 Research article
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Adjusting heterogeneous ascertainment bias for genetic association analysis with extended families
BMC Medical Genetics volume 16, Article number: 62 (2015)
Abstract
Background
In familybased association analysis, each family is typically ascertained from a single proband, which renders the effects of ascertainment bias heterogeneous among family members. This is contrary to case–control studies, and may introduce sample or ascertainment bias. Statistical efficiency is affected by ascertainment bias, and careful adjustment can lead to substantial improvements in statistical power. However, genetic association analysis has often been conducted using familybased designs, without addressing the fact that each proband in a family has had a great influence on the probability for each family member to be affected.
Method
We propose a powerful and efficient statistic for genetic association analysis that considered the heterogeneity of ascertainment bias among family members, under the assumption that both prevalence and heritability of disease are available. With extensive simulation studies, we showed that the proposed method performed better than the existing methods, particularly for diseases with large heritability.
Results
We applied the proposed method to the genomewide association analysis of Alzheimer’s disease. Four significant associations with the proposed method were found.
Conclusion
Our significant findings illustrated the practical importance of this new analysis method.
Background
Genomewide association studies (GWASs) have been used to identify many genes involved in human diseases, and during the last decade, many diseasesusceptibility variants have been identified. However, despite these successes, we have found that variants discovered from GWASs often explain only a small proportion of the heritability of diseases [1, 2]. For example, SNPs significantly associated with human height explain only about 5 % of phenotypic variance, despite studies of tens of thousands of people [3]. Many reasons, such as rare causal variants and gene/gene interactions, have been attributed to this socalled “missing heritability”. However, the low power induced by the multipletesting problem is still an intractable issue in GWASs, and further investigations of the most efficient strategies for genetic association analysis are necessary.
Careful selection of samples based on phenotypes can lead to improved power for the discovery of risk variants [4–11]. One such example is the extreme discordant sibpair design in linkage analysis, which may result in a substantial increase in statistical power when compared to other sibpair designs [11, 12]. Similarly, ascertaining the extremes of quantitative phenotypes from large population cohorts has also been shown to increase the power to identify associated variants [13–15]. In such a design, the effect of ascertainment conditions are homogeneous between individuals, and existing methods, such as the CochranArmitage(CA) trend test [16], can be an efficient choice. However, in association analysis using extended families, the effects of ascertainment bias are often heterogeneous among family members, and depending on their relationships with probands, different magnitudes of ascertainment bias may be generated. In particular, the probability of each individual being affected when his or her relatives are affected is similar to the prevalence, if the heritability is small, which indicates that the heterogeneous effect of the ascertainment bias depends on the magnitude of heritability. However, the heterogeneous effects of ascertainment conditions and the influence of heritability on it have not yet been investigated, and should therefore be taken into account for association analysis.
Recently, the CA trend test was extended for association analysis of dichotomous phenotypes with familybased samples [17, 18]. These statistics compares the genotype frequencies between affected and unaffected individuals, and the genetic association with familybased samples is tested by building a genotype correlation matrix with either kinship coefficients or an empirical correlation matrix estimated from largescale genetic data. This approach has been extended to include family members with known phenotypes and missing genotypes or vice versa. By the nature of these statistics, it performs well for ascertained familybased samples and it can be an efficient choice, even for a case–control design, if the relatives’ phenotype information is available. However, their statistical efficiency is affected by the heterogeneous effect of the ascertainment bias on family members, and for extended families, its effects on statistical efficiency can be substantial.
In this report, we consider the heterogeneous effects of the ascertainment bias on family members for dichotomous phenotypes. By the nature of the proposed methods, individuals with missing genotypes and nonmissing phenotypes can be utilized, and incorporation of the estimated kinship matrix to the proposed statistic provided robustness against the population substructure. The proposed method consists of two steps; the probability for each family member to be affected was calculated using a latent continuous liability [19], and then this probability is incorporated into a quasilikelihood score test. With an extensive simulation, we showed that the proposed method performed better than the existing methods, particularly for a disease with large heritability. Application of our method to Alzheimer’s disease (AD) demonstrated its practical use in the detection of genetic associations in ascertained familybased samples.
