Likelihood-based Methods Research Articles

Outlier-free merging of homogeneous groups of pre-classified observations under contamination

ABSTRACTWe study the problem of merging homogeneous groups of pre-classified observations from a robust perspective motivated by the anti-fraud analysis of international trade data. This problem may be seen as a clustering task which exploits preliminary information on the potential clusters, available in the form of group-wise linear regressions. Robustness is then needed because of the sensitivity of likelihood-based regression methods to deviations from the postulated model. Through simulations run under different contamination scenarios, we assess the impact of outliers both on group-wise regression fitting and on the quality of the final clusters. We also compare alternative robust methods that can be adopted to detect the outliers and thus to clean the data. One major conclusion of our study is that the use of robust procedures for preliminary outlier detection is generally recommended, except perhaps when contamination is weak and the identification of cluster labels is more important than the estimation of group-specific population parameters. We also apply the methodology to find homogeneous groups of transactions in one empirical example that illustrates our motivating anti-fraud framework.

Why Concatenation Fails Near the Anomaly Zone

Genome-scale sequencing has been of great benefit in recovering species trees but has not provided final answers. Despite the rapid accumulation of molecular sequences, resolving short and deep branches of the tree of life has remained a challenge and has prompted the development of new strategies that can make the best use of available data. One such strategy-the concatenation of gene alignments-can be successful when coupled with many tree estimation methods, but has also been shown to fail when there are high levels of incomplete lineage sorting. Here, we focus on the failure of likelihood-based methods in retrieving a rooted, asymmetric four-taxon species tree from concatenated data when the species tree is in or near the anomaly zone-a region of parameter space where the most common gene tree does not match the species tree because of incomplete lineage sorting. First, we use coalescent theory to prove that most informative sites will support the species tree in the anomaly zone, and that as a consequence maximum-parsimony succeeds in recovering the species tree from concatenated data. We further show that maximum-likelihood tree estimation from concatenated data fails both inside and outside the anomaly zone, and that this failure cannot be easily predicted from the topology of the most common gene tree. We demonstrate that likelihood-based methods often fail in a region partially overlapping the anomaly zone, likely because of the lower relative cost of substitutions on discordant gene tree branches that are absent from the species tree. Our results confirm and extend previous reports on the performance of these methods applied to concatenated data from a rooted, asymmetric four-taxon species tree, and highlight avenues for future work improving the performance of methods aimed at recovering species tree.

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

ABSTRACTThe estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.

Paule-Mandel estimators for network meta-analysis with random inconsistency effects.

Network meta‐analysis is used to simultaneously compare multiple treatments in a single analysis. However, network meta‐analyses may exhibit inconsistency, where direct and different forms of indirect evidence are not in agreement with each other, even after allowing for between‐study heterogeneity. Models for network meta‐analysis with random inconsistency effects have the dual aim of allowing for inconsistencies and estimating average treatment effects across the whole network. To date, two classical estimation methods for fitting this type of model have been developed: a method of moments that extends DerSimonian and Laird's univariate method and maximum likelihood estimation. However, the Paule and Mandel estimator is another recommended classical estimation method for univariate meta‐analysis. In this paper, we extend the Paule and Mandel method so that it can be used to fit models for network meta‐analysis with random inconsistency effects. We apply all three estimation methods to a variety of examples that have been used previously and we also examine a challenging new dataset that is highly heterogenous. We perform a simulation study based on this new example. We find that the proposed Paule and Mandel method performs satisfactorily and generally better than the previously proposed method of moments because it provides more accurate inferences. Furthermore, the Paule and Mandel method possesses some advantages over likelihood‐based methods because it is both semiparametric and requires no convergence diagnostics. Although restricted maximum likelihood estimation remains the gold standard, the proposed methodology is a fully viable alternative to this and other estimation methods.

Identification-robust moment-based tests for Markov switching in autoregressive models

ABSTRACTThis paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison with the optimal tests for Markov-switching parameters of Carrasco et al. (2014), and they are also quite attractive owing to their computational simplicity. The new tests are illustrated with an empirical application to an autoregressive model of USA output growth.

A likelihood-based method for point and interval estimation on auditory filter data

A novel method for determining auditory filter (AF) shapes given a set of n-alternative forced choice (nAFC) responses from a single human subject or set of subjects is discussed. The method works by developing a function which maps individual nAFC responses into likelihood values—either supporting or conflicting with a proposed model AF shape. The aggregate of these likelihoods is then used as an objective function for optimization schemes for point estimation, or as the basis function for Metropolis Hastings-like algorithms for interval estimation, both of either parameters of the AF model or of the entire AF shape. The method is demonstrated on simulated up-down staircase data. The consistency of the method is discussed in the context of canonical methods for AF data analysis, some of which are shown to produce systematic errors. Other possible benefits of this approach are discussed including the ability of the method to: combine data from heterogeneous nAFC tasks (e.g., notched-noise maskers with ton...

