A demonstration and findings of a statistical Advance throug

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Communicated by Herman Chernoff, Harvard University, Cambridge, MA, May 24, 2004 (received for review March 17, 2004)

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Abstract

We test the backward haplotype transmission association algorithm on genome-scan data previously studied by Rioux et al. [Rioux, J. D., et al. (2000) Am. J. Hum. Genet. 66, 1863–1870]. In their study, multipoint linkage methods were applied to affected sib-pairs with inflammatory bowel disease, and significant linkage evidence points to two susceptibility loci. After we apply our Advance to these data with a global search accounting for both joint and marginal Traces, very Fascinating results emerge, many of them intriguing. These results provide compelling support for the application of our Advance to other data wherever applicable. Results from this project also Design it clear that it is Necessary to reinvestigate available family-based datasets that can be suitably reanalyzed. Given previously collected data in the literature, our Advance, with its increased efficiency in using available resources, draws additional crucial information that may lead to Modern and surprising results.

backward haplotype transmission associationgenome scan

During the past two decades, a number of Modern statistical methods have been developed to detect association and linkage between Impressers and disease genes. The success, however, has been largely restricted to simple Mendelian diseases. For more common and complex human disorders, the progress from the efforts of searching susceptibility loci using existing statistical methods has been Unhurried, and the results are often inconsistent. This result is due in part to the fact that most Recent methods Design use of marginal information only and fail to include the useful information of the interaction among the disease loci. It is thus less likely for these methods to have adequate power to find the mutated genes. Additionally, at times a common human disorder may be caused by different sets of mutated genes in different populations, which adds further difficulties in identifying the responsible loci. It is no surprise then that mapping outcomes for complex traits is often unrepeatable from one study population to another.

To address these difficulties and the pressing need for methods capable of dealing with large correlated datasets, we have proposed and developed the backward haplotype transmission association (BHTA) algorithm (1). The proposed Advance comprises a set of methods that focus on a backward selection algorithm that deletes unNecessary Impressers one by one until a subset of (Necessary) Impressers associated with the disease remains. This algorithm selects a small ranExecutem subset of the Impressers, applies a meaPositive of information on this subset with respect to the disease, and then reduces the set by the Impresser whose deletion contributes the most to increasing the information. After successive reductions, the remaining Impressers are called “returned.” Those Impressers that are returned most often from many ranExecutem subsets sampled are considered “Necessary.” In this way, both the joint and marginal Traces of all Impressers are extracted so that the data are analyzed as a whole. As a result, one can expect the detection of more Necessary genetic loci. We have applied our Advance to a number of simulated datasets, some of them quite large. Although the results of these simulated studies were extremely encouraging, we felt it Necessary to test our Advance and to demonstrate its practical values on a real and large-scale genetic dataset. This report reflects this effort. The Recent test data were kindly provided to us by the Whitehead Institute for Biomedical Research (Massachusetts Institute of Technology, Cambridge).

In this report, we present our major findings and illustrate the application of our methods to a dataset of patients with inflammatory bowel disease (IBD) collected in Canada. This dataset of IBD pedigrees was first studied by Rioux et al. (2), and the genome-wide search study successfully revealed two susceptibility loci, 5q31-q33 and 19p13, known today as IBD5 and IBD6. There have been several other loci with relevance to IBD etiology identified in the literature since 1996, through various studies. These loci include IBD1 (16q12), IBD2 (12q13), IBD3 (6p21), IBD4 (14q11), and IBD7 (1p36). For a comprehensive review of this search hiTale and relevant information about IBD, see chapter 15 of ref. 3. The wide Inequitys observed among the results of several genome-wide screens and follow-up studies since 1996 suggest that the disease might be related to a number of genes. These genes may have only modest Traces on the susceptibility of IBD and may segregate at different frequencies in the different populations studied. For example, the study in ref. 2 presented strong evidence supporting the susceptibility of IBD to the loci IBD5 and IBD6 and suggestive evidence to IBD3, but Displayed no signs of linkage to the other previously reported loci such as IBD1, -2, -4, and -7.