Methods
Notations and statistic
We assumed that there were n families and n_{ i } family members in each family. We considered the situation where the family of size n_{ i } was ascertained because it contained a particular set of p_{ i } members, and we let q_{ i } = n_{ i } – p_{ i }. We called the members of the set of p_{ i } family members “probands”, and the remaining q_{ i } individuals “nonprobands”. To provide a clearer motivation on this concept, we randomly selected two families, family 1 and 2, from our AD data (see Fig. 1). In family 1 (Fig. 1(a)), individual 9 was diagnosed as AD and individuals 3–8 were selected as her relatives for genetic analysis. In family 2 (Fig. 1(b)), individual 3 was diagnosed as AD, and individuals 4–6 were selected. Therefore p_{1} = p_{2} = 1, q_{1} = 6 and q_{2} = 3 in this example. In real data analysis, p_{ i } is often 1 and q_{ i } = n_{ i } – 1. We assumed that N individuals were available and thus N = ∑_{ i }n_{ i }. The genotypes were coded as 0, 1, or 2, according to the number of disease alleles. x _{ ij } ^{P} and x _{i ' j '} ^{N} were defined as the genotypes of proband j and nonproband j' in family i and family i', respectively. Phenotypes were coded as 0 for an unaffected individual and 1 for an affected individual. If we let the prevalence of the disease be p, a missing phenotype was coded as p. We denoted the phenotypes of a proband and nonproband by y _{ ij } ^{P} and y _{i ' j '} ^{N} , respectively, and the vectors for genotypes and phenotypes in family i were defined by
We also denoted the w × w identity matrix by I_{ w }, and the w × 1 column vector 1_{ w } indicated a vector in which all elements were 1. Let π _{ijj '} ^{P} and π _{ijj '} ^{N} be the kinship coefficient between probands j and j' in family i, and nonproband j and j' in family i, respectively. In addition, we let π _{ijj '} ^{PN} be the kinship coefficient between proband j and nonproband j' in family i, and let d_{ ij }^{P} and d_{ ij' }^{N} be the inbreeding coefficient for proband j and nonproband j' in family i, respectively. The inbreeding coefficient is the parameter that quantifies the departure from HardyWeinberg equilibrium (HWE) and ranges from 0 to 1. Several approaches [20, 21] that can estimate d_{ ij } have been proposed. We let
and R_{ i } is defined by
If we let q_{ A } be the disease allele frequency, E(X_{ i }) was \( 2{q}_A{1}_{n_i} \), and q_{ A } is estimated with the best linear unbiased estimator (BLUE). var(X_{ i }) is expressed by σ^{2}R_{ i }, and σ^{2} is equal to 2q_{ A }(1 –q_{ A }) under HWE.
When we analyzes the distribution of genotypes as in the FBAT approach, the statistical efficiency of the test statistic could be improved by adjustments of the phenotype with the socalled offset [22]. If we let μ_{ ij }^{P} and μ_{ i'j' }^{N} be offsets for proband j and nonproband j' in family i and family i', respectively, the offset vector for family i is defined as
Setting T_{ i } = Y_{ i }–μ_{ i }, we can define
We denoted a minor allele frequency (MAF) of a variant in unaffected individuals by q. We assumed [18] that for a constant γ,
where 0 < 2p + γ < 1. Then, the score for a variant [18, 23] can be defined by
The variance of S is
and we considered the following statistic [17, 18]:
This statistic will be denoted by WL in the remainder of this report.
Adjusting the heterogeneous ascertainment bias
Families are often selected based on some probands, and the probability for family members to be affected depends on their relationship with the probands. Additional file 1 shows that the incorporation of conditional probability of each individual being affected to WL as offset lead to asymptotically smaller variance and therefore the adjustment of heterogeneous ascertainment bias is required to improve the statistical power of WL. This probability could be estimated with the liability model if the heritabilities, h^{2}, and prevalence, p, were available. We let l _{ ij } ^{P} and l _{i ' j '} ^{N} be the liability of proband j and nonproband j' in family i and family i', respectively, and let \( {\mathbf{L}}_i^P=\left({l}_{i1}^P\kern0.5em ,\dots, \kern0.5em {l}_{i{p}_i}^P\right) \) and \( {\mathbf{L}}_i^N=\left({l}_{i1}^N\kern0.5em ,\dots, \kern0.5em {l}_{i{q}_i}^N\right) \). We assumed that each liability followed the standard normal distribution, and their joint distributions were
Benchek and Morris [24] reported that significant asymptotic biases are likely to arise when the multivariate normal (MVN) liability assumption is not met and in such a case, different assumptions should be considered. We assume that M _{ i } ^{P *} and V _{ i } ^{P *} are the expectation and variances of L_{ i }^{P} when their disease statuses are conditioned. If all probands are affected, they becomes
and
They can be calculated with the numerical algorithms [25]. If p_{ i } is 1, both can be simply calculated. We denote the cumulative and probability density function of standard normal distribution by Ф(·) and ϕ(·). If we let c be the (1–p)th quantile of the standard normal distribution, M _{ i } ^{P *} and V _{ i } ^{P *} becomes
