Estimation of age of onset and progression of breast cancer by absolute risk dependent on polygenic risk score and other risk factors

BACKGROUND

Genetic, lifestyle, reproductive, and anthropometric factors are associated with the risk of developing breast cancer. However, it is not yet known whether polygenic risk score (PRS) and absolute risk based on a combination of risk factors are associated with the risk of progression of breast cancer. This study aims to estimate the distribution of sojourn time (pre-clinical screen-detectable period) and mammographic sensitivity by absolute breast cancer risk derived from polygenic profile and the other risk factors.

METHODS

The authors used data from a population-based case-control study. Six categories of 10-year absolute risk based on different combinations of risk factors were derived using the Breast and Ovarian Analysis of Disease Incidence and Carrier Estimation Algorithm. Women were classified into low, medium, and high-risk groups. The authors constructed a continuous-time multistate model. To calculate the sojourn time, they simulated the trajectories of subjects through the disease states.

RESULTS

There was little difference in sojourn time with a large overlap in the 95% confidence interval (CI) between the risk groups across the six risk categories and PRS studied. However, the age of entry into the screen-detectable state varied by risk category, with the mean age of entry of 53.4 years (95% CI, 52.2-54.1) and 57.0 years (95% CI, 55.1-57.7) in the high-risk and low-risk women, respectively.

CONCLUSION

In risk-stratified breast screening, the age at the start of screening, but not necessarily the frequency of screening, should be tailored to a woman's risk level. The optimal risk-stratified screening strategy that would improve the benefit-to-harm balance and the cost-effectiveness of the screening programs needs to be studied.

Bivariate joint models for survival and change of cognitive function

Changes in cognitive function over time are of interest in ageing research. A joint model is constructed to investigate. Generally, cognitive function is measured through more than one test, and the test scores are integers. The aim is to investigate two test scores and use an extension of a bivariate binomial distribution to define a new joint model. This bivariate distribution model the correlation between the two test scores. To deal with attrition due to death, the Weibull hazard model and the Gompertz hazard model are used. A shared random-effects model is constructed, and the random effects are assumed to follow a bivariate normal distribution. It is shown how to incorporate random effects that link the bivariate longitudinal model and the survival model. The joint model is applied to the English Longitudinal Study of Ageing data.

A multistate survival model of the natural history of cancer using data from screened and unscreened population

One of the main aims of models using cancer screening data is to determine the time between the onset of preclinical screen-detectable cancer and the onset of the clinical state of the cancer. This time is called the sojourn time. One problem in using screening data is that an individual can be observed in preclinical phase or clinically diagnosed but not both. Multistate survival models provide a method of modeling the natural history of cancer. The natural history model allows for the calculation of the sojourn time. We developed a continuous-time Markov model and the corresponding likelihood function. The model allows for the use of interval-censored, left-truncated and right-censored data. The model uses data of clinically diagnosed cancers from both screened and nonscreened individuals. Parameters of age-varying hazards and age-varying misclassification are estimated simultaneously. The mean sojourn time is calculated from a micro-simulation using model parameters. The model is applied to data from a prostate screening trial. The simulation study showed that the model parameters could be estimated accurately.

The Importance of Engaging in Physical Activity in Older Adulthood for Transitions Between Cognitive Status Categories and Death: A Coordinated Analysis of 14 Longitudinal Studies

BACKGROUND

Given increasing incidence of cognitive impairment and dementia, further understanding of modifiable factors contributing to increased healthspan is crucial. Extensive literature provides evidence that physical activity (PA) delays the onset of cognitive impairment; however, it is unclear whether engaging in PA in older adulthood is sufficient to influence progression through cognitive status categories.

METHOD

Applying a coordinated analysis approach, this project independently analyzed 14 longitudinal studies (NTotal = 52 039; mean baseline age across studies = 69.9-81.73) from North America and Europe using multistate survival models to estimate the impact of engaging in PA on cognitive status transitions (nonimpaired, mildly impaired, severely impaired) and death. Multinomial regression models were fit to estimate life expectancy (LE) based on American PA recommendations. Meta-analyses provided the pooled effect sizes for the role of PA on each transition and estimated LEs.

RESULTS

Controlling for baseline age, sex, education, and chronic conditions, analyses revealed that more PA is significantly associated with decreased risk of transitioning from nonimpaired to mildly impaired cognitive functioning and death, as well as substantially longer LE. Results also provided evidence for a protective effect of PA after onset of cognitive impairment (eg, decreased risk of transitioning from mild-to-severe cognitive impairment; increased likelihood of transitioning backward from severe-to-mild cognitive impairment), though between-study heterogeneity suggests a less robust association.

