Activity 231 a Quick Review of Hardy Weinberg Population Genetics

• Journal List
• Am J Hum Genet
• v.76(6); 2005 Jun
• PMC1196455

Am J Hum Genet. 2005 Jun; 76(6): 967–986.

Rational Inferences about Departures from Hardy-Weinberg Equilibrium

Jacqueline M. Wittke-Thompson

Departments of ¹Human Genetics and ²Medicine, The University of Chicago, Chicago

Anna Pluzhnikov

Departments of ¹Human being Genetics and ²Medicine, The Academy of Chicago, Chicago

Nancy J. Cox

Departments of ¹Human being Genetics and ²Medicine, The University of Chicago, Chicago

Received 2005 Jan 18; Accepted 2005 Mar 24.

Abstract

Previous studies have explored the use of difference from Hardy-Weinberg equilibrium (DHW) for fine mapping Mendelian disorders and for general fine mapping. Other studies accept used Hardy-Weinberg tests for genotyping quality control. To enable investigators to make rational decisions nigh whether DHW is due to genotyping error or to underlying biological science, we developed an analytic framework and software to determine the parameter values for which DHW might exist expected for common diseases. Nosotros show analytically that, for a general disease model, the divergence betwixt population and Hardy-Weinberg–expected genotypic frequencies (Δ) at the susceptibility locus is a function of the susceptibility-allele frequency (q), heterozygote relative risk (β), and homozygote relative risk (γ). For unaffected control samples, Δ is a function of run a risk in nonsusceptible homozygotes (α), the population prevalence of disease (G _P ), q, β, and γ. We used these analytic functions to calculate Δ and the number of cases or controls needed to detect DHW for a range of genetic models consequent with common diseases (1.1 ⩽ γ ⩽ 10 and 0.005 ⩽ 1000 _P ⩽ 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 example-control study is consistent with a genetic illness model. We illustrate how the analytic framework and software can be used to help investigators interpret DHW in the context of association studies of mutual diseases.

Introduction

Hardy-Weinberg equilibrium (HWE) has been used for more than a century to better sympathize genetic characteristics of populations. In 1902, William E. Castle noted that, if the selective removal from the general population of individuals who accept a recessive genotype at a particular locus ceases, then the future generation volition establish an equilibrium value of recessive alleles. His conclusions presume that one allele is ascendant with respect to the other(s), that the alleles exercise not affect fertility, that at that place is no migration into or out of the population, and that the population is large and randomly mating (Castle 1903; Li 1967). Five years afterwards, M. H. Hardy and West. Weinberg individually came to the same determination but noted that the population allele frequencies could be used to summate the equilibrium-expected genotypic proportions (Hardy 1908; Weinberg 1908). If p is the frequency of ane allele (A) and q is the frequency of the alternative allele (a) for a biallelic locus, and so the HWE-expected frequency will be p ² for the AA genotype, 2pq for the Aa genotype, and q ^two for the aa genotype. The three genotypic proportions should sum to 1, every bit should the allele frequencies (Hardy 1908; Weinberg 1908).

The most common way to assess HWE is through a goodness-of-fit χ^two exam (Weir 1996). The null hypothesis is that alleles are called randomly, and the genotypic proportions thus follow HWE-expected proportions (i.e., p ², 2pq, and q ²). Alternatively, the second allele is dependent on the first allele being selected, resulting in the genotypic proportions deviating from the HWEexpected proportions. The χ² test for HWE has k(k-1)/2 df, where thou is the number of alleles at the locus existence studied (Weir 1996). Or, more than intuitively, the degrees of freedom can be calculated by chiliad-k , where g is the number of possible genotypes and yard is the number of alleles. For example, a SNP that has two alleles and 3 possible genotypes yields 1 df for the χ² exam that assesses the significance of deviation from HWE (DHW).

Testing for HWE is normally used for quality control of large-scale genotyping and is ane of the few means to identify systematic genotyping errors in unrelated individuals (Gomes et al. 1999; Hosking et al. 2004). Yet, at that place is little consensus on the correct threshold for identifying DHW in the context of large-calibration studies or on what to do with markers that fit all other quality-command criteria but testify a pregnant DHW, given the written report-specific threshold. Some association studies exercise not consider DHW in patients to indicate genotyping error but adopt to assume a biological explanation for DHW in a patient sample, while requiring control or random samples to be in HWE (Oka et al. 1999; Levecque et al. 2003; Nejentsev et al. 2003). Others crave that both the patient and the control samples be in HWE for the marker to be used in farther analyses (Martin et al. 2000; Xu et al. 2001; Wang et al. 2003). Bonferroni correction is unremarkably used to correct for multiple testing when many markers are tested for HWE but is a conservative approach if the markers are correlated (e.g., are in linkage disequilibrium [LD]). Another widespread approach is to assign a significance level, such as five% or 1%, and to test each marker without regard for whether markers are independent. Sometimes, markers showing DHW are removed from a information set because subsequent genetic analysis requires HWE to be assumed. Yet, since the design of association studies is to sectionalisation cases and controls in the full general population on the basis of their phenotype and genetic composition, cases and controls at a disease locus may be expected to show DHW under sure conditions. There is little guidance on how to distinguish between markers whose DHW is due to hazard, genotyping mistake, or failure of one of the requisite assumptions of HWE from markers whose DHW is due to their proximity to an allele affecting a phenotype on which the sample is ascertained (Xu et al. 2002).

