Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture

Shared random effects joint models are becoming increasingly popular for investigating the relationship between longitudinal and time‐to‐event data. Although appealing, such complex models are computationally intensive, and quick, approximate methods may provide a reasonable alternative. In this paper, we first compare the shared random effects model with two approximate approaches: a naïve proportional hazards model with time‐dependent covariate and a two‐stage joint model, which uses plug‐in estimates of the fitted values from a longitudinal analysis as covariates in a survival model. We show that the approximate approaches should be avoided since they can severely underestimate any association between the current underlying longitudinal value and the event hazard. We present classical and Bayesian implementations of the shared random effects model and highlight the advantages of the latter for making predictions. We then apply the models described to a study of abdominal aortic aneurysms (AAA) to investigate the association between AAA diameter and the hazard of AAA rupture. Out‐of‐sample predictions of future AAA growth and hazard of rupture are derived from Bayesian posterior predictive distributions, which are easily calculated within an MCMC framework. Finally, using a multivariate survival sub‐model we show that underlying diameter rather than the rate of growth is the most important predictor of AAA rupture.     

Nonparametric association analysis of bivariate left‐truncated competing risks data

We develop time‐varying association analyses for onset ages of two lung infections to address the statistical challenges in utilizing registry data where onset ages are left‐truncated by ages of entry and competing‐risk censored by deaths. Two types of association estimators are proposed based on conditional cause‐specific hazard function and cumulative incidence function that are adapted from unconditional quantities to handle left truncation. Asymptotic properties of the estimators are established by using the empirical process techniques. Our simulation study shows that the estimators perform well with moderate sample sizes. We apply our methods to the Cystic Fibrosis Foundation Registry data to study the relationship between onset ages of Pseudomonas aeruginosa and Staphylococcus aureus infections.     

A note on nonparametric quantile inference for competing risks and more complex multistate models

Nonparametric quantile inference for competing risks has recentlybeen studied by Peng & Fine (2007). Their key result establishesuniform consistency and weak convergence of the inverse of theAalen–Johansen estimator of the cumulative incidence function,using the representation of the cumulative incidence estimatoras a sum of independent and identically distributed random variables.The limit process is of a form similar to that of the standardsurvival result, but with the cause-specific hazard of interestreplacing the all-causes hazard. We show that this fact is nota coincidence, but can be derived from a general Hadamard differentiationresult. We discuss a simplified proof and extensions of theapproach to more complex multistate models. As a further consequence,we find that the bootstrap works.     

Hazard ratings of pine forests to a pine wilt disease at two spatial scales (individual trees and stands) using self-organizing map and random forest

Pine wilt disease (PWD) caused by the pine wood nematode is the most serious global threat to pine forests. Hazard ratings of trees and forests to pest attacks provide important information to efficiently identify current or future hazardous conditions. However, in spite of the importance of hazard ratings for managing PWD, there are few studies on hazard ratings in this system. In this study, we evaluated the hazard ratings of pine trees and pine stands to PWD by considering environmental factors at the level of the stand and the individual tree. Our results showed that trees with larger diameter at breast height (DBH) showed a higher risk rate than those with smaller DBH, indicating that large trees have an increased probability of exposure to vector beetles because they are tall and have a large crown volume. We also found that reduced tree vigour could be related to susceptibility to PWD. In pine stands, geographical factors showed a high correlation with the occurrence of PWD. PWD occurrence was rare at high altitudes, but was more common on steep and south-facing slopes. These patterns were consistently observed in the results from 2 computational approaches: self-organizing map (SOM) and random forest models. The combination of SOM and random forest was effective to extract ecological information from the dataset. The SOM efficiently characterized relations among variables, and the random forest model was effective at predicting ecological variables, including the hazard rating of trees to disturbances.     

The Analysis of Association Between Traits When Differences Between Trait States Matter

Because of their elementary significance in almost all fields of science, measures of association between two variables or traits are abundant and multiform. One aspect of association that is of considerable interest, especially in population genetics and ecology, seems to be widely ignored. This aspect concerns association between complex traits that show variable and arbitrarily defined state differences. Among such traits are genetic characters controlled by many and potentially polyploid loci, species characteristics, and environmental variables, all of which may be mutually and asymmetrically associated. A concept of directed association of one trait with another is developed here that relies solely on difference measures between the states of a trait. Associations are considered at three levels: between individual states of two variables, between an individual state of one variable and the totality of the other variable, and between two variables. Relations to known concepts of association are identified. In particular, measures at the latter two levels turn out to be interpretable as measures of differentiation. Examples are given for areas of application (search for functional relationships, distribution of variation over populations, genomic associations, spatiogenetic structure).     

