On the estimation of average treatment effects with right-censored time to event outcome and competing risks

We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, censoring, and treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that these estimators are regular asymptotically linear estimators and estimate their variances based on estimates of their influence functions. In empirical studies, we assess the robustness of the estimators and the coverage of confidence intervals. The methods are further illustrated using data from a Danish registry study.     

Partial AUC estimation and regression

Accurate diagnosis of disease is a critical part of health care. New diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased from nondiseased states. The partial area under the receiver operating characteristic (ROC) curve is a measure of diagnostic test accuracy. We present an interpretation of the partial area under the curve (AUC), which gives rise to a nonparametric estimator. This estimator is more robust than existing estimators, which make parametric assumptions. We show that the robustness is gained with only a moderate loss in efficiency. We describe a regression modeling framework for making inference about covariate effects on the partial AUC. Such models can refine knowledge about test accuracy. Model parameters can be estimated using binary regression methods. We use the regression framework to compare two prostate-specific antigen biomarkers and to evaluate the dependence of biomarker accuracy on the time prior to clinical diagnosis of prostate cancer.     

Outlier robust model‐assisted small area estimation

Small area estimation with M‐quantile models was proposed by Chambers and Tzavidis ( 2006 ). The key target of this approach to small area estimation is to obtain reliable and outlier robust estimates avoiding at the same time the need for strong parametric assumptions. This approach, however, does not allow for the use of unit level survey weights, making questionable the design consistency of the estimators unless the sampling design is self‐weighting within small areas. In this paper, we adopt a model‐assisted approach and construct design consistent small area estimators that are based on the M‐quantile small area model. Analytic and bootstrap estimators of the design‐based variance are discussed. The proposed estimators are empirically evaluated in the presence of complex sampling designs.     

Robust estimators for expression analysis

MOTIVATION: We consider the problem of estimating values associated with gene expression from oligonucleotide arrays. Such estimates should linearly track concentration, yield non-negative results, have statistical guarantees of robustness against outliers, and allow estimates of significance and variance. RESULTS: A hierarchy of simple models is used to design robust estimators meeting these goals for both stand alone and comparative experiments. This algorithm has been validated against an extensive panel of known spike experiments, and shows comparable performance to existing standards.     

On causal mediation analysis with a survival outcome

Suppose that having established a marginal total effect of a point exposure on a time-to-event outcome, an investigator wishes to decompose this effect into its direct and indirect pathways, also known as natural direct and indirect effects, mediated by a variable known to occur after the exposure and prior to the outcome. This paper proposes a theory of estimation of natural direct and indirect effects in two important semiparametric models for a failure time outcome. The underlying survival model for the marginal total effect and thus for the direct and indirect effects, can either be a marginal structural Cox proportional hazards model, or a marginal structural additive hazards model. The proposed theory delivers new estimators for mediation analysis in each of these models, with appealing robustness properties. Specifically, in order to guarantee ignorability with respect to the exposure and mediator variables, the approach, which is multiply robust, allows the investigator to use several flexible working models to adjust for confounding by a large number of pre-exposure variables. Multiple robustness is appealing because it only requires a subset of working models to be correct for consistency; furthermore, the analyst need not know which subset of working models is in fact correct to report valid inferences. Finally, a novel semiparametric sensitivity analysis technique is developed for each of these models, to assess the impact on inference, of a violation of the assumption of ignorability of the mediator.     

VCD Robustness of the Amide‐I and Amide‐II Vibrational Modes of Small Peptide Models

The rotational strengths and the robustness values of amide‐I and amide‐II vibrational modes of For(AA)_nNHMe (where AA is Val, Asn, Asp, or Cys, n = 1–5 for Val and Asn; n = 1 for Asp and Cys) model peptides with α‐helix and β‐sheet backbone conformations were computed by density functional methods. The robustness results verify empirical rules drawn from experiments and from computed rotational strengths linking amide‐I and amide‐II patterns in the vibrational circular dichroism (VCD) spectra of peptides with their backbone structures. For peptides with at least three residues (n ≥ 3) these characteristic patterns from coupled amide vibrational modes have robust signatures. For shorter peptide models many vibrational modes are nonrobust, and the robust modes can be dependent on the residues or on their side chain conformations in addition to backbone conformations. These robust VCD bands, however, provide information for the detailed structural analysis of these smaller systems. Chirality 27:625–634, 2015 © 2015 Wiley Periodicals, Inc.     

