nips nips2012 nips2012-53 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Bonnie Kirkpatrick, Alexandre Bouchard-côté
Abstract: Pedigrees, or family trees, are directed graphs used to identify sites of the genome that are correlated with the presence or absence of a disease. With the advent of genotyping and sequencing technologies, there has been an explosion in the amount of data available, both in the number of individuals and in the number of sites. Some pedigrees number in the thousands of individuals. Meanwhile, analysis methods have remained limited to pedigrees of < 100 individuals which limits analyses to many small independent pedigrees. Disease models, such those used for the linkage analysis log-odds (LOD) estimator, have similarly been limited. This is because linkage analysis was originally designed with a different task in mind, that of ordering the sites in the genome, before there were technologies that could reveal the order. LODs are difficult to interpret and nontrivial to extend to consider interactions among sites. These developments and difficulties call for the creation of modern methods of pedigree analysis. Drawing from recent advances in graphical model inference and transducer theory, we introduce a simple yet powerful formalism for expressing genetic disease models. We show that these disease models can be turned into accurate and computationally efficient estimators. The technique we use for constructing the variational approximation has potential applications to inference in other large-scale graphical models. This method allows inference on larger pedigrees than previously analyzed in the literature, which improves disease site prediction. 1
Reference: text
sentIndex sentText sentNum sentScore
1 Meanwhile, analysis methods have remained limited to pedigrees of < 100 individuals which limits analyses to many small independent pedigrees. [sent-8, score-0.341]
2 This is because linkage analysis was originally designed with a different task in mind, that of ordering the sites in the genome, before there were technologies that could reveal the order. [sent-10, score-0.306]
3 These developments and difficulties call for the creation of modern methods of pedigree analysis. [sent-12, score-0.481]
4 Drawing from recent advances in graphical model inference and transducer theory, we introduce a simple yet powerful formalism for expressing genetic disease models. [sent-13, score-0.818]
5 We show that these disease models can be turned into accurate and computationally efficient estimators. [sent-14, score-0.414]
6 This method allows inference on larger pedigrees than previously analyzed in the literature, which improves disease site prediction. [sent-16, score-0.931]
7 1 Introduction Finding genetic correlates of disease is a long-standing important problem with potential contributions to diagnostics and treatment of disease. [sent-17, score-0.54]
8 The pedigree model for inheritance is one of the best defined models in biology, and it has been an area of active statistical and biological research for over a hundred years. [sent-18, score-0.587]
9 The most commonly used method to analyze genetic correlates of disease is quite old. [sent-19, score-0.54]
10 After Mendel introduced, in 1866, the basic model for the inheritance of genomic sites [1] Sturtevant was the first, in 1913, to provide a method for ordering the sites of the genome [2]. [sent-20, score-0.53]
11 The method of Sturtevant became the foundation for linkage analysis with pedigrees [3, 4, 5, 6]. [sent-21, score-0.335]
12 The problem can be thought of in Sturtevant’s framework as that of finding the position of a disease site relative to an map of existing sites. [sent-22, score-0.682]
13 Genomic sites are becoming considerably denser in the genome and technologies allow us to interrogate the genome for the position of sites [7]. [sent-25, score-0.512]
14 Additionally, most current pedigree 1 analysis methods are exponential either in the number of sites or in the number of individuals. [sent-26, score-0.636]
15 This is in contrast to the size of pedigrees being collected: for example the work of [8] includes a connected human pedigree containing 13 generations and 1623 individuals, and the work of [9] includes a connected non-human data set containing thousands of breeding dogs. [sent-28, score-0.766]
16 Apart from the issues of pedigree size, the LOD value is difficult to interpret, since there are few models for the distribution of the statistic. [sent-29, score-0.481]
17 These developments and difficulties call for the creation of modern methods of pedigree analysis. [sent-30, score-0.481]
18 In this work, we propose a new framework for expressing genetic disease models. [sent-31, score-0.523]
