A variable-rate quantitative trait evolution model using penalized-likelihood

Mapping Intimacies ◽

10.1101/2021.04.17.440282 ◽

2021 ◽

Author(s):

Liam J. Revell

Keyword(s):

Brownian Motion ◽

Quantitative Trait ◽

Penalized Likelihood ◽

Simulated Data ◽

Character Trait ◽

Likelihood Method ◽

Rate Variation ◽

Trait Evolution ◽

Penalty Term ◽

Rate Of Evolution

In recent years it's become increasingly popular to use phylogenetic comparative methods to investigate heterogeneity in the rate or process of quantitative trait evolution across the branches or clades of a phylogenetic tree. Here, I present a new method for modeling variability in the rate of evolution of a continuously-valued character trait on a reconstructed phylogeny. The underlying model of evolution is stochastic diffusion (Brownian motion), but in which the instantaneous diffusion rate (σ2) also evolves by Brownian motion on a log-scale. Unfortunately, it's not possible to simultaneously estimate the rates of evolution along each edge of the tree and the rate of evolution of σ2 itself using Maximum Likelihood. As such, I propose a penalized-likelihood method in which the penalty term is equal to the log-transformed probability density of the rates under a Brownian model, multiplied by a 'smoothing' coefficient, λ, selected by the user. λ determines the magnitude of penalty that's applied to rate variation between edges. Lower values of λ penalize rate variation relatively little; whereas larger λ values result in minimal rate variation among edges of the tree in our fitted model, eventually converging on a single value of σ2 for all of the branches of the tree. In addition to presenting this model here, I've also implemented it as part of my phytools R package in the function multirateBM. Using different values of the penalty coefficient, λ, I fit the model to simulated data with: Brownian rate variation among edges (the model assumption); uncorrelated rate variation; rate changes that occur in discrete places on the tree; and no rate variation at all among the branches of the phylogeny. I then compare the estimated values of σ2 to their known true values. In addition, I use the method to analyze a simple empirical dataset of body mass evolution in mammals. Finally, I discuss some applications and limitations of the approach.

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A variable-rate quantitative trait evolution model using penalized-likelihood

PeerJ ◽

10.7717/peerj.11997 ◽

2021 ◽

Vol 9 ◽

pp. e11997

Author(s):

Liam J. Revell

Keyword(s):

Brownian Motion ◽

Quantitative Trait ◽

Penalized Likelihood ◽

Comparative Methods ◽

Character Trait ◽

Likelihood Method ◽

Rate Variation ◽

Phylogenetic Comparative Methods ◽

Trait Evolution ◽

Rate Of Evolution

In recent years it has become increasingly popular to use phylogenetic comparative methods to investigate heterogeneity in the rate or process of quantitative trait evolution across the branches or clades of a phylogenetic tree. Here, I present a new method for modeling variability in the rate of evolution of a continuously-valued character trait on a reconstructed phylogeny. The underlying model of evolution is stochastic diffusion (Brownian motion), but in which the instantaneous diffusion rate (σ2) also evolves by Brownian motion on a logarithmic scale. Unfortunately, it’s not possible to simultaneously estimate the rates of evolution along each edge of the tree and the rate of evolution of σ2 itself using Maximum Likelihood. As such, I propose a penalized-likelihood method in which the penalty term is equal to the log-transformed probability density of the rates under a Brownian model, multiplied by a ‘smoothing’ coefficient, λ, selected by the user. λ determines the magnitude of penalty that’s applied to rate variation between edges. Lower values of λ penalize rate variation relatively little; whereas larger λ values result in minimal rate variation among edges of the tree in the fitted model, eventually converging on a single value of σ2 for all of the branches of the tree. In addition to presenting this model here, I have also implemented it as part of my phytools R package in the function multirateBM. Using different values of the penalty coefficient, λ, I fit the model to simulated data with: Brownian rate variation among edges (the model assumption); uncorrelated rate variation; rate changes that occur in discrete places on the tree; and no rate variation at all among the branches of the phylogeny. I then compare the estimated values of σ2 to their known true values. In addition, I use the method to analyze a simple empirical dataset of body mass evolution in mammals. Finally, I discuss the relationship between the method of this article and other models from the phylogenetic comparative methods and finance literature, as well as some applications and limitations of the approach.

