Extended Data Fig. 5: Comparison of weighted least-squares and maximum likelihood solvers.
From: Evaluating and improving heritability models using summary statistics
The plots compare likelihood ratio test (LRT) statistics (twice the improvement in log likelihood relative to the null model), computed using loglSS, our approximate model likelihood. We consider six heritability models (see Supplementary Table 13 for definitions), estimating parameters using either maximum likelihood (our recommended approach) or weighted least-squares regression (the approach used by LDSC and previously by SumHer). Note that when we estimate parameters for the Baseline and Baseline LD Models using weighted least-squares regression, we frequently obtain negative E[Sj]; so that we can compute loglSS, we replace these with 10−6. These plots show that for the GCTA, GCTA-LDMS-I, LDAK and LDAK + 24Fun Models (the simpler models), the two solvers result in near-identical model fit. However, for the Baseline and Baseline LD Models (the more complex models), weighted least-squares regression often results in a worse fit, because it does not respect that test statistics are approximately Gamma distributed. Note that the reason we observe discordance between the weighted least-squares estimates from LDSC and SumHer (mainly evident for the Baseline Model), is because the SumHer weighted least-squares solver is always iterative, whereas the LDSC solver is iterative when provided with a single-parameter heritability model, but one-step when provided with a multi-parameter model.