Hierarchical Bayesian modeling of heterogeneous cluster‐ and subject‐level associations between continuous and binary outcomes in dairy production

The augmentation of categorical outcomes with underlying Gaussian variables in bivariate generalized mixed effects models has facilitated the joint modeling of continuous and binary response variables. These models typically assume that random effects and residual effects (co)variances are homogeneous across all clusters and subjects, respectively. Motivated by conflicting evidence about the association between performance outcomes in dairy production systems, we consider the situation where these (co)variance parameters may themselves be functions of systematic and/or random effects. We present a hierarchical Bayesian extension of bivariate generalized linear models whereby functions of the (co)variance matrices are specified as linear combinations of fixed and random effects following a square‐root‐free Cholesky reparameterization that ensures necessary positive semidefinite constraints. We test the proposed model by simulation and apply it to the analysis of a dairy cattle data set in which the random herd‐level and residual cow‐level effects (co)variances between a continuous production trait and binary reproduction trait are modeled as functions of fixed management effects and random cluster effects.     

Piecewise exponential survival curves with smooth transitions

Several models of a population survival curve composed of two piecewise exponential distributions are developed. In one formulation the hazard rate changes at a point that is an unobservable random variable that varies between individuals. The population hazard function may decrease with age even when all individuals' hazards are increasing. In a second formulation, the population hazard function is modeled directly. Several models are fit to the survival history of a cohort of 5751 highly inbred male Drosophila melanogaster and the British coal mining disaster data.     

A statistical model for red blood cell survival

A statistical model for the survival time of red blood cells (RBCs) with a continuous distribution of cell lifespans is presented. The underlying distribution of RBC lifespans is derived from a probability density function with a bathtub-shaped hazard curve, and accounts for death of RBCs due to senescence (age-dependent increasing hazard rate) and random destruction (constant hazard), as well as for death due to initial or delayed failures and neocytolysis (equivalent to early red cell mortality). The model yields survival times similar to those of previously published studies of RBC survival and is easily amenable to inclusion of drug effects and haemolytic disorders.     

Analysis of a potential trigger of an acute illness

Sometimes certain short-term risk exposures are postulated to act as a trigger for the onset of a specific acute illness. When the incidence of the illness is low it is desirable to investigate this possible association using only data on cases detected during a specific observation period. Here we propose an analysis for such a study based on a model expressed in terms of the probability that the exposure triggers the illness and a random delay from a triggered illness until its diagnosis. Both the natural hazard rate for the illness and the probability that the exposure triggers the illness are assumed to be small and possibly dependent on age and covariates such as sex and duration or severity of the exposure. The method of analysis is illustrated with a study of the association between long flights and hospitalization for venous thromboembolism.     

Climatic and edaphic gradients predict variation in wildland fuel hazard in south-eastern Australia

Understanding spatial variation in wildland fuel is central to predicting wildfire behaviour as well as current and future fire regimes. Vegetation (plant material) – both live (biomass) and dead (necromass) – constitutes most aspects of wildland fuel (hereafter ‘fuel’). It therefore is likely that factors influencing vegetation structure and composition – climate, soils, disturbance – also will influence fuel structure and associated hazard. Nonetheless, these relationships are poorly understood in temperate environments. In this study, we used an extensive database of fuel hazard assessments to determine the extent to which environmental variables (climatic conditions and soil type) and disturbance (fire) can predict fuel hazard in native vegetation across south-eastern Australia. Fuel hazard ratings are based on the horizontal and vertical continuity of fine fuels (dead plant material < 6 mm thick, and live plant material < 3 mm thick) that burn in the flaming front of a fire. These scores are used widely by fire managers in Australia. We used environmental and disturbance variables to develop models to predict spatial patterns of hazard for each fuel stratum (surface, near-surface, elevated and bark) and the height of two fuel strata (near-surface, elevated). Soil, climate and time since fire were strong predictors of fuel hazard for at least one stratum, and soil predictors were the strongest predictors of fuel hazard across all strata. We used models to predict fuel hazard by stratum at a fixed time since fire in two regions with contrasting environments in south-eastern Australia to better understand the spatial arrangement of fuel hazard. Fuel hazard varied within and between regions, emphasising the complexity and heterogeneity of fuel patterns that affect fuel hazard from local to landscape extents. The models improve the basis for analysing fuel hazard patterns and therefore increase the capacity to predict fire regimes under future climates.     

