Bayesian inference on age-specific survival for censored and truncated data

1. Traditional estimation of age-specific survival and mortality rates in vertebrates is limited to individuals with known age. Although this subject has been studied extensively using effective capture-recapture and capture-recovery models, inference remains challenging because of large numbers of incomplete records (i.e. unknown age of many individuals) and because of the inadequate duration of the studies. 2. Here, we present a hierarchical model for capture-recapture/recovery (CRR) data sets with large proportions of unknown times of birth and death. The model uses a Bayesian framework to draw inference on population-level age-specific demographic rates using parametric survival functions and applies this information to reconstruct times of birth and death for individuals with unknown age. 3. We simulated a set of CRR data sets with varying study span and proportions of individuals with known age, and varying recapture and recovery probabilities. We used these data sets to compare our method to a traditional CRR model, which requires knowledge of individual ages. Subsequently, we applied our method to a subset of a long-term CRR data set on Soay sheep. 4. Our results show that this method performs better than the common CRR model when sample sizes are low. Still, our model is sensitive to the choice of priors with low recapture probability and short studies. In such cases, priors that overestimate survival perform better than those that underestimate it. Also, the model was able to estimate accurately ages at death for Soay sheep, with an average error of 0.94 years and to identify differences in mortality rate between sexes. 5. Although many of the problems in the estimation of age-specific survival can be reduced through more efficient sampling schemes, most ecological data sets are still sparse and with a large proportion of missing records. Thus, improved sampling needs still to be combined with statistical models capable of overcoming the unavoidable limitations of any fieldwork. We show that our approach provides reliable estimates of parameters and unknown times of birth and death even with the most incomplete data sets while being flexible enough to accommodate multiple recapture probabilities and covariates.     

Modeling Age-Specific Mortality for Countries with Generalized HIV Epidemics

Background

In a given population the age pattern of mortality is an important determinant of total number of deaths, age structure, and through effects on age structure, the number of births and thereby growth. Good mortality models exist for most populations except those experiencing generalized HIV epidemics and some developing country populations. The large number of deaths concentrated at very young and adult ages in HIV-affected populations produce a unique ‘humped’ age pattern of mortality that is not reproduced by any existing mortality models. Both burden of disease reporting and population projection methods require age-specific mortality rates to estimate numbers of deaths and produce plausible age structures. For countries with generalized HIV epidemics these estimates should take into account the future trajectory of HIV prevalence and its effects on age-specific mortality. In this paper we present a parsimonious model of age-specific mortality for countries with generalized HIV/AIDS epidemics.

Methods and Findings

The model represents a vector of age-specific mortality rates as the weighted sum of three independent age-varying components. We derive the age-varying components from a Singular Value Decomposition of the matrix of age-specific mortality rate schedules. The weights are modeled as a function of HIV prevalence and one of three possible sets of inputs: life expectancy at birth, a measure of child mortality, or child mortality with a measure of adult mortality. We calibrate the model with 320 five-year life tables for each sex from the World Population Prospects 2010 revision that come from the 40 countries of the world that have and are experiencing a generalized HIV epidemic. Cross validation shows that the model is able to outperform several existing model life table systems.

Conclusions

We present a flexible, parsimonious model of age-specific mortality for countries with generalized HIV epidemics. Combined with the outputs of existing epidemiological and demographic models, this model makes it possible to project future age-specific mortality profiles and number of deaths for countries with generalized HIV epidemics.     

Bayesian Modeling of Age-Specific Survival in Nesting Studies Under Dirichlet Priors

There has been much work done in nest survival analysis using the maximum likelihood (ML) method. The ML method suffers from the instability of numerical calculations when models having a large number of unknown parameters are used. A Bayesian approach of model fitting is developed to estimate age-specific survival rates for nesting studies using a large class of prior distributions. The computation is done by Gibbs sampling. Some latent variables are introduced to simplify the full conditional distributions. The method is illustrated using both a real and a simulated data set. Results indicate that Bayesian analysis provides stable and accurate estimates of nest survival rates.     