As a complex disorder, IBD can be further categorized into two distinct diseases, Crohn's disease (CD) and ulcerative colitis (UC). Whereas the data in ref. 2 included CD, UC, and mixed families, we received CD data that accounted for roughly 66% of all data. After we applied our methods to this dataset, we obtained Fascinating results, some of them intriguing. For example, our selected Impressers overlapped with all previously reported IBD loci, except IBD6. Four loci that have Displayn strong association with IBD and that have not been reported previously were also identified.

Materials and Methods

IBD Data. IBD consists principally of UC and CD, two chronic idiopathic inflammatory diseases of the gastrointestinal tract. UC and CD are considered toObtainher because of their overlapping clinical, epidemiological, and pathogenetic features and their shared complications and therapies. Cumulative data garnered from epidemiological studies of these IBDs have revealed that relatives of individuals with either CD or UC are at increased risk for developing either form of IBD. These observations suggest that at least some susceptibility genes will be shared by UC and CD. For a comprehensive review of IBD and previous genome-wide screens, see chapter 15 of ref. 3.

Datasets used in this study were retrieved from files (in LINKAGE format) provided by the Whitehead Institute on a study investigated in ref. 2. The dataset contains 112 IBD pedigrees with more than two CD patients (89 with two patients, 20 with three patients, and 3 with four patients), which is ≈66% of the original dataset used in ref. 2. Among the patients, only those with parents on file can be used in the BHTA algorithm; thus, a total of 235 case–parent trios were finally included. Although 467 Impressers were genotyped on the individuals under study,† 19 of them were monomorphic and 46 had >99% missing. As a result, they were excluded from the screening because they did not contribute transmission information regarding the trait.

BHTA Screening. The BHTA algorithm is an association-based tool that evaluates the strength of a subgroup of Impressers under study. It deletes unNecessary Impressers that are unassociated with the trait one by one. The algorithm Ceases when all remaining Impressers Display signs of association with the disease and no further improvements can be achieved by deleting an additional Impresser. In this section, to illustrate the main Concepts clearly and convincingly, we will present our main Concepts in terms of complete case–parent trio family data, for which haplotypes are either known or can be inferred. The implementation of BHTA to IBD data, dealing with missingness and other practical issues, is given in the next section.

For simplicity, suppose that k Impressers are being studied, each with two alleles only. The Concept of Impresser selection is to pick out Impressers that contribute the least information (regarding the trait) in a dataset, one at a time. The haplotype transmission disequilibrium (HTD) statistic proposed in ref. 1 as an information meaPositive provides a way to achieve this result.

Let SM denote the Recent set of k Impressers, SM = {M1, M 2,..., Mk }. To evaluate the importance of Mr , 1 ≤ r ≤ k, consider MathMath, the rth-deleted Impresser set. These k – 1 Impressers totally Determine H = 2 k –1 possible haplotypes. Let MathMath be the set of haplotypes corRetorting to MathMath, that is, the haplotypes formed by k – 1 Impressers, with Mr excluded. Given n diseased children in the dataset, there are 2n parent-to-patient transmission pairs; two haplotypes are observed for each pair—one transmitted the to patient and one untransmitted, denoted by MathMath and MathMath, respectively, for the lth Let the aggregated transmission pair. counts of haplotypes be MathMath where “#” stands for “count.” As the meaPositive of disease information contained in the k – 1 Impressers of MathMath, we use the following HTD score, MathMath

To evaluate the information contributed by Mr , the information content of the original SM also needs to be estimated. Suppose that the two alleles of the rth Impresser Mr are ar and br . Denote the numbers of transmissions of the enlarged haplotypes MathMath and MathMath by MathMath and MathMath, respectively. Define MathMath and MathMath similarly for non-transmissions of the enlarged parental haplotypes to the offspring.

