Titre : Title : PSEUDOMARKER 2 . 0 : efficient computation of likelihoods using NOMAD Auteurs :

Background: PSEUDOMARKER is a software package that performs joint linkage and linkage disequilibrium analysis between a marker and a putative disease locus. A key feature of PSEUDOMARKER is that it can combine case-controls and pedigrees of varying structure into a single unified analysis. Thus it maximizes the full likelihood of the data over marker allele frequencies or conditional allele frequencies on disease and recombination fraction. Results: The new version 2.0 uses the software package NOMAD to maximize likelihoods, resulting in generally comparable or better optima with many fewer evaluations of the likelihood functions. Conclusions: After being modified substantially to use modern optimization methods, PSEUDOMARKER version 2.0 is more robust and substantially faster than version 1.0. NOMAD may be useful in other bioinformatics problems where complex likelihood functions are optimized. Background PSEUDOMARKER [1] is a package that genomically localizes trait-predisposing loci by performing statistical tests using a putative disease locus and a series of markers. Genomic localization of genes that impact some phenotype is based on tests of independence of disease phenotypes from genotypes of a genome-spanning set of markers. Many “association tests” try to test directly for statistical relationships between disease phenotypes and marker genotypes directly by sampling large numbers of cases and controls or very small families. Such tests confound the statistical relationship between marker alleles and the genotypes at a putative nearby disease locus with the statistical relationship between the same markers and the phenotype. This confounding is unavoidable for casecontrol data because of the limited degrees of freedom, but these relationships can and should be modeled explicitly when analyzing more complex and heterogeneous pedigree sets. PSEUDOMARKER performs a full likelihood analysis under a specified model of the relationship between *Correspondence: gertz@ncbi.nlm.nih.gov 1National Center for Biotechnology Information, NIH, DHHS, Bethesda, MD, USA Full list of author information is available at the end of the article disease phenotypes and underlying genotypes. In pedigree data, one can test for genetic linkage as the preferential cosegregation of a marker or a haplotype with disease family-by-family; the marker genotype that cosegregates with the disease can differ from family to family. In either pedigree data or in case-control data, one can test for linkage disequilibrium (LD) between a marker and a putative disease locus as the preferential co-occurrence of a specific genotype at the marker with a genotype at the disease locus. By using a full likelihood model, PSEUDOMARKER can combine analysis of case-control (singletons) data and pedigree data of arbitrary size in one unified testing framework. We directly analyze linkage and LD among marker and disease genotypes, integrating over all possible genotypes at the putative two-allele disease-predisposing locus, for all individuals under an explicit model of the genotype-phenotype relationship. PSEUDOMARKER version 1 maximizes several likelihood functions [1] using a generalized pattern search (GPS) algorithm [2] implemented in a custom version of the ILINK [3] program. Previously, we showed that PSEUDOMARKER, using GPS likelihood estimates, performed well in detecting linkage and LD, outperforming several competing genetic analysis programs as measured by the power or false positive rate [4]. © 2014 Gertz et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gertz et al. BMC Bioinformatics 2014, 15:47 Page 2 of 8 http://www.biomedcentral.com/1471-2105/15/47 The running time of PSEUDOMARKER depends on the number of times the optimization algorithm evaluates any likelihood function. Each evaluation involves computation over one or, often, several pedigrees for fixed values of certain parameters that may include the recombination fraction and marker allele frequencies. ILINK computes these likelihoods using a peeling method that is a generalization of the Elston-Stewart algorithm [5]. Computation time is highly dependent on the pedigree structure and the number of untyped founders. A reduction in the number of likelihood function evaluations would allow more samples, larger and more complex pedigrees, or a greater density of markers to be analyzed in a reasonable amount of time. Although the GPS method [2] was more robust than the older line search method implemented in all previous versions of ILINK, we decided that the number of likelihood evaluationsmight be reduced by using instead a newer algorithm known to outperform GPS in some other optimization problems. Mesh Adaptive Direct Search (MADS) [6] is a framework for a class of derivative-free algorithms designed to supersede the GPS method. MADS is conceptually similar to GPS, but uses a richer set of search directions, resulting in better theoretical convergence properties. The NOMAD software package [7] is a high-quality, C++ open-source implementation of MADS algorithms in use in universities and companies around the world [8-11]. NOMAD is robust [12] and has a wide range of functionality, including handling of general nonlinear constraints, biobjective optimization, parallelism, and the restriction of variables to integer or boolean values [13] . We describe PSEUDOMARKER 2.0, which uses a customized version of ILINK that uses NOMAD tomaximize likelihoods. We show that NOMAD is more effective at finding optima than GPS, while requiring fewer evaluations of the likelihood function.