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PubMed1998

Comparison of four sib-pair linkage methods for analyzing sibships with more than two affecteds: interest of the binomial maximum likelihood approach.

Family samples collected for sib-pair linkage studies usually include some sibships with more than two affecteds (multiplex sibships). Several methods have been proposed to take into account these multiplex sibships, and four of them are discussed in this work. Two methods, which are the most widely used, are based on the number of alleles shared by the sib-pairs constitutive of the multiplex sibship, with the first using the total number of these shared alleles ("all possible pairs" method) and the second considering a weighted number of these alleles (weighted method). The two other approaches considered the sibship as a whole, with in particular a likelihood method based on a binomial distribution of parental alleles among affected offspring. We theoretically show that, in the analysis of sibships with two affecteds, this likelihood method is expected to be more powerful than the classical mean test when a common asymptotic type I error is used. The variation of the sibship informativeness (assessed by the proportion of heterozygous parents) according to the number of affected sibs is investigated under various genetic models. Simulations under the null hypothesis of no linkage indicate that the "all possible pairs" is anticonservative, especially for type I errors < or = 0.001, whereas the weighted method generally provides satisfactory results. The likelihood method shows very consistent results in terms of type I errors, whatever the sample size, and provides power levels similar to those of the other methods. This binomial likelihood approach, which accounts in a natural way for multiplex sibships and provides a simple likelihood-ratio test for linkage involving a single parameter, appears to be a quite interesting alternative to analyze sib-pair studies.

Adult↗

PubMed2006

Consistency of estimators of population scaled parameters using composite likelihood.

Composite likelihood methods have become very popular for the analysis of large-scale genomic data sets because of the computational intractability of the basic coalescent process and its generalizations: It is virtually impossible to calculate the likelihood of an observed data set spanning a large chromosomal region without using approximate or heuristic methods. Composite likelihood methods are approximate methods and, in the present article, assume the likelihood is written as a product of likelihoods, one for each of a number of smaller regions that together make up the whole region from which data is collected. A very general framework for neutral coalescent models is presented and discussed. The framework comprises many of the most popular coalescent models that are currently used for analysis of genetic data. Assume data is collected from a series of consecutive regions of equal size. Then it is shown that the observed data forms a stationary, ergodic process. General conditions are given under which the maximum composite estimator of the parameters describing the model (e.g. mutation rates, demographic parameters and the recombination rate) is a consistent estimator as the number of regions tends to infinity.

Base Sequence↗

PubMed2005

Comparison between two partial likelihood approaches for the competing risks model with missing cause of failure.

In many clinical studies where time to failure is of primary interest, patients may fail or die from one of many causes where failure time can be right censored. In some circumstances, it might also be the case that patients are known to die but the cause of death information is not available for some patients. Under the assumption that cause of death is missing at random, we compare the Goetgbebeur and Ryan (1995, Biometrika, 82, 821-833) partial likelihood approach with the Dewanji (1992, Biometrika, 79, 855-857) partial likelihood approach. We show that the estimator for the regression coefficients based on the Dewanji partial likelihood is not only consistent and asymptotically normal, but also semiparametric efficient. While the Goetghebeur and Ryan estimator is more robust than the Dewanji partial likelihood estimator against misspecification of proportional baseline hazards, the Dewanji partial likelihood estimator allows the probability of missing cause of failure to depend on covariate information without the need to model the missingness mechanism. Tests for proportional baseline hazards are also suggested and a robust variance estimator is derived.

Cause of Death↗

PubMed1993

The delay before onset of accelerated tumour cell repopulation during radiotherapy: a direct maximum-likelihood analysis of a collection of worldwide tumour-control data.

The worldwide collection of control data for head and neck tumours presented by Withers et al. (Withers, H.R., Taylor, J.M.G. and Maciejewski, B. Acta Oncol. 27: 131-146, 1988) was reanalysed using a model which includes an explicit lag phase before the onset of tumour clonogen repopulation. A direct maximum-likelihood approach was used and the methodology extended to include the computation of profile-likelihood confidence limits. A statistically significant (p = 0.02) lag of 29 days was obtained with 95% confidence limits covering the range 17-31 days. However, the confidence interval was disconnected, and excluded the period 21-23 days. The analysis gave a time factor of 0.66 Gy/day. The mean values confirm the conclusions drawn by the original authors using a two-stage (indirect) method, and the values are similar to those calculated here for another data set comprising 496 patients (lag period = 26 (19-33) days). However, the data set itself is retrospective, and potentially subject to a number of biases. Therefore any clinical conclusions can only be tentative. A new feature of the methodology is the computation of profile-likelihood confidence limits and this will be useful in future direct analyses of clinical data of this type. The more usually computed normal approximation to the confidence limits have been shown to be inadequate in this analysis, and either profile-likelihood limits or likelihood ratio tests must be employed to determine the significance of the model parameters.

