Pairwise Likelihood Research Articles

The role of copulas in random fields: Characterization and application

The spatial fluctuation of material properties is commonly modeled using random fields. However, descriptions of random fields in terms of marginal distributions and correlation structure are actually incomplete. The copulas that specify the dependence structure underlying the joint probability distribution over random fields are not provided and a Gaussian dependence structure is assumed without proper validation in practice. This study, therefore, tests the fitness of several non-Gaussian dependence structures in addition to the common use of the Gaussian dependence structure. A soil profile of cone penetration test (CPT) data is collected, and a pair-wise likelihood estimate approach is adopted to identify the best fit dependence structure. The results reveal that both qc and fs of the CPT profile exhibit the same non-Gaussian dependence characteristics, which challenges the Gaussian assumption. Generation of random fields with a non-Gaussian dependence structure is then proposed. Two slope examples where the spatial fluctuation of shear strength is respectively modeled using stationary and non-stationary random fields are used to investigate the deviation in failure probability due to different dependence structures. The effect of the dependence structure is non-trivial except when the random field is highly ragged or highly smooth. The results highlight the importance of copulas for dependence characterization in random fields.

Sparse Pairwise Likelihood Estimation for Multivariate Longitudinal Mixed Models

ABSTRACTIt is becoming increasingly common in longitudinal studies to collect and analyze data on multiple responses. For example, in the social sciences we may be interested in uncovering the factors driving mental health of individuals over time, where mental health is measured using a set of questionnaire items. One approach to analyzing such multi-dimensional data is multivariate mixed models, an extension of the standard univariate mixed model to handle multiple responses. Estimating multivariate mixed models presents a considerable challenge however, let alone performing variable selection to uncover which covariates are important in driving each response. Motivated by composite likelihood ideas, we propose a new approach for estimation and fixed effects selection in multivariate mixed models, called approximate pairwise likelihood estimation and shrinkage (APLES). The method works by constructing a quadratic approximation to each term in the pairwise likelihood function, and then augmenting this approximate pairwise likelihood with a penalty that encourages both individual and group coefficient sparsity. This leads to a relatively fast method of selection, as we can use coordinate ascent type methods to then construct the full regularization path for the model. Our method is the first to extend penalized likelihood estimation to multivariate generalized linear mixed models. We show that the APLES estimator attains a composite likelihood version of the oracle property. We propose a new information criterion for selecting the tuning parameter, which employs a dynamic model complexity penalty to facilitate aggressive shrinkage, and demonstrate that it asymptotically leads to selection consistency, that is, leads to the true model being selected. A simulation study demonstrates that the APLES estimator outperforms several univariate selection methods based on analyzing each outcome separately. Supplementary materials for this article are available online.

Sampling of pairs in pairwise likelihood estimation for latent variable models with categorical observed variables

Pairwise likelihood is a limited information estimation method that has also been used for estimating the parameters of latent variable and structural equation models. Pairwise likelihood is a special case of composite likelihood methods that uses lower-order conditional or marginal log-likelihoods instead of the full log-likelihood. The composite likelihood to be maximized is a weighted sum of marginal or conditional log-likelihoods. Weighting has been proposed for increasing efficiency, but the choice of weights is not straightforward in most applications. Furthermore, the importance of leaving out higher-order scores to avoid duplicating lower-order marginal information has been pointed out. In this paper, we approach the problem of weighting from a sampling perspective. More specifically, we propose a sampling method for selecting pairs based on their contribution to the total variance from all pairs. The sampling approach does not aim to increase efficiency but to decrease the estimation time, especially in models with a large number of observed categorical variables. We demonstrate the performance of the proposed methodology using simulated examples and a real application.