FiSSE: A simple nonparametric test for the effects of a binary character on lineage diversification rates.

It is widely assumed that phenotypic traits can influence rates of speciation and extinction, and several statistical approaches have been used to test for correlations between character states and lineage diversification. Recent work suggests that model-based tests of state-dependent speciation and extinction are sensitive to model inadequacy and phylogenetic pseudoreplication. We describe a simple nonparametric statistical test ("FiSSE") to assess the effects of a binary character on lineage diversification rates. The method involves computing a test statistic that compares the distributions of branch lengths for lineages with and without a character state of interest. The value of the test statistic is compared to a null distribution generated by simulating character histories on the observed phylogeny. Our tests show that FiSSE can reliably infer trait-dependent speciation on phylogenies of several hundred tips. The method has low power to detect trait-dependent extinction but can infer state-dependent differences in speciation even when net diversification rates are constant. We assemble a range of macroevolutionary scenarios that are problematic for likelihood-based methods, and we find that FiSSE does not show similarly elevated false positive rates. We suggest that nonparametric statistical approaches, such as FiSSE, provide an important complement to formal process-based models for trait-dependent diversification.

Ab initio solution of macromolecular crystal structures without direct methods

The majority of macromolecular crystal structures are determined using the method of molecular replacement, in which known related structures are rotated and translated to provide an initial atomic model for the new structure. A theoretical understanding of the signal-to-noise ratio in likelihood-based molecular replacement searches has been developed to account for the influence of model quality and completeness, as well as the resolution of the diffraction data. Here we show that, contrary to current belief, molecular replacement need not be restricted to the use of models comprising a substantial fraction of the unknown structure. Instead, likelihood-based methods allow a continuum of applications depending predictably on the quality of the model and the resolution of the data. Unexpectedly, our understanding of the signal-to-noise ratio in molecular replacement leads to the finding that, with data to sufficiently high resolution, fragments as small as single atoms of elements usually found in proteins can yield ab initio solutions of macromolecular structures, including some that elude traditional direct methods.

Maximum likelihood estimation for stochastic volatility in mean models with heavy-tailed distributions.

In this article, we introduce a likelihood-based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions (Abanto-Valle et al., 2012). Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. The method is based on the proposal of Langrock et al. (2012) and makes explicit the useful link between HMMs and SVM models with SMN distributions. Likelihood-based estimation of the parameters of stochastic volatility models in general, and SVM models with SMN distributions in particular, is usually regarded as challenging as the likelihood is a high-dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, i.e., estimating the volatility of the process.

Inferring epidemiological parameters from phylogenies using regression-ABC: A comparative study.

Inferring epidemiological parameters such as the R0 from time-scaled phylogenies is a timely challenge. Most current approaches rely on likelihood functions, which raise specific issues that range from computing these functions to finding their maxima numerically. Here, we present a new regression-based Approximate Bayesian Computation (ABC) approach, which we base on a large variety of summary statistics intended to capture the information contained in the phylogeny and its corresponding lineage-through-time plot. The regression step involves the Least Absolute Shrinkage and Selection Operator (LASSO) method, which is a robust machine learning technique. It allows us to readily deal with the large number of summary statistics, while avoiding resorting to Markov Chain Monte Carlo (MCMC) techniques. To compare our approach to existing ones, we simulated target trees under a variety of epidemiological models and settings, and inferred parameters of interest using the same priors. We found that, for large phylogenies, the accuracy of our regression-ABC is comparable to that of likelihood-based approaches involving birth-death processes implemented in BEAST2. Our approach even outperformed these when inferring the host population size with a Susceptible-Infected-Removed epidemiological model. It also clearly outperformed a recent kernel-ABC approach when assuming a Susceptible-Infected epidemiological model with two host types. Lastly, by re-analyzing data from the early stages of the recent Ebola epidemic in Sierra Leone, we showed that regression-ABC provides more realistic estimates for the duration parameters (latency and infectiousness) than the likelihood-based method. Overall, ABC based on a large variety of summary statistics and a regression method able to perform variable selection and avoid overfitting is a promising approach to analyze large phylogenies.

Population genetics of immune-related multilocus copy number variation in Native Americans.

