Biostatistics Advance Access published online on May 22, 2007
Biostatistics, doi:10.1093/biostatistics/kxm016
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A score test for linkage analysis of ordinal traits based on IBD sharing
Department of Biostatistics, University of Alabama at Birmingham, Birmingham, AL 35294, USA rfeng{at}ms.soph.uab.edu
Department of Epidemiology and Public Health, Yale University, New Haven, CT 06520, USA and Jiangxi Normal University, Nanchang, People's Republic of China
* To whom correspondence should be addressed.
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SUMMARY |
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 TOP SUMMARY 1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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Statistical methods for linkage analysis are well established for both binary and quantitative traits. However, numerous diseases including cancer and psychiatric disorders are rated on discrete ordinal scales. To analyze pedigree data with ordinal traits, we recently proposed a latent variable model which has higher power to detect linkage using ordinal traits than methods using the dichotomized traits. The challenge with the latent variable model is that the likelihood is usually very complicated, and as a result, the computation of the likelihood ratio statistic is too intensive for large pedigrees. In this paper, we derive a computationally efficient score statistic based on the identity-by-decent sharing information between relatives. Using simulation studies, we examined the asymptotic distribution of the test statistic and the power of our proposed test under various levels of heritability. We compared the computing time as well as power of the score test with the likelihood ratio test. We then applied our method for the Collaborative Study on the Genetics of Alcoholism and performed a genome scan to map susceptibility genes for alcohol dependence. We found a strong linkage signal on chromosome 4.
Keywords: COGA; IBD; Linkage analysis; Ordinal traits; Score test
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1. INTRODUCTION |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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Statistical methods for linkage analysis are well established for both quantitative and binary traits (Ott, 1999). However, numerous diseases including cancer and substance uses are rated on discrete ordinal scales. For many of those diseases, the genetic heritabilities or even candidate genes have been well documented (e.g. Heath and Nelson, 2002; Steinke and others, 2003). Recently, we have developed a latent variable model to map genes for ordinal traits from pedigree data, and the power for detecting linkage increased when the ordinal, rather than dichotomized, traits are directly used (Feng and others, 2004). In the model, we introduced latent variables to represent unobserved common environmental factors and inheritable genetic contributions. Considering a diallelic susceptibility locus, we assumed binary latent variables to represent presence or absence of the disease-causing allele. The likelihood involved summations over the inheritance distributions of pedigrees and integrations over the distribution of latent variables, resulting in intensive computation of the likelihood ratio statistic. In this paper, we develop a score statistic based on the identity-by-decent (IBD) sharing information within pedigrees, which greatly alleviates the computation burden. Using the new test statistic, the latent variable model for ordinal traits can test linkage for large pedigrees while adjusting for nongenetic covariates. In addition, the latent variables are allowed to be continuous to reflect the general assumption that a quantitative liability variable may underlie the observed ordinal trait.
Use of IBD is a classic approach in linkage analysis (Haseman and Elston, 1972). In the presence of covariates, a 2-stage strategy is commonly employed (Schaid and others, 2003; Wu and others, 2003). While the 2-stage approach is convenient, it is more desirable to have a joint model, as we proposed here, that can detect linkage while accommodating the covariates in the same model.
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2. METHOD |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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The regression models for linkage analysis establish the association of a trait and the inheritance pattern at genetic markers. In this section, we first illustrate how the inheritance pattern of a pedigree at a locus is inferred based on the observed marker information through the so-called inheritance vector (Kruglyak and others, 1996) and how the IBD configurations can be derived from the inheritance pattern. Then, we introduce our regression model that links the ordinal trait with the genetic susceptibility transmitted according to the inferred pattern at a putative locus. Finally, based on the regression model, we derive the score test statistic to detect linkage.