19 The key component of our models, the Haplotype-Phenotype Transducer (HPT), draws from recent advances in graphical model inference and transducer theory [10], and provides a simple and flexible formalism for building genetic disease models. [sent-32, score-0.818]
20 The output of inference over HPT models is a posterior distribution over disease sites, which is easier to interpret than LOD scores. [sent-33, score-0.449]
21 The cost of this modeling flexibility is that the graphical model corresponding to the inference problem is larger and has more loops that traditional pedigree graphical models. [sent-34, score-0.633]
22 Our framework can be specialized to create analogues of classical penetrance disease models [13]. [sent-41, score-0.486]
23 Our experiments show that even for these simpler cases, our approach can achieve significant gains in disease site identification accuracy compared to the most commonly used method, Merlin’s implementation of LOD scores [3, 5]. [sent-43, score-0.682]
24 Moreover, our inference method allows us to perform experiments on unprecedented pedigree sizes, well beyond the capacity of Merlin and other pedigree analysis tools typically used in practice. [sent-44, score-0.997]
25 While graphical models have played an important role in the development of pedigree analysis methods [14, 15], only recently were variational methods applied to the problem [6]. [sent-45, score-0.551]
26 Most current work on more advanced disease models have focused on a very different type of data, population data, for genome wide association studies (GWAS) [16]. [sent-47, score-0.5]
27 The point at which the copying of the chromosomes switches from one of the grand-maternal (grand-paternal) chromosomes to the other, is called a recombination breakpoint. [sent-50, score-0.33]
28 A site is a particular position in the genome at which we can obtain measurable values. [sent-51, score-0.354]
29 For the purposes of this paper, an allele is the nucleotide at a particular site on a particular chromosome. [sent-52, score-0.387]
30 If we had complete data, we would know the positions of all of the haplotypes, all of the recombination breakpoints as well as which allele came from which parent. [sent-54, score-0.4]
31 2 A pedigree is a directed acyclic graph with individuals as nodes, where boxes are males and circles are females, and edges directed downward from parent to child. [sent-60, score-0.68]
32 The individuals without parents in the graph are called founders, and the individuals with parents are non-founders. [sent-62, score-0.323]
33 The pedigree encodes a set of relationships that constrain the allowed inheritance options. [sent-63, score-0.587]
34 These inheritance options define a probability distribution which is investigated during pedigree analysis. [sent-64, score-0.587]
35 Assume a single-site disease model, where a diploid genotype, GD , determines the affection status (phenotype), P ∈ {’h’,’d’}, according to the penetrance probabilities: f2 = P(P = ’d’|GD = 11), f1 = P(P = ’d’|GD = 10), f0 = P(P = ’d’|GD = 00). [sent-65, score-0.507]
36 Here the disease site usually has a disease allele, 1, that confers greater risk of having the disease. [sent-66, score-1.096]
37 Let the pedigree model for n individuals be specified by a pedigree graph, a disease model f , and the minor allele frequency, µ, for a single site of interest, k. [sent-68, score-1.89]
38 Between the disease site and site k, we model the per chromosome, per generation recombination fraction, ρ, which is the frequency with which recombinations occur between those two sites. [sent-77, score-1.205]
39 Other sites linked to k can contribute to our estimate via their arrangement in single firstorder Markov chain with some sites falling to the left of the disease site and others to the right of the site of interest. [sent-78, score-1.26]
40 Previous work has shown that given a pedigree model, affection data, and genotype data, we can estimate ρ. [sent-79, score-0.603]
41 We define the likelihood as L(ρ) = P(P = p, G = g|ρ, f, µ) where ρ is the recombination probability between the disease site and the first site, p are the founder allele frequencies, and f are the penetrance probabilities. [sent-80, score-1.197]
42 To test for linkage between the disease site and the other sites, we maximize the likelihood to obtain the optimal recombination fraction ρ∗ = argmaxρ L(ρ)/L(1/2). [sent-81, score-1.039]
43 3 Methods In this section, we describe our model for inferring relationships between phenotypes and genotyped pedigree datasets. [sent-84, score-0.568]
44 The first step in this generative process consists in sampling a collection of disease model (DM) variables, which encode putative relationships between the genetic sites and the observed phenotypes. [sent-86, score-0.678]
45 There is one disease model variable for each site, s, and to a first approximation, Ds can be thought as taking values zero or one, depending on whether site s is the closest to the primary genetic factor involved in a disease (a more elaborate example is presented in the Supplement). [sent-87, score-1.224]
