Model of hidden heterogeneity in longitudinal data

AFT survival model to capture the rate of aging and age-specific mortality trajectories among first-allogeneic hematopoietic stem cells transplant patients

Accelerated failure time (AFT) model is commonly applied in engineering studies to address the failure rate of a machine. In humans, survival profile of transplant patients is among the rare scenarios whereby AFT is applicable. To date, it is uncertain whether reliable risk estimates and age-specific mortality trajectories have been published using conventional statistics approach. By investigating mortality trajectory, the rate of aging d(log(μ(x)))/dx of Hematopoietic Stem Cells Transplants (HSCTs) patients who had underwent first-allogeneic transplants can be obtained, and to unveil the possibility of elasticity of human aging rate in HSCTs. A modified parametric frailty survival model was introduced to the survival profiles of 11,160 patients who had underwent first-allogeneic HSCTs in the United States between 1995 and 2006; data was shared by Center for International Bone and Marrow Transplant Research. In comparison to stratification, the modification permits two entities in relation to time to be presented; age and calendar time. To consider its application in empirical studies, the data contains arbitrary right-censoring, a statistical condition which is preferred by choice in many transplant studies. The finalized multivariate AFT model was adjusted for clinical and demographic covariates, and age-specific mortality trajectories were presented by donor source and post-transplant time-lapse intervals. Two unexpected findings are presented: i) an inverse J-shaped hazard in unrelated donor-source t≤100-day; ii) convergence of unrelated-related hazard lines in 100-day365-day) must consider for periodic medical improvements, and transplant year as a standalone time-variable is not sufficient for statistical adjustment in the finalized multivariate model. In relevance to clinical studies, biennial event-history analysis and age-specific mortality trajectories of long-term survivors provide a more relevant intervention audit report for transplant protocols than the popular statistical presentation; i.e. survival probabilities among donor-source.

A genetic stochastic process model for genome-wide joint analysis of biomarker dynamics and disease susceptibility with longitudinal data

Unraveling the underlying biological mechanisms or pathways behind the effects of genetic variations on complex diseases remains one of the major challenges in the post-GWAS (where GWAS is genome-wide association study) era. To further explore the relationship between genetic variations, biomarkers, and diseases for elucidating underlying pathological mechanism, a huge effort has been placed on examining pleiotropic and gene-environmental interaction effects. We propose a novel genetic stochastic process model (GSPM) that can be applied to GWAS and jointly investigate the genetic effects on longitudinally measured biomarkers and risks of diseases. This model is characterized by more profound biological interpretation and takes into account the dynamics of biomarkers during follow-up when investigating the hazards of a disease. We illustrate the rationale and evaluate the performance of the proposed model through two GWAS. One is to detect single nucleotide polymorphisms (SNPs) having interaction effects on type 2 diabetes (T2D) with body mass index (BMI) and the other is to detect SNPs affecting the optimal BMI level for protecting from T2D. We identified multiple SNPs that showed interaction effects with BMI on T2D, including a novel SNP rs11757677 in the CDKAL1 gene (P = 5.77 × 10-7 ). We also found a SNP rs1551133 located on 2q14.2 that reversed the effect of BMI on T2D (P = 6.70 × 10-7 ). In conclusion, the proposed GSPM provides a promising and useful tool in GWAS of longitudinal data for interrogating pleiotropic and interaction effects to gain more insights into the relationship between genes, quantitative biomarkers, and risks of complex diseases.

stpm: an R package for stochastic process model

Background

The Stochastic Process Model (SPM) represents a general framework for modeling the joint evolution of repeatedly measured variables and time-to-event outcomes observed in longitudinal studies, i.e., SPM relates the stochastic dynamics of variables (e.g., physiological or biological measures) with the probabilities of end points (e.g., death or system failure). SPM is applicable for analyses of longitudinal data in many research areas; however, there are no publicly available software tools that implement this methodology.

Results

We developed an R package stpm for the SPM-methodology. The package estimates several versions of SPM currently available in the literature including discrete- and continuous-time multidimensional models and a one-dimensional model with time-dependent parameters. Also, the package provides tools for simulation and projection of individual trajectories and hazard functions.

Conclusion

In this paper, we present the first software implementation of the SPM-methodology by providing an R package stpm, which was verified through extensive simulation and validation studies. Future work includes further improvements of the model. Clinical and academic researchers will benefit from using the presented model and software. The R package stpm is available as open source software from the following links: https://cran.r-project.org/package=stpm(stable version) or https://github.com/izhbannikov/spm(developer version).

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-017-1538-7) contains supplementary material, which is available to authorized users.