Our focus is to empathise DHW observed in the context of association studies. When an investigator observes a marker with a meaning DHW, a primal question is "Are deviations from HWE due to genotyping error, chance, failure of assumptions underlying Hardy-Weinberg expectations, or, instead, to the underlying genetic disease model?" To accost this question, we developed an analytic framework for assessing the difference between population and HWE-expected genotypic proportions (Δ), as well as software tools for calculating expectations over a range of single-locus genetic disease models. We use a unmarried-locus context in our examination of DHW because information technology enables u.s. to represent the marginal effects of the detail susceptibility locus being examined. For full general illness models, nosotros prove that Δ in patients at the disease locus is determined by the population susceptibility-allele frequency (q), the homozygote relative take a chance (γ), and the heterozygote relative risk (β). For "true" control samples (i.e., those ascertained by being unaffected with respect to disease status), the value of Δ depends on the chance to nonsusceptible homozygotes (α), q, γ, and β. Using these formulations, nosotros have examined a wide range of parameter values for genetic models normally believed to exist consistent with observations for circuitous disorders. Our results provide back up for the notion that systematic examination of HWE has been underutilized every bit a tool for fine mapping. To aid investigators in distinguishing a DHW that is generated by the biological model underlying disease manual from one that is generated by genotyping errors, chance, or failure of the requisite assumptions of HWE, we propose a general goodness-of-fit test to make up one's mind whether the observed genotype counts are significantly different from the expected genotype counts in patients and controls for the genetic disease model that all-time fits the observed data. We provide software for both the investigation of DHW in a generalized unmarried-locus context and the exam of specific observations of DHW.

Methods

Common Illness Models

The prevalence of disease in the general population is divers as K _P =p ²α+twopqαβ+q ⁱⁱαγ, where p is the population frequency of the wild-type allele (A), q is the frequency of the disease-susceptibility allele (a), and α is the baseline penetrance of disease in homozygotes without a take a chance allele at this locus. By modeling the population prevalence of illness in the general population in this manner, nosotros explicitly assume that the susceptibility locus will be in HWE in the full general population. Note that, although this parameterization is for a single-locus model, there may be many genetic and nongenetic risk factors, in addition to the detail region being studied, that will determine values of the parameters α, β, and γ. Thus, we use this approach to model the marginal effects of a particular model, with the total recognition that, in general, in that location will be nongenetic risk factors and many other loci with genetic variation that touch the phenotype.

Nosotros explored dominant, recessive, additive, and multiplicative genetic models with Yard _P and γ values that are feature of a variety of common diseases, for which Thousand _P values range from 0.005 to 0.2 and γ is equal to one.1, i.iii, 1.5, two, five, or 10. All dominant, recessive, and additive models examined correspond to sibling relative risk (λ_south) values <2.5, although most models have λ _{due south} ⩽1.1. Multiplicative models generally have higher λ_s values than dominant, recessive, and additive models with identical parameters. The maximum λ_s for a multiplicative model is iv.20, which occurs at the lowest population prevalence (K _P =0.005) and the highest homozygote relative risk (γ=10), only near multiplicative models considered take λ_s values <1.ii.

The Deviation between Population and Expected Genotypic Frequencies (Δ)

Weir (1996) divers a variable, Δ, as the difference between population and HWE-expected genotypic frequencies. Δ is population specific (e.g., Δ_p for patients and Δ _c for unaffected controls), and it ranges from to .

To better understand the magnitude and direction of DHW for different genetic disease models, we defined Δ in terms of key parameters of the genetic model, specified separately in patients and controls, and characterized how parameters interact to touch directionality of DHW (i.e., if Δ<0, there is a deficiency of homozygotes and an excess of heterozygotes; if Δ>0, there is an excess of homozygotes and a deficiency of heterozygotes). For patients in a general disease model,

A more thorough derivation of the to a higher place equation and the simplified equations for the archetype ascendant, recessive, and additive models are given in appendix A. For a multiplicative model, Δ_p is equal to 0 and there is no expected DHW. Figure 1 highlights the management and magnitude of Δ_p every bit the susceptibility-allele frequency at the susceptibility locus varies from 0 to ane.

Δ_p plotted versus the susceptibility-allele frequency for patients. A, B, and D, Data points are every bit follows: γ=1.1 (blackened diamonds), γ=1.iii (unblackened triangles), γ=ane.5 (blackened triangles), γ=ii (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Dominant model. B, Recessive model. C, Additive model. Since γ<ii 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.three) (unblackened triangles), γ=3 (β=one.5) (blackened triangles), γ=5 (blackened squares), γ=two (unblackened diamonds). D, Multiplicative model.

Similarly, for an unaffected control sample,

and equations for the archetype recessive, classic dominant, additive, and multiplicative models are given in appendix B. Figure ii shows the direction and magnitude of Δ_c for several dominant, recessive, additive, and multiplicative models.

Δ_c plotted versus the susceptibility-allele frequency for controls. A, B, and D, Information points are as follows: γ=one.1 (blackened diamonds), γ=ane.3 (unblackened triangles), γ=one.5 (blackened triangles), γ=ii (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). A, Ascendant model, K _P =0.i. 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 equally follows: γ=4 (β=ii) (unblackened diamonds), γ=ii.ii (β=i.1) (blackened diamonds), γ=2.vi (β=1.three) (unblackened triangles), γ=3 (β=1.v) (blackened triangles), γ=2 (unblackened diamonds), γ=5 (blackened squares), and γ=10 (unblackened circles). D, Multiplicative model, K _P =0.05.

Number of Individuals Needed to Detect DHW

The χ^two test of HWE for patients tin be simplified to the following:

where North _p is the number of patients and is the susceptibility-allele frequency estimated in patients (Weir 1996). A χ² value can also be determined for controls by using the susceptibility-allele frequency in controls (), Δ _c , and the number of controls in the sample (N _c). The power of the χ² test tin can be deemed for past using a noncentral χⁱⁱ distribution with noncentrality parameter

(Weir 1996).

