Maximum likelihood estimation with missing outcomes: From simplicity to complexity.

Felsenstein, J. (Department of Genetics SK-50, University of Washington, Seattle, Washington 98195). 1973. Maximum likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Syst. Zool. 22:240-249.The general maximum likelihood approach to the statistical estimation of phylogenies is outlined, for data in which there are a number of discrete states for each character. The details of the maximum likelihood method will depend on the details of the probabilistic model of evolution assumed. There are a very large number of possible models of evolution. For a few of the simpler models, the calculation of the likelihood of an evolutionary tree is outlined. For these models, the maximum likelihood tree will be the same as the parsimonious (or minimum-steps) tree if the probability of change during the evolution of the group is assumed a priori to be very small. However, most sets of data require too many assumed state changes per character to be compatible with this assumption. Farris (1973) has argued that maximum likelihood and parsimony methods are identical under a much less restrictive set of assumptions. It is argued that the present methods are preferable to his, and a counterexample to his argument is presented. An algorithm which enables rapid calculation of the likelihood of a phylogeny is described. [Evolutionary trees: maximum likelihood.] The first systematic attempt to apply standard statistical inference procedures to the estimation of evolutionary trees was the work of Edwards and Cavalli-Sforza (1964; see also Cavalli-Sforza and Edwards, 1967). At about the same time, the parsimony' or minimum evolutionary steps method of Camin and Sokal (1965) gave a great impetus to the development of welldefined procedures for obtaining evolutionary trees. Edwards and Cavalli-Sforza concerned themselves with data from continuous variables such as gene frequencies and quantitative characters. The CaminSokal approach, on the other hand, was developed for characters which are recorded as a series of discrete states. Although some taxonomists have declared that the problem of guessing phylogenies should be viewed as a problem of statistical inference (Farris, 1967, 1968; Throckmorton, 1968), until recently there have been no attempts to explore the relationship between the statistical inference and minimum-steps approaches. Recently, Farris (1973) has presented a detailed argument that, under certain reasonable assumptions, the maximum-likelihood method of statistical inference appropriate to discrete-character data is precisely the parsimony method of Camin and Sokal. In this paper, I will examine the application of maximum likelihood methods to discrete characters, and will show that parsimony methods are not maximum likelihood methods under the assumptions made by Farris. They are maximum likelihood methods under considerably more restrictive assumptions about evolution. METHODS OF MAXIMUM LIKELIHOOD Suppose that we want to estimate the evolutionary tree, T, which is to be specified by the topological form of the tree and the times of branching. We are given a set of data, D, and a model of evolution, M, which incorporates not only the evolutionary processes, but also the processes of sampling by which we obtained the data. This model will usually be probabilistic, involving random events such as changes of the environment, occurrence of favorable

A Poisson ridge regression estimator

Relative Efficiencies of the Maximum Likelihood, Maximum Parsimony, and Neighbor-Joining Methods for Estimating Protein Phylogeny

Calculation of the uncertainty in complication probability for various dose–response models, applied to the parotid gland

The beta regression model is commonly used when analyzing data that come in the form of rates or percentages. However, a problem that may encounter when analyzing these kinds of data that has not been investigated for this model is the multicollinearity problem. It is well known that the maximum likelihood (ML) method is very sensitive to high inter-correlation among the explanatory variables. Therefore, this paper proposes some ridge estimators for the beta regression model to remedy the problem of instability of the traditional ML method and increase the efficiency of estimation. The performance of ridge estimators is compared to the ML estimator through the mean squared error (MSE) and the mean absolute error (MAE) criteria by conducting a Monte-Carlo simulation study and through an empirical application. According to the simulation and application results, the proposed estimators outperform the ML estimator in terms of MSE and MAE.

Summary We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not ‘testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have ‘passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.

