Liability threshold modeling of case–control status and family history of disease increases association power


Family history of disease can provide valuable information in case–control association studies, but it is currently unclear how to best combine case–control status and family history of disease. We developed an association method based on posterior mean genetic liabilities under a liability threshold model, conditional on case–control status and family history (LT-FH). Analyzing 12 diseases from the UK Biobank (average N = 350,000) we compared LT-FH to genome-wide association without using family history (GWAS) and a previous proxy-based method incorporating family history (GWAX). LT-FH was 63% (standard error (s.e.) 6%) more powerful than GWAS and 36% (s.e. 4%) more powerful than the trait-specific maximum of GWAS and GWAX, based on the number of independent genome-wide-significant loci across all diseases (for example, 690 loci for LT-FH versus 423 for GWAS); relative improvements were similar when applying BOLT-LMM to GWAS, GWAX and LT-FH phenotypes. Thus, LT-FH greatly increases association power when family history of disease is available.

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Fig. 1: Overview of LT-FH and other methods.
Fig. 2: LT-FH is well calibrated and increases association power in simulations.
Fig. 3: LT-FH increases association power across 12 diseases from the UK Biobank.
Fig. 4: Loci identified by LT-FH replicate in independent datasets.

Data availability

This study analyzed data from the UK Biobank, which are publicly available by application ( We have publicly released summary association statistics computed by applying our LT-FH method to UK Biobank data; LT-FH summary association statistics for 12 diseases are available at

Code availability

We have publicly released open-source software implementing our LT-FH method; LT-FH software (v1 and v2):; BOLT-LMM v2.3 software:, LTSOFT software:; and PLINK software:


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We are grateful to L. O’Connor, O. Weissbrod, N. Zaitlen, G. Kichaev and A. Gusev for helpful discussions and E.M. Pedersen for computational suggestions. This research was funded by NIH grants R01 HG006399 (N.P. and A.L.P.), R01 MH101244 (A.L.P.), R01 MH107649 (A.L.P.), NSF CAREER award DBI-1349449 (S.G. and A.L.P.) and 5T32CA009337-32 (M.L.A.H.). P.-R.L. was supported by the Next Generation Fund at the Broad Institute of MIT and Harvard and a Sloan Research Fellowship. This research was conducted using the UK Biobank resource under application no. 10438.

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M.L.A.H. and A.L.P. designed the experiments. M.L.A.H. performed the experiments and statistical analysis; N.P. assisted in proving the equivalence to a score test. M.L.A.H., S.G., P.-R.L. and A.L.P. analyzed the data. M.L.A.H. and A.L.P. wrote the manuscript with assistance from S.G., P.-R.L. and N.P.

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Correspondence to Margaux L. A. Hujoel or Alkes L. Price.

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Extended data

Extended Data Fig. 1 QQ plots from simulations with default parameter settings.

We report quantile-quantile (QQ) plots for null SNPs in simulations with default parameter settings. Results are based on 10 simulation replicates. These QQ plots compare the observed distribution of p-values with the standard uniform distribution. We plot the observed −log10(p) as a function of \(- \log _{10}\left( {\frac{{rank}}{{n + 1}}} \right)\) and the 95% confidence bands are constructed pointwise using the beta distribution.

Extended Data Fig. 2 Distribution of LT-FH phenotypes for 12 UK Biobank diseases.

We plot the distribution of the LT-FH phenotype for each disease. We also report the kurtosis for both GWAS and LT-FH; Pearson’s measure of kurtosis, \(\kappa = \frac{{E\left[ {\left( {X - \mu } \right)^4} \right]}}{{\left( {E\left[ {\left( {X - \mu } \right)^2} \right]} \right)^2}}\), is calculated using the R package moments.

Extended Data Fig. 3 Impact of modifying the LT-FH method to incorporate age information as a function of the liability threshold model parameter for age for 12 UK Biobank diseases.

We plot the increase in number of independent loci for \({\mathrm{LT}}{\hbox{-}}{\mathrm{FH}}_{no{\hbox{-}}sib,age}^{PA}\) relative to for \({\mathrm{LT}}{\hbox{-}}{\mathrm{FH}}_{no {\hbox{-}} sib}^{PA}\) (Supplementary Table 32) against the liability threshold model parameter |cage|(Supplementary Table 30).

Extended Data Fig. 4 LT-FH increases association power across 12 diseases from the UK Biobank in analyses incorporating related individuals.

We report results of GWAS using BOLT-LMM on related Europeans, GWAX using BOLT-LMM on unrelated Europeans, and LT-FH using BOLT-LMM on related Europeans using only case–control status for all sibling pairs and parent-offspring pairs within the set of target samples. Numerical results are reported in Supplementary Table 37.

Extended Data Fig. 5 Strong concordance between GWAS BOLT-LMM-inf effect sizes and transformed LT-FH BOLT-LMM-inf effect sizes.

We plot GWAS BOLT-LMM-inf effect sizes and transformed LT-FH BOLT-LMM-inf effect sizes for genome-wide significant effect sizes (P≤5*10−8 for both GWAS and LT-FH BOLT-LMM-inf). We note that BOLT-LMM only outputs effect size estimates for BOLT-LMM-inf, the BOLT-LMM approximation to the infinitesimal mixed model. Our effect size for GWAS is the outputted βGWAS,BOLTLMMinf (per-allele observed scale) and for LT-FH we estimate a (per-allele observed scale) effect size as \(\begin{array}{l}\beta = \frac{{\beta _{LT - FH,BOLT - LMM - inf}}}{{se\left( {\beta _{LT - FH,BOLT - LMM - inf}} \right)\sqrt {N_{GWAS} \ast c} }} \frac{{\sqrt {K(1 - K)} }}{{\sqrt {2\left( {MAF} \right)\left( {1 - MAF} \right)} }}\end{array}\), where c is the boost in Neff for LT-FH relative to GWAS, K is disease prevalence in GWAS and MAF is the minor allele frequency of the SNP.

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Supplementary Note and Supplementary Tables 1–45

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Hujoel, M.L.A., Gazal, S., Loh, P. et al. Liability threshold modeling of case–control status and family history of disease increases association power. Nat Genet 52, 541–547 (2020).

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