Head and Neck Neoplasms↗

PubMed2001

Power of regression and maximum likelihood methods to map QTL from sib-pair and DZ twin data.

A common study design to map quantitative trait loci (QTL) is to compare the phenotypes and marker genotypes of two or more siblings in a sample of unrelated sib groups, and to test for linkage between chromosome location and quantitative trait values. The simplest case is sib pairs only, in particular dizygotic twin pairs, and a simple and elegant regression method was proposed by Haseman & Elston in 1972 to test for linkage. Since then, several other methods have been proposed to test for linkage. In this study, we derived the statistical power of linear regression and maximum likelihood methods to map QTL from sib pair data analytically, and determined which methods are superior under which set of population parameters. In particular, we considered four regression-based and three maximum likelihood-based approaches, and derived asymptotic approximations of the mean test statistic and statistical power for each method. It was found, both analytically and by computer simulation, that the revisited or new Haseman-Elston method (based upon the mean-corrected crossproduct of the observations on sib-pairs) is less powerful than a full maximum likelihood approach and is also inferior to the Haseman-Elston method under a realistic range of values for the population parameters. We found that a simple regression method, based upon both the squared difference and the mean-corrected squared sum of the observations on sib-pairs, is as powerful as a full maximum likelihood approach. Our derivations of statistical power for regression and maximum likelihood methods provide a simple way to compare alternative methods and obviate the need to perform elaborate computer simulations. DZ twin pairs are likely to be more powerful for linkage analysis than ordinary siblings because they may share more common environmental effects, thereby increasing the proportion of within-family variance that is explained by a QTL.

Chromosome Mapping↗

PubMed2002

Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm.

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.

Algorithms↗

PubMed2001

Philosophy and phylogenetic inference: a comparison of likelihood and parsimony methods in the context of Karl Popper's writings on corroboration.

Advocates of cladistic parsimony methods have invoked the philosophy of Karl Popper in an attempt to argue for the superiority of those methods over phylogenetic methods based on Ronald Fisher's statistical principle of likelihood. We argue that the concept of likelihood in general, and its application to problems of phylogenetic inference in particular, are highly compatible with Popper's philosophy. Examination of Popper's writings reveals that his concept of corroboration is, in fact, based on likelihood. Moreover, because probabilistic assumptions are necessary for calculating the probabilities that define Popper's corroboration, likelihood methods of phylogenetic inference--with their explicit probabilistic basis--are easily reconciled with his concept. In contrast, cladistic parsimony methods, at least as described by certain advocates of those methods, are less easily reconciled with Popper's concept of corroboration. If those methods are interpreted as lacking probabilistic assumptions, then they are incompatible with corroboration. Conversely, if parsimony methods are to be considered compatible with corroboration, then they must be interpreted as carrying implicit probabilistic assumptions. Thus, the non-probabilistic interpretation of cladistic parsimony favored by some advocates of those methods is contradicted by an attempt by the same authors to justify parsimony methods in terms of Popper's concept of corroboration. In addition to being compatible with Popperian corroboration, the likelihood approach to phylogenetic inference permits researchers to test the assumptions of their analytical methods (models) in a way that is consistent with Popper's ideas about the provisional nature of background knowledge.

History, 20th Century↗

PubMedDate not supplied

A likelihood approach to estimating phylogeny from discrete morphological character data.

Evolutionary biologists have adopted simple likelihood models for purposes of estimating ancestral states and evaluating character independence on specified phylogenies; however, for purposes of estimating phylogenies by using discrete morphological data, maximum parsimony remains the only option. This paper explores the possibility of using standard, well-behaved Markov models for estimating morphological phylogenies (including branch lengths) under the likelihood criterion. An important modification of standard Markov models involves making the likelihood conditional on characters being variable, because constant characters are absent in morphological data sets. Without this modification, branch lengths are often overestimated, resulting in potentially serious biases in tree topology selection. Several new avenues of research are opened by an explicitly model-based approach to phylogenetic analysis of discrete morphological data, including combined-data likelihood analyses (morphology + sequence data), likelihood ratio tests, and Bayesian analyses.

Biometry↗

PubMed2003

Partitioned likelihood support and the evaluation of data set conflict.