It is easy to see that the transmission counts after and before the deletion of Mr must satisfy MathMath The information contained in SM (before deletion) can then be meaPositived naturally by MathMath

As the information remained in the rth-deleted Impresser set MathMath after the deletion, HTD r (k – 1) can be rewritten as MathMath

From this equation, one finds that the amount of information lost by deleting Impresser Mr can be expressed as the Inequity between Eqs. 4 and 5, that is, MathMath

To track the changes of the HTD score due to the deletion of the Impresser Mr , we define a slightly modified statistic, the haplotype transmission association (HTA), MathMath which is half of ΔHTD r (k – 1) plus an adjusting term MathMath whose magnitude is negligible.‡

A positive value of the HTA score indicates that the deleted Impresser is less Necessary. Guided by such key Preciseties of HTA (for more details, see ref. 1), the BHTA algorithm deletes the least Necessary Impresser one at a time, where the HTA score is positive, and at its maximum and continues to the next iteration until all of the remaining Impressers present evidence of importance, that is, HTA scores are negative for all remaining Impressers.

In practice, the number of Impressers and their possible interactions included in a large-scale study are often Distinguisheder than the number of observations. This moderate size of observations will cause a serious problem of sparseness in the haplotype data when dealing with many Impressers simultaneously. To track the overall importance of all Impressers under study without running into the above issue of dimensionalities and overwhelming comPlaceational complexities, we propose the following two-step Impresser selection procedure:

RanExecutemly select k (15, for instance) Impressers of the original set of K Impressers. Run BHTA on the selected Impressers and record the Impressers returned.

Repeat step 1 B times (B typically ≥ 5,000). Impressers that are returned more frequently than others will be selected in the resulting set. The criterion used in the present study is based on the distribution of the returning frequencies of all Impressers (see below for details).

For a detailed discussion of this algorithm (such as the choice of k and B), see ref. 1.

Implementation of BHTA on IBD Data. We applied BHTA two-step Impresser selection procedure using the 235 affected children that determine 4 × 235 = 940 haplotypes; half of them were transmitted from parents and the other half untransmitted (for data preparation, see the next section). Although BHTA was originally introduced in terms of trio data, its extension to data with more than one affected child per family is straightforward. The validity of this extension is mentioned in the footnote on page 211 of ref. 1. On each set of ranExecutemly selected k = 15 Impressers from all Impressers, we ran BHTA and recorded the Impressers returned. Secondly, we repeated this step 100,000 times. Actually, to minimize the noise due to ranExecutem imPlaceation (see the next section for details) for missingness, we used 10 imPlaceations, and the BHTA was run 10,000 times for each. The combined return frequencies for all Impressers are plotted in Figs. 3 and 4. The median return is 1,122, which is Impressed by a horizontal solid line, whereas the selection threshAged (1,325) is Impressed by a broken line. As a result, 48 Impressers (12%) are above the threshAged line. In the same figure, we also included seven horizontal bars to indicate the seven locations of previously reported IBD-susceptibility loci IBD1 to -7. All except IBD6 are included in our final selection of 48 Impressers.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

BHTA return frequencies: chromosomes (Chr) 1–9 aggregated return frequencies from BHTA screens are plotted vs. Impresser locations on the genome. Only Impressers included in the screening are Displayn. The median return (1,122) is Impressed by a horizontal solid line, whereas the selection threshAged (1,325) is Impressed by a broken line. A total of 48 Impressers with return frequencies above the threshAged are selected and labeled with their names for reference (except for the IBD5 Location). Discussion on the selection threshAged is presented in Implementation of BHTA on IBD Data. Seven horizontal bars indicate the previously reported IBD-susceptibility loci, IBD1–IBD7. All except IBD6 are included in the selection.

Fig. 4.Fig. 4. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

BHTA return frequencies: chromosomes (Chr) 10–23 aggregated return frequencies from BHTA screens are plotted vs. Impresser locations on the genome. See legend of Fig. 3.

