Framework For Longitudinal Data Research Articles

A review of multistate modelling approaches in monitoring disease progression: Bayesian estimation using the Kolmogorov-Chapman forward equations.

There are numerous fields of science in which multistate models are used, including biomedical research and health economics. In biomedical studies, these stochastic continuous-time models are used to describe the time-to-event life history of an individual through a flexible framework for longitudinal data. The multistate framework can describe more than one possible time-to-event outcome for a single individual. The standard estimation quantities in multistate models are transition probabilities and transition rates which can be mapped through the Kolmogorov-Chapman forward equations from the Bayesian estimation perspective. Most multistate models assume the Markov property and time homogeneity; however, if these assumptions are violated, an extension to non-Markovian and time-varying transition rates is possible. This manuscript extends reviews in various types of multistate models, assumptions, methods of estimation and data features compatible with fitting multistate models. We highlight the contrast between the frequentist (maximum likelihood estimation) and the Bayesian estimation approaches in the multistate modeling framework and point out where the latter is advantageous. A partially observed and aggregated dataset from the Zimbabwe national ART program was used to illustrate the use of Kolmogorov-Chapman forward equations. The transition rates from a three-stage reversible multistate model based on viral load measurements in WinBUGS were reported.

Nonparametric covariance estimation with shrinkage toward stationary models

AbstractEstimation of an unstructured covariance matrix is difficult because of the challenges posed by parameter space dimensionality and the positive‐definiteness constraint that estimates should satisfy. We consider a general nonparametric covariance estimation framework for longitudinal data using the Cholesky decomposition of a positive‐definite matrix. The covariance matrix of time‐ordered measurements is diagonalized by a lower triangular matrix with unconstrained entries that are statistically interpretable as parameters for a varying coefficient autoregressive model. Using this dual interpretation of the Cholesky decomposition and allowing for irregular sampling time points, we treat covariance estimation as bivariate smoothing and cast it in a regularization framework for desired forms of simplicity in covariance models. Viewing stationarity as a form of simplicity or parsimony in covariance, we model the varying coefficient function with components depending on time lag and its orthogonal direction separately and penalize the components that capture the nonstationarity in the fitted function. We demonstrate construction of a covariance estimator using the smoothing spline framework. Simulation studies establish the advantage of our approach over alternative estimators proposed in the longitudinal data setting. We analyze a longitudinal dataset to illustrate application of the methodology and compare our estimates to those resulting from alternative models.This article is categorized under: Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Algorithms and Computational Methods > Maximum Likelihood Methods

Pair Copula Constructions for Insurance Experience Rating

ABSTRACTIn nonlife insurance, insurers use experience rating to adjust premiums to reflect policyholders’ previous claim experience. Performing prospective experience rating can be challenging when the claim distribution is complex. For instance, insurance claims are semicontinuous in that a fraction of zeros is often associated with an otherwise positive continuous outcome from a right-skewed and long-tailed distribution. Practitioners use credibility premium that is a special form of the shrinkage estimator in the longitudinal data framework. However, the linear predictor is not informative especially when the outcome follows a mixed distribution. In this article, we introduce a mixed vine pair copula construction framework for modeling semicontinuous longitudinal claims. In the proposed framework, a two-component mixture regression is employed to accommodate the zero inflation and thick tails in the claim distribution. The temporal dependence among repeated observations is modeled using a sequence of bivariate conditional copulas based on a mixed D-vine. We emphasize that the resulting predictive distribution allows insurers to incorporate past experience into future premiums in a nonlinear fashion and the classic linear predictor can be viewed as a nested case. In the application, we examine a unique claims dataset of government property insurance from the state of Wisconsin. Due to the discrepancies between the claim and premium distributions, we employ an ordered Lorenz curve to evaluate the predictive performance. We show that the proposed approach offers substantial opportunities for separating risks and identifying profitable business when compared with alternative experience rating methods. Supplementary materials for this article are available online.

Exchange Rate Volatility and Macroeconomic Performance in Small Island Developing States

The purpose of the study is to provide an investigation into the impact of exchange rate volatility on the macroeconomic performance of Small Island Developing States (SIDS). Taking a sample of 15 SIDS, the present study analyses econometrically the impact of exchange rate volatility on major macroeconomic variables, viz economic growth, external trade and foreign direct investment on the SIDS. The paper first constructs the z-score measure, developed by Wolf et al (2003) as a measure of exchange rate volatility and employs data spanning the period 1999 to 2010 to analyse robust estimates in a static framework as well as in a dynamic and longitudinal data framework using the Generalised Method of Moments. It also analyses the impact of exchange rate volatility on macroeconomic performance of the economies. The OLS with robust standard errors results indicate that, exchange rate volatility impacts negatively on current account balance but positively on the growth rate of the economies studied. In a dynamic setting, however, exchange rate volatility does not influence the macroeconomic variables.

