Generalized additive models with interval-censored data and time-varying covariates: application to human immunodeficiency virus infection in hemophiliacs

We describe a method for extending smooth nonparametric modeling methods to time-to-event data where the event may be known only to lie within a window of time. Maximum penalized likelihood is used to fit a discrete proportional hazards model that also models the baseline hazard, and left-truncation and time-varying covariates are accommodated. The implementation follows generalized additive modeling conventions, allowing both parametric and smooth terms and specifying the amount of smoothness in terms of the effective degrees of freedom. We illustrate the method on a well-known interval-censored data set on time of human immunodeficiency virus infection in a multicenter study of hemophiliacs. The ability to examine time-varying covariates, not available with previous methods, allows detection and modeling of nonproportional hazards and use of a time-varying covariate that fits the data better and is more plausible than a fixed alternative.     

Parametric survival models for interval-censored data with time-dependent covariates

We present a parametric family of regression models for interval-censored event-time (survival) data that accomodates both fixed (e.g. baseline) and time-dependent covariates. The model employs a three-parameter family of survival distributions that includes the Weibull, negative binomial, and log-logistic distributions as special cases, and can be applied to data with left, right, interval, or non-censored event times. Standard methods, such as Newton-Raphson, can be employed to estimate the model and the resulting estimates have an asymptotically normal distribution about the true values with a covariance matrix that is consistently estimated by the information function. The deviance function is described to assess model fit and a robust sandwich estimate of the covariance may also be employed to provide asymptotically robust inferences when the model assumptions do not apply. Spline functions may also be employed to allow for non-linear covariates. The model is applied to data from a long-term study of type 1 diabetes to describe the effects of longitudinal measures of glycemia (HbA1c) over time (the time-dependent covariate) on the risk of progression of diabetic retinopathy (eye disease), an interval-censored event-time outcome.     

Joint analysis of panel count and interval-censored data using distribution-free frailty analysis

We propose a joint analysis of recurrent and nonrecurrent event data subject to general types of interval censoring. The proposed analysis allows for general semiparametric models, including the Box–Cox transformation and inverse Box–Cox transformation models for the recurrent and nonrecurrent events, respectively. A frailty variable is used to account for the potential dependence between the recurrent and nonrecurrent event processes, while leaving the distribution of the frailty unspecified. We apply the pseudolikelihood for interval-censored recurrent event data, usually termed as panel count data, and the sufficient likelihood for interval-censored nonrecurrent event data by conditioning on the sufficient statistic for the frailty and using the working assumption of independence over examination times. Large sample theory and a computation procedure for the proposed analysis are established. We illustrate the proposed methodology by a joint analysis of the numbers of occurrences of basal cell carcinoma over time and time to the first recurrence of squamous cell carcinoma based on a skin cancer dataset, as well as a joint analysis of the numbers of adverse events and time to premature withdrawal from study medication based on a scleroderma lung disease dataset.     

Frailty‐Based Competing Risks Model for Multivariate Survival Data

Summary In this work, we provide a new class of frailty‐based competing risks models for clustered failure times data. This class is based on expanding the competing risks model of Prentice et al. (1978, Biometrics 34 , 541–554) to incorporate frailty variates, with the use of cause‐specific proportional hazards frailty models for all the causes. Parametric and nonparametric maximum likelihood estimators are proposed. The main advantages of the proposed class of models, in contrast to the existing models, are: (1) the inclusion of covariates; (2) the flexible structure of the dependency among the various types of failure times within a cluster; and (3) the unspecified within‐subject dependency structure. The proposed estimation procedures produce the most efficient parametric and semiparametric estimators and are easy to implement. Simulation studies show that the proposed methods perform very well in practical situations.     