Given the above equations, it is straightforward to determine the number of individuals required to detect DHW for a specified level of significance and power, for patients and controls under any disease model. In figure iii, results for a multifariousness of models are graphed in terms of N _p or N _c and the overall susceptibility-allele frequency for patients or controls. The results presented in figure 3 A (patients) and ​ three B (controls) apply a cardinal χ² distribution corresponding to a significance level of 5% and 50% power, and the results presented in figure 3 C (patients) and ​ three D (controls) use a noncentral χⁱⁱ distribution corresponding to a significance level of 5% and 80% power.

A, Number of patients needed to detect DHW as the susceptibility-allele frequency changes at a significance level of 5% and 50% ability. Information points are as follows: dominant model with γ=1.3 (unblackened triangles), dominant model with γ=ten (unblackened circles), recessive model with γ=1.5 (blackened triangles), recessive model with γ=two (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 every bit follows: dominant model with Grand _P =0.2 and γ=ten (blackened circles), recessive model with 1000 _P =0.05 and γ=x (blackened circles), recessive model with K _P =0.two and γ=v (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 cantankerous), and multiplicative model with G _P =0.ii and γ=five (blackened squares with white star). C, Aforementioned information points as A but assessed at 80% power. D, Same data points equally B only assessed at fourscore% ability.

For congruence with other studies reporting the results in terms of λ_s, the risk of disease in siblings of an afflicted private relative to the risk in the full general population, are summarized in appendix C with respect to mutual affliction models. We too summarize results with respect to λ_south and Δ in figure four.

Sibling relative risk for dominant models with K _P =0.ii and varied γ values: γ=10 (unblackened circles), γ=5 (blackened squares), γ=2 (unblackened diamonds), and γ=1.v (blackened triangles).

Goodness-of-Fit Test

Given the analytic framework characterized above and a set of observations in patients and controls with DHW in one or both samples, we keep to identify the genetic affliction model with the best fit to the genotypic proportions observed in patients and controls (constrained by the known lifetime prevalence of disease). If that "all-time-fit" model is, nevertheless, a poor fit to the observed data, as assessed by a χ² examination with 1 df for a general model and ii df for a restrictive (dominant, recessive, additive, or multiplicative) model (run into appendix D), then it is unlikely that the underlying genetic illness model has generated the observed DHW; thus, alternative explanations for the DHW, including adventure, genotyping error, and/or violations of the requisite assumptions of HWE, must be considered.

Examples

We chose several examples from the literature that illustrate how our results and software can assist investigators distinguish a DHW consistent with genetic models from one that is not. The first example involves a DHW in patients but not in controls, with data derived from an association study of Crohn disease and a rare frameshift polymorphism (Leu3020fsinsC) in CARD15/NOD2 (Ogura et al. 2001; J. Cho, personal communication). The 2nd case includes data from an association study that identified a highly significant association (P=3.3×10^-6) between LTA and myocardial infarction (MI) (Ozaki et al. 2002), attributable in function to the result of significant DHW in both patients and controls in contrary directions. We also took advantage of several recent surveys of polymorphisms with DHW (Xu et al. 2002; Györffy et al. 2004; Kocsis et al. 2004a, 2004b; Osawa et al. 2004) to determine the proportion that are consistent with a genetic disease model (tabular array one).