Given a pedigree, the inheritance vector of non-founders (subjects whose parents are included in the pedigree) at a given locus t describes the ancestral origin of the DNA inherited by every non-founder at that locus (Lander and Green, 1987). For instance, in a nuclear family with 2 parents and n siblings, we may index the 2 paternal alleles at t as 1 and 2, respectively, indicating inherited grand paternal allele and grand maternal allele, and similarly maternal alleles as 3 and 4. For the jth sibling, define v2j – 1 = 1 or 2 according to which paternal allele is transmitted and v2j = 3 or 4 according to which maternal allele is transmitted. Then, the inheritance vector of the n siblings, v(t) = (v1,v2,v3,v4,...,v2n – 1,v2n)', completely specifies which of the 4 distinct paternal and maternal alleles are transmitted to every sibling. For a more complex pedigree with f founders and n non-founders, we can index the alleles of the f founders as (1,2),(3,4),(5,6),...,(2f – 1,2f) and define the inheritance vector for the n non-founders similarly. Alleles are called shared IBD if they are identical copies of the same ancestral allele. Two relatives can share 0, 1, or 2 alleles IBD at a particular locus. Given their inheritance vector, v(t) = (v1,v2,v3,v4)', the number of IBD alleles shared by them can be obtained from I(v1 = v3) + I(v1 = v4) + I(v2 = v3) + I(v2 = v4) if they are non-inbred. Here, I(·) is the indicator function.
The inheritance distribution is the conditional probability distribution over the possible 22n inheritance vectors that conform with the alleles observed at the marker locus t. In the absence of any genotypic information, all inheritance vectors of a family consistent with the family structure have equal probabilities according to Mendel's first law (Kruglyak and others, 1996). Similarly, IBD distribution is the conditional probability distribution over the numbers of IBD alleles shared by relative pairs. The distributions of the inheritance vector and IBD can be estimated by using standard software such as Genehunter (Kruglyak and others, 1996), Allegro (Gudbjartsson and others, 2000), Merlin (Abecasis and others, 2002), Simwalk2 (Sobel and Lange, 1996), etc.
Consider a trait Y taking an ordinal value from 0,1,...,K(K 
1). Let x be a p-vector of covariates that is also available for every study subject. For the ith family, we assume there exist 2 types of latent random variables U
and U
that represent, respectively, (a) the genetic susceptibility at a marker of interest and (b) the residual genetic or environmental factors in a family that are unobserved or difficult to assess through the observed data.
The latent variable Ug depends on the inheritance vector in the pedigree as follows. Let U
,...,U
be the genetic susceptibility associated with the 2fi alleles at a disease susceptibility locus (DSL) on fi pairs of chromosomes of all founders in the ith pedigree. Then, the latent variables for the jth non-founder in this family, which are completely determined by the parental latent variables and the inheritance vector, can be written as U
and U
.
If we consider the additive susceptibility due to the gene linked to locus t, then the genetic susceptibility at this locus, denoted by U
, is U
+ U
for the jth founder or U
+ U
for the kth non-founder. Through standardization, we can assume without loss of generality that E[U
] = 0 and Var[U] = 1 for j = 1,2,...,2fi. Furthermore, U is assumed to follow the standard normal distribution. Note, however, that the distribution of Ug is arbitrary except its first 2 moments.
Conditional on all the latent variables, denoted by Ui, and inheritance vectors vi within the ith family, the traits of all family members are assumed to be independent and have the following distribution:
 | (2.1) |
for i = 1,...,N,j = 1,...,ni,k = 0,...,K – 1, where N is the total number of pedigrees, ni is the number of members in the ith family, U = U + U, ß is a p-vector of parameters, reflecting the covariate effects on the trait, 
ks are the trait-level-dependent intercepts and reflect the differences between cumulative probabilities P{Y
k}, 
g measures the effect of the DSL, and e reflects the residual correlation. We must have 0 1 
K – 1 so that the category probabilities P{Y = k} are nonnegative (McCullgh and Nelder, 1989).
In the absence of linkage between marker and the trait-causing gene, the inheritance pattern at the marker locus is expected to be independent of the trait, that is, g = 0 in model (2.1). The null hypothesis H0 to be tested is g = 0.