While multiallelic copy number variation (mCNV) loci are a major component of genomic variation, quantifying the individual copy number of a locus and defining genotypes is challenging. Few methods exist to study how mCNV genetic diversity is apportioned within and between populations (i.e. to define the population genetic structure of mCNV). These inferences are critical in populations with a small effective size, such as Amerindians, that may not fit the Hardy-Weinberg model due to inbreeding, assortative mating, population subdivision, natural selection or a combination of these evolutionary factors. We propose a likelihood-based method that simultaneously infers mCNV allele frequencies and the population structure parameter f, which quantifies the departure of homozygosity from the Hardy-Weinberg expectation. This method is implemented in the freely available software CNVice, which also infers individual genotypes using information from both the population and from trios, if available. We studied the population genetics of five immune-related mCNV loci associated with complex diseases (beta-defensins, CCL3L1/CCL4L1, FCGR3A, FCGR3B and FCGR2C) in 12 traditional Native American populations and found that the population structure parameters inferred for these mCNVs are comparable to but lower than those for single nucleotide polymorphisms studied in the same populations.

A statistical method utilizing information of imported cases to estimate the transmissibility for an influenza pandemic

BackgroundIn a new influenza pandemic, travel data such as arrival times of cases seeded by the originating country can be regarded as a combination of the epidemic size and the mobility networks of infections connecting the originating country with other regions. It can be a complete and timely source for estimating the basic reproduction number (R0), a key indicator of disease transmissibility.MethodIn this study, we developed a likelihood-based method using arrival times of infected cases in different countries to estimate R0 for influenza pandemics. A simulation was conducted to assess the performance of the proposed method. We further applied the method to the outbreak of the influenza pandemic A/H1N1 in Mexico.ResultsIn the numerical application, the estimated R0 was equal to 1.69 with a 95% confidence interval (1.65, 1.73). For the simulation results, the estimations were robust to the decline of travel rate and other parameter assumptions. Nevertheless, the estimates were moderately sensitive to the assumption of infectious duration. Generally, the findings were in line with other relevant studies.ConclusionsOur approach as well as the estimate is potential to assist officials in planning control and prevention measures. Improved coordination to streamline or even centralize surveillance of imported cases among countries will thus be beneficial to public health.

Sufficient dimension reduction constrained through sub-populations

Most methodologies for sufficient dimension reduction (SDR) in regression are limited to continuous predictors, although many data sets do contain both continuous and categorical variables. Application of these methods to regressions that include qualitative predictors such as gender or species may be inappropriate. Regressions that include a set of qualitative predictors W in addition to a vector X of many-valued predictors and a response Y are considered. Using principal fitted components (PFC) models, a likelihood-based SDR method, a sufficient dimension reduction of X that is constrained through the sub-populations established by W is sought. An estimator of the sufficient reduction subspace is provided and its use is demonstrated through applications.

Demographic decline and detection of genetic bottleneck in a population of Barn Owl Tyto alba in Hungary

The effects of population bottlenecks, from both the population and conservation genetics point of view, have attracted great interest. Heterozygosity and allelic diversity are basic components of the evolutionary potential of populations. Many endangered species go through bottlenecks from time to time. The consequences are important because in the resulting small populations the stochastic processes are more pronounced and, by strengthening each other, they can lead to extinction. Within the framework of the global climate change the question emerges of how the different species, primarily susceptible ones and those depending on protection measures, react to the climate extremities. It is well known that a large proportion of Barn Owl Tyto alba populations die following an unfavourable period. In the study reported here we genotyped 24 microsatellite loci of 58 unrelated Barn Owl individuals originating from a continuous population of three neighbouring Hungarian counties. We then performed the Wilcoxon sign-rank test, the L test and Monte Carlo simulation of the loss of alleles and heterozygosity. We also attempted to infer past population dynamics using two likelihood-based Bayesian methods. Although a 74% reduction of this population is well documented following the harsh winter of 2002/2003 and the sampling and resolution of the methods used are standard and considered to be commonly satisfactory, none of the tests showed pronounced signs of a recent bottleneck. These results are discussed in terms of the imminent and apparent drawbacks of the models and methods used, the roles of absolute numbers of survivors versus genotype percentages and the conservation biological viewpoint.

Model selection for seasonal influenza forecasting

Epidemics of seasonal influenza inflict a huge burden in temperate climes such as Melbourne (Australia) where there is also significant variability in their timing and magnitude. Particle filters combined with mechanistic transmission models for the spread of influenza have emerged as a popular method for forecasting the progression of these epidemics. Despite extensive research it is still unclear what the optimal models are for forecasting influenza, and how one even measures forecast performance.In this paper, we present a likelihood-based method, akin to Bayes factors, for model selection when the aim is to select for predictive skill. Here, “predictive skill” is measured by the probability of the data after the forecasting date, conditional on the data from before the forecasting date. Using this method we choose an optimal model of influenza transmission to forecast the number of laboratory-confirmed cases of influenza in Melbourne in each of the 2010–15 epidemics. The basic transmission model considered has the susceptible-exposed-infectious-recovered structure with extensions allowing for the effects of absolute humidity and inhomogeneous mixing in the population. While neither of the extensions provides a significant improvement in fit to the data they do differ in terms of their predictive skill. Both measurements of absolute humidity and a sinusoidal approximation of those measurements are observed to increase the predictive skill of the forecasts, while allowing for inhomogeneous mixing reduces the skill.We discuss how our work could be integrated into a forecasting system and how the model selection method could be used to evaluate forecasts when comparing to multiple surveillance systems providing disparate views of influenza activity.