Table 1

Survey of DHW

			Genotype Distributions b
PubMed ID	Disease a	Cistron	Patients	Controls	K P	q	α	β	γ	χ2	P c
12221172	Alzheimer illness	A2M	61-48-23	98-77-9	.06	.264	.0513	1.085	ii.966	one.016	NS
12105308	Alzheimer disease	APOE	4-18-94	6-53-74	.06	.758	.0269	i.000	3.137	ane.013	NS
12105308	Alzheimer affliction	APOE	31-49-29	67-102-19	.06	.381	.0449	1.251	two.509	4.221	.040
12105308	Alzheimer disease	APOE	57-118-100	58-97-48	.06	.482	.0462	1.118	2.027	.547	NS
12105308	Alzheimer disease	APOE	120-361-194	206-309-142	.06	.458	.0362	1.791	ii.265	ane.485	NS
11074789	Alzheimer disease	CST3	322-166-29	260-124-6	.05	.179	.0463	1.174	1.91	iii.914	.048
11468325	Alzheimer illness	CST3	140-33-vi	179-43-6	.06	.121	.059	1.000	2.114	4.064	NS
11468325	Alzheimer disease	CST3	137-34-viii	180-40-8	.06	.13	.0587	1.000	2.343	9.671	.008
12034804	Alzheimer illness	IL-aneβ	five-40-39	15-107-95	.06	.679	.0365	1.718	one.718	four.812	NS
10869806	Alzheimer disease	IL-6 promoter	110-148-44	34-22-12	.05	.35	.043	1.257	ane.377	5.111	.024
10815136	Alzheimer affliction	PS1	3-45-45	3-24-18	.05	.66	.0141	3.803	3.94	one.929	NS
11124296	Macular degeneration	mEPHX	42-24-32	95-38-33	.01	.326	.00767	1.000	3.874	40.423	≪.001
10843185	Autoimmune hypothyrodism	IL-4	2-seven-68	3-23-75	.02	.847	.00917	1.000	2.645	ii.909	NS
10781645	Acute myocardial infarction	5-HT 2A receptor	39-133-83	43-150-62	.05	.538	.0362	ane.435	1.568	eight.829	.003
11781417	Blepharospasm	D1.i	25-51-15	42-32-16	.001	.361	.000673	1.804	1.875	iv.485	.034
11142420	Breast cancer	CYP17	39-63-10	112-132-33	.05	.352	.0412	ane.367	i.367	3.161	NS
10698474	Chest cancer	CYP1B1	37-101-48	44-127-29	.05	.468	.0357	1.504	i.678	15.736	7.iii × x−5
10698474	Breast cancer	CYP1B1	26-68-27	26-78-17	.05	.464	.0379	1.446	ane.446	xi.11	.004
10698474	Breast cancer	CYP1B1	xi-32-21	18-49-12	.05	.469	.0309	1.599	ii.46	4.774	.029
11156391	Cancer toxicity (irinotecan)	UGT1A1	14-viii-4	79-10-iii	.01	.116	.00695	2.221	15.11	vii.947	.005
12787424	Chronic allograft nephropathy	TGF-βi	0-22-viii	23-83-67	.01	.613	0	xi.76	11.76	4.204	NS
12787424	Chronic allograft nephropathy	TNFα	l-24-1	132-31-ten	.01	.164	.00949	1.179	i.179	13.577	.001
10504487	Chronic renal failure	VDR	34-118-96	29-123-68	.005	.586	.00404	1.232	1.418	1.768	NS
10189842	Coeliac illness	CTLA-4	iv-28-69	21-47-62	.01	.655	.00332	i.861	4.82	four.539	.033
10189842	Coeliac illness	CTLA-4	i-7-23	xvi-36-54	.01	.674	.003	1.731	5.43	5.198	.023
11889073	Colorectal cancer	MTHFR	28-35-12	204-229-34	.05	.324	.0415	1.231	i.995	7.478	.006
11889073	Colorectal cancer	MTHFR	28-64-94	114-560-533	.05	.669	.0455	i.000	ane.223	ten.133	.006
11889073	Colorectal cancer	MTHFR	14-66-73	39-116-148	.05	.678	.0431	1.161	1.195	4.313	.038
10794488	Colorectal polyps	MTHFR	98-72-26	297-258-71	.05	.315	.0484	1.000	1.338	4.323	NS
12631667	Crohn disease	CYP1A1	2-20-129	5-22-122	.05	.879	.0325	1.000	1.698	7.497	.024
12631667	Crohn disease	EPXH	twenty-threescore-71	59-58-32	.05	.428	.0201	2.046	6.307	vi.759	.009
12631669	Crohn disease	NOD2	248-20-three	290-10-0	.05	.0206	.0478	1.929	20.71	.454	NS
11402126	Dementia	ACE	74-239-120	98-233-144	.one	.539	.0808	1.301	1.301	2.457	NS
12221164	Epilepsy	OPRM1	193-29-8	208-20-0	.01	.0451	.0092	1.592	18.719	.399	NS
9607207	Hypertension/nephroangiosclerosis	ACE	6-10-21	8-48-xix	.05	.557	.0312	one.000	ii.933	7.18	.028
10792336	IDDM	VDR	137-xvi-4	231-16-1	.05	.046	.0482	1.236	9.315	three.482	NS
11231353	IDDM nephropathy	GLUT1	3-29-10	21-sixty-23	.005	.495	.00139	4.482	4.482	three.604	NS
11097227	Lung cancer	HRAS1	17-86-209	21-134-276	.01	.789	.00882	1.000	ane.214	three.239	NS
11097227	Lung cancer	HRAS1	8-19-49	eight-38-54	.01	.71	.00742	1.000	1.692	3.583	NS
11097227	Lung cancer	TP53	nine-16-26	57-192-197	.01	.652	.00883	1.000	1.311	4.33	NS
11045785	Lung cancer, squamous cell	p53	xvi-44-73	61-212-237	.01	.668	.00823	ane.000	one.48	3.481	NS
9844142	Macroangiopathy	PAI-i	6-28-22	22-88-42	.05	.567	.0295	1.683	2.116	4.958	.026
11122322	Myocardial infarction	ACE	184-460-217	193-393-215	.1	.508	.0884	one.174	ane.174	1.748	NS
10712418	Myocardial infarction	TF-1208	227-361-218	197-349-186	.one	.483	.0956	one.000	ane.199	6.502	.039
10680782	Multiple sclerosis	GSTM3	276-97-14	221-64-15	.05	.161	.0496	1.000	1.336	11.722	.003
15338456d	NIDDM	RETN	48-85-xvi	63-83-12	.054	.336	.0396	1.649	ane.649	6.384	.041
10231446	NIDDM nephropathy	GLUT1	43-78-ten	77-41-half-dozen	.05	.219	.0269	3.199	3.199	.0884	NS
10231446	NIDDM nephropathy	GLUT1	12-48-4	77-41-6	.05	.219	.0153	6.818	6.818	1.099	NS
9844142	NIDDM nephropathy	PAI-one	64-116-28	66-80-31	.05	.388	.0409	1.357	i.357	2.616	NS
9844142	NIDDM nephropathy	PAI-1	28-58-12	66-80-31	.05	.392	.0383	1.484	1.484	2.745	NS
12675860	NIDDM nephropathy	TGF-βi	24-24-17	27-26-5	.3	.354	.258	1.000	ii.29	.537	NS
10964048	Oral cancer	CYP1A1	96-36-13	104-54-six	.01	.186	.00943	i.000	2.732	ane.638	NS
10964048	Oral cancer	CYP1A1	56-55-31	62-65-15	.01	.332	.0087	1.02	2.275	.0473	NS
11914402	Parkinson disease	Nurr1	162-48-15	174-45-2	.02	.112	.0183	one.172	5.822	.14	NS
11786085	Renal parenchymal scarring	TGF-β1	0-27-16	16-76-79	.01	.669	.001	11.11	11.11	3.973	NS
11786085	Renal parenchymal scarring	TGF-β1	13-27-3	73-72-26	.01	.303	.00551	4.001	3.95	.0627	NS
10430441	Stroke	NOS3	109-125-31	154-203-36	.15	.353	.15	one.000	ane.004	vii.466	.024
11027931	Venous thrombosis	β-fibrogen	2-6-82	0-22-163	.01	.926	.00892	1.000	1.141	9.55	.008
11027931	Venous thrombosis	Factor 7	67-19-v	148-33-4	.01	.115	.0094	1.096	4.351	1.758	NS
12753258	Vertebral fracture	COLIA1	10-2-v	75-35-6	.01	.two	.00754	1	9.125	2.24	NS