In an urban road network, travel times are not uniquely determined by the traffic states due to stochastic properties of traffic flow, stochastic arrivals and departures at intersections and traffic signal control. As a result, for a given traffic state, a range of travel times (delays) is found. This can be represented by a distribution of travel times (delays). Calibrating a model for the travel time only for the expectation value gives a large ‘noise’ such that the model will have little value for the prediction purpose. In this paper, the delay distribution function as derived from the analytical model under different circumstances is introduced. The overflow queue distribution which is the parameter in the delay distribution function is estimated based on traffic measurements, e.g., the measured delays, flows and cycle time. The Least Squares (LS) and Maximum Likelihood (ML) Estimation are used to perform the parameter estimation in the delay distribution. The Genetic Algorithm (GA) is applied to find the optimal solution for the objective functions in terms of minimizing square error and maximizing the likelihood function. Based on the estimated model parameters, the delay distribution is reconstructed. The estimated delay distribution is compared with that obtained from VISSIM simulation. Results show that both ML and LS estimation methods perform well in the undersaturated condition. While in the oversaturated condition, the ML method performs considerably better than the LS method.

An eigenvector (EV) method for the determination of directional spectra from heave, pitch and roll buoy data is presented. Both a direct and an iterative form (based on an algorithm by Pawka) of this data-adaptive procedure are developed. The direct form outperforms the Longuet-Higgins et al. method, the cosine spread model and the maximum likelihood (ML) method for both simulated and real data. In the iterative form, the iterative EV method is superior to the other methods tested, including the iterative ML method for unimodal peaks when the noise-to-signal ratio is greater than 0.2 and for bimodal peaks at all noise levels. With real data, the direct EV method produced errors within an 80% confidence zone of the data cross-spectral matrix within the stopping criteria of Lawson and Long. Errors of the iterative EV and iterative ML methods were lower than for both the direct EV and ML results and were about the same magnitude when compared to each other. The iterative EV method, however, produced...

The objective of this paper was to predict a model for describing stand structure of tree heights (h) and diameters at breast height (dbh). The research material consisted of data collected from 64 stands of Norway spruce (Picea abies Karst.) and 91 stands of Scots pine (Pinus sylvestris L.) located in southern Finland. Both stand types contained birch (Betula pendula Roth and B. pubescent Ehrh.) admixtures. The traditional univariate approach (Model I) of using the dbh distribution (Johnson’s SB) together with a height curve (Näslund’s function) was compared against the bivariate approaches, Johnson’s SBB distribution (Model II) and Model Ie. In Model Ie within-dbh-class h-variation was included by transforming a normally distributed homogenous error of linearized Näslund’s function to concern real heights. Basal-area-weighted distributions were estimated using the maximum likelihood (ML) method. Species-specific prediction models were derived using linear regression analysis. The models were compared with Kolmogorov-Smirnov tests for marginal distributions, accuracy of stand variables and the dbh-h relationship of individual trees. The differences in the stand characteristics between the models were marginal. Model I gave a slightly better fit for spruce, but Model II was better for pine stands. The univariate Model I resulted in clearly too narrow marginal h-distribution for pine. It is recommended applying of a constrained ML method for reasonable dbh-h relationship instead of using a pure ML method when fitting the SBB model.

Recent work has developed maximum likelihood (ML) methods for track-to-track data association and fusion in a multisensor, i.e., more than two sensor, environment. In order to conserve bandwidth, only the state estimates and corresponding covariance matrices are shared amongst the nodes. The fusion engine uses this track information to determine which tracks associate to the same target and then computes a fused track to improve the accuracy of the state estimates. The simplest class of ML methods assumes that the track errors from different sensors are uncorrelated. The more computationally demanding ML methods incorporate the cross-correlations that are due to the common process noise in the kinematic model of the target. In order to account for track correlations in practice, the cross-covariance matrices must be approximated from the single sensor covariance matrices. This paper introduces new methods to approximate the cross-covariance matrices, and these approximations lead to a third class of association and estimation methods. The paper then uses simulations to assess the performance of the different association and estimation techniques. The simulations include results when the sensor tracks are produced by either a Kalman filter or an interacting multiple model (IMM) filter.