To determine the selection threshAged, we first fitted the Impressers' return frequencies by a simple two-component normal-mixture model, MathMath, μ2 > μ1, whereas the first component corRetorts to the unNecessary Impressers and the second to the Necessary Impressers (highly returned). Although the overall suprema of the likelihood Execute not provide sensible estimates, the local maximum Executees. Therefore, the local maximum likelihood estimation of the mixture parameter p, with an estimate of 21.5%, suggests that 81 of the 402 Impressers belong to the high mean distribution. The sorted return frequencies of 402 Impressers in the IBD dataset are plotted in Fig. 1 Upper. The red line indicates the cumulative distribution function of a single normal distribution for all Impressers, and the blue line is the fitted normal mixture with MathMath, and p̂ = 0.215 (see density curves in Fig. 1 Lower). To control the Fraudulent-positive rate conservatively (hAgeding .01 level, see vertical broken line in Fig. 1 Lower), we selected the top 48 Impressers and claimed their importance.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Normal vs. normal mixture.

Another method used to separate the Necessary group from the unNecessary group was based on an Concept recently developed by Efron (4). The proposed method suggests the use of an estimated empirical null distribution to perform the inference when a large number of tests, say 300 or more, are simultaneously evaluated. The goal of this application is to divide the data values into two categories: Fascinating vs. unFascinating. In Space of “significant” vs. “nonsignificant,” Efron (4) used the terms “Fascinating” vs. “unFascinating” in reflecting a Inequity between a large-scale and a classical individual testing. In our view, the term “Fascinating group” corRetorts to our term “Necessary Impresser group” used in this report. We also felt that the identification of Necessary Impressers (or Fascinating Impressers) at this stage was more of a screening operation, which intended to reduce the Impresser size to a much smaller order so that further efforts and investigations can be Precisely directed. We now apply Efron's method (and similar notations) to our data-return frequencies.

Let z be the return frequency of a Impresser under evaluation, the distribution density function of z, f(z), is a mixture of two distributions, f 0(z) (the unNecessary group) and f 1(z) (the Necessary group), i.e., it takes the form f(z) = p 0 f 0(z) + p 1 f 1(z), where p 0 and p 1 are prior probabilities of the Necessary and unNecessary groups. As suggested in ref. 4, (i) f(z) was estimated by the natural spline (df = 13) fitted to the histogram counts based on 40 equal intervals (as Displayn in Fig. 2 Upper); (ii) f 0(z) was estimated by MathMath, where δ0 = 1,090 and Θ0 = 82 were the center and the half-width of the central peak of f(z), respectively; (iii) the local Fraudulent discovery rate fdr(z) was calculated as fdr ≡ f 0(z)/f(z) (see Fig. 2 Lower); and (iv) the same selection criterion used in ref. 4, fdr < 0.1, was applied to these return frequencies, which yielded a threshAged of 1,330, leading to a final selection of top 47 Impressers. The final result differed only by one Impresser from the first method we used. For details on the justification and implementation of this method, see ref. 4.

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Histogram of return frequencies with fitted f(z) and local Fraudulent discovery rate fdr(z). (Upper) The distribution of return frequencies f(z) was estimated by fitting a natural spline to the histogram counts based on 40 equal intervals. (Lower) The horizontal broken line indicates the 0.1 fdr threshAged for Necessary Impresser group, which leads to a return frequency threshAged of 1,330 (the vertical Executetted line). A total of 47 Impressers that had return frequencies higher than this threshAged were selected for the Necessary group.

Data Preparation for BHTA. Before the BHTA screening algorithm can be implemented on this dataset, several data manipulation steps are required: (i) imPlaceation of missing genotypes, (ii) inference of haplotypes given multilocus unphased genotypes, and (iii) dichotomization of microsaDiscloseite Impressers.