Robustness and aging—A systems-level perspective

The theory of robustness describes a system level property of evolutionary systems, which predicts tradeoffs of great interest for the systems biology of aging, such as accumulation of non-heritable damage, occurrence of fragilities and limitations in performance, optimized allocation of restricted resources and confined redundancies. According to the robustness paradigm cells and organisms evolved into a state of highly optimized tolerance (HOT), which provides robustness to common perturbations, but causes tradeoffs generally characterized as “robust yet fragile”. This raises the question whether the ultimate cause of aging is more than a lack of adaptation, but an inherent fragility of complex evolutionary systems. Since robustness connects to evolutionary designs, consideration of this theory provides a deeper connection between evolutionary aspects of aging, mathematical models and experimental data. In this review several mechanisms influential for aging are re-evaluated in support of robustness tradeoffs. This includes asymmetric cell division improving performance and specialization with limited capacities to prevent and repair age-related damage, as well as feedback control mechanisms optimized to respond to acute stressors, but unable to halt nor revert aging. Improvement in robustness by increasing efficiencies through cellular redundancies in larger organisms alleviates some of the damaging effects of cellular specialization, which can be expressed in allometric relationships. The introduction of the robustness paradigm offers unique insights for aging research and provides novel opportunities for systems biology endeavors.     

A comparison of single‐sample estimators of effective population sizes from genetic marker data

In molecular ecology and conservation genetics studies, the important parameter of effective population size (N_e) is increasingly estimated from a single sample of individuals taken at random from a population and genotyped at a number of marker loci. Several estimators are developed, based on the information of linkage disequilibrium (LD), heterozygote excess (HE), molecular coancestry (MC) and sibship frequency (SF) in marker data. The most popular is the LD estimator, because it is more accurate than HE and MC estimators and is simpler to calculate than SF estimator. However, little is known about the accuracy of LD estimator relative to that of SF and about the robustness of all single‐sample estimators when some simplifying assumptions (e.g. random mating, no linkage, no genotyping errors) are violated. This study fills the gaps and uses extensive simulations to compare the biases and accuracies of the four estimators for different population properties (e.g. bottlenecks, nonrandom mating, haplodiploid), marker properties (e.g. linkage, polymorphisms) and sample properties (e.g. numbers of individuals and markers) and to compare the robustness of the four estimators when marker data are imperfect (with allelic dropouts). Extensive simulations show that SF estimator is more accurate, has a much wider application scope (e.g. suitable to nonrandom mating such as selfing, haplodiploid species, dominant markers) and is more robust (e.g. to the presence of linkage and genotyping errors of markers) than the other estimators. An empirical data set from a Yellowstone grizzly bear population was analysed to demonstrate the use of the SF estimator in practice.     

Estimating pairwise relatedness from dominant genetic markers

Knowledge of the genetic relatedness between a pair of individuals is important in many research areas of quantitative genetics, conservation genetics, evolution and ecology. Many estimators have been developed to estimate such pairwise relatedness (r) using codominant markers, such as microsatellites and enzymes. In contrast, only two estimators are proposed to use dominant markers, such as random amplified polymorphic DNAs (RAPDs) and amplified fragment length polymorphisms (AFLPs), in relatedness inference. They are both biased estimators, and their statistical properties and robustness to the sampling errors in allele frequency have not been investigated. In this short paper, I propose two new pairwise relatedness estimators for dominant markers, and compare them in precision, accuracy and robustness to sampling with the two previous estimators using simulations. It was found that the new estimator based on the least squares approach is unbiased when allele frequencies are known or estimated from a sample without correcting for sampling effects. It has, however, a low precision and as a result, an intermediate overall performance among the four estimators in terms of the mean squared deviation (MSD) of estimates from actual values of r. The new estimator based on a similarity index is slightly biased but has generally the lowest MSD among the four estimators compared, regardless of the number of loci, type of actual relationships, allele frequencies known or estimated from samples. Simulations also show that the confidence intervals estimated by bootstrapping are appropriate for different estimators provided that the number of loci used in the estimation is not small.     