Deconvolution techniques provide efficient implementations for super-resolution imaging for forward-looking scanning radar. However, deconvolution is normally an ill-posed problem, and the solution is extremely sensitive to noise. From a statistical perspective, maximum likelihood (ML) methods are able to condition the ill-posed problem into a well-posed one. Nevertheless, traditional ML methods only consider the amplitude of the echo and by ignoring the phase that do not adequately model the radar imaging system. In this letter, an I/Q-channel modeling ML method is proposed for forward-looking scanning radar. First, the probability model of the echo is deduced by jointly considering noise in the I and Q channels. Then, a probability density function of the received data is deduced and used to formulate the likelihood function. Finally, the targets can be precisely estimated by maximizing this likelihood function. The results of simulations and experiments are provided to illustrate the effectiveness of the proposed method.

Accurate determination of the effective reproduction number (Rt) is a very important strategy in the epidemiology of contagious diseases, including coronavirus disease 2019 (COVID-19). This study compares different methods of estimating the Rt of susceptible population to identify the most accurate method for estimating Rt. A secondary study. The value of Rt was estimated using attack rate (AR), exponential growth (EG), maximum likelihood (ML), time-dependent (TD), and sequential Bayesian (SB) methods, for Iran, the United States, the United Kingdom, India, and Brazil from June to October 2021. In order to accurately compare these methods, a simulation study was designed using forty scenarios. The lowest mean square error (MSE) was observed for TD and ML methods, with 15 and 12 cases, respectively. Therefore, considering the estimated values of Rt based on the TD method, it was found that Rt values in the United Kingdom (1.33; 95% CI: 1.14-1.52) and the United States (1.25; 95% CI: 1.12-1.38) substantially have been more than those in other countries, such as Iran (1.07; 95% CI: 0.95-1.19), India (0.99; 95% CI: 0.89-1.08), and Brazil (0.98; 95% CI: 0.84-1.14) from June to October 2021. The important result of this study is that TD and ML methods lead to a more accurate estimation of Rt of population than other methods. Therefore, in order to monitor and determine the epidemic situation and have a more accurate prediction of the incidence rate, as well as control COVID-19 and similar diseases, the use of these two methods is suggested to more accurately estimate Rt.

The aim of the study was to evaluate the immediate reproducibility of time domain parameters in the signal averaged electrocardiogram using a new method for endpoint determination in individual Frank leads. The method is based on a statistical model of the electrocardiogram (ECG) in which maximum likelihood (ML) estimation is employed. The reproducibility of the ML method was compared to that of conventional time domain analysis using the vector magnitude (VM) of Frank leads. Fifty-nine patients were included in the study and two consecutive ECGs were recorded for signal averaging. The results showed that the mean of the absolute difference of the filtered QRS duration (QRSD) between two consecutive recordings was significantly lower for the ML method than the conventional method when employing 60 Hz highpass filtering (2.1 +/- 2.2 ms vs 5.9 +/- 10.2 ms, P < 0.05). Moreover, the ML method resulted in a significantly longer QRSD compared to the VM-based method (P < 0.05). The terminal amplitude of the QRS complex (RMS40) showed a greater variability than the QRSD for both methods, although the ML method was associated with a higher reproducibility than the VM method for the 60 Hz filter. These findings may contribute to a better identification of patients at high risk of ventricular arrhythmias. A reduction in the number of measurement errors has important implications when QRS changes are analysed over time.

Three-parameter Weibull probability distribution function is used to represent the time-to-failure data. For parameter estimation, the errors-in-variables, maximum likelihood, and least squares methods are compared. The results obtained from five data sets show that maximum likelihood method gives the most reliable parameter estimates which are close to those obtained by errors-in-variables approach. The least squares method gave the poorest results in most cases. For parameter estimation, the Luus-Jaakola direct search optimization procedure yielded the optimal parameter values in negligible computation time, taking less than a second of computation time on a Pentium4/2.66MHz personal computer for 50 data points.

Multipoint Radiation Hybrid Mapping: Comparison of Methods, Sample Size Requirements, and Optimal Study Characteristics