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Cerqueira, Carlos H. Romanelli, Raquel C. Ali Fotouhi, Mina Majidi, M. The new approach presented in this paper is able to derive a gene-based predictive model based on SNPs data. Such model is more parsimonious than the one based on single SNPs, while preserving the capability of highlighting predictive SNPs configurations.

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The prediction performance of this approach was consistently superior to the SNP-based and the haplotype-based one in all the test sets of the evaluation procedure. The method can be then considered as an alternative way to analyze the data coming from association studies. Genetic association studies are a powerful method to assess correlations between genetic variants and traits differences occurring in a population. When a significant correlation arises with respect to a pathological trait, these studies may lead to the identification of candidate disease susceptibility genes, offering the promise of novel targets for therapeutic treatments.

Nowadays, high-throughput genotype technologies allow a genome wide approach to these studies, taking into account hundreds of thousands of different markers [ 1 , 2 ]. Standard statistics is usually applied to this data to extract univariate models and find significant markers with univariate tests. However, together with deriving the existing correlation between genetic markers and phenotypic traits it is also extremely interesting to find the relations between the markers themselves.

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Both aims can be effectively achieved by using Bayesian networks BNs [ 3 ]. BNs represent probabilistic relationships between random variables by means of a directed acyclic graph and a set of conditional probability distributions. Nodes in the graph correspond to variables and directed arcs represent dependencies between them.

A conditional probability distribution is associated with each node and quantifies the dependency of the node on its parents, i. BNs have already been successfully applied in association studies, for example to study overt stroke in sickle cell anaemia [ 4 ] and to identify the relationships between SNP variations in the human APOE gene and plasma apolipoprotein E levels [ 5 ]. When performing an association study, the data typically consist of measurements for a set of genetic markers SNPs and evidence for a certain number of phenotypic traits such as disease status, age, sex Each genetic marker is modelled as a random variable taking on one of three possible states: 'AA', which corresponds to homozygous for the minor allele, 'Aa', heterozygous, and 'aa', homozygous for the major allele.

Each phenotypic trait is also represented by a random variable, such as 'affected' and 'unaffected' for the disease status. This network not only models the relationships between the phenotype and SNPs, but it also represents conditional independence assumptions between variables. Thus, the BN can highlight potential key markers in phenotype prediction. On the left, the directed acyclic graph of the BN; on the right the conditional probabilities tables associated with each node. Both the graphical structure of a BN and the parameters of the conditional probability distributions can be learned from the available data.

However, learning these networks is often non-trivial due to the high number of variables to be taken into account in the model, with respect to the instances of the dataset. Thanks to this abstraction, a more parsimonious model might be built, whose graphical connections are also more easily interpretable. As the final aim of genetic dissection studies is to identify how genes affect the phenotype, we decided to consider the set of SNPs mapping to the same gene as a new meta-variable. In order to assign states to the meta-variables we employed an approach based on classification trees.

By learning a classification tree for the SNPs mapping to each gene, it is possible to identify the most relevant combination of SNP values to predict the phenotypic status.

Once the meta-variables have been identified, a BN is built using them and the phenotype as nodes. We applied our method to genotypic data measured in a group of patients affected by arterial hypertension and in a group of nonagenarians without history of hypertension. The ability of the BN inferred on the meta-variables to correctly predict the phenotype hypertension is quantitatively assessed and compared with that achievable with a BN built using single SNPs.

Our goal is to build a model to estimate the probability of a phenotypic trait given the genotype of an individual, represented as a suitable collection of SNPs. When learning this model from data, we also want to extract the relationships between SNPs and highlight the potential role of the genes associated to the SNPs. To this end, it is possible to resort to classification algorithms, in which the phenotype is the class and the SNPs and potentially other interesting variables, such as sex and age are the predictive attributes.

Our strategy is made of two main steps: i generation of meta-variables corresponding to each gene by using an approach based on classification trees, ii learning of a BN in which the nodes are the meta-variables and the phenotype. Classification trees CTs are one of the most largely used classification tools [ 6 ].