With PearsonAitken formula [26, 27], we could obtain the conditional mean and variancecovariance matrix of L_{ i }^{N} given \( {\mathbf{L}}_i^P>{1}_{p_i}\kern0.5em \cdot c \) as follows:
and
We denoted the jth element in M _{ i } ^{N *} by m _{ j } ^{N *} and the jth diagonal element in V _{ i } ^{N *} by v _{ j } ^{N *} . Then the probability of being affected for a nonproband under multivariate normality of the liabilities could be calculated as
and this will be incorporated into the proposed statistic as offset. Thus far, we have assumed that there was a welldesigned set of p_{ i } individuals who were “probands”, and for this situation, we calculated the statistic as indicated and denoted FQLS_{1}. However in practice, different ascertainment condition such as sequential sampling frame [28] are often utilized, and the set of p_{ i } individuals will not be well defined. For this situation, we calculated the probability for each individual to be affected under the assumption that all the other family members were “probands”, and thus p_{ i } = n_{ i } – 1 and q_{ i } = 1. The statistic calculated this way was denoted by FQLS_{2}.
Results
The simulation model
In our simulation studies, we considered two types of family structures; nuclear families with five offspring and the extended families that consist of 13 individuals along 3 generations (see Fig. 2). The latter will be called extended families in the remainder of this report. The disease allele frequency, p, was assumed to be 0.2. If we denoted the disease allele frequency by q_{ A }, the genotype frequencies for AA, Aa, and aa became q_{ A }^{2}, 2q_{ A }(1 – q_{ A }), and (1 – q_{ A })^{2} under HWE, respectively, and founders’ genotypes were generated under the corresponding multinomial distribution. The genotypes for nonfounders were generated with randomly generated Mendelian transmission. The disease status was generated with the liability threshold model. Once continuous liabilities that consisted of polygenic effects and random errors were generated, they were transformed to being affected if they were larger than the threshold; and otherwise, they were considered to be unaffected. The threshold was chosen to preserve the prevalence, and prevalence was assumed to be 0.2. Continuous liability was determined by combining the phenotypic mean, polygenic effect, main genetic effect, and random error. The main genetic effect for each individual was the product of β and the number of disease alleles. If we denoted the relative proportion of the phenotypic variance attributable to the main disease gene by h_{ a }^{2}, and h^{2} was a heritability for continuous liability, β was calculated by
For the evaluation of type1 errors and power, h_{ a }^{2} was assumed to be 0 and 0.005, respectively. Phenotypic correlations between familymembers were explained by the polygenic effects. Parental polygenic effects were generated from N(0, h^{2}), and h^{2} was assumed to be 0.2, 0.5, or 0.8. For nonfounders, the average of maternal and paternal polygenic effects was combined with the values independently sampled from N(0, 0.5 h^{2}) for the polygenic effects of offspring. Random errors were generated from N(0, σ_{ e }^{2} = 1–h^{2}). For each replicate, sampling was repeated until a given number of ascertained families was generated. Type1 error estimates were calculated with 5000 replicates, and empirical power estimates were calculated with 1000 replicates.
Evaluation of the proposed methods with simulated data
The empirical type1 errors for FQLS_{1} and FQLS_{2} were evaluated from 5000 replicates under the situation of no association (h_{ a }^{2} = 0), and 900 nuclear families with five offspring in Fig. 2 were generated for each replicate. Fig. 3 shows the quantile quantile (QQ) plots from 5000 replicates, and the nominal significance levels for both methods were preserved for various significance levels. We also estimated the empirical type1 error rates at the 0.01 and 0.05 significance levels; the empirical type1 error estimates of FQLS_{1} and FQLS_{2} preserved these nominal significance levels (Table 1). These results verified that the use of the approximation to the standard normal distribution resulted in an accurate assessment of significance for the proposed methods.
The empirical powers at the various significance levels were measured based on 1000 replicates at the 0.01 and 0.001 significance levels. The relative proportion, h_{ a }^{2}, of phenotypic variance attributable to the main disease gene, 2p_{ A }(1 – p_{ A })β^{2}, was assumed to be 0.005, and nuclear and extended families in Fig. 2 were considered for the power comparison. In the first simulation setting, the numbers of nuclear families were assumed to be 100, 300, 600, 900, 1200, and 1400, and half of the families were ascertained if the number of affected family members was larger than or equal to n_{ proband }, and the other half of the families were ascertained if the number of unaffected family members was larger than or equal to n_{ proband }. Therefore, if 100 nuclear families were generated, half of nuclear families should have more than or equal to n_{ proband } affected family members, and the other half should have at least n_{ proband } unaffected family members. We assumed that the heritabilities were 0.2, 0.5, and 0.8, and results are shown