm looking for.

First of all some details:

I'm studying 33 different mutants of a particular protein and I've partitioned these mutants in 4 small groups on the basis of their severity:

• Group A has 11 mutants
• Group B has 8 mutants
• Group C has 6 mutants
• Group D has 8 mutants

I can test these mutants for the presence/absence of a series of particular internal interactions and I want to know if there is a statistical difference among the 4 groups. These internal interactions are essentially independent binary variables: 0 the mutant does not have a particular interaction or 1 the mutant has the interaction.

Basically, what I want to do is checking if there is a significant statistical difference in the percentage of mutants of each group that sport or not a series of these interactions.

My final goal is to correlate the presence/absence of some of these interactions with the severity of the mutations and find out which of these interactions are peculiar of a given group.

This is an example with real data:

Interaction #1

• 27.3% of the mutants in Gourp A has this interaction
• 12.5% of the mutants in Gourp B has this interaction
• 83.3% of the mutants in Gourp C has this interaction
• 50.0% of the mutants in Gourp D has this interaction

My question is: Which statistical test should I use to check if the differences in these percentages are statistically significant?

Thank you

[edit]

As suggested by @AndrewM, here are some more details about what I'm trying to do.

I've ~150 interactions and only a few of them are missing solely in mutants of GroupD (highest severity), while the vast majority are variably missing by mutants in all clusters.

What I need is an unbiased way to highlight those interactions that, even if also missing in a small number of other mutants in other clusters, could be defined are typically missing in GroupD.

My final aim is to test if I can explain, at least partially, the severity of these mutants looking at their missing interactions and then correlate mutant severity with presence/absence of the interactions.

Thanks

Answer 1

Have you looked at $\chi^2$ statistics of independence?

Sounds like a classic use case for me: test whether the binary indicators you have and the mutant rate are independent.

For small sample sizes, you may need to use Yates's correction for continuity. Depending on the side of the test, you may want to do a similar adjustment the other way - to make sure you err on the wrong side (i.e. assume independence if in doubt).

Answer 2

I'm studying 33 different mutants of a particular protein and I've partitioned these mutants in 4 small groups on the basis of their severity:

Group A has 11 mutants Group B has 8 mutants Group C has 6 mutants Group D has 8 mutants

This can be modelled as:

$P(Group_A) = 11/33 \approx 0.333$

$P(Group_B) = 8/33 \approx 0.242$

$P(Group_C) = 6/33 \approx 0.182$

$P(Group_B) = 8/33 \approx 0.242$

You also give us this example:

Interaction #1

27.3% of the mutants in Gourp A has this interaction

12.5% of the mutants in Gourp B has this interaction

83.3% of the mutants in Gourp C has this interaction

50.0% of the mutants in Gourp D has this interaction

Which can be thought as:

$P(Interaction_1 | Group_A) = 0.273$

$P(Interaction_1 | Group_B) = 0.125$

$P(Interaction_1 | Group_C) = 0.833$

$P(Interaction_1 | Group_D) = 0.5$

Thus you also know that:

$P(\lnot Interaction_1 | Group_A) = 1 - P(Interaction_1 | Group_A) = 0.727$

$P(\lnot Interaction_1 | Group_B) = 1 - P(Interaction_1 | Group_B) = 0.875$

$P(\lnot Interaction_1 | Group_C) = 1 - P(Interaction_1 | Group_C) = 0.167$

$P(\lnot Interaction_1 | Group_D) = 1 - P(Interaction_1 | Group_D) = 0.5$

My ultimate goal is to find a "signature" of missing interactions strongly associated with severe mutations. In other words: I want to know which interactions, if missing, are associated with a severe condition

As I interpret it, you want to know which interactions if absent are highly to result in the case $i$ to belong to $Group_A$, which I assume here, is the group of higher severity. I don't know what you mean by "strongly associated", are you bringing in correlation?!

You can check the probability of a mutant belonging to $Group_A$(having high severity) given that he tested negative for interaction 1 as, $P(Group_A | \lnot Interaction_1)$.

This can be calculated as:

$P(Group_A | \lnot Interaction_1) = \frac{P(Group_A, \lnot Interaction_1)}{P(\lnot Interaction_1)} = \frac{P(\lnot Interaction_1 | Group_A)*P(Group_A)}{P(\lnot Interaction_1)}$

Using the same logic you can calculate $P(Group_A | \lnot Interaction_1, \lnot Interaction_1, ...)$ or any combination of existent/absent interactions.

Still I think this isn't quite what you are asking for, give us more specifications and examples of what you want to accomplish if that's the case.

You seem to want to classify the severity of a mutant by examining and studying a single variable, while I can only think of an estimate of his severity as relation of two or more interactions.