Marginal Logistic Regression for Spatially Clustered Binary Data

Summary Clustered data are often analysed under the assumption that observations from distinct clusters are independent. The assumption may not be correct when the clusters are associated with different locations within a study region, as, for example, in epidemiological studies involving subjects nested within larger units such as hospitals, districts or villages. In such cases, correct inferential conclusions critically depend on the amount of spatial dependence between locations. We develop a modification of the method of generalized estimating equations to detect and account for spatial dependence between clusters in logistic regression for binary data. The approach proposed is based on parametric modelling of the lorelogram as a function of the distance between clusters. Model parameters are estimated by the hybrid pairwise likelihood method that combines optimal estimating equations for the regression parameters and pairwise likelihood for the lorelogram parameters. The methodology is illustrated with an analysis of prevalence disease survey data.

Near universal consistency of the maximum pseudolikelihood estimator for discrete models

Maximum pseudolikelihood (MPL) estimators are useful alternatives to maximum likelihood (ML) estimators when likelihood functions are more difficult to manipulate than their marginal and conditional components. Furthermore, MPL estimators subsume a large number of estimation techniques including ML estimators, maximum composite marginal likelihood estimators, and maximum pairwise likelihood estimators. When considering only the estimation of discrete models (on a possibly countably infinite support), we show that a simple finiteness assumption on an entropy-based measure is sufficient for assessing the consistency of the MPL estimator. As a consequence, we demonstrate that the MPL estimator of any discrete model on a bounded support will be consistent. Our result is valid in parametric, semiparametric, and nonparametric settings.

An exponential–gamma mixture model for extreme Santa Ana winds

We analyze the behavior of extreme winds occurring in Southern California during the Santa Ana wind season using a latent mixture model. This mixture representation is formulated as a hierarchical Bayesian model and fit using Markov chain Monte Carlo. The two‐stage model results in generalized Pareto margins for exceedances and generates temporal dependence through a latent Markov process. This construction induces asymptotic independence in the response, while allowing for dependence at extreme, but subasymptotic, levels. We compare this model with a frequentist analogue where inference is performed via maximum pairwise likelihood. We use interval censoring to account for data quantization and estimate the extremal index and probabilities of multiday occurrences of extreme Santa Ana winds over a range of high thresholds.

A pairwise likelihood augmented Cox estimator for left-truncated data.

Survival data collected from a prevalent cohort are subject to left truncation and the analysis is challenging. Conditional approaches for left-truncated data could be inefficient as they ignore the information in the marginal likelihood of the truncation times. Length-biased sampling methods may improve the estimation efficiency but only when the underlying truncation time is uniform; otherwise, they may generate biased estimates. We propose a semiparametric method for left-truncated data under the Cox model with no parametric distributional assumption about the truncation times. Our approach is to make inference based on the conditional likelihood augmented with a pairwise likelihood, which eliminates the truncation distribution, yet retains the information about the regression coefficients and the baseline hazard function in the marginal likelihood. An iterative algorithm is provided to solve for the regression coefficients and the baseline hazard function simultaneously. By empirical process and U-process theories, it has been shown that the proposed estimator is consistent and asymptotically normal with a closed-form consistent variance estimator. Simulation studies show substantial efficiency gain of our estimator in both the regression coefficients and the cumulative baseline hazard function over the conditional approach estimator. When the uniform truncation assumption holds, our estimator enjoys smaller biases and efficiency comparable to that of the full maximum likelihood estimator. An application to the analysis of a chronic kidney disease cohort study illustrates the utility of the method.

A note on predictive densities based on composite likelihood methods

Whenever the computation of data distribution is unfeasible or inconvenient, the classical predictive procedures prove not to be useful. These rely, after all, on the conditional distribution of the future random variable, which is also unavailable. This paper considers a notion of composite likelihood for specifying composite predictive distributions, viewed as surrogates for true unknown predictive distribution. In particular, the focus is on the pairwise likelihood obtained as a weighted product of likelihood factors related to bivariate events associated with both the sample data and future observation. The specification of the weights, and more generally the evaluation of the frequentist properties of alternative pairwise predictive distributions, is performed by considering the mean square prediction error of the associated predictors and the expected Kullback–Liebler loss of the related predictive densities. Finally, simple examples concerning autoregressive models are presented.