CONCLUSIONS

These results yield evidence for the importance of engaging in PA in older adulthood for cognitive health, and a rationale for motivating older adults to engage consistently in PA.

Socio-economic inequalities in life expectancy of older adults with and without multimorbidity: a record linkage study of 1.1 million people in England

Background

Age of onset of multimorbidity and its prevalence are well documented. However, its contribution to inequalities in life expectancy has yet to be quantified.

Methods

A cohort of 1.1 million English people aged 45 and older were followed up from 2001 to 2010. Multimorbidity was defined as having 2 or more of 30 major chronic diseases. Multi-state models were used to estimate years spent healthy and with multimorbidity, stratified by sex, smoking status and quintiles of small-area deprivation.

Results

Unequal rates of multimorbidity onset and subsequent survival contributed to higher life expectancy at age 65 for the least (Q1) compared with most (Q5) deprived: there was a 2-year gap in healthy life expectancy for men [Q1: 7.7 years (95% confidence interval: 6.4–8.5) vs Q5: 5.4 (4.4–6.0)] and a 3-year gap for women [Q1: 8.6 (7.5–9.4) vs Q5: 5.9 (4.8–6.4)]; a 1-year gap in life expectancy with multimorbidity for men [Q1: 10.4 (9.9–11.2) vs Q5: 9.1 (8.7–9.6)] but none for women [Q1: 11.6 (11.1–12.4) vs Q5: 11.5 (11.1–12.2)]. Inequalities were attenuated but not fully attributable to socio-economic differences in smoking prevalence: multimorbidity onset was latest for never smokers and subsequent survival was longer for never and ex smokers.

Conclusions

The association between social disadvantage and multimorbidity is complex. By quantifying socio-demographic and smoking-related contributions to multimorbidity onset and subsequent survival, we provide evidence for more equitable allocation of prevention and health-care resources to meet local needs.

Gait speed as predictor of transition into cognitive impairment: Findings from three longitudinal studies on aging

OBJECTIVES

Very few studies looking at slow gait speed as early marker of cognitive decline investigated the competing risk of death. The current study examines associations between slow gait speed and transitions between cognitive states and death in later life.

METHODS

We performed a coordinated analysis of three longitudinal studies with 9 to 25 years of follow-up. Data were used from older adults participating in H70 (Sweden; n = 441; aged ≥70 years), InCHIANTI (Italy; n = 955; aged ≥65 years), and LASA (the Netherlands; n = 2824; aged ≥55 years). Cognitive states were distinguished using the Mini-Mental State Examination. Slow gait speed was defined as the lowest sex-specific quintile at baseline. Multistate models were performed, adjusted for age, sex and education.

RESULTS

Most effect estimates pointed in the same direction, with slow gait speed predicting forward transitions. In two cohort studies, slow gait speed predicted transitioning from mild to severe cognitive impairment (InCHIANTI: HR = 2.08, 95%CI = 1.40-3.07; LASA: HR = 1.33, 95%CI = 1.01-1.75) and transitioning from a cognitively healthy state to death (H70: HR = 3.30, 95%CI = 1.74-6.28; LASA: HR = 1.70, 95%CI = 1.30-2.21).

CONCLUSIONS

Screening for slow gait speed may be useful for identifying older adults at risk of adverse outcomes such as cognitive decline and death. However, once in the stage of more advanced cognitive impairment, slow gait speed does not seem to predict transitioning to death anymore.

Estimation of life expectancies using continuous-time multi-state models

BACKGROUND AND OBJECTIVE

There is increasing interest in multi-state modelling of health-related stochastic processes. Given a fitted multi-state model with one death state, it is possible to estimate state-specific and marginal life expectancies. This paper introduces methods and new software for computing these expectancies.

METHODS

The definition of state-specific life expectancy given current age is an extension of mean survival in standard survival analysis. The computation involves the estimated parameters of a fitted multi-state model, and numerical integration. The new R package elect provides user-friendly functions to do the computation in the R software.

RESULTS

The estimation of life expectancies is explained and illustrated using the elect package. Functions are presented to explore the data, to estimate the life expectancies, and to present results.

CONCLUSIONS

State-specific life expectancies provide a communicable representation of health-related processes. The availability and explanation of the elect package will help researchers to compute life expectancies and to present their findings in an assessable way.