Robust alternatives to the F-Test in mixed linear models based on MM-estimates

Mixed linear models are commonly used to analyze data in many settings. These models are generally fitted by means of (restricted) maximum likelihood techniques relying heavily on normality. The sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations. Very recently a highly robust estimator based on a scale estimate, that is, an S-estimator, has been proposed for general mixed linear models. It has the advantage of being easy to compute and allows the computation of a robust score test. However, this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an F-test. As the latter is usually a key tool of hypothesis testing in mixed linear models, we propose two new robust estimators that allow the desired extension. They also lead to resistant Wald-type tests useful for testing contrasts and covariate effects. We study their properties theoretically and by means of simulations. The analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.     

Robustness: confronting lessons from physics and biology

The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, self‐organisation) developed in physics are relevant to biology, or whether specific extensions and novel frameworks are required to account for the robustness properties of living systems. To clarify this issue, the different meanings covered by this unique term are discussed; it is argued that they crucially depend on the kind of perturbations that a robust system should by definition withstand. Possible mechanisms underlying robust behaviours are examined, either encountered in all natural systems (symmetries, conservation laws, dynamic stability) or specific to biological systems (feedbacks and regulatory networks). Special attention is devoted to the (sometimes counterintuitive) interrelations between robustness and noise. A distinction between dynamic selection and natural selection in the establishment of a robust behaviour is underlined. It is finally argued that nested notions of robustness, relevant to different time scales and different levels of organisation, allow one to reconcile the seemingly contradictory requirements for robustness and adaptability in living systems.     

Robustness analysis of biochemical network models

Biological systems that have been experimentally verified to be robust to significant changes in their environments require mathematical models that are themselves robust. In this context, a necessary condition for model robustness is that the model dynamics should not be sensitive to small variations in the model's parameters. Robustness analysis problems of this type have been extensively studied in the field of robust control theory and have been found to be very difficult to solve in general. The authors describe how some tools from robust control theory and nonlinear optimisation can be used to analyse the robustness of a recently proposed model of the molecular network underlying adenosine 3',5'-cyclic monophosphate (cAMP) oscillations observed in fields of chemotactic Dictyostelium cells. The network model, which consists of a system of seven coupled nonlinear differential equations, accurately reproduces the spontaneous oscillations in cAMP observed during the early development of D. discoideum. The analysis by the authors reveals, however, that very small variations in the model parameters can effectively destroy the required oscillatory dynamics. A biological interpretation of the analysis results is that correct functioning of a particular positive feedback loop in the proposed model is crucial to maintaining the required oscillatory dynamics.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Spatial capture–recapture models allowing Markovian transience or dispersal

Spatial capture–recapture (SCR) models are a relatively recent development in quantitative ecology, and they are becoming widely used to model density in studies of animal populations using camera traps, DNA sampling and other methods which produce spatially explicit individual encounter information. One of the core assumptions of SCR models is that individuals possess home ranges that are spatially stationary during the sampling period. For many species, this assumption is unlikely to be met and, even for species that are typically territorial, individuals may disperse or exhibit transience at some life stages. In this paper we first conduct a simulation study to evaluate the robustness of estimators of density under ordinary SCR models when dispersal or transience is present in the population. Then, using both simulated and real data, we demonstrate that such models can easily be described in the BUGS language providing a practical framework for their analysis, which allows us to evaluate movement dynamics of species using capture–recapture data. We find that while estimators of density are extremely robust, even to pathological levels of movement (e.g., complete transience), the estimator of the spatial scale parameter of the encounter probability model is confounded with the dispersal/transience scale parameter. Thus, use of ordinary SCR models to make inferences about density is feasible, but interpretation of SCR model parameters in relation to movement should be avoided. Instead, when movement dynamics are of interest, such dynamics should be parameterized explicitly in the model.     

Robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data

Summary .  In this article, we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function. Under some regularity conditions, the asymptotic properties of the proposed robust estimators are shown. Some simulations are carried out to investigate the performance of the proposed robust estimators. Just as expected, the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like Sinha (2004, Journal of the American Statistical Association 99, 451–460) and Qin and Zhu (2007, Journal of Multivariate Analysis 98, 1658–1683). In the end, the proposed robust method is illustrated by the analysis of a real data set.     

Review and comparison of ROC curve estimators for a time‐dependent outcome with marker‐dependent censoring

To quantify the ability of a marker to predict the onset of a clinical outcome in the future, time‐dependent estimators of sensitivity, specificity, and ROC curve have been proposed accounting for censoring of the outcome. In this paper, we review these estimators, recall their assumptions about the censoring mechanism and highlight their relationships and properties. A simulation study shows that marker‐dependent censoring can lead to important biases for the ROC estimators not adapted to this case. A slight modification of the inverse probability of censoring weighting estimators proposed by Uno et al. (2007) and Hung and Chiang (2010a) performs as well as the nearest neighbor estimator of Heagerty et al. (2000) in the simulation study and has interesting practical properties. Finally, the estimators were used to evaluate abilities of a marker combining age and a cognitive test to predict dementia in the elderly. Data were obtained from the French PAQUID cohort. The censoring appears clearly marker‐dependent leading to appreciable differences between ROC curves estimated with the different methods.     

Median percent change: A robust alternative for assessing temporal trends

A typical summary statistic for temporal trends is the average percent change (APC). The APC is estimated by using a generalized linear model, usually under the assumption of linearity on the logarithmic scale. A serious limitation of least-squares type estimators is their sensitivity to outliers. The goal of this study is twofold: firstly, we propose a robust and easy-to-compute measure of the temporal trend based on the median of the rates (median percent change – MPC), rather than their mean, under the hypothesis of constant relative change; secondly, we investigate the performance of several models for estimating the rate of change when some of the most common model assumptions are violated. We provide some guidance on the practices of the estimation of temporal trends when using different models under different circumstances. The robustness property of the median is assessed in a simulation study, which shows that the MPC provides strong reductions in estimation bias and variance in presence of outliers. We also demonstrate how a mathematical property of the median helps addressing the issue of zero counts when estimating trends on the log-scale. Finally, we analyzed an English cancer registration dataset to illustrate the proposed method. We believe that, as a good practice, both APC and MPC should be presented when sensitivity issues arise.     

Using Regression Models to Analyze Randomized Trials: Asymptotically Valid Hypothesis Tests Despite Incorrectly Specified Models

Summary .  Regression models are often used to test for cause-effect relationships from data collected in randomized trials or experiments. This practice has deservedly come under heavy scrutiny, because commonly used models such as linear and logistic regression will often not capture the actual relationships between variables, and incorrectly specified models potentially lead to incorrect conclusions. In this article, we focus on hypothesis tests of whether the treatment given in a randomized trial has any effect on the mean of the primary outcome, within strata of baseline variables such as age, sex, and health status. Our primary concern is ensuring that such hypothesis tests have correct type I error for large samples. Our main result is that for a surprisingly large class of commonly used regression models, standard regression-based hypothesis tests (but using robust variance estimators) are guaranteed to have correct type I error for large samples, even when the models are incorrectly specified. To the best of our knowledge, this robustness of such model-based hypothesis tests to incorrectly specified models was previously unknown for Poisson regression models and for other commonly used models we consider. Our results have practical implications for understanding the reliability of commonly used, model-based tests for analyzing randomized trials.