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A comparison between methods for linkage disequilibrium fine mapping of quantitative trait loci

Genetics Research ◽

10.1017/s0016672303006554 ◽

2004 ◽

Vol 83 (1) ◽

pp. 41-47 ◽

Cited By ~ 8

Author(s):

JIHAD M. ABDALLAH ◽

BRIGITTE MANGIN ◽

BRUNO GOFFINET ◽

CHRISTINE CIERCO-AYROLLES ◽

MIGUEL PÉREZ-ENCISO

Keyword(s):

Linkage Disequilibrium ◽

Quantitative Trait Loci ◽

Quantitative Trait ◽

Simulated Data ◽

Likelihood Method ◽

Trait Loci ◽

Single Marker ◽

Linkage Disequilibrium Information ◽

Marker Regression ◽

Trait Locus

We present a maximum likelihood method for mapping quantitative trait loci that uses linkage disequilibrium information from single and multiple markers. We made paired comparisons between analyses using a single marker, two markers and six markers. We also compared the method to single marker regression analysis under several scenarios using simulated data. In general, our method outperformed regression (smaller mean square error and confidence intervals of location estimate) for quantitative trait loci with dominance effects. In addition, the method provides estimates of the frequency and additive and dominance effects of the quantitative trait locus.

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Maximum likelihood mapping of quantitative trait loci using full-sib families.

Genetics ◽

10.1093/genetics/132.4.1211 ◽

1992 ◽

Vol 132 (4) ◽

pp. 1211-1222 ◽

Cited By ~ 2

Author(s):

S A Knott ◽

C S Haley

Keyword(s):

Quantitative Trait Loci ◽

Maximum Likelihood ◽

Quantitative Trait ◽

Simulated Data ◽

Likelihood Method ◽

Parameter Estimates ◽

Random Component ◽

Test Statistic ◽

Trait Loci ◽

Sib Families

Abstract A maximum likelihood method is presented for the detection of quantitative trait loci (QTL) using flanking markers in full-sib families. This method incorporates a random component for common family effects due to additional QTL or the environment. Simulated data have been used to investigate this method. With a fixed total number of full sibs power of detection decreased substantially with decreasing family size. Increasing the number of alleles at the marker loci (i.e., polymorphism information content) and decreasing the interval size about the QTL increased power. Flanking markers were more powerful than single markers. In testing for a linked QTL the test must be made against a model which allows for between family variation (i.e., including an unlinked QTL or a between family variance component) or the test statistic may be grossly inflated. Mean parameter estimates were close to the simulated values in all situations when fitting the full model (including a linked QTL and common family effect). If the common family component was omitted the QTL effect was overestimated in data in which additional genetic variance was simulated and when compared with an unlinked QTL model there was reduced power. The test statistic curves, reflecting the likelihood of the QTL at each position along the chromosome, have discontinuities at the markers caused by adjacent pairs of markers providing different amounts of information. This must be accounted for when using flanking markers to search for a QTL in an outbred population.

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A Penalized Likelihood Method for Mapping Epistatic Quantitative Trait Loci With One-Dimensional Genome Searches

Genetics ◽

10.1093/genetics/162.2.951 ◽

2002 ◽

Vol 162 (2) ◽

pp. 951-960 ◽

Cited By ~ 7

Author(s):

Martin P Boer ◽

Cajo J F ter Braak ◽

Ritsert C Jansen

Keyword(s):

Quantitative Trait Loci ◽

Quantitative Trait ◽

Statistical Tests ◽

Penalized Likelihood ◽

Likelihood Method ◽

Likelihood Methods ◽

One Dimensional ◽

Statistical Framework ◽

Multidimensional Search ◽

Trait Loci

AbstractEpistasis is a common and important phenomenon, as indicated by results from a number of recent experiments. Unfortunately, the discovery of epistatic quantitative trait loci (QTL) is difficult since one must search for multiple QTL simultaneously in two or more dimensions. Such a multidimensional search necessitates many statistical tests, and a high statistical threshold must be adopted to avoid false positives. Furthermore, the large number of (interaction) parameters in comparison with the number of observations results in a serious danger of overfitting and overinterpretation of the data. In this article we present a new statistical framework for mapping epistasis in inbred line crosses. It is based on reducing the high dimensionality of the problem in two ways. First, epistatic QTL are mapped in a one-dimensional genome scan for high interactions between QTL and the genetic background. Second, the dimension of the search is bounded by penalized likelihood methods. We use simulated backcross data to illustrate the new approach.

