# Correlation between polygenic scores and reproductive success

I was looking through this article in the journal Behavior Genetics. The researchers has investigated 400 000 people in the UK Biobank and calculated polygenic scores for 33 traits. Then they investigated how much offspring the 400 000 people had generated and coupled the traits to fecundity (number of children).

They found that people with a genetic composition that suggested that they had a high risk of (for instance) ADHD, obesity, and becoming smokers in general had better reproductive success.

I am having some problem understanding how big the effect is. If you look at the main results in figure 2, you get a list of polygenic traits with some measure of association, but how large a difference in reproductive success does the measure indicate?

If the number they get for "ADHD" is slightly above 0.03 does this mean that people with a very high genetic probability of having ADHD statistically have 3% more children than those with virtually zero risk of getting ADHD?

Question:

What is the interpretation of figure 2 in this paper? How many more children are people prone to various polygenic traits such as ADHD, high BMI etc. statistically supposed have according to the research?

Is says under the figure: "Each point represents a single bivariate regression of RLRS on a polygenic score. P value threshold is 0.05, Bonferroni-corrected for multiple comparisons."

RLRS is defined as respondent i’s number of children, divided by the mean number of children of people born in the same year

They fit a regression:

RLRSi=α+β * PGSii

In this equation, RLRSi is the normalized number of children by subject i. α, the intercept, is probably something around 1 but not necessarily exactly 1, but would account for bias in the sample in the study vs. the reference population they used to normalize number of children.

"β" is the key variable they are estimating with the equation. Note that β is multiplied by PGSi, so β is not in units of RLRS but in units of "RLRS per PGS". If PGS was, say, height in cm, you would interpret β as "change in RLRS per cm of height".

However, in Figure 2 they only state they are plotting "effect size". They do not, however, report what effect size they actually use in the paper. This is bad. Effect size is not one thing, but rather a family of approaches meant to quantify the relative size of an effect, usually relative to variability. However, unstandardized regression coefficients (i.e., "β") are also used as effect sizes.

Peaking at the code for the paper, I think this is probably what they are reporting: raw β coefficients that they then label "effect size", but I didn't go through the trouble to actually download and run the code and verify I'm reading their output correctly. It's quite a jumble.

However, they do mention:

The “selection effect”, β, reflects the strength of natural selection within the sample. In fact, since polygenic scores are normalized, β is the expected polygenic score among children of the sample (Beauchamp 2016).Footnote1

and Footnote1 is:

The selection effect β equals Cov(RLRS, PGS)/Var(PGS). Since PGS are normalized to variance 1 and mean 0, this reduces to Cov(RLRS,PGS)=E(RLRS×PGS)−E(RLRS)E(PGS)=E(RLRS×PGS) . This is the polygenic score weighted by relative lifetime reproductive success, which is the average polygenic score in the next generation (Robertson 1966).

Those conclusions seem consistent with the idea that these are indeed the unstandardized regression coefficients. It also seems like these are better numbers to use to interpret the results of the paper, rather than converting to number of children.

So, in summary, it is not trivial to convert these to "number of expected children", but if you knew the PGS for a given individual, you would multiply that by these numbers and add to α to get the RLRS. If α is exactly 1 and the PGS for an individual is 1, then yes, β=0.03 would mean their RLRS is 1.03 which would be 3% more children than the reference, not considering any of the other possible weighting or confounding factors.