Exposure,hazard, and vulnerability all contribute to Schistosoma haematobium re-infection in northern Senegal

BackgroundInfectious disease risk is driven by three interrelated components: exposure, hazard, and vulnerability. For schistosomiasis, exposure occurs through contact with water, which is often tied to daily activities. Water contact, however, does not imply risk unless the environmental hazard of snails and parasites is also present in the water. By increasing reliance on hazardous activities and environments, socio-economic vulnerability can hinder reductions in exposure to a hazard. We aimed to quantify the contributions of exposure, hazard, and vulnerability to the presence and intensity of Schistosoma haematobium re-infection.Methodology/Principal findingsIn 13 villages along the Senegal River, we collected parasitological data from 821 school-aged children, survey data from 411 households where those children resided, and ecological data from all 24 village water access sites. We fit mixed-effects logistic and negative binomial regressions with indices of exposure, hazard, and vulnerability as explanatory variables of Schistosoma haematobium presence and intensity, respectively, controlling for demographic variables. Using multi-model inference to calculate the relative importance of each component of risk, we found that hazard (Ʃw_{i =} 0.95) was the most important component of S. haematobium presence, followed by vulnerability (Ʃw_i = 0.91). Exposure (Ʃw_i = 1.00) was the most important component of S. haematobium intensity, followed by hazard (Ʃw_i = 0.77). Model averaging quantified associations between each infection outcome and indices of exposure, hazard, and vulnerability, revealing a positive association between hazard and infection presence (OR = 1.49, 95% CI 1.12, 1.97), and a positive association between exposure and infection intensity (RR 2.59–3.86, depending on the category; all 95% CIs above 1)Conclusions/SignificanceOur findings underscore the linkages between social (exposure and vulnerability) and environmental (hazard) processes in the acquisition and accumulation of S. haematobium infection. This approach highlights the importance of implementing both social and environmental interventions to complement mass drug administration.     

Bayesian LASSO,Scale Space and Decision Making in Association Genetics

Background

LASSO is a penalized regression method that facilitates model fitting in situations where there are as many, or even more explanatory variables than observations, and only a few variables are relevant in explaining the data. We focus on the Bayesian version of LASSO and consider four problems that need special attention: (i) controlling false positives, (ii) multiple comparisons, (iii) collinearity among explanatory variables, and (iv) the choice of the tuning parameter that controls the amount of shrinkage and the sparsity of the estimates. The particular application considered is association genetics, where LASSO regression can be used to find links between chromosome locations and phenotypic traits in a biological organism. However, the proposed techniques are relevant also in other contexts where LASSO is used for variable selection.

Results

We separate the true associations from false positives using the posterior distribution of the effects (regression coefficients) provided by Bayesian LASSO. We propose to solve the multiple comparisons problem by using simultaneous inference based on the joint posterior distribution of the effects. Bayesian LASSO also tends to distribute an effect among collinear variables, making detection of an association difficult. We propose to solve this problem by considering not only individual effects but also their functionals (i.e. sums and differences). Finally, whereas in Bayesian LASSO the tuning parameter is often regarded as a random variable, we adopt a scale space view and consider a whole range of fixed tuning parameters, instead. The effect estimates and the associated inference are considered for all tuning parameters in the selected range and the results are visualized with color maps that provide useful insights into data and the association problem considered. The methods are illustrated using two sets of artificial data and one real data set, all representing typical settings in association genetics.     

A semiparametric test to detect associations between quantitative traits and candidate genes in structured populations

MOTIVATION: Although population-based association mapping may be subject to the bias caused by population stratification, alternative methods that are robust to population stratification such as family-based linkage analysis have lower mapping resolution. Recently, various statistical methods robust to population stratification were proposed for association studies, using unrelated individuals to identify associations between candidate genes and traits of interest. The association between a candidate gene and a quantitative trait is often evaluated via a regression model with inferred population structure variables as covariates, where the residual distribution is customarily assumed to be from a symmetric and unimodal parametric family, such as a Gaussian, although this may be inappropriate for the analysis of many real-life datasets. RESULTS: In this article, we proposed a new structured association (SA) test. Our method corrects for continuous population stratification by first deriving population structure and kinship matrices through a set of random genetic markers and then modeling the relationship between trait values, genotypic scores at a candidate marker and genetic background variables through a semiparametric model, where the error distribution is modeled as a mixture of Polya trees centered around a normal family of distributions. We compared our model to the existing SA tests in terms of model fit, type I error rate, power, precision and accuracy by application to a real dataset as well as simulated datasets.     

