Estimation of genotype relative risks from pedigree data by retrospective likelihoods

Abstract

Pedigrees collected for linkage studies are a valuable resource that could be used to estimate genetic relative risks (RRs) for genetic variants recently discovered in case-control genome wide association studies. To estimate RRs from highly ascertained pedigrees, a pedigree "retrospective likelihood" can be used, which adjusts for ascertainment by conditioning on the phenotypes of pedigree members. We explore a variety of approaches to compute the retrospective likelihood, and illustrate a Newton-Raphson method that is computationally efficient particularly for single nucleotide polymorphisms (SNPs) modeled as log-additive effect of alleles on the RR. We also illustrate, by simulations, that a naïve "composite likelihood" method that can lead to biased RR estimates, mainly by not conditioning on the ascertainment process-or as we propose-the disease status of all pedigree members. Applications of the retrospective likelihood to pedigrees collected for a prostate cancer linkage study and recently reported risk-SNPs illustrate the utility of our methods, with results showing that the RRs estimated from the highly ascertained pedigrees are consistent with odds ratios estimated in case-control studies. We also evaluate the potential impact of residual correlations of disease risk among family members due to shared unmeasured risk factors (genetic or environmental) by allowing for a random baseline risk parameter. When modeling only the affected family members in our data, there was little evidence for heterogeneity in baseline risks across families.

Fig. 1

(A) Simulated distribution of mle β̂ from the full retrospective likelihood for pedigrees with four full sibs; simulations centered about the true, β = 0.405. (B) Simulated distribution of mle β̂ from the composite likelihood for four full sibs and (C) pairs of cousins. Results show upward bias of naïve composite likelihood over true β = 0.405.

Fig. 2

Full retrospective likelihood for simulated data for 4 affecteds sibs, and then “stepped-down” on the same data to naïve marginals for 3, 2, and 1 affected sibs. The mle β̂, printed above each marginal likelihood, is shown to be upward biased over the true simulating value of β = 0.405, with bias increasing as fewer affected subjects are used for an incorrect marginal likelihood.

Fig. 3

Cousin pedigrees used for calculation of relative efficiencies for affecteds-only vs. affecteds and unaffecteds in retrospective likelihoods.

Fig. 4

Relative efficiency of retrospective likelihood for affecteds-only vs. affecteds and unaffecteds. Relative efficiency on y-axis is larger than 1.0 when affecteds-only is more efficient than affecteds + unaffecteds for relative risk estimation. MAF = minor allele frequency; OR = odds ratio per allele; Prevalence = population disease prevalence.

Fig. 5

Results from fitting the retrospective likelihood to the Mayo Clinic pedigrees for 28 SNPs reported to be associated with prostate cancer. The upper panel illustrates the mle of the per-allele relative risk (e^β̂) and its 95% confidence interval. For four SNPs, the variance of the baseline risk was estimated to be non-zero; for these four SNPs the mle that accounts for heterogeneity is depicted as a black square. The lower panel is the −log₁₀ (P-value) from the likelihood ratio test of the null hypothesis H_o: β = 0.

Fig. 6

Contrast of results from fitting the retrospective likelihood for 28 SNPs to the Mayo Clinic linkage pedigrees vs. case-control samples. All analyses assumed log-additive effects of a risk-allele, and the broken vertical lines are 95% confidence intervals on the case-control odds ratio estimates.

Fig. 7

Standardized U scores per pedigree, vs. pedigree number, for two SNPs. The numbers in these figures give the count of the number of risk alleles in each pedigree, and the numbers are colored to make the numbers clearly distinguishable.