Let 
= (ß,0,...,K – 1,e). The log-likelihood function from the ith family is a function of and g
 | (2.2) |
where Vi is the set of all possible inheritance vectors for the ith family, F(·) and G(·) are the cumulative distribution functions of Ug and Ue, Pi(,g) = 
I(Y = k)(
(,g) – 
(,g)), and = 1 – 1/[1 + exp(Xß + k + Ue + Ug)] for k = 0,1,...,K – 1,
= 0,
= 1.
If g = 0, the log-likelihood function from the ith family reduces to 
i(,0) = log
Pi(,0)dF(u).
Then, the likelihood ratio statistic becomes
 | (2.3) |
The maximization of i(,) is computationally intensive, and the likelihood ratio statistic (2.3) generally follows mixed 
2-distribution (Feng and others, 2004).
In order to alleviate the computation burden, we derive a score test that takes advantage of the IBD sharing information.
The score for testing the hypothesis concerning g is (see Appendix for details)
 | (2.4) |
where
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and IBD
is the expected number of IBD alleles shared by the kth and lth members in the ith pedigree, given the marker information.
The asymptotic variance of S under H0 is (Cox and Hinkley, 1974)
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where
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Under H0 (g = 0), 
and 
can be calculated as follows:
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where
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Let 
and 
denote S and IS calculated, respectively, at the corresponding estimates 
under H0. Then, the score statistic equals
 | (2.5) |
Under H0, the asymptotic distribution of the score function Sn is 
(Neyman and Scott, 1966). When sample size is small, a resampling method can be used to approximate the p-value of the score statistic (Zhang and others, 2003).
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3. SIMULATION STUDIES |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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To show the performance of the score test in mapping genes for ordinal traits, we designed a series of simulations to examine the distribution under the null hypothesis, power of detecting linkage under various levels of heritability, and computing time required for 2 sample sizes.
The simulation was replicated 10 000 times, resulting in 10 000 data sets. For each data set, we generated 100 pedigrees with 2 parents and 3 offsprings in each pedigree. A latent variable U was generated from N(0,1) that is shared by all family members, and a random noise rij was also generated from N(0,1) for each individual. A liability variable Z for the jth person in the ith family was defined as U + rij. The ordinal response, Y, was 0 if Z < 0 or 1 if 0 Z < 1 or 2 if Z 1.
For each founder in a pedigree, 20 highly polymorphic markers with 10 equally likely alleles, spaced 5 cM apart, were generated on 1 chromosome. Recombination fractions were converted to map distances without interference, and there was no linkage disequilibrium (LD) among markers. After the genotypes were generated for the founders, the genotypes of non-founders were generated subsequently based on the recombination fractions.
We found that the empirical score distribution at a locus approximates the asymptotic score distribution, , very well for a modest sample size (100 pedigrees and a total of 500 subjects).
The data sets were simulated similarly to those in the previous experiment. However, a diallelic locus between the 10th and the 11th markers was set to be the disease locus, and the disease-causing allele, D, had frequency p = 0.3. The genetic contributions Ug of allele D and d in founders were set equal to a and – a, respectively. The liability variable Z was defined as eU + U
+ rrij. The ordinal response, Y, equaled 0 if Z < 0 or 1 if 0 Z < 1 or 2 if Z 1. Then, the genetic heritability h, defined as the proportion of phenotype variance explained by the major genetic variance, is 2p(1 – p)a2/(2p(1 – p)a2 + 
+ 
). We fixed the total variance 2p(1 – p)a2 + + to be 2.0 and varied the heritability from 0.05 to 0.30 at the 0.05 interval. Also we let = . For each set of parameters and 2 different numbers of pedigrees (100 and 200), we simulated 1000 data sets. The solid lines in Figure 1 display the empirical power, defined as the proportion of trials that have score statistics larger than 
at the 10th or 11th marker (these are flanking markers for the true locus). level of 0.0025 is the Bonferroni-adjusted significance level for 20 markers. As the heritability increases, the power increases quickly.
