A Bayesian Approach to Surrogacy Assessment Using Principal Stratification in Clinical Trials

Summary A surrogate marker (S) is a variable that can be measured earlier and often more easily than the true endpoint (T) in a clinical trial. Most previous research has been devoted to developing surrogacy measures to quantify how well S can replace T or examining the use of S in predicting the effect of a treatment (Z). However, the research often requires one to fit models for the distribution of T given S and Z. It is well known that such models do not have causal interpretations because the models condition on a postrandomization variable S. In this article, we directly model the relationship among T, S, and Z using a potential outcomes framework introduced by Frangakis and Rubin (2002, Biometrics 58 , 21–29). We propose a Bayesian estimation method to evaluate the causal probabilities associated with the cross‐classification of the potential outcomes of S and T when S and T are both binary. We use a log‐linear model to directly model the association between the potential outcomes of S and T through the odds ratios. The quantities derived from this approach always have causal interpretations. However, this causal model is not identifiable from the data without additional assumptions. To reduce the nonidentifiability problem and increase the precision of statistical inferences, we assume monotonicity and incorporate prior belief that is plausible in the surrogate context by using prior distributions. We also explore the relationship among the surrogacy measures based on traditional models and this counterfactual model. The method is applied to the data from a glaucoma treatment study.     

Hazard ratio estimation for two‐sample case under interval‐censored failure time data

This paper discusses two‐sample comparison in the case of interval‐censored failure time data. For the problem, one common approach is to employ some nonparametric test procedures, which usually give some p‐values but not a direct or exact quantitative measure of the survival or treatment difference of interest. In particular, these procedures cannot provide a hazard ratio estimate, which is commonly used to measure the difference between the two treatments or samples. For interval‐censored data, a few nonparametric test procedures have been developed, but it does not seem to exist as a procedure for hazard ratio estimation. Corresponding to this, we present two procedures for nonparametric estimation of the hazard ratio of the two samples for interval‐censored data situations. They are generalizations of the corresponding procedures for right‐censored failure time data. An extensive simulation study is conducted to evaluate the performance of the two procedures and indicates that they work reasonably well in practice. For illustration, they are applied to a set of interval‐censored data arising from a breast cancer study.     

Median Tests for Censored Survival Data; a Contingency Table Approach

Summary The median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function. However, existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates. In this article, we modify a K ‐sample median test for censored survival data ( Brookmeyer and Crowley, 1982 , Journal of the American Statistical Association 77, 433–440) through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa. Under censoring, this approach would generate noninteger entries for the cells in the contingency table. We propose to construct a weighted asymptotic test statistic that aggregates dependent χ² ‐statistics formed at the nearest integer points to the original noninteger entries. We show that this statistic follows approximately a χ² ‐distribution with k? 1 degrees of freedom. For a small sample case, we propose a test statistic based on combined p ‐values from Fisher’s exact tests, which follows a χ² ‐distribution with 2 degrees of freedom. Simulation studies are performed to show that the proposed method provides reasonable type I error probabilities and powers. The proposed method is illustrated with two real datasets from phase III breast cancer clinical trials.     

Capture–recapture models with heterogeneity to study survival senescence in the wild

Detecting senescence in wild populations and estimating its strength raise three challenges. First, in the presence of individual heterogeneity in survival probability, the proportion of high‐survival individuals increases with age. This increase can mask a senescence‐related decrease in survival probability when the probability is estimated at the population level. To accommodate individual heterogeneity we use a mixture model structure (discrete classes of individuals). Second, the study individuals can elude the observers in the field, and their detection rate can be heterogeneous. To account for detectability issues we use capture–mark–recapture (CMR) methodology, mixture models and data that provide information on individuals’ detectability. Last, emigration to non‐monitored sites can bias survival estimates, because it can occur at the end of the individuals’ histories and mimic earlier death. To model emigration we use Markovian transitions to and from an unobservable state. These different model structures are merged together using hidden Markov chain CMR models, or multievent models. Simulation studies illustrate that reliable evidence for survival senescence can be obtained using highly heterogeneous data from non site‐faithful individuals. We then design a tailored application for a dataset from a colony of black‐headed gull Chroicocephalus ridibundus. Survival probabilities do not appear individually variable, but evidence for survival senescence becomes significant only when accounting for other sources of heterogeneity. This result suggests that not accounting for heterogeneity leads to flawed inference and/or that emigration heterogeneity mimics survival heterogeneity and biases senescence estimates.     