Two-Stage Estimation for Seemingly Unrelated Nonparametric Regression Models

This paper is concerned with the estimating problem of seemingly unrelated (SU) nonparametric regression models. The authors propose a new method to estimate the unknown functions, which is an extension of the two-stage procedure in the longitudinal data framework. The authors show the resulted estimators are asymptotically normal and more efficient than those based on only the individual regression equation. Some simulation studies are given in support of the asymptotic results. A real data from an ongoing environmental epidemiologic study are used to illustrate the proposed procedure.

Copula credibility for aggregate loss models

This paper develops credibility predictors of aggregate losses using a longitudinal data framework. For a model of aggregate losses, the interest is in predicting both the claims number process as well as the claims amount process. In a longitudinal data framework, one encounters data from a cross-section of risk classes with a history of insurance claims available for each risk class. Further, explanatory variables for each risk class over time are available to help explain and predict both the claims number and claims amount process. For the marginal claims distributions, this paper uses generalized linear models, an extension of linear regression, to describe cross-sectional characteristics. Elliptical copulas are used to model the dependencies over time, extending prior work that used multivariate t-copulas. The claims number process is represented using a Poisson regression model that is conditioned on a sequence of latent variables. These latent variables drive the serial dependencies among claims numbers; their joint distribution is represented using an elliptical copula. In this way, the paper provides a unified treatment of both the continuous claims amount and discrete claims number processes. The paper presents an illustrative example of Massachusetts automobile claims. Estimates of the latent claims process parameters are derived and simulated predictions are provided.

Credibility Using Copulas

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes. This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction. We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.

Modeling the effects of nutritional and socioeconomic factors on the growth and morbidity of Kenyan school children.

This paper estimates dynamic models for the height, head circumference, weight, and morbidity of approximately 110 Kenyan school children (6-9 years) in a multivariate longitudinal data framework. Dynamic models allow anthropometric dimensions to depend on the respective measurements in the previous period. The system of 4 equations specified for height, head circumference, weight, and morbidity incorporates the inter-relationships among these variables; explanatory variables in the model consist of nutritional, socioeconomic, demographic, and environmental factors. The model parameters are estimated using the principle of maximum-likelihood, while controlling for the unobserved between-children differences. The main findings are, first, that calcium intakes are positively associated with height while protein and energy intakes are associated with weight. Vitamin A intakes are negatively associated with morbidity. Second, socioeconomic status plus the cash income of the household is a significant predictor of height, head circumference, and morbidity. Third, maternal height is positively associated with children's height and maternal body mass index (BMI) is positively associated with children's weight. Fourth, parents' scores on psychological tests, mother's age, and children's hemoglobin concentration are negatively associated with morbidity while mothers' morbidity is positively associated with children's morbidity. Implications of the modeling results are discussed. Am. J. Hum. Biol. 11:317-326, 1999. Copyright 1999 Wiley-Liss, Inc.

Modeling the effects of nutritional and socioeconomic factors on the growth and morbidity of Kenyan school children

This paper estimates dynamic models for the height, head circumference, weight, and morbidity of approximately 110 Kenyan school children (6–9 years) in a multivariate longitudinal data framework. Dynamic models allow anthropometric dimensions to depend on the respective measurements in the previous period. The system of 4 equations specified for height, head circumference, weight, and morbidity incorporates the inter-relationships among these variables; explanatory variables in the model consist of nutritional, socioeconomic, demographic, and environmental factors. The model parameters are estimated using the principle of maximum-likelihood, while controlling for the unobserved between-children differences. The main findings are, first, that calcium intakes are positively associated with height while protein and energy intakes are associated with weight. Vitamin A intakes are negatively associated with morbidity. Second, socioeconomic status plus the cash income of the household is a significant predictor of height, head circumference, and morbidity. Third, maternal height is positively associated with children's height and maternal body mass index (BMI) is positively associated with children's weight. Fourth, parents' scores on psychological tests, mother's age, and children's hemoglobin concentration are negatively associated with morbidity while mothers' morbidity is positively associated with children's morbidity. Implications of the modeling results are discussed. Am. J. Hum. Biol. 11:317–326, 1999. © 1999 Wiley-Liss, Inc.

A General Framework for Longitudinal Data through Marked Point Processes

AbstractThis paper reviews a general framework for the modelling of longitudinal data with random measurement times based on marked point processes and presents a worked example. We construct a quite general regression models for longitudinal data, which may in particular include censoring that only depend on the past and outside random variation, and dependencies between measurement times and measurements. The modelling also generalises statistical counting process models. We review a non‐parametric Nadarya‐Watson kernel estimator of the regression function, and a parametric analysis that is based on a conditional least squares (CLS) criterion. The parametric analysis presented, is a conditional version of the generalised estimation equations of LIANG and ZEGER (1986). We conclude that the usual nonparametric and parametric regression modelling can be applied to this general set‐up, with some modifications. The presented framework provides an easily implemented and powerful tool for model building for repeated measurements.