Pseudo-likelihood Method Research Articles

Improved contact prediction in proteins: Using pseudolikelihoods to infer Potts models

Spatially proximate amino acids in a protein tend to coevolve. A protein's three-dimensional (3D) structure hence leaves an echo of correlations in the evolutionary record. Reverse engineering 3D structures from such correlations is an open problem in structural biology, pursued with increasing vigor as more and more protein sequences continue to fill the data banks. Within this task lies a statistical inference problem, rooted in the following: correlation between two sites in a protein sequence can arise from firsthand interaction but can also be network-propagated via intermediate sites; observed correlation is not enough to guarantee proximity. To separate direct from indirect interactions is an instance of the general problem of inverse statistical mechanics, where the task is to learn model parameters (fields, couplings) from observables (magnetizations, correlations, samples) in large systems. In the context of protein sequences, the approach has been referred to as direct-coupling analysis. Here we show that the pseudolikelihood method, applied to 21-state Potts models describing the statistical properties of families of evolutionarily related proteins, significantly outperforms existing approaches to the direct-coupling analysis, the latter being based on standard mean-field techniques. This improved performance also relies on a modified score for the coupling strength. The results are verified using known crystal structures of specific sequence instances of various protein families. Code implementing the new method can be found at http://plmdca.csc.kth.se/.

Variable selection in high-dimensional double generalized linear models

In this paper we are concerned with the problems of variable selection and estimation in double generalized linear models in which both the mean and the dispersion are allowed to depend on explanatory variables. We propose a maximum penalized pseudo-likelihood method when the number of parameters diverges with the sample size. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and asymptotic properties of the resulting estimators are established. We also carry out simulation studies and a real data analysis to assess the finite sample performance of the proposed variable selection procedure, showing that the proposed variable selection method works satisfactorily.

Robust Inference for Incomplete Binary Longitudinal Data

Missing data occur in many longitudinal studies. When data are nonignorably missing, it is necessary to incorporate the missing data mechanism into the observed data likelihood function. A full likelihood analysis of nonignorable missing data is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. To avoid such computational difficulties, pseudo-likelihood methods have been proposed in the literature under minimal parametric assumptions. However, like the classical maximum likelihood estimators, these pseudo-likelihood estimators are also sensitive to potential outliers in the data. In this article, we propose and explore a robust method in the framework of a pseudo-likelihood function that is derived under the working assumption that the longitudinal responses are independent over time. The performance of the proposed robust method is investigated in simulations. The method is also illustrated in an example using actual data on CD4 counts from clinical trials of HIV-infected patients.

Conditional pseudolikelihood methods for clustered ordinal, multinomial, or count outcomes with complex survey data

In order to adjust individual-level covariate effects for confounding due to unmeasured neighborhood characteristics, we have recently developed conditional pseudolikelihood methods to estimate the parameters of a proportional odds model for clustered ordinal outcomes with complex survey data. The methods require sampling design joint probabilities for each within-neighborhood pair. In the present article, we develop a similar methodology for a baseline category logit model for clustered multinomial outcomes and for a loglinear model for clustered count outcomes. All of the estimators and asymptotic sampling distributions we present can be conveniently computed using standard logistic regression software for complex survey data, such as sas proc surveylogistic. We demonstrate validity of the methods theoretically and also empirically by using simulations. We apply the new method for clustered multinomial outcomes to data from the 2008 Florida Behavioral Risk Factor Surveillance System survey in order to investigate disparities in frequency of dental cleaning both unadjusted and adjusted for confounding by neighborhood.

Fitting Variance Components Model and Fixed Effects Model for One-Way Analysis of Variance to Complex Survey Data

Under complex survey sampling, in particular when selection probabilities depend on the response variable (informative sampling), the sample and population distributions are different, possibly resulting in selection bias. This article is concerned with this problem by fitting two statistical models, namely: the variance components model (a two-stage model) and the fixed effects model (a single-stage model) for one-way analysis of variance, under complex survey design, for example, two-stage sampling, stratification, and unequal probability of selection, etc. Classical theory underlying the use of the two-stage model involves simple random sampling for each of the two stages. In such cases the model in the sample, after sample selection, is the same as model for the population; before sample selection. When the selection probabilities are related to the values of the response variable, standard estimates of the population model parameters may be severely biased, leading possibly to false inference. The idea behind the approach is to extract the model holding for the sample data as a function of the model in the population and of the first order inclusion probabilities. And then fit the sample model, using analysis of variance, maximum likelihood, and pseudo maximum likelihood methods of estimation. The main feature of the proposed techniques is related to their behavior in terms of the informativeness parameter. We also show that the use of the population model that ignores the informative sampling design, yields biased model fitting.