Hidden three-state survival model for bivariate longitudinal count data

A model is presented that describes bivariate longitudinal count data by conditioning on a progressive illness-death process where the two living states are latent. The illness-death process is modelled in continuous time, and the count data are described by a bivariate extension of the binomial distribution. The bivariate distributions for the count data approach include the correlation between two responses even after conditioning on the state. An illustrative data analysis is discussed, where the bivariate data consist of scores on two cognitive tests, and the latent states represent two stages of underlying cognitive function. By including a death state, possible association between cognitive function and the risk of death is accounted for.

Duration of preclinical, prodromal, and dementia stages of Alzheimer's disease in relation to age, sex, and APOE genotype

INTRODUCTION

We estimated the age-specific duration of the preclinical, prodromal, and dementia stages of Alzheimer's disease (AD) and the influence of sex, setting, apolipoprotein E (APOE) genotype, and cerebrospinal fluid tau on disease duration.

METHODS

We performed multistate modeling in a combined sample of 6 cohorts (n = 3268) with death as the end stage and estimated the preclinical, prodromal, and dementia stage duration.

RESULTS

The overall AD duration varied between 24 years (age 60) and 15 years (age 80). For individuals presenting with preclinical AD, age 70, the estimated preclinical AD duration was 10 years, prodromal AD 4 years, and dementia 6 years. Male sex, clinical setting, APOE ε4 allele carriership, and abnormal cerebrospinal fluid tau were associated with a shorter duration, and these effects depended on disease stage.

DISCUSSION

Estimates of AD disease duration become more accurate if age, sex, setting, APOE, and cerebrospinal fluid tau are taken into account. This will be relevant for clinical practice and trial design.

Flexible parametric multistate modelling of employment history

A multistate model is used to describe employment history. Transition-specific rates are defined using generalized gamma distributions and Gompertz distributions. This flexible parametric modelling of the rate of change is combined with latent classes for unobserved propensity to change jobs. The propensity is described by two latent classes which can be interpreted as consisting of movers and stayers. The modelling is illustrated by analysing longitudinal data from the German Life History Study.

Changes in working life expectancy with disability in the Netherlands, 1992-2016

Objectives Like other western countries, the Netherlands has abolished early retirement schemes and is currently increasing the statutory retirement age. It is likely that also older workers with disabilities will be required to work longer. We examine the change in working life expectancy (WLE) with disability of older workers by comparing data from three periods: 1992-1996, 2002-2006 and 2012-2016. Methods Data are from the Longitudinal Aging Study Amsterdam (LASA). Respondents aged 55-65 with a paid job at baseline were included (N=1074). Disability was measured using the Global Activity Limitations Indicator (GALI). First, a continuous-time three-state survival model was created. Second, WLE with and without disability were estimated using MSM and ELECT in R. The modifying effects of gender and educational level were examined. Results Among those initially in paid employment, total WLE increased over 20 years. For example at age 58, total WLE increased from 3.7 to 5.5 years. WLE with disability at age 58 increased from 0.8 to 1.5 years. There was no difference in WLE with disability between male and female workers or low- and highly educated workers. Conclusions Between the 1990s and the 2010s, subsequent generations of older workers with disabilities have extended their working lives. The findings emphasize the importance of workplace interventions that facilitate older workers with disabilities to maintain well-being and work ability. In addition, the question arises whether current exit routes out of the workforce are still adequate.

A dynamic Bayesian Markov model for health economic evaluations of interventions in infectious disease

BACKGROUND

Health economic evaluations of interventions in infectious disease are commonly based on the predictions of ordinary differential equation (ODE) systems or Markov models (MMs). Standard MMs are static, whereas ODE systems are usually dynamic and account for herd immunity which is crucial to prevent overestimation of infection prevalence. Complex ODE systems including distributions on model parameters are computationally intensive. Thus, mainly ODE-based models including fixed parameter values are presented in the literature. These do not account for parameter uncertainty. As a consequence, probabilistic sensitivity analysis (PSA), a crucial component of health economic evaluations, cannot be conducted straightforwardly.

METHODS

We present a dynamic MM under a Bayesian framework. We extend a static MM by incorporating the force of infection into the state allocation algorithm. The corresponding output is based on dynamic changes in prevalence and thus accounts for herd immunity. In contrast to deterministic ODE-based models, PSA can be conducted straightforwardly. We introduce a case study of a fictional sexually transmitted infection and compare our dynamic Bayesian MM to a deterministic and a Bayesian ODE system. The models are calibrated to simulated time series data.