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Hyperbolastic Models from a Stochastic Differential Equation Point of View

Mathematics ◽

10.3390/math9161835 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1835

Author(s):

Antonio Barrera ◽

Patricia Román-Román ◽

Francisco Torres-Ruiz

Keyword(s):

Differential Equation ◽

Simulated Data ◽

Real Data ◽

Point Of View ◽

Likelihood Method ◽

Numerical Resolution ◽

The Family ◽

The Common ◽

Linear Differential ◽

Mean Function

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

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Consequences of Recombination Rate Variation on Quantitative Trait Locus Mapping Studies: Simulations Based on the Drosophila melanogaster Genome

Genetics ◽

10.1093/genetics/159.2.581 ◽

2001 ◽

Vol 159 (2) ◽

pp. 581-588

Author(s):

Mohamed A F Noor ◽

Aimee L Cunningham ◽

John C Larkin

Keyword(s):

Quantitative Trait Locus ◽

Drosophila Melanogaster ◽

Qtl Mapping ◽

Quantitative Trait ◽

Recombination Rate ◽

Genome Project ◽

Gene Density ◽

Rate Variation ◽

Trait Locus ◽

Recombination Rate Variation

Abstract We examine the effect of variation in gene density per centimorgan on quantitative trait locus (QTL) mapping studies using data from the Drosophila melanogaster genome project and documented regional rates of recombination. There is tremendous variation in gene density per centimorgan across this genome, and we observe that this variation can cause systematic biases in QTL mapping studies. Specifically, in our simulated mapping experiments of 50 equal-effect QTL distributed randomly across the physical genome, very strong QTL are consistently detected near the centromeres of the two major autosomes, and few or no QTL are often detected on the X chromosome. This pattern persisted with varying heritability, marker density, QTL effect sizes, and transgressive segregation. Our results are consistent with empirical data collected from QTL mapping studies of this species and its close relatives, and they explain the “small X-effect” that has been documented in genetic studies of sexual isolation in the D. melanogaster group. Because of the biases resulting from recombination rate variation, results of QTL mapping studies should be taken as hypotheses to be tested by additional genetic methods, particularly in species for which detailed genetic and physical genome maps are not available.

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Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering

Applied Sciences ◽

10.3390/app8091674 ◽

2018 ◽

Vol 8 (9) ◽

pp. 1674

Author(s):

Wengang Chen ◽

Wenzheng Xiu ◽

Jin Shen ◽

Wenwen Zhang ◽

Min Xu ◽

...

Keyword(s):

Particle Size ◽

Light Scattering ◽

Dynamic Light Scattering ◽

Simulated Data ◽

Bimodal Distribution ◽

Scattering Data ◽

Inversion Algorithm ◽

Penalty Term ◽

Peak Resolution ◽

Time Period

By using different weights to deal with the autocorrelation function data of every delay time period, the information utilization of dynamic light scattering can be obviously enhanced in the information-weighted constrained regularization inversion, but the denoising ability and the peak resolution under noise conditions for information-weighted inversion algorithm are still insufficient. On the basis of information weighting, we added a penalty term with the function of flatness constraints to the objective function of the regularization inversion, and performed the inversion of multiangle dynamic light scattering data, including the simulated data of bimodal distribution particles (466/915 nm, 316/470 nm) and trimodal distribution particles (324/601/871 nm), and the measured data of bimodal distribution particles (306/974 nm, 300/502 nm). The results of the inversion show that multiple-penalty-weighted regularization inversion can not only improve the utilization of the particle size information, but also effectively eliminate the false peaks and burrs in the inversed particle size distributions, and further improve the resolution of peaks in the noise conditions, and then improve the weighting effects of the information-weighted inversion.

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Reliable confidence intervals for RelTime estimates of evolutionary divergence times

10.1101/677286 ◽

2019 ◽

Cited By ~ 1

Author(s):

Qiqing Tao ◽

Koichiro Tamura ◽

Beatriz Mello ◽

Sudhir Kumar

Keyword(s):

Confidence Intervals ◽

Divergence Time ◽

Simulated Data ◽

Molecular Dating ◽

Divergence Times ◽

Rate Variation ◽

Evolutionary Divergence ◽

Posterior Density ◽

Divergence Time Estimates ◽

Highest Posterior Density

AbstractConfidence intervals (CIs) depict the statistical uncertainty surrounding evolutionary divergence time estimates. They capture variance contributed by the finite number of sequences and sites used in the alignment, deviations of evolutionary rates from a strict molecular clock in a phylogeny, and uncertainty associated with clock calibrations. Reliable tests of biological hypotheses demand reliable CIs. However, current non-Bayesian methods may produce unreliable CIs because they do not incorporate rate variation among lineages and interactions among clock calibrations properly. Here, we present a new analytical method to calculate CIs of divergence times estimated using the RelTime method, along with an approach to utilize multiple calibration uncertainty densities in these analyses. Empirical data analyses showed that the new methods produce CIs that overlap with Bayesian highest posterior density (HPD) intervals. In the analysis of computer-simulated data, we found that RelTime CIs show excellent average coverage probabilities, i.e., the true time is contained within the CIs with a 95% probability. These developments will encourage broader use of computationally-efficient RelTime approach in molecular dating analyses and biological hypothesis testing.

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