Orthogonalized residuals for estimation of marginally specified association parameters in multivariate binary data.

This paper focuses on marginal regression models for correlated binary responses when estimation of the association structure is of primary interest. A new estimating function approach based on orthogonalized residuals is proposed. A special case of the proposed procedure allows a new representation of the alternating logistic regressions method through marginal residuals. The connections between second-order generalized estimating equations, alternating logistic regressions, pseudo-likelihood and other methods are explored. Eficiency comparisons are presented, with emphasis on variable cluster size and on the role of higher-order assumptions. The new method is illustrated with an analysis of data on impaired pulmonary function.

Semiparametric Pseudo Likelihoods for Longitudinal Data with Outcome-Dependent Nonmonotone Nonresponse

In a study with longitudinal outcomes, the outcome nonresponse mech- anism often depends on the observed or unobserved value of the outcome. When nonresponse is monotone in the sense that a subject having a missing outcome at time t is not observed after time t, Tang, Little, and Raghunathan (2003) devel- oped a semiparametric pseudo-likelihood method for the estimation of parameters of interest. In practice, however, nonresponse is often not monotone and a direct application of their method discards observed data from subjects having nonmono- tone nonresponse, which may result in inefficient estimators. We extend the idea in Tang, Little, and Raghunathan (2003) to nonmonotone nonresponse and construct a semiparametric pseudo-likelihood that utilizes all observed data. Asymptotic normality of the maximum pseudo-likelihood estimators is established. An applica- tion is made to the household income data from the Health and Retirement Study. Simulation results are also presented to examine finite sample properties of the proposed estimators.

Comparisons of Methods for Analysis of Repeated Binary Responses with Missing Data

It is important yet challenging to choose an appropriate analysis method for the analysis of repeated binary responses with missing data. The conventional method using the last observation carried forward (LOCF) approach can be biased in both parameter estimates and hypothesis tests. The generalized estimating equations (GEE) method is valid only when missing data are missing completely at random, which may not be satisfied in many clinical trials. Several random-effects models based on likelihood or pseudo-likelihood methods and multiple-imputation-based methods have been proposed in the literature. In this paper, we evaluate the random-effects models with full- or pseudo-likelihood methods, GEE, and several multiple-imputation approaches. Simulations are used to compare the results and performance among these methods under different simulation settings.

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

This article deals with quasi- and pseudo-likelihood estimation for a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.

Mathematical and Experimental Approaches to Identify and Predict the Effects of Chemotherapy on Neuroglial Precursors

The adverse effects of chemotherapy on normal cells of the body create substantial clinical problems for many cancer patients. However, relatively little is known about the effects, other than promotion of cell death, of such agents on the function of normal precursor cells critical in tissue homeostasis and repair. We have combined mathematical and experimental analyses to identify the effects of sublethal doses of chemotherapy on glial precursor cells of the central nervous system. We modeled the temporal development of a population of precursor and terminally differentiated cells exposed to sublethal doses of carmustine (BCNU), a classic alkylating chemotherapeutic agent used in treatment of gliomas and non-Hodgkin's lymphomas, as a multitype age-dependent branching process. We fitted our model to data from in vitro clonal experiments using the method of pseudo-likelihood. This approach identifies several novel drug effects, including modification of the cell cycle length, the time between division and differentiation, and alteration in the probability of undergoing self-renewal division in precursor cells. These changes of precursor cell function in the chemotherapy-exposed brain may have profound clinic implications. We applied our computational approach to analyze the effects of BCNU on clonal cultures of oligodendrocyte progenitor cells-one of the best-characterized neural progenitor cells in the mammalian brain. Our analysis reveals that transient exposures to BCNU increased the cell cycle length of progenitor cells and decreased their time to differentiation, while also decreasing the likelihood that they will undergo self-renewing divisions. By investigating the behavior of our mathematical model, we demonstrate that precursor cell populations should recover spontaneously from transient modifications of the timing of division and of differentiation, but such recovery will not happen after alteration of cell fate. These studies identify means by which precursor cell function can be critically compromised by transient exposure to chemotherapy with long-term consequences on the progenitor cell pool even in the absence of drug-induced apoptosis. These analyses also provide novel tools that apply broadly to identify effects of chemotherapeutic agents and other physiological stressors.