How Genes Modulate Patterns of Aging-Related Changes on the Way to 100: Biodemographic Models and Methods in Genetic Analyses of Longitudinal Data

BACKGROUND AND OBJECTIVE

To clarify mechanisms of genetic regulation of human aging and longevity traits, a number of genome-wide association studies (GWAS) of these traits have been performed. However, the results of these analyses did not meet expectations of the researchers. Most detected genetic associations have not reached a genome-wide level of statistical significance, and suffered from the lack of replication in the studies of independent populations. The reasons for slow progress in this research area include low efficiency of statistical methods used in data analyses, genetic heterogeneity of aging and longevity related traits, possibility of pleiotropic (e.g., age dependent) effects of genetic variants on such traits, underestimation of the effects of (i) mortality selection in genetically heterogeneous cohorts, (ii) external factors and differences in genetic backgrounds of individuals in the populations under study, the weakness of conceptual biological framework that does not fully account for above mentioned factors. One more limitation of conducted studies is that they did not fully realize the potential of longitudinal data that allow for evaluating how genetic influences on life span are mediated by physiological variables and other biomarkers during the life course. The objective of this paper is to address these issues.

DATA AND METHODS

We performed GWAS of human life span using different subsets of data from the original Framingham Heart Study cohort corresponding to different quality control (QC) procedures and used one subset of selected genetic variants for further analyses. We used simulation study to show that approach to combining data improves the quality of GWAS. We used FHS longitudinal data to compare average age trajectories of physiological variables in carriers and non-carriers of selected genetic variants. We used stochastic process model of human mortality and aging to investigate genetic influence on hidden biomarkers of aging and on dynamic interaction between aging and longevity. We investigated properties of genes related to selected variants and their roles in signaling and metabolic pathways.

RESULTS

We showed that the use of different QC procedures results in different sets of genetic variants associated with life span. We selected 24 genetic variants negatively associated with life span. We showed that the joint analyses of genetic data at the time of bio-specimen collection and follow up data substantially improved significance of associations of selected 24 SNPs with life span. We also showed that aging related changes in physiological variables and in hidden biomarkers of aging differ for the groups of carriers and non-carriers of selected variants.

CONCLUSIONS

. The results of these analyses demonstrated benefits of using biodemographic models and methods in genetic association studies of these traits. Our findings showed that the absence of a large number of genetic variants with deleterious effects may make substantial contribution to exceptional longevity. These effects are dynamically mediated by a number of physiological variables and hidden biomarkers of aging. The results of these research demonstrated benefits of using integrative statistical models of mortality risks in genetic studies of human aging and longevity.

Examining mortality risk and rate of ageing among Polish Olympic athletes: a survival follow-up from 1924 to 2012

OBJECTIVES

Population-based studies have shown that an active lifestyle reduces mortality risk. Therefore, it has been a longstanding belief that individuals who engage in frequent exercise will experience a slower rate of ageing. It is uncertain whether this widely-accepted assumption holds for intense wear-and-tear. Here, using the 88 years survival follow-up data of Polish Olympic athletes, we report for the first time on whether frequent exercise alters the rate of ageing.

DESIGN

Longitudinal survival data of male elite Polish athletes who participated in the Olympic Games from year 1924 to 2010 were used. Deaths occurring before the end of World War II were excluded for reliable estimates.

SETTING AND PARTICIPANTS

Recruited male elite athletes N=1273 were preassigned to two categorical birth cohorts--Cohort I 1890-1919; Cohort II 1920-1959--and a parametric frailty survival analysis was conducted. An event-history analysis was also conducted to adjust for medical improvements from year 1920 onwards: Cohort II.

RESULTS

Our findings suggest (1) in Cohort I, for every threefold reduction in mortality risk, the rate of ageing decelerates by 1%; (2) socioeconomic transitions and interventions contribute to a reduction in mortality risk of 29% for the general population and 50% for Olympic athletes; (3) an optimum benefit gained for reducing the rate of ageing from competitive sports (Cohort I 0.086 (95% CI 0.047 to 0.157) and Cohort II 0.085 (95% CI 0.050 to 0.144)).

CONCLUSIONS

This study further suggests that intensive physical training during youth should be considered as a factor to improve ageing and mortality risk parameters.

Joint Analyses of Longitudinal and Time-to-Event Data in Research on Aging: Implications for Predicting Health and Survival