We also calculated the likelihood ratio statistic in the above simulations. Using the asymptotic distribution of mixed and 
(Self and Liang, 1987), we also estimated the empirical power by the likelihood ratio test (dashed lines in Figure 1). The score test has slight higher power than the likelihood ratio test at level of 0.05, but much higher power at level of 0.0025. However, the power differences decrease when the sample size or heritability increases. Even though we expect the likelihood ratio test to be more powerful than the score test, numerical simulations show the opposite due to the computational complexity in calculating the likelihood ratio statistic, which includes finding the maximum likelihood estimator of the parameters under the alternative hypothesis and computing the conditional inheritance distributions given the observed marker information.
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As a main incentive to develop our score statistic, we also recorded the CPU time of the score tests and the likelihood ratio tests on all the simulations carried out in Section 3.2. The average execution CPU times of 1000 simulations required by the likelihood ratio test range from 1379.4 to 2401.2 s for 100 nuclear families and heritabilities of 0.05 to 0.3 on a SOLARIS processor. When the number of nuclear families increases to 200, the average time required by likelihood ratio test ranges from 13060.5 to 15951.0 s for different heritabilities. On average, the score test only takes 0.008 and 0.016 s for 100 and 200 nuclear families, respectively. The score test reduced the computation time by hundreds of folds, and the reduction becomes more magnificent when the numbers of pedigrees increase. In the score test, the calculation of the score and its asymptotic variance S only involves the moderate number of additions, proportional to the total number of individuals in the study. Therefore, the computation time for the score test increases linearly with the number of pedigrees and pedigree sizes, and it can be applied on pedigrees of any size provided that their IBD matrices are available.
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4. APPLICATION TO A GENETIC STUDY OF ALCOHOLISM |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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The Collaborative Study on the Genetics Alcoholism (COGA) aims to identify genes that affect the alcohol dependence and related phenotypes (Edenberg, 2002). The COGA data set includes 143 families with a total of 1614 members (including non-genotyped founders). The family sizes vary from 5 to 32 members and the generations range from 2 to 5. Among all individuals, 1388 members have some discrete and quantitative phenotypes, covariates, and microsatellite genotypes for a 10-cM genome scan.
Of particular interest to us, the phenotype ALDX1 defines the severity of the alcohol dependence based on the DSM-III-R (1987) criteria. It is thoroughly evaluated through various symptoms including craving, binging, desires to stop drinking, giving up activities, blackouts due to drinking, physical health problems, and emotional/psychological problems from drinking. Various linkage analyses for alcohol dependence have been performed using the microsatellite genotypes either in whole genome (e.g. Wiener and others, 2005; Zhu and others, 2005; Williams and others, 2005) or on selected chromosomes (e.g. Bartlett and Vieland, 2005; Dunn and others, 2005; Martin and others, 2005; Zhao, 2005). However, all the previous analyses treated ALDX1 as a binary outcome while the original ALDX1 is a 4-level ordinal-scaled variable including pure unaffected, never drunk, unaffected with some symptoms, and affected.
Some linkage studies used sib-pairs only in order to apply classic sib-pair methods (e.g. Doan and others, 2005; Shih and others, 2005; Xu and others, 2005; Yoon and others, 2005). Others dissected large pedigrees into small families due to the constraint of the computation (e.g. Reich and others, 1998; Zhang and others, 2005). Only a few have taken advantage of the relationship within large complex pedigrees, which contain more linkage information and thus can provide more reliable results.
Our genome-wide analysis is adjusted for age at interview and sex, which were found significant in predicting the alcohol-related phenotype. The appropriate adjustment has been found to increase power of detecting linkage relative to the unadjusted analyses (Doan and others, 2005).
In the first step, we used SOLAR (Almasy and Blangero, 1998) to calculate the multi-point IBD distribution of all the pedigrees. All the individuals including those with missing phenotypes or covariates or genotypes were included to derive the maximum inheritance information. In the second step, the individuals with missing response or covariate were excluded. We excluded the 29 subjects who had never drunk due to the concern that they might not have been exposed to alcohol as were the others. The trait (Y) values are 0, 1, and 2 for purely unaffected, unaffected with some symptoms, and affected, respectively. We first fitted the data using the model without polygenic or familial environmental effect Ue and then fitted the data again using the complete model (2.1).