in Tables 2, 3, 4, respectively. In the second simulation setting, the numbers of extended families were assumed to be 100, 300, 600, and 900, and all families were ascertained if the number of affected amily members was larger than or equal to n_{ proband }. Empirical power estimates for scenario 2 were calculated when h^{2} = 0.2, 0.5, and 0.8, and the data are shown in Tables 5, 6, 7, respectively. Our results showed that either FQLS_{1} or FQLS_{2} was usually the most efficient statistic, and the least efficiency was provided from WL. In particular, the power gap between the proposed methods and WL was largest if h^{2} was 0.8, which indicates that power improvement may be proportional to the heritability. If h^{2} was 0.2, the proposed methods were only slightly better than WL. While all methods in our power comparison focused on the distribution of genotypes to calculate statistics, the proposed methods uniquely considered the heterogeneous effects of ascertainment bias among family members which were proportional to the magnitude of heritability; this explained the power improvement of the proposed methods. Furthermore the differences of empirical power estimates from WL and the proposed methods are larger for Tables 5, 6, 7 than Tables 2, 3, 4, which indicates that the heterogeneity of ascertainment condition may be positively related with family size and the proposed methods become more efficient for large families. Last our simulation results show that FQLS_{2} was slightly better than FQLS_{1}, and this may be induced by the uncertainty of probands in our simulation studies. Therefore, we concluded that the incorporation of a sampling scheme to the offset could make a substantial difference, and test statistic should be carefully selected depending on type of sampling scheme.
The bold text indicates the highest empirical estimate of the power for each situation
The bold text indicates the highest empirical estimate of the power for each situation
The bold text indicates the highest empirical estimate of the power for each situation
The bold text indicates the highest empirical estimate of the power for each situation
The bold text indicates the highest empirical estimate of the power for each situation
The bold text indicates the highest empirical estimate of the power for each situation
Robustness of the proposed methods against the misspecification of prevalence
The statistical powers of the proposed methods may depend on the accuracy of the prevalence and we evaluated the sensitivity of the proposed method to the misspecified prevalence with simulated data. h_{ a }^{2} and h^{2} were assumed to be 0.05 and 0.8, and nuclear families (Fig. 2(a)) were considered in this simulation. The number of nuclear families was assumed to be 900 and n_{ proband } was assumed to be 1, 2, or 3. Prevalence was assumed to be 0.2 for phenotype generation, and the offset for which we recommended prevalence was set to be 0.1, 0.2 or 0.3 for calculation of FQLS_{1} and FQLS_{2}. In particular, y_{ ij } for individuals with missing phenotypes are coded by the assumed prevalence, and sensitivity of the proposed methods can be substantial when there are individuals with missing phenotypes. Therefore, individuals were randomly selected from nonprobands, and their phenotypes were assumed to be unknown for calculation of the proposed statistics. The number of family members with missing phenotypes in each family was denoted by n_{ missing }. The empirical powers were calculated at the 0.01 significance level with 1000 replicates. Table 8 shows that the results obtained by setting prevalence to be 0.1 and 0.3 are similar to the results when the prevalence was set to be 0.2, which indicates that the power loss attributable to the misspecified prevalence is not substantial. Furthermore the empirical power estimates are positively related with n_{ proband } and inversely related with n_{ missing }. If n_{ missing } is larger than 3, the power loss may be more substantial.
The bold text indicates the reference for the proposed statistics to compare with the misspecified prevalence.
Application of the proposed method to AD
AD is an irreversible, progressive brain disorder characterized by genetic heterogeneity. However, the genetic variations that contribute to AD still remain elusive. Thus, we applied the proposed method for identification of the disease susceptibility loci for AD. The heritability and prevalence of AD are approximately 0.8 [29, 30] and 0.1, respectively; therefore, we chose heritabilities of 0.8, and a prevalence of 0.1 for the calculation of proposed methods. Samples were collected as part of the National Institute of Mental Health Genetics Initiative (NIMH). The NIMH Alzheimer’s Disease Genetics Family Sample was used along with the information about the genotype platform (Affy 6.0) [20, 31], and ethical approach and participant approval were obtained through the NIMH IRB panel. Families were selected based on the disease status of a certain family member. However the proband for each family was not clear, and FQLS_{2} was uniquely applied for the proposed method. 1376 individuals from 410 families were available, and all families were nuclear. All individuals were of selfreported European ancestry. HWE for each single nucleotide polymorphism (SNP) was tested, and MAFs were estimated. SNPs for which pvalues for HWE were less than 10^{−6} or MAFs less than 0.05 were excluded, and therefore, 417,680 SNPs were analyzed for genetic association analysis.