Evolution of a plastic quantitative trait in an age-structured population in a fluctuating environment

We analyze weak fluctuating selection on a quantitative character in an age-structured population not subject to density regulation. We assume that early in the first year of life before selection, during a critical state of development, environments exert a plastic effect on the phenotype, which remains constant throughout the life of an individual. Age-specific selection on the character affects survival and fecundity, which have intermediate optima subject to temporal environmental fluctuations with directional selection in some age classes as special cases. Weighting individuals by their reproductive value, as suggested by Fisher, we show that the expected response per year in the weighted mean character has the same form as for models with no age structure. Environmental stochasticity generates stochastic fluctuations in the weighted mean character following a first-order autoregressive model with a temporally autocorrelated noise term and stationary variance depending on the amount of phenotypic plasticity. The parameters of the process are simple weighted averages of parameters used to describe age-specific survival and fecundity. The "age-specific selective weights" are related to the stable distribution of reproductive values among age classes. This allows partitioning of the change in the weighted mean character into age-specific components.     

Age-specific Effects of Novel Mutations in Drosophila Melanogaster II. Fecundity and Male Mating Ability

Evolutionary theories of senescence assume that mutations with age-specific effects exist, yet until now, there has been little experimental evidence to support this assumption. In this study, we allowed mutations to accumulate in an outbred, wild population of Drosophila melanogaster to test for age-specific differences in both male mating ability and fecundity. We assayed for age-specific effects of mutations after 10, 20, and 30 generations of mutation accumulation. For mating ability, we found the strongest effects of mutations in the first half of the life span after 20 generations, and at nearly all ages by generation 30. These results are qualitatively consistent with results from a companion study in which age-specific mortality was assayed on the same lines of D. melanogaster. By contrast, effects of fecundity were confined to late ages after 20 generations of mutation accumulation, but by generation 30, as with male mating ability, effects of novel mutations were distributed across all age classes. We discuss several possible explanations for the differences that we observe between generations within traits, and among traits, and the relevance for these patterns to models of aging as well as models of mate choice and sexual selection. This revised version was published online in July 2006 with corrections to the Cover Date.     

Social contacts and the locations in which they occur as risk factors for influenza infection

The interaction of human social behaviour and transmission is an intriguing aspect of the life cycle of respiratory viral infections. Although age-specific mixing patterns are often assumed to be the key drivers of the age-specific heterogeneity in transmission, the association between social contacts and biologically confirmed infection has not previously been tested at the individual level. We administered a questionnaire to participants in a longitudinal cohort survey of influenza in which infection was defined by longitudinal paired serology. Using a variety of statistical approaches, we found overwhelming support for the inclusion of individual age in addition to contact variables when explaining odds of infection: the best model not including age explained only 15.7% of the deviance, whereas the best model with age explained 23.6%. However, within age groups, we did observe an association between contacts, locations and infection: median numbers of contacts (or locations) reported by those infected were higher than those from the uninfected group in every age group other than the youngest. Further, we found some support for the retention of location and contact variables in addition to age in our regression models, with excess odds of infection of approximately 10% per additional 10 contacts or one location. These results suggest that, although the relationship between age and incidence of respiratory infection at the level of the individual is not driven by self-reported social contacts, risk within an age group may be.     

Multiple imputation for estimating the risk of developing dementia and its impact on survival

Dementia, Alzheimer's disease in particular, is one of the major causes of disability and decreased quality of life among the elderly and a leading obstacle to successful aging. Given the profound impact on public health, much research has focused on the age-specific risk of developing dementia and the impact on survival. Early work has discussed various methods of estimating age-specific incidence of dementia, among which the illness-death model is popular for modeling disease progression. In this article we use multiple imputation to fit multi-state models for survival data with interval censoring and left truncation. This approach allows semi-Markov models in which survival after dementia depends on onset age. Such models can be used to estimate the cumulative risk of developing dementia in the presence of the competing risk of dementia-free death. Simulations are carried out to examine the performance of the proposed method. Data from the Honolulu Asia Aging Study are analyzed to estimate the age-specific and cumulative risks of dementia and to examine the effect of major risk factors on dementia onset and death.     

Maternal Age-Specific Rates for Trisomy 21 and Common Autosomal Trisomies in Fetuses from a Single Diagnostic Center in Thailand

To provide maternal age-specific rates for trisomy 21 (T21) and common autosomal trisomies (including trisomies 21, 18 and 13) in fetuses. We retrospectively reviewed prenatal cytogenetic results obtained between 1990 and 2009 in Songklanagarind Hospital, a university teaching hospital, in southern Thailand. Maternal age-specific rates of T21 and common autosomal trisomies were established using different regression models, from which only the fittest models were used for the study. A total of 17,819 records were included in the statistical analysis. The fittest models for predicting rates of T21 and common autosomal trisomies were regression models with 2 parameters (Age and Age²). The rate of T21 ranged between 2.67 per 1,000 fetuses at the age of 34 and 71.06 per 1,000 at the age of 48. The rate of common autosomal trisomies ranged between 4.54 per 1,000 and 99.65 per 1,000 at the same ages. This report provides the first maternal age-specific rates for T21 and common autosomal trisomies fetuses in a Southeast Asian population and the largest case number of fetuses have ever been reported in Asians.     