In simultaneous analyses of multiple data partitions, the trees relevant when measuring support for a clade are the optimal tree, and the best tree lacking the clade (i.e., the most reasonable alternative). The parsimony-based method of partitioned branch support (PBS) "forces" each data set to arbitrate between the two relevant trees. This value is the amount each data set contributes to clade support in the combined analysis, and can be very different to support apparent in separate analyses. The approach used in PBS can also be employed in likelihood: a simultaneous analysis of all data retrieves the maximum likelihood tree, and the best tree without the clade of interest is also found. Each data set is fitted to the two trees and the log-likelihood difference calculated, giving "partitioned likelihood support" (PLS) for each data set. These calculations can be performed regardless of the complexity of the ML model adopted. The significance of PLS can be evaluated using a variety of resampling methods, such as the Kishino-Hasegawa test, the Shimodiara-Hasegawa test, or likelihood weights, although the appropriateness and assumptions of these tests remains debated.

Animals↗

PubMed2003

A simple, fast, and accurate algorithm to estimate large phylogenies by maximum likelihood.

The increase in the number of large data sets and the complexity of current probabilistic sequence evolution models necessitates fast and reliable phylogeny reconstruction methods. We describe a new approach, based on the maximum- likelihood principle, which clearly satisfies these requirements. The core of this method is a simple hill-climbing algorithm that adjusts tree topology and branch lengths simultaneously. This algorithm starts from an initial tree built by a fast distance-based method and modifies this tree to improve its likelihood at each iteration. Due to this simultaneous adjustment of the topology and branch lengths, only a few iterations are sufficient to reach an optimum. We used extensive and realistic computer simulations to show that the topological accuracy of this new method is at least as high as that of the existing maximum-likelihood programs and much higher than the performance of distance-based and parsimony approaches. The reduction of computing time is dramatic in comparison with other maximum-likelihood packages, while the likelihood maximization ability tends to be higher. For example, only 12 min were required on a standard personal computer to analyze a data set consisting of 500 rbcL sequences with 1,428 base pairs from plant plastids, thus reaching a speed of the same order as some popular distance-based and parsimony algorithms. This new method is implemented in the PHYML program, which is freely available on our web page: http://www.lirmm.fr/w3ifa/MAAS/.

Algorithms↗

PubMed2003

Comparison of likelihood and Bayesian methods for estimating divergence times using multiple gene Loci and calibration points, with application to a radiation of cute-looking mouse lemur species.

Divergence time and substitution rate are seriously confounded in phylogenetic analysis, making it difficult to estimate divergence times when the molecular clock (rate constancy among lineages) is violated. This problem can be alleviated to some extent by analyzing multiple gene loci simultaneously and by using multiple calibration points. While different genes may have different patterns of evolutionary rate change, they share the same divergence times. Indeed, the fact that each gene may violate the molecular clock differently leads to the advantage of simultaneous analysis of multiple loci. Multiple calibration points provide the means for characterizing the local evolutionary rates on the phylogeny. In this paper, we extend previous likelihood models of local molecular clock for estimating species divergence times to accommodate multiple calibration points and multiple genes. Heterogeneity among different genes in evolutionary rate and in substitution process is accounted for by the models. We apply the likelihood models to analyze two mitochondrial protein-coding genes, cytochrome oxidase II and cytochrome b, to estimate divergence times of Malagasy mouse lemurs and related outgroups. The likelihood method is compared with the Bayes method of Thorne et al. (1998, Mol. Biol. Evol. 15:1647-1657), which uses a probabilistic model to describe the change in evolutionary rate over time and uses the Markov chain Monte Carlo procedure to derive the posterior distribution of rates and times. Our likelihood implementation has the drawbacks of failing to accommodate uncertainties in fossil calibrations and of requiring the researcher to classify branches on the tree into different rate groups. Both problems are avoided in the Bayes method. Despite the differences in the two methods, however, data partitions and model assumptions had the greatest impact on date estimation. The three codon positions have very different substitution rates and evolutionary dynamics, and assumptions in the substitution model affect date estimation in both likelihood and Bayes analyses. The results demonstrate that the separate analysis is unreliable, with dates variable among codon positions and between methods, and that the combined analysis is much more reliable. When the three codon positions were analyzed simultaneously under the most realistic models using all available calibration information, the two methods produced similar results. The divergence of the mouse lemurs is dated to be around 7-10 million years ago, indicating a surprisingly early species radiation for such a morphologically uniform group of primates.