46 We will define the distribution of Pi conditionally on the haplotype of the individual in question, Hi , and on the global disease model D. [sent-95, score-0.644]
47 ) Figure 1: (a) The pedigree graphical model for independent sites. [sent-108, score-0.531]
48 The nodes are labeled as follows: M for the marriage node which enforces the Mendelian inheritance constraints, H for haplotype, L and L for the two alleles, D(1) for the disease site indicator, and D(2) for the disease allele value. [sent-110, score-1.364]
49 This transducer for HPT(·) models a recessive disease where the input at each state is the disease (top) and haplotype alleles (bottom). [sent-114, score-1.349]
50 The remaining variables (the non-founder individuals’ haplotype variables) are obtained deterministically from the values of the founders and the inheritance: Hi,s,x = Hx(i),s,Ri,s,x , where x(i) denotes the index of the father (mother) of i if x = ‘father’ (‘mother’). [sent-116, score-0.304]
51 The distribution on the founder haplotypes is a product of independent Bernoulli distributions, one for each site (the parameters of these Bernoulli distributions is not restricted to be identically distributed and can be estimated [3]). [sent-117, score-0.563]
52 Having generated all the haplotypes and disease variables, we denote the conditional distribution of the phenotypes as follows: Pi |(D, Hi ) ∼ HPT( · ; D, Hi ), where HPT stands for a Haplotype-Phenotype Transducer. [sent-119, score-0.689]
53 We also make two simplifications to facilitate exposition: first, that the disease site is one of the observed sites, and second, that the disease allele is the less frequent (minor) allele (we show in the Supplement a slightly more complicated transducer that does not make these assumptions). [sent-125, score-1.522]
54 Under the two above assumptions, we claim that the state diagrams in Figure 1(b) specify an HPT transducer for a recessive disease model. [sent-126, score-0.682]
55 0 denotes that a disease indicator is emitted with weight one. [sent-129, score-0.414]
56 The set of valid paths along with their weights can be thought of as encoding a parametric disease model. [sent-136, score-0.456]
57 For example, with a recessive disease, shown in Figure 1(b), we can see that if the transducer is at the site of the disease (encoded as the current symbol in c being equal to 1) then only an input homozygous haplotype ‘AA’ will lead to an output disease phenotype ‘d. [sent-137, score-1.605]
58 ’ This formalism gives a considerable amount of flexibility to the modeler, who can go beyond simple Mendelian disease models by constructing different transducers. [sent-138, score-0.436]
59 After explaining in more detail the graphical model of interest, we discuss in this section the approximation algorithm that we have used to infer haplotypes, disease loci, and other disease statistics. [sent-146, score-0.878]
60 Note that our graphical model has more cycles than standard pedigree graphical models [19]; even if we assumed the sites to be independent and the pedigree to be acyclic, our graphical model would still be cyclic. [sent-149, score-1.267]
61 Our inference method is based on the following observation: if we kept only one subtype of factors in the Supplement, say only those connected to the recombination variables R, then inference could be done easily. [sent-150, score-0.359]
62 The main difficulty arises when attempting to resample D: because of the deterministic constraints that arise even in 5 the simplest disease model, it is necessary to sample D in a block also containing a large subset of R and H. [sent-157, score-0.414]
63 Disease predictions were used to validate the HPT disease model. [sent-196, score-0.414]
64 The pedigree is built starting from the oldest generation. [sent-200, score-0.481]
65 20 (a) Forest-Cover Factors Figure 2: The pedigree was generated with the following parameters, number of generations 20 and n = 15 which resulted in a pedigree with 424 individuals, 197 marriage nodes, 47 founders. [sent-231, score-1.019]
66 Panel (a) shows the effect of removing factors from the forest cover of the pedigree where the lines are labeled with the number of factors that each experiment contains. [sent-234, score-0.627]
67 This panel shows that the recombination parameter can be off by an order of magnitude and the haplotype reconstruction is robust. [sent-243, score-0.47]
68 Genotype data were simulated in the simulated pedigree graph. [sent-244, score-0.539]
69 The founder haplotypes were drawn from an empirical distribution (see Supplement for details). [sent-245, score-0.295]
70 The recombination parameters used for inheritance are given in the Supplement. [sent-246, score-0.342]