Toward reconciling inferences concerning genetic variation in senescence in Drosophila melanogaster.

Standard models for senescence predict an increase in the additive genetic variance for log mortality rate late in the life cycle. Variance component analysis of age-specific mortality rates of related cohorts is problematic. The actual mortality rates are not observable and can be estimated only crudely at early ages when few individuals are dying and at late ages when most are dead. Therefore, standard quantitative genetic analysis techniques cannot be applied with confidence. We present a novel and rigorous analysis that treats the mortality rates as missing data following two different parametric senescence models. Two recent studies of Drosophila melanogaster, the original analyses of which reached different conclusions, are reanalyzed here. The two-parameter Gompertz model assumes that mortality rates increase exponentially with age. A related but more complex three-parameter logistic model allows for subsequent leveling off in mortality rates at late ages. We find that while additive variance for mortality rates increases for late ages under the Gompertz model, it declines under the logistic model. The results from the two studies are similar, with differences attributable to differences between the experiments.     

Below-threshold mortality: implications for studies in evolution,ecology and demography

Evolutionary biologists, ecologists and experimental gerontologists have increasingly used estimates of age-specific mortality as a critical component in studies of a range of important biological processes. However, the analysis of age-specific mortality rates is plagued by specific statistical challenges caused by sampling error. Here we discuss the nature of this ‘demographic sampling error’, and the way in which it can bias our estimates of (1) rates of ageing, (2) age at onset of senescence, (3) costs of reproduction and (4) demographic tests of evolutionary models of ageing. We conducted simulations which suggest that using standard statistical techniques, we would need sample sizes on the order of tens of thousands in most experiments to effectively remove any bias due to sampling error. We argue that biologists should use much larger sample sizes than have previously been used. However, we also present simple maximum likelihood models that effectively remove biases due to demographic sampling error even at relatively small sample sizes.     

Convergence of body mass with aging: The longitudinal interrelationship of health, weight, and survival

There has been ongoing debate about the health risks associated with increased body weight among the elderly population. One issue has not been investigated thoroughly is that body weight changes over time, as both the reasons and results of, the development of chronic diseases and functional disabilities. Structural models have the ability to unravel the complicated simultaneous relationship between body weight, disability, and mortality along the aging process. Using longitudinal data from the Medicare Current Beneficiary Survey from 1992 to 2001, we constructed a structural model to estimate the longitudinal dynamic relationship between weight, chronic diseases, functional status, and mortality among the aging population. A simulation of an age cohort from 65 to 100 was conducted to show the changes in weight and health outcomes among the cohorts with different baseline weight based on the parameters estimated by the model. The elderly with normal weight at age 65 experience higher life expectancy and lower disability rates than the same age cohorts in other weight categories. The interesting prediction of our model is that the average body size of an elderly cohort will converge to the normal weight range through a process of survival, senescence, and behavioral adjustment.     

Tobacco smoking and sex ratios in the United States

Results of an investigation into the effect of smoking on U.S. age-specific sex ratios indicate that the contribution of smoking to imablances in these ratios in substantial. 1st, the mathematics of the stationary population model is used to analyze the overall trend in age-specific stationary population sex ratios for 1910 and 1962, without reference to smoking. As age-specific and sex-specific mortality rates at the younger ages declined to nearly zero during this period, most male-female differences between those rates also declined. As a result, sex ratios at the younger ages decreased more slowly with age in 1962 than in 1910. Average U.S. annual cigarette consumption per person for ages 15 and over increased from 49 in 1910 to 3958 in 1962. Splitting overall age-specific and sex-specific mortality rates into smoking and nonsmoking components, the analysis is extended to examine the effects of smoking, utilizing age-specific mortality rates derived by linearly interpolating and extrapolating American Cancer Society mortality data. Percent reductions in age-specific sex ratios due to smoking probably represent fairly accurately the true values in the U.S. population in 1962. Reductions in sex ratios at each age from what they would have been if the total population had been nonsmoking varies from 0% at age 35 to 20% at age 85. It is apparent that smoking has a cumulative effect on the sex ratio with increased age.     