Residuals for proportional hazards models with interval-censored survival data

We develop diagnostic tools for use with proportional hazards models for interval-censored survival data. We propose counterparts to the Cox-Snell, Lagakos (or martingale), deviance, and Schoenfeld residuals. Many of the properties of these residuals carry over to the interval-censored case. In particular, the interval-censored versions of the Lagakos and Schoenfeld residuals may be derived as components of suitable score statistics. The Lagakos residuals may be used to check regression relationships, while the Schoenfeld residuals can help to detect nonproportional hazards in semiparametric models. The methods apply to parametric models and to the semiparametric model with discrete observation times.     

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan–Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.     

Effects of climate on fall migration phenology of monarch butterflies departing the northeastern breeding grounds in Canada

Monarch butterflies (Danaus plexippus) undergo an iconic multi-generational migration, traveling thousands of kilometers from the summer breeding grounds in southern Canada to overwintering sites in central Mexico. This migration phenomena can be affected by climate change, which may have important implications on fitness and ultimately populations status. We investigated the long-term trends in fall migration phenology of monarchs using a 25-year dataset collected along the coast of Lake Erie in Ontario, Canada. We also investigated local long-term trends in weather covariates that have the potential to influence migration phenology at this site. Patterns in standardized daily counts of monarchs were compared with local weather covariates using two methods (i.e., monthly averages and moving windows) to assess difference in outputs between analytical approaches. Our results suggest that monarch migration timing (migration midpoint, average peak, first peak, and late passage) and weather covariates have been consistent over time, in direct contrast to a similar study in Cape May, New Jersey, which showed a significant increase in both fall temperature and a 16- to 19-day shift in monarch migration timing. Furthermore, our results differed between analytical approaches. With respect to annual variability in air temperature, our monthly average analysis suggested that for each degree increase in September air temperature, late season passage would advance 4.71 days (±1.59 SE, p = .01). However, the moving window analysis suggested that this result is likely spurious and found no significant correlations between migration timing and any weather covariates. Importantly, our results caution against extrapolating the effects of climate change on the migration phenology of the monarch across study regions and the need for more long-term monitoring efforts to better understand regional drivers of variability in migration timing.     

Proximate Cues for a Short-Distance Migratory Species: an Application of Survival Analysis

ABSTRACT Investigation of bird migration has often highlighted the importance of external factors in determining timing of migration. However, little distinction has been made between short- and long-distance migrants and between local and flight birds (passage migrants) in describing migration chronology. In addition, measures of food abundance as a proximate factor influencing timing of migration are lacking in studies of migration chronology. To address the relationship between environmental variables and timing of migration, we quantified the relative importance of proximate external factors on migration chronology of local American woodcock (Scolopax minor), a short distance migrant, using event-time analysis methods (survival analysis). We captured 1,094 woodcock local to our study sites in Michigan, Minnesota, and Wisconsin (USA) during autumn 2002–2004 and documented 786 departure dates for these birds. Photoperiod appeared to provide an initial proximate cue for timing of departure. Moon phase was important in modifying timing of departure, which may serve as a navigational aid in piloting and possibly orientation. Local synoptic weather variables also contributed to timing of departure by changing the rate of departure from our study sites. We found no evidence that food availability influenced timing of woodcock departure. Our results suggest that woodcock use a conservative photoperiod-controlled strategy with proximate modifiers for timing of migration rather than relying on abundance of their primary food, earthworms. Managing harvest pressure on local birds by adjusting season lengths may be an effective management tool with consistent migration patterns from year to year based on photoperiod.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.     

Cormack–Jolly–Seber model with environmental covariates: A P‐spline approach

In capture–recapture models, survival and capture probabilities can be modelled as functions of time‐varying covariates, such as temperature or rainfall. The Cormack–Jolly–Seber (CJS) model allows for flexible modelling of these covariates; however, the functional relationship may not be linear. We extend the CJS model by semi‐parametrically modelling capture and survival probabilities using a frequentist approach via P‐splines techniques. We investigate the performance of the estimators by conducting simulation studies. We also apply and compare these models with known semi‐parametric Bayesian approaches on simulated and real data sets.     