Pairwise likelihood inference for the random effects probit model

This paper proposes a pairwise likelihood estimator based on an analytic approximation method for the random effects probit model. It is widely known that the standard approach for the random effects probit model relies on numerical integration and that its likelihood function does not have a closed form. When the number of time periods or the serial correlation across periods is large, the resulting estimator is likely to become biased. This study derives an analytic approximation for the likelihood function of one pair of time periods without relying on typical numerical-integral procedures. We then apply this formula in a pairwise likelihood estimation procedure to derive our estimator, which is obtained as the product of the analytic approximation of the likelihood function for all possible pairs of time periods. A simulation study is conducted for the comparison of our proposed estimator with the estimators for the pooled probit model and Gaussian quadrature procedure. The evidence shows that our proposed estimator enjoys desirable asymptotic properties. In addition, compared to the estimator based on the Gaussian quadrature procedure, our proposed estimator exhibits comparable performances in all the configurations considered in the simulation study and shows superiority for the cases of a large number of time periods and high serial correlation across periods. We apply our proposed estimator to British Household Panel Survey data so as to characterize the trend of working probabilities.

Total least‐square‐based receiver for asymmetrically clipped optical‐orthogonal frequency divisional multiplexing visible light communication system

This study proposes a receiver for an asymmetrically clipped optical orthogonal frequency-division multiplexing system that utilises the total least-squares (TLS) technique. Commonly used estimation techniques such as least squares, diversity combining, minimum-mean-square-error, and pairwise maximum likelihood ignore the perturbation present while estimating the received signal, which causes the estimated signal to be inaccurate. In this study, a TLS-based receiver is proposed, which takes into consideration the errors present in both the data matrix (channel) and the observation matrix (received signal). While this technique is suboptimal, it is shown that with reasonable computational complexity it has an acceptable performance in a scenario with low processing capability and low battery power. Simulations are carried out for both line-of-sight and non-line-of-sight optical channels and the performance is analysed on the basis of bit error rate versus E b/N 0 where E b is the energy received per information bit and N 0 is the noise power spectral density.

Parameter Estimation of Smith Model Max-Stable Process Spatial Extreme Value (Case-Study: Extreme Rainfall Modelling in Ngawi Regency)

The unpredictable extreme rainfall can affect flood. Prediction of extreme rainfall is needed to do, so that the efforts to preventing the flood can be effective. One of the methods that can predict the extreme rainfall is the Spatial Extreme Value (SEV) with the Max-Stable Process (MSP) approach. The important purpose of SEV is calculated of return level (the extreme value prediction). The calculation of return level depends on parameter estimation in that method. This research discusses about parameter estimation of the Spatial Extreme Value Max-Stable Process especially Smith model. Parameter estimation was performed using Maximum Composite Likelihood Estimation (MCLE) method and Maximum Pairwise Likelihood Estimation (MPLE) method. The result of estimation using this method is not closed form, it must be continued by using numerical iteration method. The iteration method used in this research is Broyden-Fletcher Goldfarb-Shanno (BFGS) Quasi Newton, which is faster than other methods to achieve convergence. The result of parameter estimation applied to the rainfall data of Ngawi Regency which is the Regency with the largest rice production in East Java Province (the province with the largest rice farm in Indonesia). Based on the results of data analysis obtained trend surface model (s) = 2,794 + 0,242 v(s); (s) = 1,8196 + 0,1106 v(s); (s) = 1,012 with goodness criterion model Takeuchi Information Criterion (TIC) 26237,62. Root Mean Square Error (RMSE) based on 20 testing data is 32,078 and Mean Absolute Percentage Error (MAPE) is 27,165%

Sequential change-point detection in time series models based on pairwise likelihood

Sequential change-point detection in time series models based on pairwise likelihood

Analysis of progressive multi-state models with misclassified states: likelihood and pairwise likelihood methods

ABSTRACTMulti-state models are commonly used in studies of disease progression. Methods developed under this framework, however, are often challenged by misclassification in states. In this article, we investigate issues concerning continuous-time progressive multi-state models with state misclassification. We develop inference methods using both the likelihood and pairwise likelihood methods that are based on joint modelling of the progressive and misclassification processes. We assess the performance of the proposed methods by simulation studies, and illustrate their use by the application to the data arising from a coronary allograft vasculopathy study.