Given a database of n cases, each containing the values for v attributes and a class c , a CT learned from this database graphically represents a set of rules that allow the classification of each case on the basis of its attribute values Figure 2. A test on the value of an attribute is associated with every non-leaf node of the tree and a branch descends from this node for every possible value taken by the attribute; leaf nodes are instead associated with a class value. Classification tree for meta-variable state assignment.

Example of classification tree used to infer the possible states of the meta-variable associated with gene C, represented by two SNPs, C1 and C2. Bayesian networks [ 7 ] are a formalism for the representation and use of probabilistic knowledge widely employed in various fields, such as Artificial Intelligence, Statistics, and more recently Bioinformatics.

As mentioned in the Background Section, a BN consists of two main components, a directed acyclic graph and a set of probability distributions. While the graph qualitatively describes dependence relationships between variables, a conditional probability distribution is associated with each node X i and quantifies the probabilistic dependence of the node on its parents pa X i.

A very interesting property of BNs is the fact that the joint probability distribution of all variables can be expressed as the product of these conditional distributions chain rule : P X 1 , Once a BN is learned it is possible to use it to perform probabilistic inference, i. A BN can thus be employed for classification purposes, allowing the prediction of the most probable value for a class node once the values of some attributes are known.

In the following of this section we describe how we employ CTs to generate meta-variables and how we learn BNs on the generated variables. There are different available algorithms to learn a classification tree from a dataset. Partitioning algorithms recursively split the tree by choosing the "most informative" attribute, i. These algorithms usually implement some "pruning" strategies, i. Pruning helps avoiding overfitting and thus helps improving the tree's ability to classify new instances not used to generate the tree. CTs allow us to find rules to assign state values to meta-variables.

Our procedure is performed with the following steps:. Learn a classification tree using the phenotype to be forecast as class and the set S i as attributes. To this aim, we employed the C4. Apply minimal error pruning with m-estimate [ 10 ] and equal prior probability for each class. Check the total number of meta-variable states that remain after pruning steps a and b: if there are more than 5 states, cut the subtree with the lowest number of instances. Create a discrete variable G i with states corresponding to the leaves of the final pruned tree.

As an example, suppose having a gene C represented by two SNPs C1 and C2 , each taking three possible values "AA" and "BB" stand for homozygous for the minor allele, "Aa" and "Bb" stand for heterozygous, "aa" and "bb" for homozygous for the major allele. Suppose also that the classification tree corresponding to gene C is shown in Figure 2. The classification trees were learned using the software Orange [ 11 ]. Learning BNs can be approached as a model selection problem, in which different network models are compared on the basis of their posterior probability with respect to the available data.

Thanks to the decomposability of the joint probability of all variables, the network with highest posterior can be learned by learning local models, i. However, the number of possible models to be explored grows exponentially with respect to the number of candidate parents. For this reason, an exhaustive search is unfeasible and a heuristic strategy must be employed. An effective one is the greedy search strategy known as K2 algorithm [ 12 ].

This algorithm requires the specification of an ordering of the analyzed variables, so that the parents of each variable are searched only among those variables that precede it in the ordering. We decided to use the gain ratio of variables i.

## Bioinformatics

In this way, variables with higher gain ratio were tested as parents of those with lower ratios. Moreover, we focused on networks in which the genotypes are dependent on the phenotype, in accordance with Sebastiani et al. In order to infer BNs from data we employed the software Bayesware Discoverer [ 13 ], which implements the K2 algorithm for the search. We applied our approach to data coming from a genome-wide scan on individuals affected by arterial hypertension AH and nonagenarians without history of AH. Arterial hypertension is considered a polygenic disease, resulting from the combination of a number of genetic risk factors, whose expression depends on their interaction with environmental factors such as high dietary intake of sodium, alcohol, obesity and stress [ 14 ].

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