Flexible modeling via a hybrid estimation scheme in generalized mixed models for longitudinal data

To circumvent the computational complexity of likelihood inference in generalized mixed models that assume linear or more general additive regression models of covariate effects, Laplace's approximations to multiple integrals in the likelihood have been commonly used without addressing the issue of adequacy of the approximations for individuals with sparse observations. In this article, we propose a hybrid estimation scheme to address this issue. The likelihoods for subjects with sparse observations use Monte Carlo approximations involving importance sampling, while Laplace's approximation is used for the likelihoods of other subjects that satisfy a certain diagnostic check on the adequacy of Laplace's approximation. Because of its computational tractability, the proposed approach allows flexible modeling of covariate effects by using regression splines and model selection procedures for knot and variable selection. Its computational and statistical advantages are illustrated by simulation and by application to longitudinal data from a fecundity study of fruit flies, for which overdispersion is modeled via a double exponential family.     

Inference on Multiple Endpoints in Clinical Trials with Adaptive Interim Analyses

Planned interim analyses which permit early stopping or sample size adaption of a trial are desirable for ethical and scientific reasons. Multiple test procedures allow inference about several hypotheses within a single clinical trial. In this paper, a method which combines multiple testing with adaptive interim analyses whilst controlling the experimentwise error rate is proposed. The general closed testing principle, the situation of a priori ordered hypotheses, and application of the Bonferroni-Holm method are considered. The practical application of the method is demonstrated by an example.     

Combining multiple comparisons and modeling techniques in dose-response studies

The analysis of data from dose-response studies has long been divided according to two major strategies: multiple comparison procedures and model-based approaches. Model-based approaches assume a functional relationship between the response and the dose, taken as a quantitative factor, according to a prespecified parametric model. The fitted model is then used to estimate an adequate dose to achieve a desired response but the validity of its conclusions will highly depend on the correct choice of the a priori unknown dose-response model. Multiple comparison procedures regard the dose as a qualitative factor and make very few, if any, assumptions about the underlying dose-response model. The primary goal is often to identify the minimum effective dose that is statistically significant and produces a relevant biological effect. One approach is to evaluate the significance of contrasts between different dose levels, while preserving the family-wise error rate. Such procedures are relatively robust but inference is confined to the selection of the target dose among the dose levels under investigation. We describe a unified strategy to the analysis of data from dose-response studies which combines multiple comparison and modeling techniques. We assume the existence of several candidate parametric models and use multiple comparison techniques to choose the one most likely to represent the true underlying dose-response curve, while preserving the family-wise error rate. The selected model is then used to provide inference on adequate doses.     

Robust Inferences for Covariate Effects on Survival Time with Censored Linear Regression Models

Various inference procedures for linear regression models with censored failure times have been studied extensively. Recent developments on efficient algorithms to implement these procedures enhance the practical usage of such models in survival analysis. In this article, we present robust inferences for certain covariate effects on the failure time in the presence of “nuisance” confounders under a semiparametric, partial linear regression setting. Specifically, the estimation procedures for the regression coefficients of interest are derived from a working linear model and are valid even when the function of the confounders in the model is not correctly specified. The new proposals are illustrated with two examples and their validity for cases with practical sample sizes is demonstrated via a simulation study.     

Successful Reconstruction of a Physiological Circuit with Known Connectivity from Spiking Activity Alone

Identifying the structure and dynamics of synaptic interactions between neurons is the first step to understanding neural network dynamics. The presence of synaptic connections is traditionally inferred through the use of targeted stimulation and paired recordings or by post-hoc histology. More recently, causal network inference algorithms have been proposed to deduce connectivity directly from electrophysiological signals, such as extracellularly recorded spiking activity. Usually, these algorithms have not been validated on a neurophysiological data set for which the actual circuitry is known. Recent work has shown that traditional network inference algorithms based on linear models typically fail to identify the correct coupling of a small central pattern generating circuit in the stomatogastric ganglion of the crab Cancer borealis. In this work, we show that point process models of observed spike trains can guide inference of relative connectivity estimates that match the known physiological connectivity of the central pattern generator up to a choice of threshold. We elucidate the necessary steps to derive faithful connectivity estimates from a model that incorporates the spike train nature of the data. We then apply the model to measure changes in the effective connectivity pattern in response to two pharmacological interventions, which affect both intrinsic neural dynamics and synaptic transmission. Our results provide the first successful application of a network inference algorithm to a circuit for which the actual physiological synapses between neurons are known. The point process methodology presented here generalizes well to larger networks and can describe the statistics of neural populations. In general we show that advanced statistical models allow for the characterization of effective network structure, deciphering underlying network dynamics and estimating information-processing capabilities.     