Demographic window to aging in the wild: constructing life tables and estimating survival functions from marked individuals of unknown age

Summary We address the problem of establishing a survival schedule for wild populations. A demographic key identity is established, leading to a method whereby age-specific survival and mortality can be deduced from a marked cohort life table established for individuals that are randomly sampled at unknown age and marked, with subsequent recording of time-to-death. This identity permits the construction of life tables from data where the birth date of subjects is unknown. An analogous key identity is established for the continuous case in which the survival schedule of the wild population is related to the density of the survival distribution in the marked cohort. These identities are explored for both life tables and continuous lifetime data. For the continuous case, they are implemented with statistical methods using non-parametric density estimation methods to obtain flexible estimates for the unknown survival distribution of the wild population. The analytical model provided here serves as a starting point to develop more complex models for residual demography, i.e. models for estimating survival of wild populations in which age-at-entry is unknown and using remaining information in randomly encountered individuals. This is a first step towards a broad new concept of 'expressed demographic information content of marked or captured individuals'.     

Morbidity and mortality in acute viral hepatitides in a hyperendemic region. The role of viral hepatitis E

Official annual statistical data on morbidity in acute viral hepatitides (AVH), including the number of lethal cases, for 1985-1995 were analyzed. Mortality rates per 100,000 of the population at the period of 11 years were calculated for different age groups, sex and the place of residence. 396 and 99 patients were examined for the presence of serological markers of hepatitides A, B and E, respectively, at the periods of epidemic rises in morbidity and satisfactory epidemic situation. In the course of 11 years AVH caused the death of 22,405 persons. In 1985-1987 the average mortality level (ML) reached 12.3-17.8 per 100,000 of the population (with morbidity being 1,200-1,400 and was essentially higher among the rural population in comparison with the urban population. During these years the highest ML, was registered among children aged 0-2 years (190-50 per 100,000) and, among adults, mainly among women aged 20-29 years (21.4-19.6 per 100,000). During the years when the epidemic of AVH was absent, ML among these groups was essentially lower: 40-20 among children aged 0-2 years and 4-5 among women aged 20-29 years. In 1987 in the Fergana Valley hepatitis E was detected in 72.2% of all examined patients, and in the southern areas of the country in 68.7%. A sharp rise on mortality among women of the productive age at the period of the epidemic rise of AVH morbidity in the endemic region indicated that this epidemic was linked with hepatitis E. High ML among young children may be indicative of a highly unfavorable course of hepatitis E in the group of infants, which had never been registered before. This newly established regularity may be used for the retrospective diagnostics of the outbreak of hepatitis E.     

An agent-based modelling approach to estimate error in gyrodactylid population growth

Comparative studies of gyrodactylid monogeneans on different host species or strains rely upon the observation of growth on individual fish maintained within a common environment, summarised using maximum likelihood statistical approaches. Here we describe an agent-based model of gyrodactylid population growth, which we use to evaluate errors due to stochastic reproductive variation in such experimental studies. Parameters for the model use available fecundity and mortality data derived from previously published life tables of Gyrodactylus salaris, and use a new data set of fecundity and mortality statistics for this species on the Neva stock of Atlantic salmon, Salmo salar. Mortality data were analysed using a mark-recapture analysis software package, allowing maximum-likelihood estimation of daily survivorship and mortality. We consistently found that a constant age-specific mortality schedule was most appropriate for G. salaris in experimental datasets, with a daily survivorship of 0.84 at 13°C. This, however, gave unrealistically low population growth rates when used as parameters in the model, and a schedule of constantly increasing mortality was chosen as the best compromise for the model. The model also predicted a realistic age structure for the simulated populations, with 0.32 of the population not yet having given birth for the first time (pre-first birth). The model demonstrated that the population growth rate can be a useful parameter for comparing gyrodactylid populations when these are larger than 20-30 individuals, but that stochastic error rendered the parameter unusable in smaller populations. It also showed that the declining parasite population growth rate typically observed during the course of G. salaris infections cannot be explained through stochastic error and must therefore have a biological basis. Finally, the study showed that most gyrodactylid-host studies of this type are too small to detect subtle differences in local adaptation of gyrodactylid monogeneans between fish stocks.     

