Rational inferences about departures from Hardy-Weinberg equilibrium

Abstract

Previous studies have explored the use of departure from Hardy-Weinberg equilibrium (DHW) for fine mapping Mendelian disorders and for general fine mapping. Other studies have used Hardy-Weinberg tests for genotyping quality control. To enable investigators to make rational decisions about whether DHW is due to genotyping error or to underlying biology, we developed an analytic framework and software to determine the parameter values for which DHW might be expected for common diseases. We show analytically that, for a general disease model, the difference between population and Hardy-Weinberg expected genotypic frequencies (delta) at the susceptibility locus is a function of the susceptibility-allele frequency (q), heterozygote relative risk (beta), and homozygote relative risk (gamma). For unaffected control samples, is a function of risk in nonsusceptible homozygotes (alpha), the population prevalence of disease (KP), q, beta, and gamma. We used these analytic functions to calculate and the number of cases or controls needed to detect DHW for a range of genetic models consistent with common diseases (1.1 < or = gamma < or = 10 and 0.005 < or = KP < or = 0.2). Results suggest that significant DHW can be expected in relatively small samples of patients over a range of genetic models. We also propose a goodness-of-fit test to aid investigators in determining whether a DHW observed in the context of a case-control study is consistent with a genetic disease model. We illustrate how the analytic framework and software can be used to help investigators interpret DHW in the context of association studies of common diseases.

Figure 1

Δ_p plotted versus the susceptibility-allele frequency for patients. A, B, and D, Data points are as follows: γ=1.1 (blackened diamonds), γ=1.3 (unblackened triangles), γ=1.5 (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Dominant model. B, Recessive model. C, Additive model. Since γ<2 would not satisfy our definition of an additive model as γ=2β and β>1, the data points in C are as follows: γ=2.2 (β=1.1) (blackened diamonds), γ=2.6 (β=1.3) (unblackened triangles), γ=3 (β=1.5) (blackened triangles), γ=5 (blackened squares), γ=2 (unblackened diamonds). D, Multiplicative model.

Figure 1

Δ_p plotted versus the susceptibility-allele frequency for patients. A, B, and D, Data points are as follows: γ=1.1 (blackened diamonds), γ=1.3 (unblackened triangles), γ=1.5 (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Dominant model. B, Recessive model. C, Additive model. Since γ<2 would not satisfy our definition of an additive model as γ=2β and β>1, the data points in C are as follows: γ=2.2 (β=1.1) (blackened diamonds), γ=2.6 (β=1.3) (unblackened triangles), γ=3 (β=1.5) (blackened triangles), γ=5 (blackened squares), γ=2 (unblackened diamonds). D, Multiplicative model.

Figure 2

Δ_c plotted versus the susceptibility-allele frequency for controls. A, B, and D, Data points are as follows: γ=1.1 (blackened diamonds), γ=1.3 (unblackened triangles), γ=1.5 (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Dominant model, K_P=0.1. B, Recessive model, K_P=0.2. C, Additive model, K_P=0.01. As in figure 1, because of our definition of an additive model (γ=2β and β>1), the data points in C are as follows: γ=4 (β=2) (unblackened diamonds), γ=2.2 (β=1.1) (blackened diamonds), γ=2.6 (β=1.3) (unblackened triangles), γ=3 (β=1.5) (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). D, Multiplicative model, K_P=0.05.

Figure 2

Δ_c plotted versus the susceptibility-allele frequency for controls. A, B, and D, Data points are as follows: γ=1.1 (blackened diamonds), γ=1.3 (unblackened triangles), γ=1.5 (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Dominant model, K_P=0.1. B, Recessive model, K_P=0.2. C, Additive model, K_P=0.01. As in figure 1, because of our definition of an additive model (γ=2β and β>1), the data points in C are as follows: γ=4 (β=2) (unblackened diamonds), γ=2.2 (β=1.1) (blackened diamonds), γ=2.6 (β=1.3) (unblackened triangles), γ=3 (β=1.5) (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). D, Multiplicative model, K_P=0.05.

Figure 3

A, Number of patients needed to detect DHW as the susceptibility-allele frequency changes at a significance level of 5% and 50% power. Data points are as follows: dominant model with γ=1.3 (unblackened triangles), dominant model with γ=10 (unblackened circles), recessive model with γ=1.5 (blackened triangles), recessive model with γ=2 (unblackened diamonds), additive model with γ=2.2 (blackened diamonds), and additive model with γ=5 (blackened squares). B, Number of controls needed to detect DHW as the susceptibility-allele frequency changes at a significance level of 5% and 50% power. Data points are as follows: dominant model with K_P=0.2 and γ=10 (blackened circles), recessive model with K_P=0.05 and γ=10 (blackened circles), recessive model with K_P=0.2 and γ=5 (unblackened squares), additive model with K_P=0.2 and γ=5 (blackened squares), multiplicative model with K_P=0.1 and γ=10 (blackened squares with white cross), and multiplicative model with K_P=0.2 and γ=5 (blackened squares with white star). C, Same data points as A but assessed at 80% power. D, Same data points as B but assessed at 80% power.

Figure 3

A, Number of patients needed to detect DHW as the susceptibility-allele frequency changes at a significance level of 5% and 50% power. Data points are as follows: dominant model with γ=1.3 (unblackened triangles), dominant model with γ=10 (unblackened circles), recessive model with γ=1.5 (blackened triangles), recessive model with γ=2 (unblackened diamonds), additive model with γ=2.2 (blackened diamonds), and additive model with γ=5 (blackened squares). B, Number of controls needed to detect DHW as the susceptibility-allele frequency changes at a significance level of 5% and 50% power. Data points are as follows: dominant model with K_P=0.2 and γ=10 (blackened circles), recessive model with K_P=0.05 and γ=10 (blackened circles), recessive model with K_P=0.2 and γ=5 (unblackened squares), additive model with K_P=0.2 and γ=5 (blackened squares), multiplicative model with K_P=0.1 and γ=10 (blackened squares with white cross), and multiplicative model with K_P=0.2 and γ=5 (blackened squares with white star). C, Same data points as A but assessed at 80% power. D, Same data points as B but assessed at 80% power.

Figure 4

Sibling relative risk for dominant models with K_P=0.2 and varied γ values: γ=10 (unblackened circles), γ=5 (blackened squares), γ=2 (unblackened diamonds), and γ=1.5 (blackened triangles).

Figure 5

Simulations with the goodness-of-fit test. A, 1,000 simulations of a disease locus constructed under a general model (q=0.20, α=0.12, β=2.67, and γ=4.33), a dominant model (q=0.20, α=0.11, and β=γ=3.27), and a recessive model (q=0.20, α=β=0.18, and γ=3.78), in which the population prevalence is high (K_P=0.20). Each simulation, in which DHW was observed in patients and/or controls, was assessed using the goodness-of-fit test and was compared with a simulated distribution of 1,000 χ² values, with 1 df for a general model and 2 df for a dominant or recessive model. B, 1,000 simulations of a disease locus at a lower population prevalence (K_P=0.005) than A constructed under a general model (q=0.10, α=0.003, β=4.17, and γ=10.67), a dominant model (q=0.10, α=0.0027, and β=γ=5.48), and a recessive model (q=0.10, α=β=0.0048, and γ=5.17).