Evaluation of community-intervention trials via generalized linear mixed models

In community-intervention trials, communities, rather than individuals, are randomized to experimental arms. Generalized linear mixed models offer a flexible parametric framework for the evaluation of community-intervention trials, incorporating both systematic and random variations at the community and individual levels. We propose here a simple two-stage inference method for generalized linear mixed models, specifically tailored to the analysis of community-intervention trials. In the first stage, community-specific random effects are estimated from individual-level data, adjusting for the effects of individual-level covariates. This reduces the model approximately to a linear mixed model with the unit of analysis being community. Because the number of communities is typically small in community-intervention studies, we apply the small-sample inference method of Kenward and Roger (1997, Biometrics53, 983-997) to the linear mixed model of second stage. We show by simulation that, under typical settings of community-intervention studies, the proposed approach improves the inference on the intervention-effect parameter uniformly over both the linearized mixed-effect approach and the adaptive Gaussian quadrature approach for generalized linear mixed models. This work is motivated by a series of large randomized trials that test community interventions for promoting cancer preventive lifestyles and behaviors.     

In defence of model‐based inference in phylogeography

Recent papers have promoted the view that model‐based methods in general, and those based on Approximate Bayesian Computation (ABC) in particular, are flawed in a number of ways, and are therefore inappropriate for the analysis of phylogeographic data. These papers further argue that Nested Clade Phylogeographic Analysis (NCPA) offers the best approach in statistical phylogeography. In order to remove the confusion and misconceptions introduced by these papers, we justify and explain the reasoning behind model‐based inference. We argue that ABC is a statistically valid approach, alongside other computational statistical techniques that have been successfully used to infer parameters and compare models in population genetics. We also examine the NCPA method and highlight numerous deficiencies, either when used with single or multiple loci. We further show that the ages of clades are carelessly used to infer ages of demographic events, that these ages are estimated under a simple model of panmixia and population stationarity but are then used under different and unspecified models to test hypotheses, a usage the invalidates these testing procedures. We conclude by encouraging researchers to study and use model‐based inference in population genetics.     

Pitfalls of heterogeneous processes for phylogenetic reconstruction

Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work of Kolaczkowski and Thornton (2004, Nature 431: 980-984) and Chang (1996, Math. Biosci. 134: 189-216), we show examples where maximum likelihood (under a homogeneous model) is an inconsistent estimator of the tree. We then explore the prospects of phylogenetic inference under a heterogeneous model. In some models, there are examples where phylogenetic inference under any method is impossible - despite the fact that there is a common tree topology. In particular, there are nonidentifiable mixture distributions, i.e., multiple topologies generate identical mixture distributions. We address which evolutionary models have nonidentifiable mixture distributions and prove that the following duality theorem holds for most DNA substitution models. The model has either: (i) nonidentifiability - two different tree topologies can produce identical mixture distributions, and hence distinguishing between the two topologies is impossible; or (ii) linear tests - there exist linear tests which identify the common tree topology for character data generated by a mixture distribution. The theorem holds for models whose transition matrices can be parameterized by open sets, which includes most of the popular models, such as Tamura-Nei and Kimura's 2-parameter model. The duality theorem relies on our notion of linear tests, which are related to Lake's linear invariants.     

Joint Regression Analysis of Survival and Quality-Adjusted Survival

In this paper, a semiparametric bivariate linear regression model for survival and quality-adjusted survival is investigated. Even with a parametric specification for the joint, distribution, maximum likelihood is not applicable because of induced informative censoring. We propose inference procedures based on estimating functions. The estimators are consistent and asymptotically normal. Hypothesis tests and confidence intervals may be constructed with easy-to-implement resampling techniques. Simultaneous regression modeling of survival and quality-adjusted survival has not been studied formally. Our methodology gives parameter estimates that are highly interpretable in the context of a cost-effectiveness analysis. The usefulness of the proposal is illustrated with a breast cancer dataset.     

Penalized likelihood and multiple testing

The classical multiple testing model remains an important practical area of statistics with new approaches still being developed. In this paper we develop a new multiple testing procedure inspired by a method sometimes used in a problem with a different focus. Namely, the inference after model selection problem. We note that solutions to that problem are often accomplished by making use of a penalized likelihood function. A classic example is the Bayesian information criterion (BIC) method. In this paper we construct a generalized BIC method and evaluate its properties as a multiple testing procedure. The procedure is applicable to a wide variety of statistical models including regression, contrasts, treatment versus control, change point, and others. Numerical work indicates that, in particular, for sparse models the new generalized BIC would be preferred over existing multiple testing procedures.     

Is WIC effective in improving pregnancy-related outcomes? An empirical reassessment

This study provides new evidence on how prenatal WIC participation influences pregnancy-related outcomes, using a large dataset of Medicaid mothers with two or more singleton births. Our analysis suggests there is negative selection by maternal unobserved factors even with a relatively homogenous sample and a rich set of observed characteristics. The conservative estimates from multiple regression which doesn’t address maternal unobserved heterogeneity already demonstrate beneficial effects on a range of outcomes. The concern of mis-specification or extrapolation in the linear model is also ruled out. Controlling for the mother fixed effects, we find more statistically significant estimates which are usually larger in size. The within-mother estimates are robust in a series of sensitivity checks especially multiple inference adjustments. Overall, we find WIC does work to improve infant health and maternal health behaviors as well as reduce usage of costly maternity care.     