Estimation with correlated censored survival data with missing covariates

Incomplete covariate data are a common occurrence in studies in which the outcome is survival time. Further, studies in the health sciences often give rise to correlated, possibly censored, survival data. With no missing covariate data, if the marginal distributions of the correlated survival times follow a given parametric model, then the estimates using the maximum likelihood estimating equations, naively treating the correlated survival times as independent, give consistent estimates of the relative risk parameters Lipsitz et al. 1994 50, 842-846. Now, suppose that some observations within a cluster have some missing covariates. We show in this paper that if one naively treats observations within a cluster as independent, that one can still use the maximum likelihood estimating equations to obtain consistent estimates of the relative risk parameters. This method requires the estimation of the parameters of the distribution of the covariates. We present results from a clinical trial Lipsitz and Ibrahim (1996b) 2, 5-14 with five covariates, four of which have some missing values. In the trial, the clusters are the hospitals in which the patients were treated.     

Flexible parametric model for survival data subject to dependent censoring

When modeling survival data, it is common to assume that the (log-transformed) survival time (T) is conditionally independent of the (log-transformed) censoring time (C) given a set of covariates. There are numerous situations in which this assumption is not realistic, and a number of correction procedures have been developed for different models. However, in most cases, either some prior knowledge about the association between T and C is required, or some auxiliary information or data is/are supposed to be available. When this is not the case, the application of many existing methods turns out to be limited. The goal of this paper is to overcome this problem by developing a flexible parametric model, that is a type of transformed linear model. We show that the association between T and C is identifiable in this model. The performance of the proposed method is investigated both in an asymptotic way and through finite sample simulations. We also develop a formal goodness-of-fit test approach to assess the quality of the fitted model. Finally, the approach is applied to data coming from a study on liver transplants.     

Effect of weather conditions on the spring migration of Eurasian Woodcock and consequences for breeding

Migration is a critical period of time with fitness consequences for birds. The development of tracking technologies now allows researchers to examine how different aspects of bird migration affect population dynamics. Weather conditions experienced during migration are expected to influence movements and, subsequently, the timing of arrival and the energetic costs involved. We analysed satellite‐tracking data from 68 Eurasian Woodcock Scolopax rusticola fitted with Argos satellite tags in the British Isles and France (2012–17). First, we evaluated the effect of weather conditions (temperature, humidity, wind speed and direction, atmospheric stability and visibility) on migration movements of individuals. Then we investigated the consequences for breeding success (age ratio) and brood precocity (early‐brood ratio) population‐level indices while accounting for climatic variables on the breeding grounds. Air temperature, wind and relative humidity were the main variables related to migration movements, with high temperatures and northward winds greatly increasing the probability of onward flights, whereas a trend towards greater humidity over 4 days decreased the probability of movement. Breeding success was mostly affected by climatic variables on the breeding grounds. The proportion of juveniles in autumn was negatively correlated with temperature in May, but positively correlated with precipitation in June and July. Brood precocity was poorly explained by the covariates used in this study. Our data for the Eurasian Woodcock indicate that, although weather conditions during spring migration affect migration movements, they do not have a major influence on subsequent breeding success.     

Inference for set-based effects in genetic association studies with interval-censored outcomes

The rapid acceleration of genetic data collection in biomedical settings has recently resulted in the rise of genetic compendiums filled with rich longitudinal disease data. One common feature of these data sets is their plethora of interval-censored outcomes. However, very few tools are available for the analysis of genetic data sets with interval-censored outcomes, and in particular, there is a lack of methodology available for set-based inference. Set-based inference is used to associate a gene, biological pathway, or other genetic construct with outcomes and is one of the most popular strategies in genetics research. This work develops three such tests for interval-censored settings beginning with a variance components test for interval-censored outcomes, the interval-censored sequence kernel association test (ICSKAT). We also provide the interval-censored version of the Burden test, and then we integrate ICSKAT and Burden to construct the interval censored sequence kernel association test—optimal (ICSKATO) combination. These tests unlock set-based analysis of interval-censored data sets with analogs of three highly popular set-based tools commonly applied to continuous and binary outcomes. Simulation studies illustrate the advantages of the developed methods over ad hoc alternatives, including protection of the type I error rate at very low levels and increased power. The proposed approaches are applied to the investigation that motivated this study, an examination of the genes associated with bone mineral density deficiency and fracture risk.     