The choice of kinship coefficient matrix for the proposed method depends on the presence of population substructure. To confirm the presence of population substructure, multidimensionalscaling [32] analysis was performed with PLINK1.07 [20], and we constructed a multidimensionalscaling plot (Fig. 4) to provide evidence concerning the presence of population substructure. Therefore, the genomic control method [31, 33, 34] was used for explicit detection and correction of population stratification with common SNPs, and they were incorporated into both FQLS_{2} and WL. Fig. 5 shows QQ plots for FQLS_{2} and WL; these plots revealed that the presence of population substructure was appropriately adjusted. Results showed four genomewide significant results for FQLS_{2} and one significant result for WL.
Detailed information for these significant results is provided in Table 9. We also considered FBAT statistics [35]. FQLS_{2} identified four genomewide significant SNPs, while WL and FBAT identified one genomewide significant SNP. In addition, one SNP that was significant according to WL was more significant according to FQLS_{2}. The most significant result acquired using FQLS_{2} was SNP4 (p = 2.10× 10^{−10}). The other three SNPs, i.e., SNP1, SNP2 and SNP3, reached the genomewide significance level.
Discussion
Although major advances in highdensity genome scans have enabled the genetic association analysis of more than 10,000 individuals, disappointing results in the mapping of many common diseases have illustrated the need for more powerful methods for detecting disease susceptibility loci. Statistical efficiency is known to being affected by the ascertainment bias, and its careful adjustment can lead to substantial improvement of statistical power [36–38]. In particular, genetic association analysis has often been conducted using familybased designs, but without addressing the fact that the probability for each family member to be affected is inversely related with the familial relationship with affected probands. In this report, we proposed new methods to adjust this heterogeneity with known prevalence and heritability. Our simulation studies showed that the proposed methods provided substantial power improvement. In particular, the misspecified heritability and prevalence can lead to the statistical power loss for the proposed methods, but it was found to be not substantial at least in our simulation studies. FQLS_{1} and FQLS_{2} were suggested, and FQLS_{1} is an efficient choice if probands for each family are clearly defined and all remaining family members are incorporated to the genetic analysis. However, these conditions are often not satisfied, and different methods such as sequential sampling frame [28] are usually utilized. Simulation studies showed that FQLS_{2} is usually better than FQLS_{1} if the ascertaining condition is not clearly defined and thus we recommend FQLS_{2} unless probands are clearly defined. However we considered the limited ascertainment conditions and comprehensive simulation studies are still necessary.
Furthermore, the proposed method was conceptually simple and can be applied to the large families. Our methods require only a single calculation of offset for all markers, and the real data analysis could be completed with a single CPU in a few hours. For M markers and N individuals, the time complexity is O(N^{3} + MN^{2}) for the proposed method. The proposed method was implemented with C++, and can be downloaded from http://healthstat.snu.ac.kr/mfqls/.
Heterogeneity between samples is an important issue in largescale genetic analysis, and the proposed method can likely be applied to various additional scenarios with some modifications. For instance, the disease status of relatives reveals the importance of genetic components for each individual, and for this reason, such information has been used, albeit only on occasion, in genetic association analysis. The effect of relatives’ disease statuses is dependent on prevalence and heritability, and the probability for each individual to be affected could be calculated with the proposed method. This probability can be used to improve the statistical efficiency of genetic association analysis. In addition, the heterogeneity of the ascertainment bias is often an important issue for genomewide metaanalysis because samples are collected from multiple medical centers [39, 40], and different sampling schemes among studies need to be adjusted to improve statistical efficiency. Therefore, we believe that proposed method can be extended to provide a statistical framework that adjusts the heterogeneity between samples.
Conclusions
We proposed FQLS method to adjust this heterogeneity with known prevalence and heritability and the software was implemented with C++. We identified several significant associations between AD and SNPs, and their potential functional information will provide the better understanding of the pathogenesis of AD. Although this study has some limitations, our proposed methods illustrated important features required for genetic analysis with familybased samples, and an extension of the proposed method to rare variant association analysis such as FARVAT [41] will be investigated in future studies.
Abbreviations
 GWAS:

