The large sample size of UK Biobank, coupled with the large number of phenotypes only completed by a subset of participants and the number of rarer binary phenotypes with lower statistical power, provides and excellent opportunity to observe how LDSR results behave across GWAS sample sizes. We consider that relationship here, including providing estimated approximate power curves for detecting \(h^2_g\) in LD Score regression (more details in Methods).
In looking at these trends we largely focus on effective sample, defined for binary phenotypes as
\[N_{eff} = \frac{4}{\frac{1}{N_{cases}}+\frac{1}{N_{controls}}}\] and as the standard sample size for non-binary phenotypes. We expect this effective N to better capture statistical power for rare binary phenotypes. We compare how \(N\) and \(N_{eff}\) relate to the significance of \(h^2_g\) results below.
The relationship between the SNP heritability estimate and sample size is of interest given we previously observed evidence that \(h^2_g\) estimates may be downwardly biased at low sample sizes. We’ve revisited that question in our evaluation of confidence in the current results, and reproduce the relevant figures here.
Note: Gray line is loess fit of liability scale \(h^2_g\) as a function of effective sample size. Colors indicate our confidence rating as described in the Methods section.
Note: Plot restricted to \(N_{eff} <\) 200,000 and \(h^2_g\) between -0.2 and 0.4 for visibility.
Under the expectation of the LDSR model that the intercept equals \(1 + N\alpha\), we estimate \(\alpha = \frac{\text{intercept}-1}{n}\) as the amount of confounding or model misspecification adjusted for sample size. Compared to the ratio, this quantity shouldn’t depending on the amount of polygenic signal contributing to the mean \(\chi^2\). Then among phenotypes with at least some confidence in the LDSR results, we split the phenotypes according to deciles of the estimated \(\alpha\) and fit loess curves for the relationship between effective sample size \(N_{eff}\) and estimated SNP hertiability \(h^2_g\). Read more about this analysis in the Methods section.
As with most statistics, we expect potentially stronger significance and more precise estimates of the SNP heritability as the GWAS sample size increases. As shown below we generally observe that trend, but also note that phenotypes with higher \(h^2_g\) estimates tend to have less precise estimates. One possible explanation is that the regression weights used by LDSR are iterated with the \(h^2_g\) estimates and optimized assuming an infinitesimal model, and so LDSR estimates may become less efficient as causal variants become more sparse and have stronger effects.
We can also confirm here our expectation that the power of LDSR depends primarily on the effective sample size, rather than total sample size or the prevalence.
Note: Colors reflect the magnitude of the \(h^2_g\) estimate truncated to values between 0 and 0.5.