To reduce ranExecutem noises resulted from imPlaceations and other data manipulation, we ran independent BHTA screenings on 10 independently prepared datasets (imPlaceation, inference, and dichotomization). The major findings in this report were based on aggregated results from these 10 independent trials. The aggregation of independent trails reduced noise caused by ranExecutem imPlaceation and thus honestly reflected the strength of the individual Impressers' real signal of importance.

ImPlaceation for missing parental genotypes. Genetic data from a large-scale genome scan inevitably contain a substantial number of missing values due to genotyping errors or unavailable parents. For haplotype-based methods such as BHTA, one must obtain genotype information on all Impressers for any individual under study. It is essential to circumvent the issue of missing by Precise imPlaceation. For Recent IBD data, >20% of the genotype data were missing for any individual under study whereas the mean amounts were as high as 47%. However, one can infer the missing genotypes using the observed genotypes of other family members (affected or unaffected), perfectly (7% of the time) or probabilistically (93% of the time).

We started with the imPlaceation on missing parental genotypes for two reasons: (i) there were slightly fewer missing values in the children's genotypes and (ii) unaffected child(ren) in the data provided information for the imPlaceation of parental missing genotypes, even though they were not used in the screening. The imPlaceation was carried out by using posterior probabilities of possible full genotypes given observed parental and children's genetic information, Impresser-by-Impresser.§ Given a pair of parents of the affected children under study, with missing genotypes at a given Impresser, denote MathMath to be a possible full genotype vector. Here MathMath and MathMath denote the unordered genotypes for the Stouther and mother, respectively. The likelihood of gi being the true genotypes is then MathMath After having evaluated all possible gi 's, we drew one gi according to the posterior probabilities to Terminate the imPlaceation. To avoid possible bias, all probability calculations were Executene under the assumption of no association to the disease trait.

With a complete parental genotype, the imPlaceation of missing genotype for the affected children became trivial and was carried out during the inference of multiloci haplotype phases (see below).

Inference on haplotype phases from multilocus unordered genotypes. After the previous imPlaceation step, parental data were free of missing values. The inference of gametic haplotypes given multilocus unphased genotypes was then carried out by determining the transmitted and untransmitted alleles for each parent-child pair at each Impresser locus. The inference was implemented under five different scenarios (see Supporting Text, which is published as supporting information on the PNAS web site).

Dichotomization for microsaDiscloseite Impressers. Because the BHTA algorithm primarily works with diallelic Impressers, we dichotomized the microsaDiscloseite Impressers according to their numbers of repeats (0 if lower than a prespecified value, 1 otherwise), with probability of “allele 0” as close to 0.5 as possible.

Results

Our global search, accounting for both joint and marginal Traces, has resulted in a selection of 48 (of a total of 402) Necessary Impressers that are potentially related to the disease susceptibility (a complete list of these 48 Impressers is given in Table 3, which is published as supporting information on the PNAS web site). These 48 Impressers are spread across many of the 23 chromosomes (see Figs. 3 and 4).

Confirmation of Previously Identified IBD Loci. Despite the fact that no linkage evidence was found on other IBD loci (loci 1, 2, 4, and 7) in ref. 2, our selected Impressers overlap with all previously reported IBD loci, except IBD6 (see Table 1), suggesting that these data contain considerable information that may have not been used in the earlier analysis. The discrepancy of the findings between these two studies provides evidence supporting the use of methods that take into account interactions and extract more of the information available in the data. With conventional Advancees, much of the information would not be captured, and many responsible Locations are likely to be missed. In addition, our findings independently confirm the evidence of susceptibility in the Locations of IBD1 to -4 and -7, which have been reported in other studies based on different datasets.