CHARACTER DISCRIMINATORY POWER,CHARACTER-SET CONGRUENCE,AND THE CLASSIFICATION OF INDIVIDUALS FROM HYBRID ZONES: AN EXAMPLE USING STONE CRABS (MENIPPE)

Many investigators categorize individuals from hybrid zones to facilitate comparisons among genotypic classes (e.g., parental, F₁, backcross) for comparative studies in which components of fitness or geographic variation are being analyzed. Frequently, multiple character sets representing genetically independent traits are used to classify these individuals and various methodologies are employed to combine the classifications obtained from the different character sets. We adapted the principles of total evidence and taxonomic congruence (two formalized approaches used by systematists in formulating phylogenetic hypotheses) to address the problem of discriminating hybridizing species and classifying individuals from hybrid zones. As our model, we used two morphological (coloration and morphometric) and two molecular (allozyme and mitochondrial DNA restriction-fragment-length polymorphism) character sets that differentiate two stone crab species (Menippe adina and M. mercenaria). Using principal-components analysis, we determined that combining character sets and eliminating characters or character sets that did not have large eigenvector coefficients for the principal component that best separated the two species yielded the highest level of discrimination between species and allowed us to classify a broad range of morpho-genotypes as hybrids. For the stone crabs, three diagnostic allozyme loci and five diagnostic coloration characters best separated the species. The two character sets were not completely congruent, but they agreed in their classification of 50% of the individuals from the hybrid zone and rarely strongly disagreed in their classifications. Classification discrepancies between the two character sets probably represent variation between traits in interspecific gene flow rather than intraspecific, ecologically mediated variation. Our results support the assertions of previous investigators who espoused the benefits associated with using multiple character sets to classify individuals from hybrid zones and demonstrate that, if character sets are reasonably congruent and numerically balanced, combining diagnostic characters from multiple character sets (a total-evidence approach) can enhance discriminatory power between species and facilitate the assignment of hybrid-zone individuals to genotypic classes. On the contrary, classifying hybrid-zone individuals using character sets separately (a taxonomic-congruence approach) provides the opportunity to compare levels of introgression between species and to assess reasons for discordance among the data sets.     

Estimation for the optimal combination of markers without modeling the censoring distribution

Summary .  In the time-dependent receiver operating characteristic curve analysis with several baseline markers, research interest focuses on seeking appropriate composite markers to enhance the accuracy in predicting the vital status of individuals over time. Based on censored survival data, we proposed a more flexible estimation procedure for the optimal combination of markers under the validity of a time-varying coefficient generalized linear model for the event time without restrictive assumptions on the censoring pattern. The consistency of the proposed estimators is also established in this article. In contrast, the inverse probability weighting (IPW) approach might introduce a bias when the selection probabilities are misspecified in the estimating equations. The performance of both estimation procedures are examined and compared through a class of simulations. It is found from the simulation study that the proposed estimators are far superior to the IPW ones. Applying these methods to an angiography cohort, our estimation procedure is shown to be useful in predicting the time to all-cause and coronary artery disease related death.     

Estimating survival in the Apennine brown bear accounting for uncertainty in age classification

For most rare and elusive species, estimating age-specific survival is a challenging task, although it is an important requirement to understand the drivers of population dynamics, and to inform conservation actions. Apennine brown bears Ursus arctos marsicanus are a small, isolated population under a severe risk of extinction, for which the main demographic mechanisms underlying population dynamics are still unknown, and population trends have not been formally assessed. We present a 12-year analysis of their survival rates using non-invasive genetic sampling data collected through four different sampling techniques. By using multi-event capture–recapture models, we estimated survival probabilities for two broadly defined age classes (cubs and older individuals), even though the age of the majority of sampled bears was unknown. We also applied the Pradel model to provide a preliminary assessment of population trend during the study period. Survival was different between cubs [ϕ = 0.51, 95% CI (0.22, 0.79)], adult males [ϕ = 0.85, 95% CI (0.76, 0.91)] and adult females [ϕ = 0.92, 95% CI (0.87, 0.95)], no temporal variation in survival emerged, suggesting that bear survival remained substantially stable throughout the study period. The Pradel analysis of population trend yielded an estimate of λ = 1.009 [SE = 0.018; 95% CI (0.974, 1.046)]. Our results indicate that, despite the status of full legal protection, the basically stable demography of this relict population is compatible with the observed lack of range expansion, and that a relatively high cub mortality could be among the main factors depressing recruitment and hence population growth.     

