Using Hierarchical Likelihood for Missing Data Problems

Most statistical solutions to the problem of statistical inferencewith missing data involve integration or expectation. This canbe done in many ways: directly or indirectly, analytically ornumerically, deterministically or stochastically. Missing-dataproblems can be formulated in terms of latent random variables,so that hierarchical likelihood methods of Lee & Nelder(1996) can be applied to missing-value problems to provide onesolution to the problem of integration of the likelihood. Theresulting methods effectively use a Laplace approximation tothe marginal likelihood with an additional adjustment to themeasures of precision to accommodate the estimation of the fixedeffects parameters. We first consider missing at random caseswhere problems are simpler to handle because the integrationdoes not need to involve the missing-value mechanism and thenconsider missing not at random cases. We also study tobit regressionand refit the missing not at random selection model to the antidepressanttrial data analyzed in Diggle & Kenward (1994).     

Consistency of estimators of the population-scaled recombination rate

We consider (approximate) likelihood methods for estimating the population-scaled recombination rate from population genetic data. We show that the dependence between the data from two regions of a chromosome decays inversely with the amount of recombination between the two regions. We use this result to show that the maximum likelihood estimator (mle) for the recombination rate, based on the composite likelihood of Fearnhead and Donnelly, is consistent. We also consider inference based on the pairwise likelihood of Hudson. We consider two approximations to this likelihood, and prove that the mle based on one of these approximations is consistent, while the mle based on the other approximation (which is used by McVean, Awadalla and Fearnhead) is not.     

Analysis of current status data with missing covariates

Statistical inference based on right-censored data for the proportional hazards (PH) model with missing covariates has received considerable attention, but interval-censored or current status data with missing covariates has not yet been investigated. Our study is partly motivated by the analysis of fracture data from the 2005 National Health Interview Survey Original Database in Taiwan, where the occurrence of fractures was interval censored and the covariate osteoporosis was not reported for all residents. We assume that the data are realized from a PH model. A semiparametric maximum likelihood estimate implemented by a hybrid algorithm is proposed to analyze current status data with missing covariates. A comparison of the performance of our method with full-cohort analysis, complete-case analysis, and surrogate analysis is made via simulation with moderate sample sizes. The fracture data are then analyzed.     

A note on profile likelihood for exponential tilt mixture models

Suppose that independent observations are drawn from multipledistributions, each of which is a mixture of two component distributionssuch that their log density ratio satisfies a linear model witha slope parameter and an intercept parameter. Inference forsuch models has been studied using empirical likelihood, andmixed results have been obtained. The profile empirical likelihoodof the slope and intercept has an irregularity at the null hypothesisso that the two component distributions are equal. We derivea profile empirical likelihood and maximum likelihood estimatorof the slope alone, and obtain the usual asymptotic propertiesfor the estimator and the likelihood ratio statistic regardlessof the null. Furthermore, we show the maximum likelihood estimatorof the slope and intercept jointly is consistent and asymptoticallynormal regardless of the null. At the null, the joint maximumlikelihood estimator falls along a straight line through theorigin with perfect correlation asymptotically to the firstorder.     

A likelihood-based approach to estimating and testing for isolation by distance

Simple regression of genetic similarities between pairs of populations on their corresponding geographic distances is frequently used to detect the presence of isolation by distance (IBD). However, these pairwise values are obviously not independent and there is no parametric procedure for estimating and testing for the IBD intercepts and slopes based on standard regression theory. Nonparametric tests, such as the Mantel test, and resampling techniques, such as bootstrapping, have been exploited with limited success. Here, I describe a likelihood-based analysis to allow for simultaneously detecting patterns of correlated residuals and estimating and testing for the presence of IBD. It is shown, through the analysis of two molecular datasets in pine species, that different covariance structures of the residuals exist. More over, the likelihood ratio tests under these covariance structures are less sensitive to the presence of IBD than the Mantel test and the simple regression analysis but more sensitive than the bootstrap and jackknife samples over independent populations or population pairs. Because the likelihood analysis directly models and accounts for nonindependence of residuals, it should legitimately detect the presence of IBD, thereby allowing for accurate inferences about evolutionary and demographic processes influencing the extent and patterns of IBD.     

Discussion on “Adaptive enrichment designs with a continuous biomarker” by Nigel Stallard

We congratulate Dr. Nigel Stallard on his stimulating paper on adaptive enrichment designs with a continuous biomarker. Dr. Stallard details a framework for a large and interesting class of enrichment procedures. His work has motivated us to offer some thoughts in response. Dr. Stallard's strategy is to use the maximum of a test statistic over a set of possible threshold values to define the enriched population to be sampled in a second stage. This reminds us of procedures for identifying a change point, a biomarker value beyond which the effect of treatment is increased. For simplicity we focus our comments on Dr. Stallard's Rule 1 for selecting the second-stage sampling threshold. Using this rule, we present the likelihood ratio approach for adaptive testing and compare it to Dr. Stallard's approach for a few scenarios.     

Investigating deep phylogenetic relationships among cyanobacteria and plastids by small subunit rRNA sequence analysis.

Small subunit rRNA sequence data were generated for 27 strains of cyanobacteria and incorporated into a phylogenetic analysis of 1,377 aligned sequence positions from a diverse sampling of 53 cyanobacteria and 10 photosynthetic plastids. Tree inference was carried out using a maximum likelihood method with correction for site-to-site variation in evolutionary rate. Confidence in the inferred phylogenetic relationships was determined by construction of a majority-rule consensus tree based on alternative topologies not considered to be statistically significantly different from the optimal tree. The results are in agreement with earlier studies in the assignment of individual taxa to specific sequence groups. Several relationships not previously noted among sequence groups are indicated, whereas other relationships previously supported are contradicted. All plastids cluster as a strongly supported monophyletic group arising near the root of the cyanobacterial line of descent.     

Statistical inference for stochastic simulation models--theory and application

Statistical models are the traditional choice to test scientific theories when observations, processes or boundary conditions are subject to stochasticity. Many important systems in ecology and biology, however, are difficult to capture with statistical models. Stochastic simulation models offer an alternative, but they were hitherto associated with a major disadvantage: their likelihood functions can usually not be calculated explicitly, and thus it is difficult to couple them to well-established statistical theory such as maximum likelihood and Bayesian statistics. A number of new methods, among them Approximate Bayesian Computing and Pattern-Oriented Modelling, bypass this limitation. These methods share three main principles: aggregation of simulated and observed data via summary statistics, likelihood approximation based on the summary statistics, and efficient sampling. We discuss principles as well as advantages and caveats of these methods, and demonstrate their potential for integrating stochastic simulation models into a unified framework for statistical modelling.