Results

Magnitude of Δ

As noted elsewhere, the larger the value of Δ, the more than the genotypic frequency departs from HWE expectations (Weir 1996). In patients, the magnitude of Δ_p is determined by q, γ, and β, as shown in equation (1) in a higher place. The maximum value for Δ_p, given a genetic illness model, occurs at the susceptibility-allele frequency that maximizes the HWE-expected proportion of heterozygotes affected with disease (one thousand _Aa ) and can be calculated past

For instance, the maximum Δ_p for a ascendant model with γ=ane.5 occurs when the allele frequency is 0.45, which corresponds to the highest proportion of HWE-expected heterozygotes among patients (g _Aa =0.5505) nether this model.

In controls, Δ_c is a office of M _P, α, β, γ, and q. Larger Δ_c values occur equally the prevalence of disease in the general population increases, whereas, in patients, K _P has no direct effect on the size of Δ_p. For example, a dominant model with γ=1.5 and q=0.33 leads to a Δ_c of 0.00209 when K _P =0.one and a Δ_c of 0.000193 when K _P =0.01. Larger Δ_c values also occur with higher γ values, as shown in figure 2. The susceptibility-allele frequency also affects the magnitude of Δ _c , with common allele frequencies (0.ane⩽q⩽0.8) corresponding to larger Δ_c values and with allele frequencies at the tails of the frequency spectrum respective to smaller Δ_c values.

Direction of Δ

The direction of Δ is vital to agreement whether an observed DHW can exist due to the underlying genetic model, and, for patients, the disease model specifies the management of Δ_p (table ii). The additive model is different than the ascendant, recessive, or multiplicative models, in that Δ_p tin be positive, negative, or zero. Δ_p is positive when 2 < γ < 4 and negative when γ>4. Withal, when γ=four, the additive model is equivalent to the multiplicative model (β=2 and γ=four), and Δ_p is equal to 0 (i.east., no DHW is expected). The general model is dependent on the value associated with β, where Δ_p is positive if γ>β² and negative if γ<β².

Tabular array 2

Direction of DHW, Given a Genetic Disease Model and Parameter Values^[Note]

	Management of DHW in
Model	Patients	Controls
Ascendant	−	+
Recessive	+	−
Additive	If γ<2, +;if γ>2, −;if γ=2, 0	−
Multiplicative	0	−
General	If γ>βtwo, +;if γ<βii, −	If γ+1+αβii>2β+αγ, −; if γ+1+αβii<2β+αγ, +

The management of Δ in controls is too dependent on the underlying genetic illness model, and, for dominant and recessive models, Δ_c is in the direction opposite that observed in patients. Interestingly, Δ_c for controls under the condiment model does not change with γ, equally observed for patients, but always shows an backlog of heterozygotes (Δ _c <0). However, at that place is an disproportion in the homozygote classes when Δ≠0 for both patients and controls, with patients showing an excess of affliction-susceptible homozygotes and controls showing an backlog of wild-type homozygotes. The multiplicative model generates Δ values for controls like to those generated past the recessive and additive models, with Δ _c <0. The direction of DHW for a full general illness model in controls is negative if γ+1+αβ²>2β+αγ and positive if the inequality is reversed (i.eastward., γ+1+αβ²<2β+αγ).

Number of Individuals Needed to Observe DHW for a Specified Significance and Power

As might be expected, fewer patients are needed to detect DHW as the genotype relative risk increases. Information technology is notable, yet, that depression γ values (due east.g., 1.v) tin produce DHW with relatively small samples of patients. For example, <500 patients are needed to detect DHW (at a significance level of 5% and l% power) under a recessive model with γ=1.5 and over a range of susceptibility-allele frequencies from 0.29 to 0.63. Every bit γ increases to 2, the range of susceptibility-allele frequencies increases to 0.11–0.81, with a minimum sample size of ∼130 patients when q=0.4 with 50% ability (fig. 3 A) and 267 patients with fourscore% power (fig. three C). An condiment model, all the same, requires a minimum of ∼750 patients at q=0.38 to observe DHW with γ=three, but the required sample size decreases to 176 patients with γ=2.2. Every bit shown above, Δ_p increases for condiment models as γ increases or decreases from iv. Therefore, at a significance level of 5% and 50% power, an additive model can lead to detection of DHW in 176 patients when either γ=2.2 or γ=vii.56. The difference between the ii observations, however, is the management of the DHW, in which the former (γ=2.two) shows an excess of homozygotes and the latter (γ=7.56) shows a deficiency of homozygotes.