RESULTS

By means of the case study, we show that our methodology produces outcome which is comparable to the "gold standard" of the Bayesian ODE system.

CONCLUSIONS

In contrast to ODE systems in the literature, the dynamic MM includes distributions on all model parameters at manageable computational effort (including calibration). The run time of the Bayesian ODE system is 15 times longer.

Flexible multistate models for interval-censored data: Specification, estimation, and an application to ageing research

Continuous-time multistate survival models can be used to describe health-related processes over time. In the presence of interval-censored times for transitions between the living states, the likelihood is constructed using transition probabilities. Models can be specified using parametric or semiparametric shapes for the hazards. Semiparametric hazards can be fitted using P-splines and penalised maximum likelihood estimation. This paper presents a method to estimate flexible multistate models that allow for parametric and semiparametric hazard specifications. The estimation is based on a scoring algorithm. The method is illustrated with data from the English Longitudinal Study of Ageing.

Transitions across cognitive states and death among older adults in relation to education: A multistate survival model using data from six longitudinal studies

INTRODUCTION

This study examines the role of educational attainment, an indicator of cognitive reserve, on transitions in later life between cognitive states (normal Mini-Mental State Examination (MMSE), mild MMSE impairment, and severe MMSE impairment) and death.

METHODS

Analysis of six international longitudinal studies was performed using a coordinated approach. Multistate survival models were used to estimate the transition patterns via different cognitive states. Life expectancies were estimated.

RESULTS

Across most studies, a higher level of education was associated with a lower risk of transitioning from normal MMSE to mild MMSE impairment but was not associated with other transitions. Those with higher levels of education and socioeconomic status had longer nonimpaired life expectancies.

DISCUSSION

This study highlights the importance of education in later life and that early life experiences can delay later compromised cognitive health. This study also demonstrates the feasibility and benefit in conducting coordinated analysis across multiple studies to validate findings.

Longitudinal mixed-effects models for latent cognitive function

A mixed-effects regression model with a bent-cable change-point predictor is formulated to describe potential decline of cognitive function over time in the older population. For the individual trajectories, cognitive function is considered to be a latent variable measured through an item response theory model given longitudinal test data. Individual-specific parameters are defined for both cognitive function and the rate of change over time, using the change-point predictor for non-linear trends. Bayesian inference is used, where the Deviance Information Criterion and the L-criterion are investigated for model comparison. Special attention is given to the identifiability of the item response parameters. Item response theory makes it possible to use dichotomous and polytomous test items, and to take into account missing data and survey-design change during follow-up. This will be illustrated in an application where data stem from the Cambridge City over-75s Cohort Study.

Investigating terminal decline: results from a UK population-based study of aging

The terminal decline hypothesis states that in the proximity of death, an individual's decline in cognitive abilities accelerates. We aimed at estimating the onset of faster rate of decline in global cognition using Mini Mental State Examination (MMSE) scores from participants of the Cambridge City over 75 Cohort Study (CC75C), a U.K. population-based longitudinal study of aging where almost all participants have died. The random change point model fitted to MMSE scores structured as a function of distance to death allowed us to identify a potentially different onset of change in rate of decline before death for each individual in the sample. Differences in rate of change before and after the onset of change in rate of decline by sociodemographic variables were investigated. On average, the onset of a faster rate of change occurred about 7.7 years before death and varied across individuals. Our results show that most individuals experience a period of slight decline followed by a much sharper decline. Education, age at death, and cognitive impairment at study entry were identified as modifiers of rate of change before and after change in rate of decline. Gender differences were found in rate of decline in the final stages of life. Our study suggests that terminal decline is a heterogeneous process, with its onset varying between individuals.

Active cognitive lifestyle is associated with positive cognitive health transitions and compression of morbidity from age sixty-five

BACKGROUND

Three factors commonly used as measures of cognitive lifestyle are education, occupation, and social engagement. This study determined the relative importance of each variable to long term cognitive health in those with and without severe cognitive impairment.

METHODS

Data came from 12,470 participants from a multi-centre population-based cohort (Medical Research Council Cognitive Function and Ageing Study). Respondents were aged 65 years and over and were followed-up over 16 years. Cognitive states of no impairment, slight impairment, and moderate/severe impairment were defined, based on scores from the Mini-Mental State Examination. Multi-state modelling was used to investigate links between component cognitive lifestyle variables, cognitive state transitions over time, and death.