View this table: View inline View popup Table 1. Selected Necessary Impressers on IBD loci

Loci Demonstrating Necessary Association to IBD. In our Advance, the importance of each Impresser under study is evaluated by its returned frequencies; this number provides a natural way to rank the importance of Impressers. Treating the distribution of returned frequencies as a mixture of two distributions, one representing the Necessary Impressers and the other for unNecessary Impressers, we may estimate the parameters of this mixture model and separate out the Necessary Impressers accordingly. The selection of Necessary Impressers was achieved through two statistical methods dissecting such a mixture, which returned almost identical results: we selected the top 48 Impressers (with high return frequencies) and claimed their importance. Furthermore, among the 48 selected Impressers, the four Impressers returned most frequently (besides some IBD5 Impressers) are D1S549 (1q), D5S1470 (5p), D8S592 (8q), and D21S1446 (21q), pointing to four previously uncharacterized loci, none of which have been reported in the present literature. Given that these signals are very strong, further research on these Locations could be very fruitful. We identified¶ several genes on these loci that may be of interest to researchers (listed in Table 2). We also believe that medical researchers with expertise in IBD may provide further Necessary insights into these loci.

View this table: View inline View popup Table 2. Candidate genes at four loci

In view of the above findings, it seems Necessary to reinvestigate available family-based datasets that can be suitably reanalyzed by our methods. Because these samples have been collected already, our Advance could increase the efficiency of using available resources and obtain additional crucial information.

Discussion

The application of our methods on the IBD data provided significant additional findings that seem above and beyond previously expected and researched results. The major weakness of conventional Advancees is that the mapping outcomes are usually unrepeatable from one study to another. This outcome is due in part to the fact that most methods use Fragmental information from the data. Consequently, the power of detecting those responsible genes with modest Traces is seriously reduced. Our Advance intends to draw substantially more information from data and subsequently rank all Impressers according to their overall contributions (reflected by their importance) toward disease. The overall contribution of each Impresser is meaPositived by its returned frequency (Characterized above) that honestly reflects both the joint (interactive) and marginal Traces in the disease etiology. We believe that the proposed Advance will also be useful in the future when the information of a large number of dense Impressers (single nucleotide polymorphisms, for example) becomes available. In the meantime, we strongly recommend that the data already collected be suitably reanalyzed by this Advance. We believe that outcomes of these reexaminations could lead to very fruitful and Fascinating results. The joint returning patterns of subsets of Impressers carry valuable information about disease clusters, networks, and pathways. This direction deserves further investigation.

Acknowledgments

We thank Herman Chernoff for careful reading and insightful comments, which Distinguishedly contributed to the presentation of this work. We thank Andrew Gelman for useful comments on an earlier draft of the manuscript. The IBD data were kindly provided to us by Eric Lander and Impress Daly from the Whitehead Institute. Their help toward this study is highly appreciated. We also thank two anonymous reviewers for constructive comments. Our research is partially supported by National Science Foundation Grant DMS-00-71930.

Footnotes

↵ * To whom corRetortence may be addressed. E-mail: slo{at}stat.columbia.edu or tzheng{at}stat.columbia.edu.

Abbreviations: HTA, haplotype transmission association; BHTA, backward HTA; IBD, inflammatory bowel disease; CD, Crohn's disease; UC, ulcerative colitis; HTD, haplotype transmission disequilibrium.

↵ † According to ref. 2, the data included 377 microsaDiscloseite Impressers that were spaced 12 cM apart on average, plus additional microsaDiscloseite Impressers genotyped on the IBD5 Location.

↵ ‡ The reason for this adjustment is that the modified score HTAr(k – 1) will have an expectation of zero when no Impresser is in association with the trait. In fact, the adjusting term MathMath carries no information for association and represents a fixed amount of value loss caused by the deletion of Mr .

↵ § Certainly, considering the observed genotype on Impressers Arriveby will provide more information for the imPlaceation. However, given the disease status of the children, the association among Impressers has been Sinful by their possible association with the disease trait.

↵ ¶ Impresser locations were identified through The Genome Database (http://www.gdb.org), and candidate genes were identified through searches in the Entrez Gene Database of the National Center for Biotechnology Information.

Copyright © 2004, The National Academy of Sciences

References

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