The number of controls needed to observe DHW is generally much larger but is often still within the range of usually collected sample sizes for a wide range of models. Every bit with patients, the number of controls required to detect DHW decreases equally γ increases. But, unlike in the calculations for patients, K _P too determines the number of controls needed to detect DHW. For dominant models, <1,000 unaffected individuals are needed to notice DHW when Thou _P =0.two and γ⩾5 with 50% power (fig. 3 B), and <2,000 controls with 80% power (fig. 3 D). Considering of the specifications of the additive model, the same upshot is observed in controls as in patients, whereby a subtract in the number of controls needed to notice DHW occurs as γ increases or decreases from 4. Recessive models with γ⩾v can pb to detection of DHW in <200 controls with 50% power when Chiliad _P =0.two. When K _P decreases to as low as 0.05, ∼i,000 controls are needed to observe DHW for the recessive model with γ⩾10 and q=0.23 (fig. 3 B). Multiplicative models show DHW in a sample of <2,000 controls when K _P ⩾0.1 and γ⩾x and in a sample of <1,200 controls when K _P ⩾0.two and γ⩾5 (at 50% power and a significance level of v%). Generally, larger Δ_c values are associated with models specified by loftier G _P values (e.k., K _P ⩾0.2) and loftier γ values (e.g., γ⩾five).

Sibling Relative Risk Values that Represent to Expected DHW

We calculated λ_s for all models studied, to empathize the results for these common affliction models relative to those of previous studies that assessed power by using this parameter. Equally expected, λ_s increases as M _P decreases, and, overall, the resulting λ_s values (including those that are expected to give DHW in a small sample) are quite small. For ascendant models or recessive models (γ=i.5), in which DHW is expected for samples of <500 patients, λ_s is <1.03 (fig. 4). For additive models, in which <1,400 patients are needed to detect DHW, λ_southward values do not exceed 1.18. Thus, DHW can be observed for strikingly low λ_s values in sample sizes currently being used for large-scale association studies.

Example 1: DHW in Patients Merely

Ogura et al. (2001) conducted a case-command clan study of Crohn illness to fine map IBD1 in the pericentromeric region of chromosome 16. They identified a cytosine insertion (Leu3020fsinsC) in CARD15/NOD2 that causes a truncated NOD2 protein and that results in deficient activation of NF-κB and the immune response. We calculated HWE for the Leu3020fsinsC (Cins) polymorphism in unrelated Jewish and European American patients with Crohn disease and in unrelated controls. Because of the low proportion of HWE-expected homozygotes, we used Fisher'south verbal exam to determine DHW (Weir 1996). The distribution of genotypes in Jewish patients was 121 wild-blazon (wt)/wt homozygotes, 15 Cins/wt heterozygotes, and 4 Cins/Cins homozygotes, which yields P=.0059. The genotype distribution in European American patients was 314 wt/wt homozygotes, 41 Cins/wt heterozygotes, and 9 Cins/Cins homozygotes. Fisher's exact examination generates P=1.28×10^-iv, which is also highly pregnant. When all unrelated patients are tested for DHW (456 wt/wt homozygotes, 61 Cins/wt heterozygotes, and thirteen Cins/Cins homozygotes), P=6.83×10^-6. All 3 populations of patients showed significant DHW, with an excess of homozygotes. The distribution of genotypes in all controls was 311 wt/wt homozygotes, 25 Cins/wt heterozygotes, and 0 Cins/Cins homozygotes, which is consistent with HWE.

If the allele frequencies in both the patients (q _p =0.082) and the controls (q _c =0.037) are used to help delineate the general genetic disease model that all-time fits the observed genotypic distributions, we find that it can be characterized by Thousand _P =0.002, q=0.0379, α=0.00186, β=1.696, and γ=18.338. If a χⁱⁱ examination is performed to determine the goodness-of-fit of the expected number of patients and controls, given the model that best fits the observed genotypic counts, χ^two=0.478 (P=.827 with i df); thus, the best-fit model is a good fit to the observed data. Moreover, the model is consistent with what has been obtained elsewhere for other types of information (Hampe et al. 2001; Ogura et al. 2001).

Example 2: DHW in Contrary Directions in Patients and Controls

Ozaki et al. (2002) published a large-scale association study designed to identify genetic variation affecting take chances of MI in a Japanese sample. The study involved an initial screen of >92,000 SNPs in a small sample and follow-up studies for those SNPs with evidence of association (P<.01 for a dominant or recessive model) performed in a larger sample (one,133 patients, one,006 random controls, and 872 age-matched random controls). This led to more intensive studies of the LTA region (virtually HLA on 6p21), with 5 of 26 SNPs genotyped in the region showing highly meaning association with MI (east.grand., for LTA exon 1 polymorphism, P=three.3×10^-vi).

Although Ozaki et al. (2002) reported that all 26 SNPs were in HWE, with P>.01, we notation that each of the 5 SNPs for which genotype distributions are published show a pregnant DHW in opposite directions for patients and controls. The LTA exon 1 polymorphism is the near significantly associated with MI, with a genotype distribution among the i,133 patients of 416 GG homozygotes, 504 GA heterozygtoes, and 213 AA homozygotes, which yields χⁱⁱ=7.40 (P=.0065)—an excess of homozygotes relative to HWE-expected genotype proportions. The genotype distribution among the 1,006 random controls is 378 GG homozygotes, 512 GA heterozygotes, and 116 AA homozygotes. This sample of controls generates χ²=8.51 (P=.0035) merely shows a difference in the opposite direction of patients, with a deficiency of homozygotes. The age-matched random control group of 872 individuals has a genotype distribution of 344 GG homozygotes, 428 GA heterozygotes, and 100 AA homozygotes, respective to a nonsignificant χ²=3.69 (P=.055) merely also showing a deficiency of homozygotes like to that seen for the 1,006 random controls.