RESULTS

Higher educational attainment and a more complex mid-life occupation were associated with a lower risk of moving from a non-impaired to a slightly impaired state (hazard ratios 0.5 and 0.8), but with increased mortality from a severely impaired state (1.3 and 1.1). More socially engaged individuals had a decreased risk of moving from a slightly impaired state to a moderately/severely impaired state (0.7). All three cognitive lifestyle variables were linked to an increased chance of cognitive recovery back to the non-impaired state.

CONCLUSIONS

In those without severe cognitive impairment, different aspects of cognitive lifestyle predict positive cognitive transitions over time, and in those with severe cognitive impairment, a reduced life-expectancy. An active cognitive lifestyle is therefore linked to compression of cognitive morbidity in late life.

Analysing cognitive test data: Distributions and non-parametric random effects

An important assumption in many linear mixed models is that the conditional distribution of the response variable is normal. This assumption is violated when the models are fitted to an outcome variable that counts the number of correctly answered questions in a questionnaire. Examples include investigations of cognitive decline where models are fitted to Mini Mental State Examination scores, the most widely used test to measure global cognition. Mini Mental State Examination scores take integer values in the 0–30 range, and its distribution has strong ceiling and floor effects. This article explores alternative distributions for the outcome variable in mixed models fitted to mini mental state examination scores from a longitudinal study of ageing. Model fit improved when a beta-binomial distribution was chosen as the distribution for the response variable.

Active cognitive lifestyle associates with cognitive recovery and a reduced risk of cognitive decline

Education and lifestyle factors linked with complex mental activity are thought to affect the progression of cognitive decline. Collectively, these factors can be combined to create a cognitive reserve or cognitive lifestyle score. This study tested the association between cognitive lifestyle score and cognitive change in a population-based cohort of older persons from five sites across England and Wales. Data came from 13,004 participants of the Medical Research Council Cognitive Function and Ageing Study who were aged 65 years and over. Cognition was assessed at multiple waves over 16 years using the Mini-Mental State Examination. Subjects were grouped into four cognitive states (no impairment, slight impairment, moderate impairment, severe impairment) and cognitive lifestyle score was assessed as a composite measure of education, mid-life occupation, and current social engagement. A multi-state model was used to test the effect of cognitive lifestyle score on cognitive transitions. Hazard ratios for cognitive lifestyle score showed significant differences between those in the upper compared to the lower tertile with a more active cognitive lifestyle associating with: a decreased risk of moving from no to slight impairment (0.58, 95% CI (0.45, 0.74)); recovery from a slightly impaired state back to a non-impaired state (2.93 (1.35, 6.38)); but an increased mortality risk from a severely impaired state (1.28 (1.12, 1.45)). An active cognitive lifestyle is associated with a more favorable cognitive trajectory in older persons. Future studies would ideally incorporate neuroradiological and neuropathological data to determine if there is causal evidence for these associations.

A semi-Markov model for stroke with piecewise-constant hazards in the presence of left, right and interval censoring

This paper presents a parametric method of fitting semi-Markov models with piecewise-constant hazards in the presence of left, right and interval censoring. We investigate transition intensities in a three-state illness–death model with no recovery. We relax the Markov assumption by adjusting the intensity for the transition from state 2 (illness) to state 3 (death) for the time spent in state 2 through a time-varying covariate. This involves the exact time of the transition from state 1 (healthy) to state 2. When the data are subject to left or interval censoring, this time is unknown. In the estimation of the likelihood, we take into account interval censoring by integrating out all possible times for the transition from state 1 to state 2. For left censoring, we use an Expectation–Maximisation inspired algorithm. A simulation study reflects the performance of the method. The proposed combination of statistical procedures provides great flexibility. We illustrate the method in an application by using data on stroke onset for the older population from the UK Medical Research Council Cognitive Function and Ageing Study. Copyright © 2012 John Wiley & Sons, Ltd.

Bayesian inference for an illness-death model for stroke with cognition as a latent time-dependent risk factor

Longitudinal data can be used to estimate the transition intensities between healthy and unhealthy states prior to death. An illness-death model for history of stroke is presented, where time-dependent transition intensities are regressed on a latent variable representing cognitive function. The change of this function over time is described by a linear growth model with random effects. Occasion-specific cognitive function is measured by an item response model for longitudinal scores on the Mini-Mental State Examination, a questionnaire used to screen for cognitive impairment. The illness-death model will be used to identify and to explore the relationship between occasion-specific cognitive function and stroke. Combining a multi-state model with the latent growth model defines a joint model which extends current statistical inference regarding disease progression and cognitive function. Markov chain Monte Carlo methods are used for Bayesian inference. Data stem from the Medical Research Council Cognitive Function and Ageing Study in the UK (1991-2005).