A molecular tool for parentage analysis in Indian major carp, Labeo rohita (Hamilton, 1822)

Determination of genealogical relationship between broodstock and progeny is important to estimate selection response in aquaculture breeding programs. Currently, the application of microsatellite markers for parentage determination is gaining popularity in aquaculture. In the present study, 17 simple sequence repeat markers were evaluated for parentage assignment on rohu. The allele frequencies, exclusion probabilities and polymorphic information contents (PIC) were calculated using Cervus. The polymorphic information content values showed that most of the microsatellite markers were highly informative (PIC > 0.7). Based on the exclusion power, eight microsatellite markers were chosen for parentage assignment. The robustness of this suite was assessed in five full-sib and four half-sib families using two contrasting methods, a pair-wise likelihood comparison approach in Cervus and a full-pedigree likelihood method in the COLONY program. Using real data set, the correct matching rate was more than 98%, comparable to the simulated study (99.9%) in the case of Cervus 3.0 and 100% in the case of COLONY. The result of the present study implies that this set of microsatellites would be an effective tool for parentage and sibship identification, testing performance of families and estimating genetic parameters in the ongoing selective breeding program of rohu.

Asymptotic Expansion of the Posterior Based on Pairwise Likelihood

This paper provides an asymptotic expansion of the posterior based on pairwise likelihood instead of the regular likelihood. The celebrated Bernstein-von Mises theorem is derived as a special case. A multiparameter version of the asymptotic expansion is also given involving nuisance parameters. As a direct application of these expansions, one can obtain moment matching priors and quantile matching priors with or without nuisance parameters. A simulation study is provided verifying this agreement between frequentist quantiles and Bayesian quantiles using quantile matching priors. One of the major tools used in this paper is strong consistency of the maximum pairwise likelihood estimator (MPLE).

A semiparametric model for vQTL mapping.

Quantitative trait locus analysis has been used as an important tool to identify markers where the phenotype or quantitative trait is linked with the genotype. Most existing tests for single locus association with quantitative traits aim at the detection of the mean differences across genotypic groups. However, recent research has revealed functional genetic loci that affect the variance of traits, known as variability-controlling quantitative trait locus. In addition, it has been suggested that many genotypes have both mean and variance effects, while the mean effects or variance effects alone may not be strong enough to be detected. The existing methods accounting for unequal variances include the Levene's test, the Lepage test, and the D-test, but suffer from their limitations of lack of robustness or lack of power. We propose a semiparametric model and a novel pairwise conditional likelihood ratio test. Specifically, the semiparametric model is designed to identify the combined differences in higher moments among genotypic groups. The pairwise likelihood is constructed based on conditioning procedure, where the unknown reference distribution is eliminated. We show that the proposed pairwise likelihood ratio test has a simple asymptotic chi-square distribution, which does not require permutation or bootstrap procedures. Simulation studies show that the proposed test performs well in controlling Type I errors and having competitive power in identifying the differences across genotypic groups. In addition, the proposed test has certain robustness to model mis-specifications. The proposed test is illustrated by an example of identifying both mean and variances effects in body mass index using the Framingham Heart Study data.

Pairwise Likelihood Ratio Tests and Model Selection Criteria for Structural Equation Models with Ordinal Variables.