Effects of water discharge on fledging time,growth, and survival of piping plovers on the Missouri River

River flow management and modification is a global issue, and its effects on river-dependent organisms are pervasive. Flow modification can directly affect avian species through mortality or habitat loss, but less is known about indirect and sublethal effects of flow modification on reproductive output in these species. Young birds are more vulnerable to predation between hatching and fledging than after flight is achieved, but tradeoffs must be made to balance growth and survival. Predation pressure appears to be a significant factor affecting the time to fledging in altricial birds, but less is known about this threat for precocial birds. Birds reaching fledging earlier should have greater rates of survival to migration because their predator escape repertoire includes flight at an earlier age. We evaluated the effect of varying outflows from the Gavins Point Dam on the growth, age at fledging, and survival of piping plover (Charadrius melodus) chicks on the Missouri River (2006–2009). The study was characterized by 2 relatively high flow years (2006 and 2009) and 2 relatively low flow years (2007 and 2008). We used success rate in recapturing chicks in capture–mark–recapture models as an index for fledging. We attempted to recapture all chicks (n = 1,099) by hand every 3–4 days throughout the season to acquire morphological measurements. Models indicated that as flows from the dam increased, age at fledging increased. We also found that increasing flows were associated with decreasing daily survival rates (β_flow = −2.401, 95% CI: −4.351 to −0.452). Flow was also negatively related to chick mass gain, but we found less evidence for an effect on wing-chord length. Increased flows covered wet-substrate foraging habitat, and likely affected plover reproductive output directly through chick survival and indirectly through decreased growth and increased fledging times. © The Wildlife Society, 2013     

Generalized Rank Transformations for Tests of Survival

The asymptotic equivalence of nonparametric tests and parametric tests based on rank-transformed data (CONOVER and IMAN , 1981) can be extended to the case of censoring. This paper presents generalized rank transformations for analyses of censored data, of interval-censored data and of survival data with uncertain causes of death. A Monte Carlo study and an analysis of leukemia remission times demonstrate excellent agreement of suggested procedures with GEHAN 'S (1965) and PRENTICE 'S (1978) tests.     

Statistical Analysis of Illness–Death Processes and Semicompeting Risks Data

Summary In many instances, a subject can experience both a nonterminal and terminal event where the terminal event (e.g., death) censors the nonterminal event (e.g., relapse) but not vice versa. Typically, the two events are correlated. This situation has been termed semicompeting risks (e.g., Fine, Jiang, and Chappell, 2001 , Biometrika 88, 907–939; Wang, 2003 , Journal of the Royal Statistical Society, Series B 65, 257–273), and analysis has been based on a joint survival function of two event times over the positive quadrant but with observation restricted to the upper wedge. Implicitly, this approach entertains the idea of latent failure times and leads to discussion of a marginal distribution of the nonterminal event that is not grounded in reality. We argue that, similar to models for competing risks, latent failure times should generally be avoided in modeling such data. We note that semicompeting risks have more classically been described as an illness–death model and this formulation avoids any reference to latent times. We consider an illness–death model with shared frailty, which in its most restrictive form is identical to the semicompeting risks model that has been proposed and analyzed, but that allows for many generalizations and the simple incorporation of covariates. Nonparametric maximum likelihood estimation is used for inference and resulting estimates for the correlation parameter are compared with other proposed approaches. Asymptotic properties, simulations studies, and application to a randomized clinical trial in nasopharyngeal cancer evaluate and illustrate the methods. A simple and fast algorithm is developed for its numerical implementation.     

Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data

In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum–Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum‐likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real‐world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.