Accounting for bias due to a non-ignorable tracing mechanism in a retrospective breast cancer cohort study

We consider the analysis of competing risks in a retrospective breast cancer cohort study where tracing of patients is dependent on survival to a pre-specified truncation time. We demonstrate that if ignored, the observed cause-specific hazards will become distorted before the truncation time. Two approaches to account for the tracing bias are considered. First, a likelihood-based method using piecewise constant transition intensities under a Markov assumption. Second, a pseudo-likelihood method using inverse probability of tracing weights. For the breast cancer example, both methods improve the precision of estimates compared with a conventional approach based on excluding patients.

Genotype‐based association mapping of complex diseases: gene‐environment interactions with multiple genetic markers and measurement error in environmental exposures

With the advent of dense single nucleotide polymorphism genotyping, population-based association studies have become the major tools for identifying human disease genes and for fine gene mapping of complex traits. We develop a genotype-based approach for association analysis of case-control studies of gene-environment interactions in the case when environmental factors are measured with error and genotype data are available on multiple genetic markers. To directly use the observed genotype data, we propose two genotype-based models: genotype effect and additive effect models. Our approach offers several advantages. First, the proposed risk functions can directly incorporate the observed genotype data while modeling the linkage disequilibrium information in the regression coefficients, thus eliminating the need to infer haplotype phase. Compared with the haplotype-based approach, an estimating procedure based on the proposed methods can be much simpler and significantly faster. In addition, there is no potential risk due to haplotype phase estimation. Further, by fitting the proposed models, it is possible to analyze the risk alleles/variants of complex diseases, including their dominant or additive effects. To model measurement error, we adopt the pseudo-likelihood method by Lobach et al. [2008]. Performance of the proposed method is examined using simulation experiments. An application of our method is illustrated using a population-based case-control study of association between calcium intake with the risk of colorectal adenoma development.

Confidence intervals for the mean of a population containing many zero values under unequal-probability sampling

In many applications, a finite population contains a large proportion of zero values that make the population distribution severely skewed. An unequal-probability sampling plan compounds the problem, and as a result the normal approximation to the distribution of various estimators has poor precision. The central-limit-theorem-based confidence intervals for the population mean are hence unsatisfactory. Complex designs also make it hard to pin down useful likelihood functions, hence a direct likelihood approach is not an option. In this paper, we propose a pseudo-likelihood approach. The proposed pseudo-log-likelihood function is an unbiased estimator of the log-likelihood function when the entire population is sampled. Simulations have been carried out. When the inclusion probabilities are related to the unit values, the pseudo-likelihood intervals are superior to existing methods in terms of the coverage probability, the balance of non-coverage rates on the lower and upper sides, and the interval length. An application with a data set from the Canadian Labour Force Survey-2000 also shows that the pseudo-likelihood method performs more appropriately than other methods. The Canadian Journal of Statistics 38: 582–597; 2010 © 2010 Statistical Society of Canada

Hierarchical Modeling for Estimating Relative Risks of Rare Genetic Variants: Properties of the Pseudo-Likelihood Method

Many major genes have been identified that strongly influence the risk of cancer. However, there are typically many different mutations that can occur in the gene, each of which may or may not confer increased risk. It is critical to identify which specific mutations are harmful, and which ones are harmless, so that individuals who learn from genetic testing that they have a mutation can be appropriately counseled. This is a challenging task, since new mutations are continually being identified, and there is typically relatively little evidence available about each individual mutation. In an earlier article, we employed hierarchical modeling (Capanu et al., 2008, Statistics in Medicine 27, 1973-1992) using the pseudo-likelihood and Gibbs sampling methods to estimate the relative risks of individual rare variants using data from a case-control study and showed that one can draw strength from the aggregating power of hierarchical models to distinguish the variants that contribute to cancer risk. However, further research is needed to validate the application of asymptotic methods to such sparse data. In this article, we use simulations to study in detail the properties of the pseudo-likelihood method for this purpose. We also explore two alternative approaches: pseudo-likelihood with correction for the variance component estimate as proposed by Lin and Breslow (1996, Journal of the American Statistical Association 91, 1007-1016) and a hybrid pseudo-likelihood approach with Bayesian estimation of the variance component. We investigate the validity of these hierarchical modeling techniques by looking at the bias and coverage properties of the estimators as well as at the efficiency of the hierarchical modeling estimates relative to that of the maximum likelihood estimates. The results indicate that the estimates of the relative risks of very sparse variants have small bias, and that the estimated 95% confidence intervals are typically anti-conservative, though the actual coverage rates are generally above 90%. The widths of the confidence intervals narrow as the residual variance in the second-stage model is reduced. The results also show that the hierarchical modeling estimates have shorter confidence intervals relative to estimates obtained from conventional logistic regression, and that these relative improvements increase as the variants become more rare.