The get-go problem easily identified is that a meaning DHW is observed in a sample that may be better characterized equally a random sample than a control sample and thus would non be expected to bear witness DHW, regardless of the underlying genetic disease model. If we presume that the control sample is a control sample, rather than a random sample, the best-fit model is Grand _P =0.01, q=0.371, α=0.00932, β=1.024, and γ=1.469. The goodness-of-fit to the observed data is significantly poor, with χ^two=5.024 (P=.025 with 1 df). Given that the best-fit model is non a adept fit to what is observed in the 1,133 patients and 1,006 controls, the underlying genetic disease model for a susceptibility locus in that region is therefore an unlikely explanation for the observed DHW.

Survey of DHW

We identified 41 association studies with 60 polymorphisms that depart from HWE, from several recent reviews (Xu et al. 2002; Györffy et al. 2004; Kocsis et al. 2004a, 2004b; Osawa et al. 2004) (tabular array 1). All Thousand _P values were either given past the association study or were identified on the World Health Organization Web site. There are 35 polymorphisms that depart from HWE in patients only, 21 that depart in controls merely, two that depart in the same direction in patients and controls, and 1 that departs in the opposite direction in patients and controls. Of those that have a dominant or recessive model every bit the best-fit genetic model, 35.3% (12 of 34) did not accept a genetic model that fit the observed genotype distributions. Of those for which the all-time-fit model was a general model, 53.8% (14 of 26) did non have a model that fit well the observed data. Therefore, 43.3% of the polymorphisms nosotros identified from the reviews are inconsistent with a biological reason for the observed DHW. This highlights the importance not merely of correctly assessing HWE for genotype data but also of understanding whether an observed DHW is consistent with a genetic model of affliction susceptibility.

Give-and-take

Investigators have taken surprisingly inconsistent pathways in the interpretation and use of DHW. In some cases, investigators report significant association between variation at candidate genes and circuitous disorders that is quite dependent on the existence of DHW (i.e., the associations are genotypic rather than allelic, and the genotypic differences are driven by DHW). In at least some situations, the observed DHW is implausible from a biological perspective, which calls into question the association result. On the other hand, some investigators believe that whatever marker showing a DHW is probable to be erroneous and/or misleading; as a result, they may be throwing away information valuable for mapping and identifying causal polymorphisms. With the analytic framework and software we accept adult, we provide a way for investigators to assess markers with DHW in a more logical and systematic way by distinguishing those that could be attributed to the underlying genetic model at the susceptibility locus from those due to genotyping errors, chance, and/or violations of the assumptions of HWE, thereby improving the quality of scientific inferences.

Given the increasing sample sizes contemplated for large-scale association studies, it would seem that DHW is a neglected and potentially fruitful avenue for further research to improve signal localization. Results of studies of the CARD15/NOD2 region provide back up for this notion. A common coding polymorphism (P268S) (q=0.35 in patients with Crohn disease, and q=0.29 in unaffected controls) shows meaning DHW, with an excess of homozygotes in both Jewish patients (P=.0277) and in patients overall (P=.00546) (J. Cho, personal advice). The best-fit model for this polymorphism provides a good fit to the observed genotypic distributions (χⁱⁱ=1.501×10^-x; P=ane.0 with 1 df). Although this polymorphism is non thought to affect the chance of Crohn disease, three of the known susceptibility alleles (Leu3020fsinsC, Gly908Arg, and Arg702Trp) are in the same haplotype as the more rare allele at this site. If patients with any of those mutations are removed from the patient population and HWE is reassessed, the DHW is no longer significant (P=.2115 for Jewish patients, and P=.632 for patients overall). If none of the causative SNPs had been previously identified, the ascertainment that the common polymorphism departed from HWE in a style that was consistent with a genetic model could accept been used to back up farther investigation of the local region. This observation highlights the value of using DHW when fine mapping circuitous diseases, equally originally hypothesized past Nielsen et al. (1998).

Given the relatively large sample sizes used in modernistic candidate-gene studies every bit well as in fine-mapping and positional cloning studies, and the fact that DHW should be expected for a broad range of genetic illness models (consistent with the modest genetic contributions likely to be relevant for circuitous disorders), it is perhaps surprising that we have not seen a larger number of markers reported to testify DHW. Of 60 SNPs from association studies in which DHW has been identified in patients and/or controls and in which both patient and command genotype distributions are available, 34 (56.7%) have genotype distributions consistent with the expectations from a best-fit model (tabular array 1). DHW in patients is never expected for a multiplicative model and is less likely to be detected as significant for a model with susceptibility-allele frequencies at the tails of the frequency spectrum. We have non examined the consequences for HWE in more-complex disease models with allelic or locus epistasis, profound sex effects, etc., and, if such models were standard for circuitous disorders, it is unclear what the consequences would exist. Thus, the failure to more often observe DHW at polymorphisms hypothesized to bear upon susceptibility to complex disorders may reverberate biological characteristics of the genetic disease model that make such DHW impossible or at least less probable.

