Random Change Point Research Articles

Using a Bayesian change-point statistical model with autoregressive terms to study the monthly number of dispensed asthma medications by public health services

In this paper, it is proposed a Bayesian analysis of a time series in the presence of a random change-point and autoregressive terms. The development of this model was motivated by a data set related to the monthly number of asthma medications dispensed by the public health services of Ribeirao Preto, Southeast Brazil, from 1999 to 2011. A pronounced increase trend has been observed from 1999 to a specific change-point, with a posterior decrease until the end of the series. In order to obtain estimates for the parameters of interest, a Bayesian Markov Chain Monte Carlo (MCMC) simulation procedure using the Gibbs sampler algorithm was developed. The Bayesian model with autoregressive terms of order 1 fits well to the data, allowing to estimate the change-point at July 2007, and probably reflecting the results of the new health policies and previously adopted programs directed toward patients with asthma. The results imply that the present model is useful to analyse the monthly number of dispensed asthma medications and it can be used to describe a broad range of epidemiological time series data where a change-point is present.

A note on estimating single-class piecewise mixed-effects models with unknown change points

Piecewise mixed-effects models are useful for analyzing longitudinal educational and psychological data sets to model segmented change over time. These models offer an attractive alternative to commonly used quadratic and higher-order polynomial models because the coefficients obtained from fitting the model have meaningful substantive interpretation. The current study thus focuses on the estimation of piecewise mixed-effects model with unknown random change points using maximum likelihood (ML) as described in Du Toit and Cudeck (2009). Previous simulation work (Wang & McArdle, 2008) showed that Bayesian estimation produced reliable parameter estimates for the piecewise model in comparison to frequentist procedures (i.e., first-order Taylor expansion and the adaptive Gaussian quadrature) across all simulation conditions. In the current article a small Monte Carlo simulation study was conducted to assess the performance of the ML approach, a frequentist procedure, and the Bayesian approach for fitting linear–linear piecewise mixed-effects model. The obtained findings show that ML estimation approach produces reliable and accurate estimates under the conditions of small residual variance of the observed variables, and that the size of the residual variance had the most impact on the quality of model parameter estimates. Second, neither ML nor Bayesian estimation procedures performed well under all manipulated conditions with respect to the accuracy and precision of the estimated model parameters.

A MOSUM procedure for the estimation of multiple random change points

In this work, we investigate statistical properties of change point estimators based on moving sum statistics. We extend results for testing in a classical situation with multiple deterministic change points by allowing for random exogenous change points that arise in Hidden Markov or regime switching models among others. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates are strict by deriving the limit distribution of properly scaled estimators. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples showing that its performance is very good compared to existing methods.

Detecting Multiple Random Changepoints in Bayesian Piecewise Growth Mixture Models.

Piecewise growth mixture models are a flexible and useful class of methods for analyzing segmented trends in individual growth trajectory over time, where the individuals come from a mixture of two or more latent classes. These models allow each segment of the overall developmental process within each class to have a different functional form; examples include two linear phases of growth, or a quadratic phase followed by a linear phase. The changepoint (knot) is the time of transition from one developmental phase (segment) to another. Inferring the location of the changepoint(s) is often of practical interest, along with inference for other model parameters. A random changepoint allows for individual differences in the transition time within each class. The primary objectives of our study are as follows: (1) to develop a PGMM using a Bayesian inference approach that allows the estimation of multiple random changepoints within each class; (2) to develop a procedure to empirically detect the number of random changepoints within each class; and (3) to empirically investigate the bias and precision of the estimation of the model parameters, including the random changepoints, via a simulation study. We have developed the user-friendly package BayesianPGMM for R to facilitate the adoption of this methodology in practice, which is available at https://github.com/lockEF/BayesianPGMM . We describe an application to mouse-tracking data for a visual recognition task.

Bayesian Piecewise Linear Mixed Models With a Random Change Point: An Application to BMI Rebound in Childhood.