Generalized pairwise comparisons for censored data: An overview

The method of generalized pairwise comparisons (GPC) is an extension of the well-known nonparametric Wilcoxon–Mann–Whitney test for comparing two groups of observations. Multiple generalizations of Wilcoxon–Mann–Whitney test and other GPC methods have been proposed over the years to handle censored data. These methods apply different approaches to handling loss of information due to censoring: ignoring noninformative pairwise comparisons due to censoring (Gehan, Harrell, and Buyse); imputation using estimates of the survival distribution (Efron, Péron, and Latta); or inverse probability of censoring weighting (IPCW, Datta and Dong). Based on the GPC statistic, a measure of treatment effect, the “net benefit,” can be defined. It quantifies the difference between the probabilities that a randomly selected individual from one group is doing better than an individual from the other group. This paper aims at evaluating GPC methods for censored data, both in the context of hypothesis testing and estimation, and providing recommendations related to their choice in various situations. The methods that ignore uninformative pairs have comparable power to more complex and computationally demanding methods in situations of low censoring, and are slightly superior for high proportions (>40%) of censoring. If one is interested in estimation of the net benefit, Harrell's c index is an unbiased estimator if the proportional hazards assumption holds. Otherwise, the imputation (Efron or Peron) or IPCW (Datta, Dong) methods provide unbiased estimators in case of proportions of drop-out censoring up to 60%.     

Prediction errors for state occupation and transition probabilities in multi‐state models

In this paper, we consider the estimation of prediction errors for state occupation probabilities and transition probabilities for multistate time‐to‐event data. We study prediction errors based on the Brier score and on the Kullback–Leibler score and prove their properness. In the presence of right‐censored data, two classes of estimators, based on inverse probability weighting and pseudo‐values, respectively, are proposed, and consistency properties of the proposed estimators are investigated. The second part of the paper is devoted to the estimation of dynamic prediction errors for state occupation probabilities for multistate models, conditional on being alive, and for transition probabilities. Cross‐validated versions are proposed. Our methods are illustrated on the CSL1 randomized clinical trial comparing prednisone versus placebo for liver cirrhosis patients.     

A hierarchical Bayesian approach for handling missing classification data

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Modeling stream fish distributions using interval‐censored detection times

Controlling for imperfect detection is important for developing species distribution models (SDMs). Occupancy‐detection models based on the time needed to detect a species can be used to address this problem, but this is hindered when times to detection are not known precisely. Here, we extend the time‐to‐detection model to deal with detections recorded in time intervals and illustrate the method using a case study on stream fish distribution modeling. We collected electrofishing samples of six fish species across a Mediterranean watershed in Northeast Portugal. Based on a Bayesian hierarchical framework, we modeled the probability of water presence in stream channels, and the probability of species occupancy conditional on water presence, in relation to environmental and spatial variables. We also modeled time‐to‐first detection conditional on occupancy in relation to local factors, using modified interval‐censored exponential survival models. Posterior distributions of occupancy probabilities derived from the models were used to produce species distribution maps. Simulations indicated that the modified time‐to‐detection model provided unbiased parameter estimates despite interval‐censoring. There was a tendency for spatial variation in detection rates to be primarily influenced by depth and, to a lesser extent, stream width. Species occupancies were consistently affected by stream order, elevation, and annual precipitation. Bayesian P‐values and AUCs indicated that all models had adequate fit and high discrimination ability, respectively. Mapping of predicted occupancy probabilities showed widespread distribution by most species, but uncertainty was generally higher in tributaries and upper reaches. The interval‐censored time‐to‐detection model provides a practical solution to model occupancy‐detection when detections are recorded in time intervals. This modeling framework is useful for developing SDMs while controlling for variation in detection rates, as it uses simple data that can be readily collected by field ecologists.     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.     