Animals↗

PubMed2004

Column sorting: rapid calculation of the phylogenetic likelihood function.

Likelihood applications have become a central approach for molecular evolutionary analyses since the first computationally tractable treatment two decades ago. Although Felsenstein's original pruning algorithm makes likelihood calculations feasible, it is usually possible to take advantage of repetitive structure present in the data to arrive at even greater computational reductions. In particular, alignment columns with certain similarities have components of the likelihood calculation that are identical and need not be recomputed if columns are evaluated in an optimal order. We develop an algorithm for exploiting this speed improvement via an application of graph theory. The reductions provided by the method depend on both the tree and the data, but typical savings range between 15%and 50%. Real-data examples with time reductions of 80%have been identified. The overhead costs associated with implementing the algorithm are minimal, and they are recovered in all but the smallest data sets. The modifications will provide faster likelihood algorithms, which will allow likelihood methods to be applied to larger sets of taxa and to include more thorough searches of the tree topology space.

Algorithms↗

PubMed2005

Hidden likelihood support in genomic data: can forty-five wrongs make a right?

Combined analysis of multiple phylogenetic data sets can reveal emergent character support that is not evident in separate analyses of individual data sets. Previous parsimony analyses have shown that this hidden support often accounts for a large percentage of the overall phylogenetic signal in cladistic studies. Here, reanalysis of a large comparative genomic data set for yeast (genus Saccharomyces) demonstrates that hidden support can be an important factor in maximum likelihood analyses of multiple data sets as well. Emergent signal in a concatenation of 106 genes was responsible for up to 64% of the likelihood support at a particular node (the difference in log likelihood scores between optimal topologies that included and excluded a supported clade). A grouping of four yeast species (S. cerevisiae, S. paradoxus, S. mikatae, and S. kudriavzevii) was robustly supported by combined analysis of all 106 genes, but separate analyses of individual genes suggested numerous conflicts. Forty-eight genes strictly contradicted S. cerevisiae + S. paradoxus + S. mikatae + S. kudriavzevii in separate analyses, but combined likelihood analyses that included up to 45 of the "wrong" data sets supported this group. Extensive hidden support also emerged in a combined likelihood analysis of 41 genes that each recovered the exact same topology in separate analyses of the individual genes. These results show that isolated analyses of individual data sets can mask congruence and distort interpretations of clade stability, even in strictly model-based phylogenetic methods. Consensus and supertree procedures that ignore hidden phylogenetic signals are, at best, incomplete.

Classification↗

PubMed2004

DPRml: distributed phylogeny reconstruction by maximum likelihood.

MOTIVATION: In recent years there has been increased interest in producing large and accurate phylogenetic trees using statistical approaches. However for a large number of taxa, it is not feasible to construct large and accurate trees using only a single processor. A number of specialized parallel programs have been produced in an attempt to address the huge computational requirements of maximum likelihood. We express a number of concerns about the current set of parallel phylogenetic programs which are currently severely limiting the widespread availability and use of parallel computing in maximum likelihood-based phylogenetic analysis. RESULTS: We have identified the suitability of phylogenetic analysis to large-scale heterogeneous distributed computing. We have completed a distributed and fully cross-platform phylogenetic tree building program called distributed phylogeny reconstruction by maximum likelihood. It uses an already proven maximum likelihood-based tree building algorithm and a popular phylogenetic analysis library for all its likelihood calculations. It offers one of the most extensive sets of DNA substitution models currently available. We are the first, to our knowledge, to report the completion of a distributed phylogenetic tree building program that can achieve near-linear speedup while only using the idle clock cycles of machines. For those in an academic or corporate environment with hundreds of idle desktop machines, we have shown how distributed computing can deliver a 'free' ML supercomputer.

Algorithms↗

PubMed2005

Improving the efficiency of SPR moves in phylogenetic tree search methods based on maximum likelihood.

MOTIVATION: Maximum likelihood (ML) methods have become very popular for constructing phylogenetic trees from sequence data. However, despite noticeable recent progress, with large and difficult datasets (e.g. multiple genes with conflicting signals) current ML programs still require huge computing time and can become trapped in bad local optima of the likelihood function. When this occurs, the resulting trees may still show some of the defects (e.g. long branch attraction) of starting trees obtained using fast distance or parsimony programs. METHODS: Subtree pruning and regrafting (SPR) topological rearrangements are usually sufficient to intensively search the tree space. Here, we propose two new methods to make SPR moves more efficient. The first method uses a fast distance-based approach to detect the least promising candidate SPR moves, which are then simply discarded. The second method locally estimates the change in likelihood for any remaining potential SPRs, as opposed to globally evaluating the entire tree for each possible move. These two methods are implemented in a new algorithm with a sophisticated filtering strategy, which efficiently selects potential SPRs and concentrates most of the likelihood computation on the promising moves. RESULTS: Experiments with real datasets comprising 35-250 taxa show that, while indeed greatly reducing the amount of computation, our approach provides likelihood values at least as good as those of the best-known ML methods so far and is very robust to poor starting trees. Furthermore, combining our new SPR algorithm with local moves such as PHYML's nearest neighbor interchanges, the time needed to find good solutions can sometimes be reduced even more.