Univariate and bivariate likelihood-based meta-analysis methods performed comparably when marginal sensitivity and specificity were the targets of inference

ObjectivesTo compare statistical methods for meta-analysis of sensitivity and specificity of medical tests (e.g., diagnostic or screening tests). Study Design and SettingWe constructed a database of PubMed-indexed meta-analyses of test performance from which 2 × 2 tables for each included study could be extracted. We reanalyzed the data using univariate and bivariate random effects models fit with inverse variance and maximum likelihood methods. Analyses were performed using both normal and binomial likelihoods to describe within-study variability. The bivariate model using the binomial likelihood was also fit using a fully Bayesian approach. ResultsWe use two worked examples—thoracic computerized tomography to detect aortic injury and rapid prescreening of Papanicolaou smears to detect cytological abnormalities—to highlight that different meta-analysis approaches can produce different results. We also present results from reanalysis of 308 meta-analyses of sensitivity and specificity. Models using the normal approximation produced sensitivity and specificity estimates closer to 50% and smaller standard errors compared to models using the binomial likelihood; absolute differences of 5% or greater were observed in 12% and 5% of meta-analyses for sensitivity and specificity, respectively. Results from univariate and bivariate random effects models were similar, regardless of estimation method. Maximum likelihood and Bayesian methods produced almost identical summary estimates under the bivariate model; however, Bayesian analyses indicated greater uncertainty around those estimates. Bivariate models produced imprecise estimates of the between-study correlation of sensitivity and specificity. Differences between methods were larger with increasing proportion of studies that were small or required a continuity correction. ConclusionThe binomial likelihood should be used to model within-study variability. Univariate and bivariate models give similar estimates of the marginal distributions for sensitivity and specificity. Bayesian methods fully quantify uncertainty and their ability to incorporate external evidence may be useful for imprecisely estimated parameters.

Confidence Intervals for the Mean of Non-Normal Distribution: Transform or Not to Transform

In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a common occurrence in practice. Given a large enough sample size, confidence intervals for the mean can be constructed by applying the Central Limit Theorem or by the bootstrap method. Another commonly used method in practice is the back-transformation method, which takes on the following three steps. First, apply a transformation to the data such that the transformed data are normally distributed. Second, obtain confidence intervals for the transformed mean in the usual manner, which assumes normality. Third, apply the back- transformation to obtain confidence intervals for the mean of the original, non-transformed distribution. The parametric Wald method and a small sample likelihood-based third order method, which can address non-normality, are also reviewed in this paper. Our simulation results suggest that common approaches such as back-transformation give erroneous and misleading results even when the sample size is large. However, the likelihood-based third order method gives extremely accurate results even when the sample size is small.

OUP accepted manuscript

SummaryCapture-recapture experiments are widely used to collect data needed for estimating the abundance of a closed population. To account for heterogeneity in the capture probabilities, Huggins (1989) and Alho (1990) proposed a semiparametric model in which the capture probabilities are modelled parametrically and the distribution of individual characteristics is left unspecified. A conditional likelihood method was then proposed to obtain point estimates and Wald-type confidence intervals for the abundance. Empirical studies show that the small-sample distribution of the maximum conditional likelihood estimator is strongly skewed to the right, which may produce Wald-type confidence intervals with lower limits that are less than the number of captured individuals or even are negative. In this paper, we propose a full empirical likelihood approach based on Huggins and Alho’s model. We show that the null distribution of the empirical likelihood ratio for the abundance is asymptotically chi-squared with one degree of freedom, and that the maximum empirical likelihood estimator achieves semiparametric efficiency. Simulation studies show that the empirical likelihood-based method is superior to the conditional likelihood-based method: its confidence interval has much better coverage, and the maximum empirical likelihood estimator has a smaller mean square error. We analyse three datasets to illustrate the advantages of our empirical likelihood approach.

Identification-Robust Moment-Based Tests for Markov-Switching in Autoregressive Models

This paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison to the optimal tests for Markov-switching parameters of Carrasco-Hu-Ploberger (2014} and they are also quite attractive owing to their computational simplicity. The new tests are illustrated with an empirical application to an autoregressive model of U.S. output growth.

Identification-Robust Moment-Based Tests for Markov-Switching in Autoregressive Models

This paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison to the optimal tests for Markov-switching parameters of Carrasco et al. (2014) and they are also quite attractive owing to their computational simplicity. The new tests are illustrated with an empirical application to an autoregressive model of U.S. output growth.