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The recurrence risk ratio is calculated as follows:

$$ \lambda_s = \frac{\text{risk of sibling}}{\text{risk in population}}$$

where the risk of sibling refers to the risk that a sibling has to get the disease and the risk in population is the risk that the disease occurs in the population.

I do not understand how it is possible to calculate the numerator. I saw a video here that discusses how it's calculated for simple diseases that follow Mendel's laws. However, my lecturer was discussing this in the context of multifactorial traits.

How would we calculate the risk of the disease occurring in a sibling without simply forcing the parents to breed thousands of times?

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If we look at web resources, what we will notice is that this ratio is not deployed at the family level but at the population level.

In other words, when we look at a population, we can ask:

What proportion of these people have the disease?

That is the denominator.

Taking that group of people, we can then ask:

Of those people that have the disease, what proportion of this affected group has a full sibling that also has the disease?

That is the numerator.

So if we have a population of 100 people, say 10 people have the disease. The denominator is then $0.1$. We might then observe that 4 of those cases are two pairs of siblings (and there are no other siblings). So

$$risk\ in\ sibling = Pr(sib\_has\_disease | person\_has\_disease) = \frac{4}{10} = 0.4$$

And thus

$$ \lambda_s = \frac{\text{risk of sibling}}{\text{risk in population}} \ = \frac{0.4}{0.1} = 4$$

It's worth pointing out that we are not controlling for environment or for pedigree or anything like that here. The null hypothesis of what $\lambda_s$ should be without any genetic risk is a complex function of prevalence and family structure (what if everyone is an only child, or the whole population is all siblings?). So it's not even clear how to interpret it, except to say that you might intuitively guess that a disease that strikes purely randomly might have $lambda_s = 1$. But I'm not even sure that that's true! All that this ratio is measuring is a rough estimate of how clustered the disease is within families within a population.

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