Background:Body mass index (BMI) rebound refers to the beginning of the second rise in BMI during childhood. Accurate estimation of an individual’s timing of BMI rebound is important because it is associated with health outcomes in later life.Methods:We estimated BMI trajectories for 6545 children from the Avon Longitudinal Study of Parents and Children. We used a novel Bayesian two-phase piecewise linear mixed model where the “change point” was an individual-level random effect corresponding to the individual-specific timing of BMI rebound. The model’s individual-level random effects (intercept, prechange slope, postchange slope, change point) were multivariate normally distributed with an unstructured variance–covariance matrix, thereby, allowing for correlation between all random effects.Results:Average age at BMI rebound (mean change point) was 6.5 (95% credible interval: 6.4 to 6.6) years. The standard deviation of the individual-specific timing of BMI rebound (random effects) was 2.0 years for females and 1.6 years for males. Correlation between the prechange slope and change point was 0.57, suggesting that faster rates of decline in BMI prior to rebound were associated with rebound occurring at an earlier age. Simulations showed that estimates from the model were less biased than those from models, assuming a common change point for all individuals or a nonlinear trajectory based on fractional polynomials.Conclusions:Our model flexibly estimated the individual-specific timing of BMI rebound, while retaining parameters that are meaningful and easy to interpret. It is applicable in any situation where one wishes to estimate a change-point process which varies between individuals.

TRIGLYCERIDES AS A MODERATOR OF ACCELERATED COGNITIVE LOSS PRIOR TO DEATH AND DEMENTIA

Cognitive aging is frequently characterized by accelerated loss with an onset some years before death (terminal decline) or dementia diagnosis (pre-clinical dementia). Recent evidence suggest that an elevated level of serum triglycerides may contribute to accumulation of cerebral amyloids (Aβ) and potentially also cognitive decline. We evaluated this issue using data from the H70 study. The sample consisted of 392 individuals, systematically selected from the general population of 70 years-olds living in Gothenburg, Sweden, born in 1901–1902, and repeatedly measured on three cognitive outcomes (i.e., speed, spatial, verbal) at ages 70, 75, 79, 81, 85, 88, 90, 92, 95, 97, 99, and 100, or until death. Triglycerides levels were derived from blood samples taken at age 70 and complete date of death was retrieved from population register. Dementia diagnosis was made according to DSM-III-R. We fitted several random change point, and polynomial, quadratic mixed effects models to the data. Our findings revealed no evidence that triglycerides levels moderate the onset of the acceleration in either the terminal or the pre-clinical dementia models. Elevated levels of triglycerides were however consistently associated with poorer cognitive performance and, against expectation, also with less steep decline. When we compared the effect sizes to results using serum total cholesterol as moderator we found that the cognitive trajectory was more strongly associated with the levels of triglycerides than total cholesterol. These findings support the claim that triglycerides have a role in cognitive aging but not to the degree that it moderates the onset of the acceleration.

A Bayesian screening approach for hepatocellular carcinoma using multiple longitudinal biomarkers.

Advanced hepatocellular carcinoma (HCC) has limited treatment options and poor survival, therefore early detection is critical to improving the survival of patients with HCC. Current guidelines for high-risk patients include ultrasound screenings every six months, but ultrasounds are operator dependent and not sensitive for early HCC. Serum α-Fetoprotein (AFP) is a widely used diagnostic biomarker, but it has limited sensitivity and is not elevated in all HCC cases so, we incorporate a second blood-based biomarker, des'γ carboxy-prothrombin (DCP), that has shown potential as a screening marker for HCC. The data from the Hepatitis C Antiviral Long-term Treatment against Cirrhosis (HALT-C) Trial is a valuable source of data to study biomarker screening for HCC. We assume the trajectories of AFP and DCP follow a joint hierarchical mixture model with random changepoints that allows for distinct changepoint times and subsequent trajectories of each biomarker. The changepoint indicators are jointly modeled with a Markov Random Field distribution to help detect borderline changepoints. Markov chain Monte Carlo methods are used to calculate posterior distributions, which are used in risk calculations among future patients and determine whether a patient has a positive screen. The screening algorithm was compared to alternatives in simulations studies under a range of possible scenarios and in the HALT-C Trial using cross-validation.