Figure 2 displays the Logarithm of the odds (LOD) value, defined as Sn/2ln10, along the genome. The solid curves are the LODs calculated from complete model, and the dashed curves are from the model without Ue.
To compare with the available methods for binary traits, we also dichotomized ordinal traits into binary traits as (a) 0 versus {1,2} and (b) {0,1} versus 2. The LODs from the SOLAR analyses of 2 binary traits are displayed as 2 dotted curves in Figure 2. We also applied our proportional-odds model (2.2), which degenerates to a logistic model, to both dichotomized traits.
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The LODs from the complete model with Ue (in the solid curve in Figure 2) identified one most significant region near the marker ADH3 on chromosome 4 (LOD = 3.092, position 94 cM, p-value = 0.0002). Without taking the common environmental effect into account, the reduced model without Ue (in the dashed curve) gives the reduced LOD values compared with the complete model. SOLAR found a different peak between GABRB1 and D4S1645 (LOD = 2.592, position 55 cM, p-value = 0.0004) and also a lower peak around ADH3 on chromosome 4 (LOD = 2.464, position 94 cM, p-value = 0.0008) by the first dichotomization approach. For the second dichotomization approach, SOLAR failed to detect any linkage. Our model using either dichotomized trait did not find any significant linkage, and the highest peaks are around the marker D3S1311 on chromosome 3 (LOD = 1.129, position 236 cM) and around ADH3 on chromosome 4 (LOD = 1.358, position = 95cM) for 2 dichotomizations, respectively.
According to the method of Feingold and others (1993), a LOD of 2.77 is equivalent to a genome-wide p-value of 0.05 for the 315 microsatellite markers in the COGA data. Hence, other approaches did not identify significant linkage with ALDX1.
A comparison of similar findings in the same COGA data set not only enhances the validity of our finding but also confirms the higher power using ordinal traits instead of dichotomized traits (Feng and others, 2004). Reich and others (1998) suggested ADH3 as one of the markers linked to ALDX1 using sib-pair methods for Caucasian families and reported a maximum LOD of 2.5 at 86.6 cM after redefining the phenotype.
The highest peak around GABRB1-D4S1645(51–62 cM) on chromosome 4 found by SOLAR is one of the most common findings using another discrete trait ALDX2 and a continuous trait ebc21 (resting electroencephalogram), in the COGA data (Bartlett and Vieland, 2005; Xu and others, 2005; Reich and others, 1998; Kraja and others, 2005; Zhong and Zhang, 2005). The results from our score tests also show the peak but not very significant.
The covariate coefficients are estimated at 0.027 for age (OR = 1.027 for an older person versus a person 1 year younger to be in a higher ordinal scale, p < 0.001) and 1.650 for sex (OR = 5.207 for male versus female to be in a higher ordinal scale, p < 0.001).
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5. DISCUSSION |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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We proposed a latent variable, proportional-odds model to map susceptibility genes for ordinal traits. We developed a score statistic to take advantages of the IBD sharing information between relatives, which greatly alleviates the computational burden for large sample sizes. The score test has higher power than the likelihood ratio test, and the gain of power might be significant when sample size is small or heritability is low. Our method was implemented in a C program and will be available from http://peace. med.yale.edu/pub/LOT. The current Linkage for Ordinal Traits program (Feng and others, 2004) assumes that all latent variables follow Bernoulli distribution to represent presence or absence of a familial factor or a disease-causing allele. That assumption is intuitive yet restrictive. In this paper, we extend the familial variables to be normally distributed and the genetic susceptible variables to have distributions with mean 0 and variance 1.