Longitudinal data on aging, health, and longevity provide a wealth of information to investigate different aspects of the processes of aging and development of diseases leading to death. Statistical methods aimed at analyses of time-to-event data jointly with longitudinal measurements became known as the "joint models" (JM). An important point to consider in analyses of such data in the context of studies on aging, health, and longevity is how to incorporate knowledge and theories about mechanisms and regularities of aging-related changes that accumulate in the research field into respective analytic approaches. In the absence of specific observations of longitudinal dynamics of relevant biomarkers manifesting such mechanisms and regularities, traditional approaches have a rather limited utility to estimate respective parameters that can be meaningfully interpreted from the biological point of view. A conceptual analytic framework for these purposes, the stochastic process model of aging (SPM), has been recently developed in the biodemographic literature. It incorporates available knowledge about mechanisms of aging-related changes, which may be hidden in the individual longitudinal trajectories of physiological variables and this allows for analyzing their indirect impact on risks of diseases and death. Despite, essentially, serving similar purposes, JM and SPM developed in parallel in different disciplines with very limited cross-referencing. Although there were several publications separately reviewing these two approaches, there were no publications presenting both these approaches in some detail. Here, we overview both approaches jointly and provide some new modifications of SPM. We discuss the use of stochastic processes to capture biological variation and heterogeneity in longitudinal patterns and important and promising (but still largely underused) applications of JM and SPM to predictions of individual and population mortality and health-related outcomes.

The quadratic hazard model for analyzing longitudinal data on aging, health, and the life span

A better understanding of processes and mechanisms linking human aging with changes in health status and survival requires methods capable of analyzing new data that take into account knowledge about these processes accumulated in the field. In this paper, we describe an approach to analyses of longitudinal data based on the use of stochastic process models of human aging, health, and longevity which allows for incorporating state of the art advances in aging research into the model structure. In particular, the model incorporates the notions of resistance to stresses, adaptive capacity, and "optimal" (normal) physiological states. To capture the effects of exposure to persistent external disturbances, the notions of allostatic adaptation and allostatic load are introduced. These notions facilitate the description and explanation of deviations of individuals' physiological indices from their normal states, which increase the chances of disease development and death. The model provides a convenient conceptual framework for comprehensive systemic analyses of aging-related changes in humans using longitudinal data and linking these changes with genotyping profiles, morbidity, and mortality risks. The model is used for developing new statistical methods for analyzing longitudinal data on aging, health, and longevity.

New stochastic carcinogenesis model with covariates: an approach involving intracellular barrier mechanisms

In this paper we present a new multiple-pathway stochastic model of carcinogenesis with potential of predicting individual incidence risks on the basis of biomedical measurements. The model incorporates the concept of intracellular barrier mechanisms in which cell malignization occurs due to an inefficient operation of barrier cell mechanisms, such as antioxidant defense, repair systems, and apoptosis. Mathematical formalism combines methodological innovations of mechanistic carcinogenesis models and stochastic process models widely used in studying biodemography of aging and longevity. An advantage of the modeling approach is in the natural combining of two types of measures expressed in terms of model parameters: age-specific hazard rate and means of barrier states. Results of simulation studies allow us to conclude that the model parameters can be estimated in joint analyses of epidemiological data and newly collected data on individual biomolecular measurements of barrier states. Respective experimental designs for such measurements are suggested and discussed. An analytical solution is obtained for the simplest design when only age-specific incidence rates are observed. Detailed comparison with TSCE model reveals advantages of the approach such as the possibility to describe decline in risk at advanced ages, possibilities to describe heterogeneous system of intermediate cells, and perspectives for individual prognoses of cancer risks. Application of the results to fit the SEER data on cancer risks demonstrates a strong predictive power of the model. Further generalizations of the model, opportunities to measure barrier systems, biomedical and mathematical aspects of the new model are discussed.

Individual heterogeneity and senescence in silvereyes on Heron Island

Individual heterogeneity and correlations between life history traits play a fundamental role in life history evolution and population dynamics. Unobserved individual heterogeneity in survival can be a nuisance for estimation of age effects at the individual level by causing bias due to mortality selection. We jointly analyze survival and breeding output from successful breeding attempts in an island population of Silvereyes (Zosterops lateralis chlorocephalus) by fitting models that incorporate age effects and individual heterogeneity via random effects. The number of offspring produced increased with age of parents in their first years of life but then eventually declined with age. A similar pattern was found for the probability of successful breeding. Annual survival declined with age even when individual heterogeneity was not accounted for. The rate of senescence in survival, however, depends on the variance of individual heterogeneity and vice versa; hence, both cannot be simultaneously estimated with precision. Model selection supported individual heterogeneity in breeding performance, but we found no correlation between individual heterogeneity in survival and breeding performance. We argue that individual random effects, unless unambiguously identified, should be treated as statistical nuisance or taken as a starting point in a search for mechanisms rather than given direct biological interpretation.