Beyond the proportional frailty model: Bayesian estimation of individual heterogeneity on mortality parameters

Today, we know that demographic rates can be greatly influenced by differences among individuals in their capacity to survive and reproduce. These intrinsic differences, commonly known as individual heterogeneity, can rarely be measured and are thus treated as latent variables when modeling mortality. Finite mixture models and mixed effects models have been proposed as alternative approaches for inference on individual heterogeneity in mortality. However, in general models assume that individual heterogeneity influences mortality proportionally, which limits the possibility to test hypotheses on the effect of individual heterogeneity on other aspects of mortality such as ageing rates. Here, we propose a Bayesian model that builds upon the mixture models previously developed, but that facilitates making inferences on the effect of individual heterogeneity on mortality parameters other than the baseline mortality. As an illustration, we apply this framework to the Gompertz–Makeham mortality model, commonly used in human and wildlife studies, by assuming that the Gompertz rate parameter is affected by individual heterogeneity. We provide results of a simulation study where we show that the model appropriately retrieves the parameters used for simulation, even for low variances in the heterogeneous parameter. We then apply the model to a dataset on captive chimpanzees and on a cohort life table of 1751 Swedish men, and show how model selection against a null model (i.e., without heterogeneity) can be carried out.     

MODELING AGE-SPECIFIC MORTALITY FOR MARINE MAMMAL POPULATIONS

A method is presented for estimating age-specific mortality based on minimal information: a model life table and an estimate of longevity. This approach uses expected patterns of mammalian survivorship to define a general model of age-specific mortality rates. One such model life table is based on data for northern fur seals (Callorhinus ursinus) using Siler's (1979) 5-parameter competing risk model. Alternative model life tables are based on historical data for human females and on a published model for Old World monkeys. Survival rates for a marine mammal species are then calculated by scaling these models by the longevity of that species. By using a realistic model (instead of assuming constant mortality), one can see more easily the real biological limits to population growth. The mortality estimation procedure is illustrated with examples of spotted dolphins (Stenella attenuata) and harbor porpoise (Phocoena phocoena).     

Mortality from cardiovascular diseases in the German uranium miners cohort study, 1946–1998

An increased risk of cardiovascular diseases after exposure to low doses of ionizing radiation has been suggested among the atomic bomb survivors. Few and inconclusive results on this issue are available from miner studies. A positive correlation between coronary heart disease mortality and radon exposure has been reported in the Newfoundland fluorspar miners study, yet low statistical power due to small sample size was of concern. To get further insight into this controversial issue, data from the German uranium miners cohort study were used, which is by far the largest miner study up to date. The cohort includes 59,001 male subjects who were employed for at least six months between 1946 and 1989 at the former Wismut uranium company in Eastern Germany. Exposure to radon, long-lived radionuclides and external gamma radiation was estimated by using a detailed job-exposure matrix. About 16,598 cohort members were deceased until 31 December 1998, including 5,417 deaths from cardiovascular diseases. Linear Poisson regression models were used to estimate the excess relative risk (ERR) per unit of cumulative radiation exposure after adjusting for attained age and calendar period. No trend in risk of circulatory diseases with increasing cumulative exposure to either radon [ERR per 100 working level month: 0.0006; 95% confidence limit (CI): −0.004 to 0.006], external gamma radiation (ERR per Sv: −0.26, 95% CI: −0.6 to 0.05) or long-lived radionuclides (ERR per 100 kBqh/m³: −0.2, 95% CI: −0.5 to 0.06), respectively, was observed. This was also true for the sub-group heart disease and stroke. Our findings do not support an association between cardiovascular disease mortality and exposure to radiation among miners, yet low doses and uncontrolled confounding hamper interpretation.     

Age-specific mortality rates in reef fishes: Evidence and implications

Abstract Mortality is a fundamental demographic rate, the nature of which has profound consequences for both the dynamics of populations and the life-history evolution of species. For example, if per capita mortality rates are age- or stage-specific, life-history traits should evolve in response to age- and stage-specific differences in selection arising from these temporally variable rates. Similarly, variation in the average mortality rate across ages and/or stages can also select for shifts in life history. Mortality rates of recently settled reef fishes can be very high and per capita mortality is commonly assumed to decrease with increasing age. A review of evidence for age-specific per capita mortality rates in reef fishes from early postsettlement up to 13 months postsettlement suggests that during this period these rates are often age invariant. The data on which these interpretations are based, however, are extremely limited both in terms of the proportion of the life cycle over which mortality rates have been sampled and the quality of these data. Nonetheless, these data do suggest that selective pressures associated with patterns of mortality may vary among species of reef fishes and that these species therefore could be more effectively used in the study of life-history evolution. At present, reef fishes are under-represented in the study of life-history evolution compared with other vertebrate taxa.