Using the proposed model, we scanned the microsatellite genes on 22 autosomes for alcohol dependence in the COGA data set. We reported a larger LOD at alcohol dehydrogenase (ADH) 3 on chromosome 4 than did the previous study based on dichotomized ALDX1. Our finding on chromosome 4 is supported by other human genetic research and experiments on animals. For example, some alleles at the surrounding region of ADH2 and ADH3 were found to reduce the risk of alcoholism in Asian populations (Thomasson and others, 1991; Shen and others, 1997). These alleles encode isozymes that either metabolize ethanol to acetaldehyde more rapidly or slow the conversion of acetaldehyde to acetate. Accumulated acetaldehyde may cause an aversive reaction that discourages further alcohol intake (Reich and others, 1998). Mouse experiments found an alcohol preference locus, which is homologous with the region close to ADH gene on human chromosome 4 (Markel and others, 1996).
We used the asymptotic distribution of the score to evaluate the p-values, but the theoretical p-value might not be the best choice for the small number of pedigrees and several large pedigrees in the COGA data. We thus performed a permutation test to obtain the empirical genome-wise p-value of our highest score 14.240 using the real data. We permuted the phenotypes of individuals without missing phenotype and covariates and then selected the largest scores calculated at 1 cM distanced loci along 22 chromosomes. The empirical p-value is 0.172 based on 5000 permutations, which may suggest a departure from the asymptotic distribution. But the empirical and asymptotic p-values are very similar using simulated data sets with similar sample and pedigree sizes. The discrepancy in the real data may be caused by restricting the permutation among individuals with complete phenotypes. Further investigation will be helpful.
The multi-point IBD calculations are often based on the linkage equilibrium assumption, which is expected for sparse markers with inter-distances exceeding a few centimorgans for most population. However, the IBD calculation using the denser markers such as single nucleotide polymorphisms should take into account LD which is likely to exist among the markers. Ignoring LD in linkage analysis may cause oversharing IBD estimates and thus false-positive evidence of linkage when founders are absent (Goode and others, 2005). The new IBD calculation method that incorporates LD information (Abecasis and Wigginton, 2005) can be used for linkage analysis based on dense markers. Since tremendous efforts have been devoted to infer the inheritance patterns through IBD estimation and new IBD estimation approaches are becoming available (Abecasis and Wigginton, 2005; Gao and Hoeschele, 2005), we only focused on modeling the disease through derived inheritance patterns in this paper.
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APPENDIX A |
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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Let
= sign(g)
in (2.1). Then, g = sign()
and testing the hypothesis g = 0 in (2.1) is equivalent to testing = 0 in the following model:
 | (A.1) |
Let 
= gU
and i = (
,...,
)T. A straightforward calculation shows
 | (A.2) |
 | (A.3) |
 | (A.4) |
Since Pi(,) has a sectionally continuous derivative with respect to in a closed interval 
and the 2 integrals Pi(,)dG(u) and 
converge uniformly in , the order of the integration and differentiation in (A.3) can be interchanged (Courant, 1992).
Also for non-inbred individuals k and l within a pedigree, we have
 | (A.5) |
Then, we can obtain the score function for testing the hypothesis concerning as follows:
 | (A.6) |
 | (A.7) |
where
 |
Replacing 
in (A.6) by (A.7), we have the score function expressed in (equ-4). An important ingredient EUU
in score function can be obtained through IBD sharing information of kth and lth individuals in the family.
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ACKNOWLEDGMENTS |
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This research is supported in part by grants K02DA017713 and R01DA016750 from the National Institutes of Health. Data were provided by the COGA (U10AA008401). The authors thank Raymond Crowe and Jean MacCluer for facilitating the use of the COGA data. The authors also would like to thank Prof. Hongtu Zhu at University of North Carolina at Chapel Hill for helpful advice, Dr Guimin Gao at University of Alabama for insightful comments and suggestions, and Dr Yuanqing Ye at Yale for preparing the data in SOLAR format and reviewing the manuscript. Conflict of Interest: None declared.
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TOPSUMMARY1. INTRODUCTION2. METHOD3. SIMULATION STUDIES4. APPLICATION TO A...5. DISCUSSIONAPPENDIX AREFERENCES |
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Received October 27, 2006; revised March 27, 2007; accepted for publication April 11, 2007.
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