Modeling hematopoietic system response caused by chronic exposure to ionizing radiation

A new model of the hematopoietic system response in humans chronically exposed to ionizing radiation describes the dynamics of the hematopoietic stem cell compartment as well as the dynamics of each of the four blood cell types (lymphocytes, neutrophiles, erythrocytes, and platelets). The required model parameters were estimated based on available results of human and experimental animal studies. They include the steady-state number of hematopoietic stem cells and peripheral blood cell lines in an unexposed organism, amplification parameters for each blood line, parameters describing proliferation and apoptosis, parameters of feedback functions regulating the steady-state numbers, and characteristics of radiosensitivity related to cell death and non-lethal cell damage. The model predictions were tested using data on hematological measurements (e.g., blood counts) performed in 1950-1956 in the Techa River residents chronically exposed to ionizing radiation since 1949. The suggested model of hematopoiesis is capable of describing experimental findings in the Techa River Cohort, including: (1) slopes of the dose-effect curves reflecting the inhibition of hematopoiesis due to chronic ionizing radiation, (2) delay in effect of chronic exposure and accumulated character of the effect, and (3) dose-rate patterns for different cytopenic states (e.g., leukopenia, thrombocytopenia).

Dynamic determinants of longevity and exceptional health

It is well known from epidemiology that values of indices describing physiological state in a given age may influence human morbidity and mortality risks. Studies of connection between aging and life span suggest a possibility that dynamic properties of age trajectories of the physiological indices could also be important contributors to morbidity and mortality risks. In this paper we use data on longitudinal changes in body mass index, diastolic blood pressure, pulse pressure, pulse rate, blood glucose, hematocrit, and serum cholesterol in the Framingham Heart Study participants, to investigate this possibility in depth. We found that some of the variables describing individual dynamics of the age-associated changes in physiological indices influence human longevity and exceptional health more substantially than the variables describing physiological state. These newly identified variables are promising targets for prevention aiming to postpone onsets of common elderly diseases and increase longevity.

Dynamic heterogeneity and life histories

Biodemography is increasingly focused on the large and persistent differences between individuals within populations in fitness components (age at death, reproductive success) and fitness-related components (health, biomarkers) in humans and other species. To study such variation we propose the use of dynamic models of observable phenotypes of individuals. Phenotypic change in turn determines variation among individuals in their fitness components over the life course. We refer to this dynamic accumulation of fitness differences as dynamic heterogeneity and illustrate it for an animal population in which longitudinal data are studied using multistate capture-mark-recapture models. Although our approach can be applied to any characteristic, for our empirical example we use reproduction as the phenotypic character to define stages. We indicate how our stage-structured model describes the nature of the variation among individual characteristics that is generated by dynamic heterogeneity. We conclude by discussing our ongoing and planned work on animals and humans. We also discuss the connections between our work and recent work on human mortality, disability and health, and life course theory.

Genetic model for longitudinal studies of aging, health, and longevity and its potential application to incomplete data

Many longitudinal studies of aging collect genetic information only for a sub-sample of participants of the study. These data also do not include recent findings, new ideas and methodological concepts developed by distinct groups of researchers. The formal statistical analyses of genetic data ignore this additional information and therefore cannot utilize the entire research potential of the data. In this paper, we present a stochastic model for studying such longitudinal data in joint analyses of genetic and non-genetic sub-samples. The model incorporates several major concepts of aging known to date and usually studied independently. These include age-specific physiological norms, allostasis and allostatic load, stochasticity, and decline in stress resistance and adaptive capacity with age. The approach allows for studying all these concepts in their mutual connection, even if respective mechanisms are not directly measured in data (which is typical for longitudinal data available to date). The model takes into account dependence of longitudinal indices and hazard rates on genetic markers and permits evaluation of all these characteristics for carriers of different alleles (genotypes) to address questions concerning genetic influence on aging-related characteristics. The method is based on extracting genetic information from the entire sample of longitudinal data consisting of genetic and non-genetic sub-samples. Thus it results in a substantial increase in the accuracy of statistical estimates of genetic parameters compared to methods that use only information from a genetic sub-sample. Such an increase is achieved without collecting additional genetic data. Simulation studies illustrate the increase in the accuracy in different scenarios for datasets structurally similar to the Framingham Heart Study. Possible applications of the model and its further generalizations are discussed.

Cumulative deficits and physiological indices as predictors of mortality and long life

We evaluated the predictive potential for long-term (24-year) survival and longevity (85+ years) of an index of cumulative deficits (DI) and six physiological indices (pulse pressure, diastolic blood pressure, pulse rate, serum cholesterol, blood glucose, and hematocrit) measured in mid- to late life (44-88 years) for participants of the 9th and 14th Framingham Heart Study examinations. For all ages combined, the DI, pulse pressure, and blood glucose are the strongest determinants of both long-term survival and longevity, contributing cumulatively to their explanation. Diastolic blood pressure and hematocrit are less significant determinants of both of these outcomes. The pulse rate is more relevant to survival, whereas serum cholesterol is more relevant to longevity. Only the DI is a significant predictor of longevity and mortality for each 5-year age group ranging from 45 to 85 years. The DI appears to be a more important determinant of long-term risks of death and longevity than are the physiological indices.