Estimating the prevalence of sensitive behaviour and cheating with a dual design for direct questioning and randomized response

Randomized response is a misclassification design to estimate the prevalence of sensitive behaviour. Respondents who do not follow the instructions of the design are considered to be cheating. A mixture model is proposed to estimate the prevalence of sensitive behaviour and cheating in the case of a dual sampling scheme with direct questioning and randomized response. The mixing weight is the probability of cheating, where cheating is modelled separately for direct questioning and randomized response. For Bayesian inference, Markov chain Monte Carlo sampling is applied to sample parameter values from the posterior. The model makes it possible to analyse dual sample scheme data in a unified way and to assess cheating for direct questions as well as for randomized response questions. The research is illustrated with randomized response data concerning violations of regulations for social benefit.

Smooth random change point models

Change point models are used to describe processes over time that show a change in direction. An example of such a process is cognitive ability, where a decline a few years before death is sometimes observed. A broken-stick model consists of two linear parts and a breakpoint where the two lines intersect. Alternatively, models can be formulated that imply a smooth change between the two linear parts. Change point models can be extended by adding random effects to account for variability between subjects. A new smooth change point model is introduced and examples are presented that show how change point models can be estimated using functions in R for mixed-effects models. The Bayesian inference using WinBUGS is also discussed. The methods are illustrated using data from a population-based longitudinal study of ageing, the Cambridge City over 75 Cohort Study. The aim is to identify how many years before death individuals experience a change in the rate of decline of their cognitive ability.

The randomized response log linear model as a composite link model

In randomized response (RR) designs, misclassification is used to protect the privacy of respondents when sensitive questions are asked. A generalized linear model with a composite link function is presented to formulate log linear models that take the RR design into account. The approach is extended to model the situation where some respondents do not follow the instructions of the RR design. For example, if there are three binary RR variables with regard to practicing fraud, the 2 × 2 × 2 cross-classification of the true answers is latent due to the misclassification. Using composite link functions, log linear models can be specified for the latent table to investigate possible association between the variables. Fast iteratively re-weighted least squares algorithms are presented.

Estimating stroke-free and total life expectancy in the presence of non-ignorable missing values

A continuous time three-state model with time-dependent transition intensities is formulated to describe transitions between healthy and unhealthy states before death. By using time continuously, known death times can be taken into account. To deal with possible non-ignorable missing states, a selection model is proposed for the joint distribution of both the state and whether or not the state is observed. To estimate total life expectancy and its subdivision into life expectancy in health and ill health, the three-state model is extrapolated beyond the follow-up of the study. Estimation of life expectancies is illustrated by analysing data from a longitudinal study of aging where individuals are in a state of ill health if they have ever experienced a stroke. Results for the selection model are compared with results for a model where states are assumed to be missing at random and with results for a model that ignores missing states.

Estimating dementia-free life expectancy for Parkinson's patients using Bayesian inference and microsimulation

Interval-censored longitudinal data taken from a Norwegian study of individuals with Parkinson's disease are investigated with respect to the onset of dementia. Of interest are risk factors for dementia and the subdivision of total life expectancy (LE) into LE with and without dementia. To estimate LEs using extrapolation, a parametric continuous-time 3-state illness–death Markov model is presented in a Bayesian framework. The framework is well suited to allow for heterogeneity via random effects and to investigate additional computation using model parameters. In the estimation of LEs, microsimulation is used to take into account random effects. Intensities of moving between the states are allowed to change in a piecewise-constant fashion by linking them to age as a time-dependent covariate. Possible right censoring at the end of the follow-up can be incorporated. The model is applicable in many situations where individuals are followed over a long time period. In describing how a disease develops over time, the model can help to predict future need for health care.

A piecewise-constant Markov model and the effects of study design on the estimation of life expectancies in health and ill health

Multi-state models are frequently applied to describe transitions over time between three states: healthy, not healthy and death. The three-state model can be used to estimate life expectancies in health and ill health. In this article, continuous-time Markov models are specified for the transitions between the three states. Transition intensities are regressed on age as a time-dependent covariate. The covariate is handled in a piecewise-constant fashion where the time interval between two consecutive observations is divided into subintervals of fixed and equal lengths. Study design choices such as sample size, length of follow-up, and time intervals between observations are investigated in a simulation study. The effects on parameter estimation are discussed as well as the effects on the estimation of life expectancies. In addition, data taken from the UK Cognitive Functioning and Ageing Study are analysed.