#liability threshold model

[doctorveera]

Making peace with the liability threshold model

I’ve been struggling with understanding a concept for while now. But I think I am gradually getting to understand it.

Many studies have shown that cases with disease causing rare variants (e.g. individuals with 22q11.2 deletion with schizophrenia) have lesser polygenic score than cases without. It is suggested that this can be explained by the liability threshold hypothesis. According to this hypothesis, one has to accumulate a certain amount of liability (karma, if you will) and cross a threshold (called liability threshold) to develop the disease. One can do so either by accumulating a lot of small effect common variants or a single or multiple large effect rare variants. The beauty of this concept is that it can be leveraged to isolate monogenic cases from polygenic cases of a specific condition (e.g. myocardial infarction) in a population as monogenic cases are likely to fall at the lower tail of the polygenic score distribution in the cases. This approach is starting to pick up, thanks to Prof. Brent Richards, who, I think, is the first to empirically demonstrate this (see this, and this).

That being explained, I’ll now state my problem. Although I understood the liability threshold hypothesis, one question kept dripping inside my brain constantly: why we don’t see cases who have both rare variants and also, high polygenic score? Rare variants enter a population as denovo variants. Such variants occur randomly in the population, hence can hit anyone in the population. If so, why are we not seeing cases with both high polygenic score and rare variants?  The answer is, I think, simple: because polygenic cases are only a small subset of a large population, hence, the chances that they are hit by a rare variant (that causes the same disease that they already have due to common risk variants) are extremely low.  Consider a company with 100 employees comprising of 80 men and 20 women. The company decides to fire 20 people and they would want to do so using a lottery system. Then, what are the chances that more women will be fired than men?

I hope, I am now relieved of the troubles that this question has caused me for a long time. But who knows. Some odd aspect of this problem will strike me again and will suspend me in limbo for eternity. It bothers me that this is probably a simple concept that others might have grabbed easily, but I find it difficult to understand and make peace with it. Every one’s brain has their own strengths and weaknesses. My weakness is my inability to understand certain mathematical and statistical concepts from a formulaic description.

[doctorveera]

Would you like a latte of common variants or an expresso shot of rare variants?

A preprint on the association of Alzheimer’s PRS with cognitive functions in the UK biobank has some interesting results. It seems that the effect sizes of the association of Alzheimer’s PRS with certain cognitive functions such as fluid intelligence and matrix completion become larger after adjusting for APOE-e4 allele status (albeit with larger standard errors, I assume). Although these differences were not discussed in the paper, I think this is interesting, and it suggests that the polygenic effect of the Alzheimer’s PRS get attenuated when other strong sources of variations such as APOE-e4 allele exist. I have seen this in my own work. In an analysis in the iPSYCH cohort, I found that the association of educational attainment polygenic score with school performance is weaker in individuals with ADHD and ASD compared to individuals from the general population, and I reasoned that this might be due to that many of the ADHD and ASD individuals included in my analysis might have large effect rare variants (e.g. deletion CNVs or rare protein truncating variants) that might have attenuated the polygenic effect of the common variants. A similar example I came across recently: heritability of cardiomyopathy is higher in individuals without family history compared to individuals with family history, and this might be due to that the family history here serves as a proxy for large effect mutations.

Another way to look at these results is through the lens of the liability threshold model. According to this model, individuals with rare variants carry significantly lower polygenic risk score than individuals without rare variants (as has been shown in many studies, e.g. schizophrenia). If you take the cardiomyopathy example above, it might be that the common variant heritability in family history positive individuals is low because such individuals might carry less number of common risk variants (under the liability threshold model).

Either way, it’s clear that there exists a negative relationship among cases between common and rare variants (yes, I can hear you. It’s not a true negative relationship, it’s just a collider bias). This negative relationship can be leveraged to identify rare variants. For example, you have a cohort of 100,000 individuals who were all genotyped; you’d like to sequence a subset of the cohort, say 20,000 individuals, to identify rare variants for a disease. You could construct polygenic risk score for the disease and pick up the ones who have low polygenic risk score, but still have the disease. Under the liability threshold model, these individuals have the disease despite low polygenic score because they hold other risk factors that increased their liability for the disease, one of which might be rare variants. This has been indeed recently demonstrated in the UK Biobank sample.