Algorithms↗

PubMed2005

Quasi-likelihood estimation for relative risk regression models.

For a prospective randomized clinical trial with two groups, the relative risk can be used as a measure of treatment effect and is directly interpretable as the ratio of success probabilities in the new treatment group versus the placebo group. For a prospective study with many covariates and a binary outcome (success or failure), relative risk regression may be of interest. If we model the log of the success probability as a linear function of covariates, the regression coefficients are log-relative risks. However, using such a log-linear model with a Bernoulli likelihood can lead to convergence problems in the Newton-Raphson algorithm. This is likely to occur when the success probabilities are close to one. A constrained likelihood method proposed by Wacholder (1986, American Journal of Epidemiology 123, 174-184), also has convergence problems. We propose a quasi-likelihood method of moments technique in which we naively assume the Bernoulli outcome is Poisson, with the mean (success probability) following a log-linear model. We use the Poisson maximum likelihood equations to estimate the regression coefficients without constraints. Using method of moment ideas, one can show that the estimates using the Poisson likelihood will be consistent and asymptotically normal. We apply these methods to a double-blinded randomized trial in primary biliary cirrhosis of the liver (Markus et al., 1989, New England Journal of Medicine 320, 1709-1713).

Antibiotics, Antineoplastic↗

PubMedDate not supplied

Comparisons of likelihood and machine learning methods of individual classification.

Classification methods used in machine learning (e.g., artificial neural networks, decision trees, and k-nearest neighbor clustering) are rarely used with population genetic data. We compare different nonparametric machine learning techniques with parametric likelihood estimations commonly employed in population genetics for purposes of assigning individuals to their population of origin ("assignment tests"). Classifier accuracy was compared across simulated data sets representing different levels of population differentiation (low and high F(ST)), number of loci surveyed (5 and 10), and allelic diversity (average of three or eight alleles per locus). Empirical data for the lake trout (Salvelinus namaycush) exhibiting levels of population differentiation comparable to those used in simulations were examined to further evaluate and compare classification methods. Classification error rates associated with artificial neural networks and likelihood estimators were lower for simulated data sets compared to k-nearest neighbor and decision tree classifiers over the entire range of parameters considered. Artificial neural networks only marginally outperformed the likelihood method for simulated data (0-2.8% lower error rates). The relative performance of each machine learning classifier improved relative likelihood estimators for empirical data sets, suggesting an ability to "learn" and utilize properties of empirical genotypic arrays intrinsic to each population. Likelihood-based estimation methods provide a more accessible option for reliable assignment of individuals to the population of origin due to the intricacies in development and evaluation of artificial neural networks.

Animals↗

PubMed2001

Maximum-likelihood approach for gene family evolution under functional divergence.

According to the observed alignment pattern (i.e., amino acid configuration), we studied two basic types of functional divergence of a protein family. Type I functional divergence after gene duplication results in altered functional constraints (i.e., different evolutionary rate) between duplicate genes, whereas type II results in no altered functional constraints but radical change in amino acid property between them (e.g., charge, hydrophobicity, etc.). Two statistical approaches, i.e., the subtree likelihood and the whole-tree likelihood, were developed for estimating the coefficients of (type I or type II) functional divergence. Numerical algorithms for obtaining maximum-likelihood estimates are also provided. Moreover, a posterior-based site-specific profile is implemented to predict critical amino acid residues that are responsible for type I and/or type II functional divergence after gene duplication. We compared the current likelihood with a fast method developed previously by examples; both show similar results. For handling altered functional constraints (type I functional divergence) in the large gene family with many member genes (clusters), which appears to be a normal case in postgenomics, the subtree likelihood provides a solution that is computationally feasible and robust against the uncertainty of the phylogeny. The cost of this feasibility is the approximation when frequencies of amino acids are very skewed. The potential bias and correction are discussed.

Amino Acid Sequence↗