It is also possible that investigators mistrust data with DHW and may sometimes ignore such polymorphisms with DHW in their studies. Systematic errors in genotyping or nonrandom patterns of missing information may generate a relatively consistent design of DHW (e.g., disproportionate missing data in heterozygotes may lead to a consistent blueprint of DHW, with an backlog of homozygotes). Other types of error may be more sporadic. An unrecognized polymorphism in primer sequences used in subsequent PCR may lead to DHW, with an excess of homozygotes, particularly when the primer polymorphism is in LD with i of the test alleles. Genomic duplications or deletions can also lead to DHW. Considering fifty-fifty the systematic errors will non be universal, error and nonrandom patterns of missing information may be detectable in a single marker within a fix that are in strong LD. In contrast, when markers take DHW due to violations of assumptions of HWE or to run a risk, markers in LD may evidence similar patterns of DHW. Independent samples volition tend not to replicate chance DHW but may replicate DHW due to violations of HWE assumptions. Conspicuously, genomewide or big-scale studies offer additional information for estimation of DHW, including, for example, the power to examine potential population substructure. Many of the studies summarized in tabular array ane, however, focus on ane or a modest number of candidate genes, and the approaches we propose offer a unique opportunity to improve the scientific inferences about observed DHW. It should likewise be noted that just considering an observed DHW is consequent with genetic models does not mean that errors, missing data patterns, or violations of HWE assumptions did not generate or contribute to the observed DHW, and DHW tin can no doubt exist attributable to a combination of factors. Finally, it should too be noted that investigators ofttimes underestimate the significance of observed DHW, probably because of a misunderstanding of the degrees of freedom associated with the examination.

Acknowledgments

We acknowledge William Wen, for creating the DHW software, and Dr. Judy Cho, for giving us access to the association data on NOD2/CARD15 and Crohn affliction. We also acknowledge Drs. Mark Abney, Carole Ober, Dan Nicolae, Jonathan Pritchard, and Daniel Schaid, for their helpful discussions. This work was supported by grants DK-55889, DK-58026, and U01-GM61393. The DHW software is freely available and tin be downloaded from the Web site given below.

Appendix A: Δ in Patients

Weir (1996) defines Δ as the difference betwixt the population and HWE-expected genotypic frequencies. Therefore, we commencement ascertain these genotypic frequencies in terms of q, the susceptibility-allele frequency in the general population; α, the baseline penetrance for wild-type homozgyotes; β, the relative run a risk to heterozygotes; γ, the relative take chances to homozygotes for the susceptibility allele; and K _P, the population prevalence of disease. The expected genotypic proportions in patients, under the supposition of HWE in the full general population, are as follows:

The expected susceptibility-allele frequency for patients is

The frequency of the wild-type allele in patients is

Therefore,

which then simplifies to equation (1).

For a classic recessive model, where β=1 and γ>i,

For a classic ascendant model, where β=γ and γ>1,

For an additive model, where γ=2β and β>1,

As explained in the "Methods" department, the multiplicative model for patients, where βⁱⁱ=γ and β>1, simplifies to Δ _p =0.

Appendix B: Δ in Controls

Equally defined for patients in appendix A, we ascertain the genotypic proportions in controls for a general disease model equally

The expected susceptibility-allele frequency in controls is

The frequency of the wild-blazon allele in controls is

Therefore,

which simplifies to equation (two).

For a classic recessive model, where β=1 and γ>ane,

For a archetype dominant model, where β=γ and γ>1,

For an condiment model, where γ=2β and β>1,

Finally, for a multiplicative model, where γ=β² and β>one,

Appendix C: λ _s Values

Given that

(Risch 1990), where 5 _A is the additive variance that tin can exist calculated by

and V _D is the dominance variance that can be calculated by

λ_s under the archetype ascendant model can then be calculated every bit

λ_s under the classic recessive model can be calculated as

λ_southward under the additive model tin be calculated equally

and, finally, λ_southward under the multiplicative model tin can be calculated equally

Appendix D: Goodness-of-Fit Test

The expected genotypic proportions for patients nether a genetic disease model are m _AA-p, yard _Aa−p, and g _aa-p, as plant in appendix A. The expected genotypic proportions for controls under a genetic disease model are 1000 _AA-c, g _Aa-c, and g _aa-c, every bit found in appendix B.

For a particular set up of disease model parameter values (α, β, γ, and q), where the total number of patients is N _p, the total number of controls denoted is Northward _c, and the observed number of patients (or controls), given a specific genotype, is N _genotype-p (or N _genotype-c), the goodness-of-fit test is as follows:

Minimizing the resulting test statistic over the entire parameter space, subject to appropriate constraints on α, β, γ, and q and where K _P is fixed, yields parameter estimates that are approximately maximum-likelihood estimates. The minimal value of the test statistic is approximately distributed as a χ^two with one df for a general model and ii df for restricted models (i.e., dominant, recessive, additive, and multiplicative models) (Cramer 1946). We verify through simulations that applying this test to data showing a DHW does not affect the resulting distribution of the test statistic. We generated 1,000 replicates of 1,000 patients and i,000 controls under a ascendant, recessive, or general illness model as specified past the penetrances, which offer an advantage in being bound between 0 and 1. We kept for further analysis merely replicates that showed DHW in patients, in controls, or in both patients and controls. Nosotros used the goodness-of-fit examination to find the best-fit genetic model for each simulation and compared the resulting χ^two value to 1,000 simulated χⁱⁱ values, given the specified degrees of freedom (fig. 5).

Simulations with the goodness-of-fit test. A, one,000 simulations of a disease locus synthetic under a full 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.xx, α=β=0.xviii, and γ=3.78), in which the population prevalence is high (M _P =0.20). Each simulation, in which DHW was observed in patients and/or controls, was assessed using the goodness-of-fit exam and was compared with a imitation distribution of 1,000 χ² values, with i df for a general model and 2 df for a ascendant or recessive model. B, 1,000 simulations of a disease locus at a lower population prevalence (1000 _P =0.005) than A constructed under a general model (q=0.10, α=0.003, β=iv.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).

If the goodness-of-fit examination, with ane df for a general model or 2 df for a restrictive model, corresponds to a P value less than the specified threshold, the genetic disease model can exist rejected as a poor fit to the observed data. Note that rare-allele frequencies, which may produce low genotype counts, may produce approximations from the goodness-of-fit exam that do not perform equally expected.

Electronic-Database Information

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