An application of Dirichlet process in clustering subjects via variance shift models: A course-evaluation study

In this article, the Dirichlet process (DP) is applied to cluster subjects with longitudinal observations. The basis of clustering is the ability of subjects to adapt themselves to new circumstances. Indeed, the basis of clustering depends on the time of changing response variability. This is done by providing a random change-point time in the variance structure of mixed-effects models. The DP is assumed as a prior for the distribution of the random change point. The discrete nature of the DP is utilized to cluster subjects according to the time of adaption. The proposed model is useful to identify groups of subjects with distinctive time-based progressions or declines. Transition mixed-effects models are also used to account for the serial correlation among observations over time. A joint modelling approach is utilized to handle the bias created in these models. The Gibbs sampling technique is adopted to achieve parameter estimates. Performance of the proposed method is evaluated via conducting a simulation study. The usefulness of the proposed model is assessed on a course-evaluation dataset.

IQ as moderator of terminal decline in perceptual and motor speed, spatial, and verbal ability: Testing the cognitive reserve hypothesis in a population-based sample followed from age 70 until death.

Terminal decline (TD) refers to acceleration in within-person cognitive decline prior to death. The cognitive reserve hypothesis postulates that individuals with higher IQ are able to better tolerate age-related increase in brain pathologies. On average, they will exhibit a later onset of TD, but once they start to decline, their trajectory is steeper relative to those with lower IQ. We tested these predictions using data from initially nondemented individuals (n = 179) in the H70-study repeatedly measured at ages 70, 75, 79, 81, 85, 88, 90, 92, 95, 97, 99, and 100, or until death, on cognitive tests of perceptual-and-motor-speed and spatial and verbal ability. We quantified IQ using the Raven's Coloured Progressive Matrices (RCPM) test administrated at age 70. We fitted random change point TD models to the data, within a Bayesian framework, conditioned on IQ, age of death, education, and sex. In line with predictions, we found that 1 additional standard deviation on the IQ scale was associated with a delay in onset of TD by 1.87 (95% highest density interval [HDI; 0.20, 4.08]) years on speed, 1.96 (95% HDI [0.15, 3.54]) years on verbal ability, but only 0.88 (95% HDI [-0.93, 3.49]) year on spatial ability. Higher IQ was associated with steeper rate of decline within the TD phase on measures of speed and verbal ability, whereas results on spatial ability were nonconclusive. Our findings provide partial support for the cognitive reserve hypothesis and demonstrate that IQ can be a significant moderator of cognitive change trajectories in old age. (PsycINFO Database Record

A two-stage model in a Bayesian framework to estimate a survival endpoint in the presence of confounding by indication.

Background Biomarker series can indicate disease progression and predict clinical endpoints. When a treatment is prescribed depending on the biomarker, confounding by indication might be introduced if the treatment modifies the marker profile and risk of failure. Objective Our aim was to highlight the flexibility of a two-stage model fitted within a Bayesian Markov Chain Monte Carlo framework. For this purpose, we monitored the prostate-specific antigens in prostate cancer patients treated with external beam radiation therapy. In the presence of rising prostate-specific antigens after external beam radiation therapy, salvage hormone therapy can be prescribed to reduce both the prostate-specific antigens concentration and the risk of clinical failure, an illustration of confounding by indication. We focused on the assessment of the prognostic value of hormone therapy and prostate-specific antigens trajectory on the risk of failure. Methods We used a two-stage model within a Bayesian framework to assess the role of the prostate-specific antigens profile on clinical failure while accounting for a secondary treatment prescribed by indication. We modeled prostate-specific antigens using a hierarchical piecewise linear trajectory with a random changepoint. Residual prostate-specific antigens variability was expressed as a function of prostate-specific antigens concentration. Covariates in the survival model included hormone therapy, baseline characteristics, and individual predictions of the prostate-specific antigens nadir and timing and prostate-specific antigens slopes before and after the nadir as provided by the longitudinal process. Results We showed positive associations between an increased prostate-specific antigens nadir, an earlier changepoint and a steeper post-nadir slope with an increased risk of failure. Importantly, we highlighted a significant benefit of hormone therapy, an effect that was not observed when the prostate-specific antigens trajectory was not accounted for in the survival model. Conclusion Our modeling strategy was particularly flexible and accounted for multiple complex features of longitudinal and survival data, including the presence of a random changepoint and a time-dependent covariate.