Alzheimer's disease,oral function and nutritional status

Objectives: To study differences in nutritional, dental status and oral function between institutionalised patients with Alzheimer's disease and cognitively healthy elderly people living in the community. Design: Comparison was made between two groups, Alzheimer's disease sufferers and healthy controls, using established criteria for anthropometric, mental and dental state. Setting: An institution and residential area in Stockholm, Sweden. Subjects: Forty patients with Alzheimer's disease living in a nursing home and 40 age-and gender-matched control subjects living independently. Intervention: Dental status and anthropometric variables. Results: Overnutrition was less frequent among the demented than the controls and more demented were undernourished. Dental status was similar in the two groups with few edentulous subjects but only 2 of 7 edentulous subjects with Alzheimer's disease wore dentures. Having natural teeth and many functional oral zones is important for food consistency choice, but not for nutritional status. In the Alzheimer group, the stage of dementia has a strong association to the ability to eat unaided and an association with dental status. Conclusion: There are differences in nutritional status between Alzheimer's patients in institutions and cognitively healthy elderly living at home. The choice of food consistency is correlated to dental status but nutritional status is not shown to be influenced by dental status. However, the ability to eat unaided is strongly correlated to cognitive status.     

Alzheimer's disease,oral function and nutritional status

Objectives: To study differences in nutritional, dental status and oral function between institutionalised patients with Alzheimer's disease and cognitively healthy elderly people living in the community. Design: Comparison was made between two groups, Alzheimer's disease sufferers and healthy controls, using established criteria for anthropometric, mental and dental state. Setting: An institution and residential area in Stockholm, Sweden. Subjects: Forty patients with Alzheimer's disease living in a nursing home and 40 age-and gender-matched control subjects living independently. Intervention: Dental status and anthropometric variables. Results: Overnutrition was less frequent among the demented than the controls and more demented were undernourished. Dental status was similar in the two groups with few edentulous subjects but only 2 of 7 edentulous subjects with Alzheimer's disease wore dentures. Having natural teeth and many functional oral zones is important for food consistency choice, but not for nutritional status. In the Alzheimer group, the stage of dementia has a strong association to the ability to eat unaided and an association with dental status. Conclusion: There are differences in nutritional status between Alzheimer patients in institutions and cognitively healthy elderly living at home. The choice of food consistency is correlated to dental status but nutritional status is not shown to be influenced by dental status. However, the ability to eat unaided is strongly correlated to cognitive status.     

Bayes's Theorem and the Analysis of Binomial Random Variables

A very practical application of Bayes's theorem, for the analysis of binomial random variables, is presented. Previous papers (Walters, 1985; Walters, 1986a) have already demonstrated the reliability of the technique for one, or two random variables, and the extension of the approach to several random variables is described. Two biometrical examples are used to illustrate the method.     

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

The mixtures of Gompertz random variables (Gompertz , 1825) X₁ and X₂ are identified in terms of relations between the conditional expectation of [exp (αX_2:2) — exp (αX_1:2)]^k given X_1:2 and the hazard rate function of the distribution, k is a positive integer and α < 0. Here X_1:2 and X_2:2 denote the corresponding order statistics. In addition, we also mention some related theorems to characterize the mixtures of Gompertz distributions. Finally, when the sample size is n, the above results are also valid and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems).     

Testing Pairwise Association between Spatially Autocorrelated Variables: A New Approach Using Surrogate Lattice Data

Background

Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson''s r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data.

Methodology/Principal Findings

The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods.

Conclusions/Significance

The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material.     

Description of an approach based on maximum likelihood to adjust an excess hazard model with a random effect

Objective: To adjust an excess hazard regression model with a random effect associated with a geographical level, the Département in France, and compare its parameter estimates with those obtained using a “fixed-effect” excess hazard regression model. Methods: An excess hazard regression model with a piecewise constant baseline hazard was used and a normal distribution was assumed for the random effect. Likelihood maximization was performed using a numerical integration technique, the Quadrature of Gauss–Hermite. Results were obtained with colon-rectum and thyroid cancer data from the French network of cancer registries. Result: The results were in agreement with what was theoretically expected. We showed a greater heterogeneity of the excess hazard in thyroid cancers than in colon-rectum cancers. The hazard ratios for the covariates as estimated with the mixed-effect model were close to those obtained with the fixed-effect model. However, unlike the fixed-effect model, the mixed-effect model allowed the analysis of data with a large number of clusters. The shrinkage estimator associated with Département is an optimal measure of Département-specific excess risk of death and the variance of the random effect gave information on the within-cluster correlation. Conclusion: An excess hazard regression model with random effect can be used for estimating variation in the risk of death due to cancer between many clusters of small sizes.     

Probability of response models and DNA repair: a statistical-biological approach

This paper develops a simple statistical model, the weighted hazard model, which incorporates the toxicological idea of DNA repair and its role in chemical carcinogenesis. We restrict attention to a small segment of DNA that migrates in and out of the high risk states; it is shown that random hazard functions play an important role in the distributional properties of the time to first detectable tumor. Included in the many shapes of the weighted hazard model is one that has a shape in the low dose region similar to that of the probit model, a model that many toxicologists favor. The analyses of two data sets are presented and interpreted, and suggestions for further research are given.