Correlated multivariate ordinal data can be analysed with structural equation models. Parameter estimation has been tackled in the literature using limited-information methods including three-stage least squares and pseudo-likelihood estimation methods such as pairwise maximum likelihood estimation. In this paper, two likelihood ratio test statistics and their asymptotic distributions are derived for testing overall goodness-of-fit and nested models, respectively, under the estimation framework of pairwise maximum likelihood estimation. Simulation results show a satisfactory performance of type I error and power for the proposed test statistics and also suggest that the performance of the proposed test statistics is similar to that of the test statistics derived under the three-stage diagonally weighted and unweighted least squares. Furthermore, the corresponding, under the pairwise framework, model selection criteria, AIC and BIC, show satisfactory results in selecting the right model in our simulation examples. The derivation of the likelihood ratio test statistics and model selection criteria under the pairwise framework together with pairwise estimation provide a flexible framework for fitting and testing structural equation models for ordinal as well as for other types of data. The test statistics derived and the model selection criteria are used on data on 'trust in the police' selected from the 2010 European Social Survey. The proposed test statistics and the model selection criteria have been implemented in the R package lavaan.

Missing Data Mechanisms for Analysing Longitudinal Data with Incomplete Observations in Both Responses and Covariates

Summary Missing observations in both responses and covariates arise frequently in longitudinal studies. When missing data are missing not at random, inferences under the likelihood framework often require joint modelling of response and covariate processes, as well as missing data processes associated with incompleteness of responses and covariates. Specification of these four joint distributions is a nontrivial issue from the perspectives of both modelling and computation. To get around this problem, we employ pairwise likelihood formulations, which avoid the specification of third or higher order association structures. In this paper, we consider three specific missing data mechanisms which lead to further simplified pairwise likelihood (SPL) formulations. Under these missing data mechanisms, inference methods based on SPL formulations are developed. The resultant estimators are consistent, and enjoy better robustness and computation convenience. The performance is evaluated empirically though simulation studies. Longitudinal data from the National Population Health Survey and Waterloo Smoking Prevention Project are analysed to illustrate the usage of our methods.

Prior event rate ratio adjustment for hidden confounding in observational studies of treatment effectiveness: a pairwise Cox likelihood approach.

Observational studies provide a rich source of information for assessing effectiveness of treatment interventions in many situations where it is not ethical or practical to perform randomized controlled trials. However, such studies are prone to bias from hidden (unmeasured) confounding. A promising approach to identifying and reducing the impact of unmeasured confounding is prior event rate ratio (PERR) adjustment, a quasi‐experimental analytic method proposed in the context of electronic medical record database studies. In this paper, we present a statistical framework for using a pairwise approach to PERR adjustment that removes bias inherent in the original PERR method. A flexible pairwise Cox likelihood function is derived and used to demonstrate the consistency of the simple and convenient alternative PERR (PERR‐ALT) estimator. We show how to estimate standard errors and confidence intervals for treatment effect estimates based on the observed information and provide R code to illustrate how to implement the method. Assumptions required for the pairwise approach (as well as PERR) are clarified, and the consequences of model misspecification are explored. Our results confirm the need for researchers to consider carefully the suitability of the method in the context of each problem. Extensions of the pairwise likelihood to more complex designs involving time‐varying covariates or more than two periods are considered. We illustrate the application of the method using data from a longitudinal cohort study of enzyme replacement therapy for lysosomal storage disorders. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Semiparametric density ratio modeling of survival data from a prevalent cohort.

In this article, we consider methods for assessing covariate effects on survival outcome in the target population when data are collected under prevalent sampling. We investigate a flexible semiparametric density ratio model without the constraints of the constant disease incidence rate and discrete covariates as required in Shen and others 2012. For inference, we introduce two likelihood approaches with distinct computational algorithms. We first develop a full likelihood approach to obtain the most efficient estimators by an iterative algorithm. Under the density ratio model, we exploit the invariance property of uncensored failure times from the prevalent cohort and also propose a computationally convenient estimation procedure that uses a conditional pairwise likelihood. The empirical performance and efficiency of the two approaches are evaluated through simulation studies. The proposed methods are applied to the Surveillance, Epidemiology, and End Results Medicare linked data for women diagnosed with stage IV breast cancer.