Momentum liquidation under partial information

Abstract Momentum is the notion that an asset that has performed well in the past will continue to do so for some period. We study the optimal liquidation strategy for a momentum trade in a setting where the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the trader, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. Comparisons with existing results for momentum trading under incomplete information show that the assumption that the disappearance of the momentum effect is triggered by observable external shocks significantly improves the optimal strategy.

Educational inequalities in aging-related declines in fluid cognition and the onset of cognitive pathology

BackgroundEducation has been robustly associated with cognitive reserve and dementia, but not with the rate of cognitive aging, resulting in some confusion about the mechanisms of cognitive aging. This study uses longitudinal data to differentiate between trajectories indicative of healthy versus pathologic cognitive aging. MethodsParticipants included 9401 Health and Retirement Study respondents aged ≥55 years who completed cognitive testing regularly over 17.3 years until most recently in 2012. Individual-specific random change-point modeling was used to identify age of incident pathologic decline; acceleration is interpreted as indicating likely onset of pathologic decline when it is significant and negative. ResultsThese methods detect incident dementia diagnoses with specificity/sensitivity of 89.3%/44.3%, 5.6 years before diagnosis. Each year of education was associated with 0.09 (95% confidence interval [CI], 0.087–0.096; P < .001) standard deviation higher baseline cognition and delayed onset of cognitive pathology (hazard ratio, 0.98; 95% CI, 0.96–0.99; P = .006). ConclusionsLongitudinal random change-point modeling was able to reliably identify incident dementia. Accounting for incident cognitive pathology, we find that education predicts cognitive capability and delayed onset pathologic declines.

When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One

To solve the selling problem which is resembled to the buying problem in [1], in this paper we solve the problem of determining the optimal time to sell a property in a location the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the investor, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. The asset price is modeled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time and thus, we model the drift as a two-state Markov chain. Using filtering and martingale techniques, stochastic analysis transform measurement, we reduce the problem to a one-dimensional optimal stopping problem. We also establish the optimal boundary at which the investor should liquidate the asset when the price process hit the boundary at first time.

Modeling longitudinal data with a random change point and no time-zero: applications to inference and prediction of the labor curve.

In some longitudinal studies the initiation time of the process is not clearly defined, yet it is important to make inference or do predictions about the longitudinal process. The application of interest in this article is to provide a framework for modeling individualized labor curves (longitudinal cervical dilation measurements) where the start of labor is not clearly defined. This is a well-known problem in obstetrics where the benchmark reference time is often chosen as the end of the process (individuals are fully dilated at 10 cm) and time is run backwards. This approach results in valid and efficient inference unless subjects are censored before the end of the process, or if we are focused on prediction. Providing dynamic individualized predictions of the longitudinal labor curve prospectively (where backwards time is unknown) is of interest to aid obstetricians to determine if a labor is on a suitable trajectory. We propose a model for longitudinal labor dilation that uses a random-effects model with unknown time-zero and a random change point. We present a maximum likelihood approach for parameter estimation that uses adaptive Gaussian quadrature for the numerical integration. Further, we propose a Monte Carlo approach for dynamic prediction of the future longitudinal dilation trajectory from past dilation measurements. The methodology is illustrated with longitudinal cervical dilation data from the Consortium of Safe Labor Study.

Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study

We present a simple and effective iterative procedure to estimate segmented mixed models in a likelihood based framework. Random effects and covariates are allowed for each model parameter, including the changepoint. The method is practical and avoids the computational burdens related to estimation of nonlinear mixed effects models. A conventional linear mixed model with proper covariates that account for the changepoints is the key to our estimating algorithm. We illustrate the method via simulations and using data from a randomized clinical trial focused on change in depressive symptoms over time which characteristically show two separate phases of change.