Genomewide association studies
 CA:

CochranArmitage
 AD:

Alzheimer’s diseas
 PD:

Parkinson’s disease
 HWE:

HardyWeinberg equilibrium
 BLUE:

Best linear unbiased estimator
 QQ:

Quantile quantile
 MVN:

Multivariate normal
 MAF:

Minor allele frequency
 SNP:

Single nucleotide polymorphism
 FBAT:

Family Based Association Testing software
 FQLS:

Family based quasilikelihood score test
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Acknowledgements
SP, SL, YL, and SW were supported by the Industrial Core Technology Development Program (10040176, Development of Various Bioinformatics Software Using Next Generation BioData) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea), and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2010437). CP was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2011220C00004). BH, KM, CL and RT were supported by NIMH RO1 grant (2R01MH060009) and the Cure Alzheimer’s Fund.
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The authors declare that they have no competing interests.
Authors’ contributions
SP participated in the design of the study, and performed the simulation studies and statistical analysis. SL and TP developed the program. YL conducted the statistical analysis. CH, BH, KM, CP, LB, CL and RT conceived of the study. SW conceived of the study, and participated in data analysis. All authors read and approved the final manuscript.
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Additional file 1:
Webbased Supporting Materials for “Adjusting heterogeneous ascertainment bias for genetic association analysis with extended families” by Suyeon Park, Sungyoung Lee, Young Lee, Christine Herold , Basavaraj Hooli, Kristina Mullin, Lars Bertram, Taesung Park, Changsoon Park, Christoph Lange, Rudolph Tanzi , and Sungho Won.
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Park, S., Lee, S., Lee, Y. et al. Adjusting heterogeneous ascertainment bias for genetic association analysis with extended families. BMC Med Genet 16, 62 (2015) doi:10.1186/s1288101501986
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Keywords
 Familybased association analysis
 Ascertainment
 Liability model