Reproducing kernel hilbert spaces regression methods for genomic assisted prediction of quantitative traits

Reproducing kernel Hilbert spaces regression procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are discussed from a theoretical perspective. It is argued that a nonparametric treatment may be needed for capturing the multiple and complex interactions potentially arising in whole-genome models, i.e., those based on thousands of single-nucleotide polymorphism (SNP) markers. After a review of reproducing kernel Hilbert spaces regression, it is shown that the statistical specification admits a standard mixed-effects linear model representation, with smoothing parameters treated as variance components. Models for capturing different forms of interaction, e.g., chromosome-specific, are presented. Implementations can be carried out using software for likelihood-based or Bayesian inference.     

Gaussian Process Based Bayesian Semiparametric Quantitative Trait Loci Interval Mapping

Summary .  In linkage analysis, it is often necessary to include covariates such as age or weight to increase power or avoid spurious false positive findings. However, if a covariate term in the model is specified incorrectly (e.g., a quadratic term misspecified as a linear term), then the inclusion of the covariate may adversely affect power and accuracy of the identification of quantitative trait loci (QTL). Furthermore, some covariates may interact with each other in a complicated fashion. We implement semiparametric models for single and multiple QTL mapping. Both mapping methods include an unspecified function of any covariate found or suspected to have a more complex than linear but unknown relationship with the response variable. They also allow for interactions among different covariates. This analysis is performed in a Bayesian inference framework using Markov chain Monte Carlo. The advantages of our methods are demonstrated via extensive simulations and real data analysis.     

Computing likelihoods for coalescents with multiple collisions in the infinitely many sites model

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (Genetics 172:2621-2633, 2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (Ann Probab 15(2):515-545, 1987) and Griffiths and Tavaré (Theor Pop Biol 46:131-159, 1994a, Stat Sci 9:307-319, 1994b, Philos Trans Roy Soc Lond Ser B 344:403-410, 1994c, Math Biosci 12:77-98, 1995) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta(2 - alpha, alpha)-coalescents, where alpha in (1, 2], are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.     

A Note on Balanced Sampling and Robust Genetic Predictors

The concept of balanced sampling is applied to prediction in finite samples using model based inference procedures. Necessary and sufficient conditions are derived for a general linear model with arbitrary covariance structure to yield the expansion estimator as the best linear unbiased predictor for the mean. The analysis is extended to produce a robust estimator for the mean squared error under balanced sampling and the results are discussed in the context of statistical genetics where appropriate sampling produces simple efficient and robust genetic predictors free from unnecessary genetic assumptions.     

Statistical inference from single channel records: two-state Markov model with limited time resolution

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little consideration. This paper describes techniques of statistical inference, in particular likelihood methods, suitable for Markov models incorporating limited time resolution by means of a discrete detection limit. To simplify the analysis, attention is restricted to two-state models, although the methods have more general applicability. Non-uniqueness of the mean open-time and mean closed-time estimators obtained by moment methods based on single exponential approximations to the apparent open-time and apparent closed-time distributions has been reported. The present study clarifies and extends this previous work by proving that, for such approximations, the likelihood equations as well as the moment equations (usually) have multiple solutions. Such non-uniqueness corresponds to non-identifiability of the statistical model for the apparent quantities. By contrast, higher-order approximations yield theoretically identifiable models. Likelihood-based estimation procedures are developed for both single exponential and bi-exponential approximations. The methods and results are illustrated by numerical examples based on literature and simulated data, with consideration given to empirical distributions and model control, likelihood plots, and point estimation and confidence regions.     

Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles

A mixed model is a flexible tool for joint modeling purposes, especially when the gathered data are unbalanced. However, computational problems due to the dimension of the joint covariance matrix of the random effects arise as soon as the number of outcomes and/or the number of used random effects per outcome increases. We propose a pairwise approach in which all possible bivariate models are fitted, and where inference follows from pseudo-likelihood arguments. The approach is applicable for linear, generalized linear, and nonlinear mixed models, or for combinations of these. The methodology will be illustrated for linear mixed models in the analysis of 22-dimensional, highly unbalanced, longitudinal profiles of hearing thresholds.     

A Note on Probabilistic Models over Strings: The Linear Algebra Approach

Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the “TKF91” model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.     

Functional mixed effects models

In this article, a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced. The linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects, which are modeled as realizations of zero-mean stochastic processes. The fixed functional effects and the random functional effects are modeled in the same functional space, which guarantee the population-average and subject-specific curves have the same smoothness property. These models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures, can include continuous covariates as well as dummy factors in both the fixed or random design matrices, and include the nested curves models as special cases. Two estimation procedures are proposed. The first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software. The second procedure is a sequential estimation procedure using Kalman filtering. This algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets. A generalized maximum likelihood (GML) ratio test is proposed for inference and model selection. An application to comparison of cortisol profiles is used as an illustration.