71 We then simulated the inheritance and recombination process to obtain the haplotypes of the descendants using the external program [27]. [sent-247, score-0.559]
72 An independent 50% of individuals have missing phenotypes for the disease prediction comparison. [sent-251, score-0.659]
73 For the haplotype reconstruction, the inference being scored is, for each individual, the maximum a posteriori haplotype predicted by the marginal haplotype distribution. [sent-253, score-0.653]
74 These haplotypes are not necessarily Mendelian consistent, meaning that it is possible for a child to have an allele on the maternal haplotype that could not possibly be inherited from the mother according to the mother’s marginal distribution. [sent-254, score-0.581]
75 However, transforming the posterior distribution over haplotypes into a set of globally consistent haplotypes is somewhat orthogonal to the methods in this paper, and there exist methods for this task [28]. [sent-255, score-0.376]
76 The goal of this comparison is threefold: 1) to see if adding more factors improves inference, 2) to see if more iterations of the measure factorization algorithm help, and 3) to see if there is robustness of the results to the recombination parameters. [sent-256, score-0.301]
77 Synthetic founder haplotypes were simulated, see Supplement for details. [sent-257, score-0.295]
78 Each experiment was replicated 10 times where for each replicate the founder haplotypes were sampled with a different random seed. [sent-258, score-0.313]
79 We computed a metric φ which is a normalized count of the number of sites that differ between the held-out haplotype and the predicted haplotype. [sent-259, score-0.361]
80 For disease prediction, the inference being scored is the ranking of the sites given by our Bayesian method as compared with LOD estimates computed by Merlin [3]. [sent-265, score-0.604]
81 The disease models we consider are recessive f = (0. [sent-266, score-0.494]
82 The disease site is one of the sites chosen uniformly at random. [sent-273, score-0.837]
83 The goal of this comparison is to see whether our disease model performs at least as well as the LOD estimator used by Merlin. [sent-274, score-0.414]
84 16) Table 1: This table gives the performance of our method and Merlin for recessive and dominant diseases as measured by the disease prediction metric. [sent-317, score-0.494]
85 The sizes of the simulated pedigrees are given in the first three columns, the disease model in the next three columns, and the performance of our method and that of Merlin in the final four columns. [sent-318, score-0.657]
86 The founder haplotypes were taken from the phased haplotypes of the JPT+CHB HapMap [29] populations, see Supplement for details. [sent-322, score-0.483]
87 Each experiment was replicated 10 times where for each replicate the founder haplotypes were sampled with a different random seed. [sent-323, score-0.313]
88 We computed a metric ψ which is roughly the rank of the disease site in the sorted list of predictions given by each method. [sent-324, score-0.682]
89 Between delineated rows of the table, we can compare the effect of pedigree size, and we observe that larger pedigrees aid in disease site prediction. [sent-329, score-1.404]
90 Indeed, the largest pedigree of 1276 individuals reaches an accuracy of 6e−4 . [sent-330, score-0.608]
91 This pedigree is the largest pedigree that we know of being analyzed in the literature. [sent-331, score-0.962]
92 6 Discussion This paper introduces a new disease model and a new variational inference method which are applied to find a Bayesian solution to the disease-site correlation problem. [sent-332, score-0.469]
93 This is in contrast to traditional linkage analysis where a likelihood ratio statistic is computed to find the position of the disease site relative to a map of existing sites. [sent-333, score-0.803]
94 Instead, our approach is to use a Haplotype-Phenotype Transducer to obtain a posterior for the probability of each site to be the disease site. [sent-334, score-0.682]
95 Particularly with sequencing data, it is likely that either the disease site or a nearby site will be observed. [sent-336, score-0.983]
96 Our method performs well in practice both for genotype prediction and for disease site prediction. [sent-337, score-0.764]
97 In the presence of missing data, where for some individuals the whole genome is missing, our method is able to infer the missing genotypes with high accuracy. [sent-338, score-0.297]
98 As compared with LOD linkage analysis method, our method was better able to predict the disease site when one observed site was responsible for the disease. [sent-339, score-1.071]
99 A general model for the analysis of pedigree data. [sent-426, score-0.481]
100 On the complexity of fundamental computational problems in pedigree analysis. [sent-454, score-0.481]
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