Temporal Patterns of Apparent Leg Band Retention in North American Geese

ABSTRACT An important assumption of mark—recapture studies is that individuals retain their marks, which has not been assessed for goose reward bands. We estimated aluminum leg band retention probabilities and modeled how band retention varied with band type (standard vs. reward band), band age (1–40 months), and goose characteristics (species and size class) for Canada (Branta canadensis), cackling (Branta hutchinsii), snow (Chen caerulescens), and Ross's (Chen rossii) geese that field coordinators double-leg banded during a North American goose reward band study (N = 40,999 individuals from 15 populations). We conditioned all models in this analysis on geese that were encountered with ≥1 leg band still attached (n = 5,747 dead recoveries and live recaptures). Retention probabilities for standard aluminum leg bands were high ( = 0.9995, SE < 0.001) and constant over 1–40 months. In contrast, apparent retention probabilities for reward bands demonstrated an interactive relationship between 5 size and species classes (small cackling, medium Canada, large Canada, snow, and Ross's geese). In addition, apparent retention probabilities for each of the 5 classes varied quadratically with time, being lower immediately after banding and at older age classes. The differential retention probabilities among band type (reward vs. standard) that we observed suggests that 1) models estimating reporting probability should incorporate differential band loss if it is nontrivial, 2) goose managers should consider the costs and benefits of double-banding geese on an operational basis, and 3) the United States Geological Survey Bird Banding Lab should modify protocols for receiving recovery data.     

The Accelerated Failure Time Model Under Biased Sampling

Summary Chen (2009, Biometrics) studies the semi‐parametric accelerated failure time model for data that are size biased. Chen considers only the uncensored case and uses hazard‐based estimation methods originally developed for censored observations. However, for uncensored data, a simple linear regression on the log scale is more natural and provides better estimators.     

Parameter Identifiability and Model Selection in Capture-Recapture Models: A Numerical Approach

Capture-recapture models are a powerful tool for estimating and comparing survival probabilities among groups of individuals in wild animal populations. One of the remaining problems is the calculation of the number of independently estimable parameters in the models, which is necessary in using model selection tools such as Likelihood ratio tests or the Akaike's Information Criterion. We show that the number of separately identifiable parameters in a model is equal to the rank of the Hessian matrix (second derivatives of the maximum likelihood relative to the parameters). We present the numerical problems involved in computing the Hessian and its numerical rank, and we apply the technique to data on nesting swifts (Apus apus).     

Extrinsic and intrinsic factors influence fitness in an avian hybrid zone

The effects of hybridization on evolutionary processes are primarily determined by the differential between hybrid and parental species fitness. Assessing the impacts of hybridization can be challenging, however, as determining the relationship between individual fitness and the extent of introgression in wild populations is difficult. We evaluated the fitness consequences of hybridization for pure and hybrid females in a hybrid zone between two tidal marsh birds, the saltmarsh sparrow (Ammodramus caudacutus), a salt marsh obligate, and Nelson's sparrow (A. nelsoni), which has a broader ecological niche and a much younger evolutionary association with salt marshes. Biotic stressors associated with nesting in tidal environments suggest an important role for differential adaptation in shaping hybrid zone dynamics, with saltmarsh sparrows predicted to be better adapted to nesting in salt marshes. We collected DNA samples from adults (n = 394) and nestlings (n = 431) to determine the extent of introgression using 12 microsatellite loci and tested for the influence of extrinsic (nest placement) and intrinsic (genotype) factors on female reproductive success. We monitored nests (n = 228), collected data on reproductive output, and estimated daily nest survival rates using female genotype and nest characteristics as covariates. To test for reduced survival of hybrid females, we also used capture data to assess the distribution of admixed male and female individuals across age classes. Reproductive success of females varied by genotypic class, but hybrids did not have intermediate success as predicted. Instead, we found that pure Nelson's sparrows had, on average, 33% lower hatching success than any other genotype, whereas F1/F2 hybrids, backcrossed Nelson's sparrows, and backcrossed and pure saltmarsh sparrows all had similar hatching success. We found no effect of genotype or nest placement on daily nest survival probabilities. However, hybrid individuals with a higher proportion of saltmarsh sparrow alleles exhibit nesting behaviours better suited to nesting successfully in tidal marshes. Further, while the proportion of F1/F2 individuals was similar between nestling and adult males, we found that the proportion of F1/F2 individuals was 2.3 times greater in nestling females compared with adult females, indicating reduced survival of F1 females. We conclude that differences in reproductive success among pure and admixed individuals coupled with intrinsic mechanisms (reduced survival in F1 females) shape hybrid zone dynamics in this system.     