Postmodel selection estimators of variance function for nonlinear autoregression

We consider a problem of estimating a conditional variance function of an autoregressive process. A finite collection of parametric models for conditional density is studied when both regression and variance are modelled by parametric functions. The proposed estimators are defined as the maximum likelihood estimators in the models chosen by penalized selection criteria. Consistency properties of the resulting estimator of the variance when the conditional density belongs to one of the parametric models are studied as well as its behaviour under mis-specification. The autoregressive process does not need to be stationary but only existence of a stationary distribution and ergodicity is required. Analogous results for the pseudolikelihood method are also discussed. A simulation study shows promising behaviour of the proposed estimator in the case of heavy-tailed errors in comparison with local linear smoothers.

A copula model for bivariate hybrid censored survival data with application to the MACS study

A copula model for bivariate survival data with hybrid censoring is proposed to study the association between survival time of individuals infected with HIV and persistence time of infection with an additional virus. Survival with HIV is right censored and the persistence time of the additional virus is subject to interval censoring case 1. A pseudo-likelihood method is developed to study the association between the two event times under such hybrid censoring. Asymptotic consistency and normality of the pseudo-likelihood estimator are established based on empirical process theory. Simulation studies indicate good performance of the estimator with moderate sample size. The method is applied to a motivating HIV study which investigates the effect of GB virus type C (GBV-C) co-infection on survival time of HIV infected individuals.

Estimation of poverty measures with auxiliary information in sample surveys

The analysis of poverty measures has been receiving increased attention in recent years. This paper contributes to the literature by developing percentile ratio estimators based on the pseudo empirical likelihood method. In practice, variances of poverty measures could be not expressible by simple formulae and consequently other techniques should be used in the variance estimation stage. Assuming percentile ratios, resampling techniques are investigated in this paper. A numerical example based on data from the Spanish Household Panel Survey is taken up to illustrate how suggested procedures can perform better than existing ones. The effect of a model-misspecification on the proposed estimators is also evaluated by using simulated populations.

Modelling Tree Roots in Mixed Forest Stands by Inhomogeneous Marked Gibbs Point Processes

The aim of the paper is to apply point processes to root data modelling. We propose a new approach to parametric inference when the data are inhomogeneous replicated marked point patterns. We generalize Geyer's saturation point process to a model, which combines inhomogeneity, marks and interaction between the marked points. Furthermore, the inhomogeneity influences the definition of the neighbourhood of points. Using the maximum pseudolikelihood method, this model is then fitted to root data from mixed stands of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.) to quantify the degree of root aggregation in such mixed stands. According to the analysis there is no evidence that the two root systems are not independent.

Semiparametric inference on a class of Wiener processes

Abstract. This article studies the estimation of a nonhomogeneous Wiener process model for degradation data. A pseudo-likelihood method is proposed to estimate the unknown parameters. An attractive algorithm is established to compute the estimator under this pseudo-likelihood formulation. We establish the asymptotic properties of the estimator, including consistency, convergence rate and asymptotic distribution. Random effects can be incorporated into the model to represent the heterogeneity of degradation paths by letting the mean function be random. The Wiener process model is extended naturally to a normal inverse Gaussian process model and similar pseudo-likelihood inference is developed. A score test is used to test the presence of the random effects. Simulation studies are conducted to validate the method and we apply our method to a real data set in the area of health structure monitoring.

Flexible modeling based on copulas in nonparametric median regression

Consider the model Y = m ( X ) + ε , where m ( ⋅ ) = med ( Y | ⋅ ) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e. ( ε , X ) ∼ C θ ( F ε ( ⋅ ) , F X ( ⋅ ) ) , where C θ is a copula function depending on an unknown parameter θ , and F ε and F X are the marginals of ε and X . Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model. We estimate the parameter θ via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of Y given X . The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.