Frailty models

Background: The gamma-Gompertz multiplicative frailty model is the most common parametric model applied to human mortality data at adult and old ages. The resulting life expectancy has been calculated so far only numerically. Objective: Properties of the gamma-Gompertz distribution have not been thoroughly studied. The focus of the paper is to shed light onto its first moment or, demographically speaking, characterize life expectancy resulting from a gamma-Gompertz force of mortality. The paper provides an exact formula for gamma-Gompertz life expectancy at birth and a simpler high-accuracy approximation that can be used in practice for computational convenience. In addition, the article compares actual (life-table) to model-based (gamma-Gompertz) life expectancy to assess on aggregate how many years of life expectancy are not captured (or overestimated) by the gamma-Gompertz mortality mechanism. Comments: A closed-form expression for gamma-Gomeprtz life expectancy at birth contains a special (the hypergeometric) function. It aids assessing the impact of gamma-Gompertz parameters on life expectancy values. The paper shows that a high-accuracy approximation can be constructed by assuming an integer value for the shape parameter of the gamma distribution. A historical comparison between model-based and actual life expectancy for Swedish females reveals a gap that is decreasing to around 2 years from 1950 onwards. Looking at remaining life expectancies at ages 30 and 50, we see this gap almost disappearing.

In this paper, a joint model is presented for analyzing longitudinal continuous and count mixed responses. The frequency distribution of continuous longitudinal response variable for each subject at any time has a skewed and or multi-modal form. Then, a suitable finite mixture of normals is used as its distribution. It seems that the continuous response comes from several distinct sub-populations. The number of zeros of the count response is inflated. Also, a zero-inflated power series (ZIPS) distribution is applied as its distribution in order to model the count response. The correlation of longitudinal responses through time and that of mixed continuous and count responses are modeled by utilizing the random-effects vectors in the finite mixtures of regression (FMR) models. Further, a full likelihood-based approach is used to obtain the maximum likelihood estimates of parameters via the EM algorithm. Then, some simulation studies are performed for assessing the performance of the model. Additionally, an application is illustrated for joint analysis of the number of days during the last month that the individual drank alcohol, as well as the respondents' weight. Finally, the two first times of the Americans Changing Lives survey are evaluated.

'Frailty' is increasingly used as a clinical term to refer and respond to a particular bodily presentation, with numerous scores and measures to support its clinical determination. While these tools are typically quantitative in nature and based primarily on physical capacity, qualitative research has revealed that frailty is also associated with a range of social, economic and environmental factors. Here, we progress the understanding of frailty in older people via a new materialist synthesis of recent qualitative studies of frailty and ageing. We replace a conception of frailty as a bodily attribute with a relational understanding of a 'frailty assemblage'. Within this more-than-human assemblage, materialities establish the ongoing 'becoming' of the frail body. What clinicians refer to as 'frailty' is one becoming among many, produced during the daily activities and interactions of older people. Acknowledging the complexity of these more-than-human becomings is essential to make sense of frailty, and how to support and enhance the lives of frail older people.

The objective of this paper is to develop models for the estimation of the temporal and spatial extent of congestion impact caused by accidents. Although there have been various approaches based on the deterministic queuing diagrams and kinematic wave (or shockwave) theory, only a few studies have been able to estimate the spatiotemporal congested region based on field data, such as ubiquitous loop detector data. Accordingly, this paper applies a previously developed procedure to capture the spatiotemporal accident impacts based on binary integer programming (BIP). The procedure provides a foundation for models of the following: 1) maximum spatial distance to the end of the congestion region affected by each accident and 2) maximum time affected by congestion resulting from each accident. Based on these models, the objective of this paper is to estimate two statistical models for providing maximum congested distance and time information due to freeway accidents. Since various observations from BIP were censored with respect to time and space, survival analysis—specifically, frailty models to account for unobserved heterogeneity—is applied to identify factors critical to spatiotemporal congestion impacts of freeway accidents.

The article aims at describing in a unified framework all plateau-generating random effects models in terms of i) plausible distributions for the hazard (baseline mortality) and the random effect (unobserved heterogeneity, frailty), as well as ii) impact of frailty on the baseline hazard. Mortality plateaus result from multiplicative (proportional) and additive hazards, but not from accelerated failure time models. Frailty can have any distribution with regularly-varying-at- 0 density and the distribution of frailty among survivors to each subsequent age converges to a gamma. In a multiplicative setting the baseline cumulative hazard can be represented as the inverse of the negative logarithm of any completely monotone function. If the plateau is reached, the only meaningful solution at the plateau is provided by the gamma-Gompertz model.

ABSTRACT In this paper, a joint model is presented for analyzing longitudinal continuous and count mixed responses. The frequency distribution of continuous longitudinal response variable for each subject at any time has a skewed and or multi-modal form. Then, a suitable finite mixture of normals is used as its distribution. It seems that the continuous response comes from several distinct sub-populations. The number of zeros of the count response is inflated. Also, a zero-inflated power series (ZIPS) distribution is applied as its distribution in order to model the count response. The correlation of longitudinal responses through time and that of mixed continuous and count responses are modeled by utilizing the random-effects vectors in the finite mixtures of regression (FMR) models. Further, a full likelihood-based approach is used to obtain the maximum likelihood estimates of parameters via the EM algorithm. Then, some simulation studies are performed for assessing the performance of the model. Additionally, an application is illustrated for joint analysis of the number of days during the last month that the individual drank alcohol, as well as the respondents’ weight. Finally, the two first times of the Americans Changing Lives survey are evaluated.

The best policy for an insurance company is that which lasts for a long
period and it is less uncertain with reference to its claims. In information
theory, entropy is a measure of the uncertainty associated with a random
variable. It is a descriptive quantity as it belongs to the class of variability
measures, such as the variance and the standard deviation. The purpose of
this paper is to investigate the effect of inflation, truncation and censoring
from below (use of a deductible) and truncation and censoring from above
(use of a policy limit) on the entropy. Losses are differentiated between per-
payment and per-loss (franchise deductible). In this context we study the
properties of the resulting entropies such as the residual loss entropy and the
past loss entropy which are the result of use of a deductible and a policy limit,
respectively. Interesting relationships between these entropies are presented.
The combined effect of a deductible and a policy limit is also studied. Finally,
we investigate residual and past entropies for survival models.

Adult human mortality is well captured by a gamma-Gompertz-Makeham model that accounts for the exponential increase in individual hazards, the existence of age-independent mortality component, and unobserved heterogeneity in the study population. The article studies the impact of neglecting statistically significant extrinsic mortality or frailty on human mortality measures: aggregate indicators like life expectancy, life disparity, entropy, and the Gini coefficient are little affected while the rates of individual and population aging, the modal age at death, and temporary life expectancy can be substantially distorted.