Dealing with uncertainty in landscape genetic resistance models: a case of three co‐occurring marsupials

Landscape genetics lacks explicit methods for dealing with the uncertainty in landscape resistance estimation, which is particularly problematic when sample sizes of individuals are small. Unless uncertainty can be quantified, valuable but small data sets may be rendered unusable for conservation purposes. We offer a method to quantify uncertainty in landscape resistance estimates using multimodel inference as an improvement over single model‐based inference. We illustrate the approach empirically using co‐occurring, woodland‐preferring Australian marsupials within a common study area: two arboreal gliders (Petaurus breviceps, and Petaurus norfolcensis) and one ground‐dwelling antechinus (Antechinus flavipes). First, we use maximum‐likelihood and a bootstrap procedure to identify the best‐supported isolation‐by‐resistance model out of 56 models defined by linear and non‐linear resistance functions. We then quantify uncertainty in resistance estimates by examining parameter selection probabilities from the bootstrapped data. The selection probabilities provide estimates of uncertainty in the parameters that drive the relationships between landscape features and resistance. We then validate our method for quantifying uncertainty using simulated genetic and landscape data showing that for most parameter combinations it provides sensible estimates of uncertainty. We conclude that small data sets can be informative in landscape genetic analyses provided uncertainty can be explicitly quantified. Being explicit about uncertainty in landscape genetic models will make results more interpretable and useful for conservation decision‐making, where dealing with uncertainty is critical.     

K‐Sample comparisons using propensity analysis

In this paper, we investigate K‐group comparisons on survival endpoints for observational studies. In clinical databases for observational studies, treatment for patients are chosen with probabilities varying depending on their baseline characteristics. This often results in noncomparable treatment groups because of imbalance in baseline characteristics of patients among treatment groups. In order to overcome this issue, we conduct propensity analysis and match the subjects with similar propensity scores across treatment groups or compare weighted group means (or weighted survival curves for censored outcome variables) using the inverse probability weighting (IPW). To this end, multinomial logistic regression has been a popular propensity analysis method to estimate the weights. We propose to use decision tree method as an alternative propensity analysis due to its simplicity and robustness. We also propose IPW rank statistics, called Dunnett‐type test and ANOVA‐type test, to compare 3 or more treatment groups on survival endpoints. Using simulations, we evaluate the finite sample performance of the weighted rank statistics combined with these propensity analysis methods. We demonstrate these methods with a real data example. The IPW method also allows us for unbiased estimation of population parameters of each treatment group. In this paper, we limit our discussions to survival outcomes, but all the methods can be easily modified for any type of outcomes, such as binary or continuous variables.     

Predicting tree death for Fagus sylvatica and Abies alba using permanent plot data

Question: How well can mortality probabilities of deciduous trees(Fagus sylvatica) and conifers (Abies alba) be predicted using permanent plot data that describe growth patterns, tree species, tree size and site conditions? Location: Fagus forests in the montane belt of the Jura folds (Switzerland). Method: Permanent plot data were used to develop and validate logistic regression models predicting survival probabilities of individual trees. Backward model selection led to a reduced model containing the growth‐related variable ‘relative basal area increment’ (growth‐dependent mortality) and variables not directly reflecting growth such as species, size and site (growth‐independent mortality). Results: The growth‐mortality relationship was the same for both species (growth‐dependent mortality). However, species, site and tree size also influenced mortality probabilities (growth‐independent mortality). The predicted survival probabilities of the final model were well calibrated, and the model showed an excellent discriminatory power (area under the receiver operating characteristic curve = 0.896). Conclusion: Mortality probabilities of Fagus sylvatica and Abies alba can be predicted with high discriminatory power using a well calibrated logistic regression model. Extending this case study to a larger number of tree species and sites could provide species‐ and site‐specific tree mortality models that allow for more realistic projections of forest succession.