Education associated with a delayed onset of terminal decline

Background: the terminal decline hypothesis suggests an acceleration in the rate of loss of cognitive function before death. Evidence about the association of educational attainment and the onset of terminal decline is scarce.Objective: to investigate the association of education with the onset of terminal decline in global cognitive function measured by Mini Mental State Exam (MMSE) scores.Subjects: deceased participants of the Cambridge City over 75 Cohort Study who were interviewed at about 2, 7, 9, 13, 17 and 21 years after baseline.Methods: regular and Tobit random change point growth models were fitted to MMSE scores to identify the onset of terminal decline and assess the effect of education on this onset.Results: people who left school at an older age had a delayed onset of terminal decline. Thus better educated individuals experience a slightly shorter period of faster decline before death.Conclusion: an important finding emerging from our work is that education does appear to delay the onset of terminal decline, although only by a limited amount.

Filtering and change point estimation for hidden Markov-modulated Poisson processes

A continuous-time Markov chain which is partially observed in Poisson noise is considered, where a structural change in the dynamics of the hidden process occurs at a random change point. Filtering and change point estimation of the model is discussed. Closed-form recursive estimates of the conditional distribution of the hidden process and the random change point are obtained, given the Poisson process observations

Much of late life cognitive decline is not due to common neurodegenerative pathologies.

The pathologic indices of Alzheimer disease, cerebrovascular disease, and Lewy body disease accumulate in the brains of older persons with and without dementia, but the extent to which they account for late life cognitive decline remains unknown. We tested the hypothesis that these pathologic indices account for the majority of late life cognitive decline. A total of 856 deceased participants from 2 longitudinal clinical-pathologic studies, Rush Memory and Aging Project and Religious Orders Study, completed a mean of 7.5 annual evaluations, including 17 cognitive tests. Neuropathologic examinations provided quantitative measures of global Alzheimer pathology, amyloid load, tangle density, macroscopic infarcts, microinfarcts, and neocortical Lewy bodies. Random coefficient models were used to examine the linear relation of pathologic indices with global cognitive decline. In subsequent analyses, random change point models were used to examine the relation of the pathologic indices with the onset of terminal decline and rates of preterminal and terminal decline (ie, nonlinear decline). Cognition declined a mean of about 0.11 U per year (estimate = -0.109, standard error [SE] = 0.004, p < 0.001), with significant individual differences in rates of decline; the variance estimate for the individual slopes was 0.013 (SE = 0.112, p < 0.001). In separate analyses, global Alzheimer pathology, amyloid, tangles, macroscopic infarcts, and neocortical Lewy bodies were associated with faster rates of decline and explained 22%, 6%, 34%, 2%, and 8% of the variation in decline, respectively. When analyzed simultaneously, the pathologic indices accounted for a total of 41% of the variation in decline, and the majority remained unexplained. Furthermore, in random change point models examining the influence of the pathologic indices on the onset of terminal decline and the preterminal and terminal components of the cognitive trajectory, the common pathologic indices accounted for less than a third of the variation in the onset of terminal decline and rates of preterminal and terminal decline. The pathologic indices of the common causes of dementia are important determinants of cognitive decline in old age and account for a large proportion of the variation in late life cognitive decline. Surprisingly, however, much of the variation in cognitive decline remains unexplained, suggesting that other important determinants of cognitive decline remain to be identified. Identification of the mechanisms that contribute to the large unexplained proportion of cognitive decline is urgently needed to prevent late life cognitive decline.

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.

Particle Smoothing Algorithms for Variable Rate Models

Standard state-space methods assume that the latent state evolves uniformly over time, and can be modeled with a discrete-time process synchronous with the observations. This may be a poor representation of some systems in which the state evolution displays discontinuities in its behavior. For such cases, a variable rate model may be more appropriate; the system dynamics are conditioned on a set of random changepoints which constitute a marked point process. In this paper, new particle smoothing algorithms are presented for use with conditionally linear-Gaussian and conditionally deterministic dynamics. These are demonstrated on problems in financial modelling and target tracking. Results indicate that the smoothing approximations provide more accurate and more diverse representations of the state posterior distributions.