Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsA study of compound Poisson and compound negative binomial frailty models
Volume 3(2) · Pages 41–44
Research areasInsurance, Mortality, Demography, Risk Management
Abstract
Author-writtenIn this paper we have introduced the Compound Poisson Frailty model and Compound Negative Binomial Frailty Model and we have tried to get maximum likelihood estimators. We developed the distribution by use of Laplace transform. The likelihood equations have intrinsic in nature.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- In survival analysis, accounting for unknown differences in individual risks (heterogeneity) is critical because ignoring it can distort hazard rates and survival curves. View evidence ↗
- Incidence rates often exhibit a unimodal shape over time, rising to a maximum before declining, which necessitates specific modeling approaches. View evidence ↗
- The compound Poisson distribution is used as a frailty distribution to account for subgroups with zero frailty who never experience the event being studied. View evidence ↗
The question
- The authors aim to introduce the Compound Poisson and Compound Negative Binomial frailty models and derive their maximum likelihood estimators. View evidence ↗
What they did
- The authors used Laplace transforms to develop the distributions for the proposed frailty models. View evidence ↗
- The compound Poisson distribution is constructed by summing a Poisson-distributed number of independent and identical gamma-distributed random variables. View evidence ↗
- The compound negative binomial distribution is derived where the number of variables (N) follows a geometric or negative binomial distribution and each variable follows a gamma distribution. View evidence ↗
What they found
- The parameter 'r' categorizes the distributions: r ≥ 0 results in a power variance function distribution, r < 0 yields the compound Poisson distribution, and r = 0 represents the gamma distribution. View evidence ↗
- When r < 0, the integral of the hazard function over [0,∞) is finite, and the survival function is incomplete because some individuals have zero frailty and never experience the event. View evidence ↗
- The compound negative binomial distribution consists of a discrete part, where the probability of zero frailty is P(Z=0) = p^r, and a continuous part defined by a probability density function. View evidence ↗
- The authors found that the resulting likelihood equations for the estimators are intrinsic in nature. View evidence ↗
What it means
- The authors conclude that they have successfully introduced both the Compound Poisson and Compound Negative Binomial frailty models and derived their corresponding maximum likelihood estimators. View evidence ↗
Visual summary
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Distribution Classification by Parameter r
AI-generated conceptual illustration (Abstract mathematical probability distributions represented as geometric shapes). It explains the idea and does not show study data.
View source evidence ↗Introduction
To properly interpret result of survival analysis, we have to consider the fact that individual risks differ in, possibly, unknown ways. This heterogeneity may be difficult to assess, but it nevertheless of great importance. It may distort observed survival curves and hazard rates. This has been discussed by many authors e.g., Manton et.al, (1986) [9] , Vaupel and Yashin (1985) [11] and Hougaard (1984, 1986a, 1986b) [4,5,6] . When intensities are estimated, for instance by incidence rates, one may be interested in how they change as a function of time. Quite often they are seen to be rising at the start, reaching a maximum and then declining. Hence the intensity has a unimodel shape with a finite mode. Aalen (1988, 1992) [1,2] introduced the compound Poisson distribution as a frailty distribution. This model yields a subgroup of zero frailty, which survives forever. In spite of the fact that the density of the continuous part is only given as an infinite series which has to be calculated numerically, the distribution is mathematically convenient. It may also be seen as a natural choice. Similarly Compound Negative Binomial frailty Model is also useful in demography and medicine. Here also two parts have been studied separately.
In section 2 we study the Compound Poisson frailty model with the use of Laplace transform. In section 3 estimation of Compound Poisson frailty model has been done. In section 4 Compound Negative Binomial frailty model has been introduced. Section 5 discusses estimation of Compound Negative Binomial frailty model.
Compound Poisson frailty model
The compound Poisson distribution can be constructed as the sum of a Poisson distributed number of independent and identical gamma distributed random variables. When each individual experiences a random number of hits, each of a random size, this distribution can be viewed as a hit model.
where N is Poisson distributed with parameter , where is expected value of N , while , , … are independent and gamma distributed with ~ , . The Laplace transforms of gamma and Poisson distributions are given by 1 and respectively.
Let us make parameterization of , , as
Parameter divides the class of distributions in two major parts. The distribution is a power variance function (PVF) distribution, for 0. Aalen (1988) [1] It should be noted that when < 0 the integral of over [0,∞) is finite. Also note that the survival function is incomplete because a fraction of individuals has zero frailty that will never experience the event under study. Aalen (1992) [2] has used model to model the incidence of marriage of women born in Denmark. Marriage is an example of an event that does not happen to everybody. A certain percentage of individuals never marry and models of marriage incidence have to account for this. The fact that the observed incidence peaks around age 23 and becomes rather low after age 30 is therefore interpreted to be a selection phenomenon due to heterogeneity, meaning that those who are most prone to marriage will marry quite early, and the reminder will have a lower tendency to marry. A second application of Aalen (1992) [2] deals with fertility data in Norwegian woman. It is known that around 5% to 10% of all couples are unable to conceive children. Naturally, P Z 0 (where Z denotes the frailty to conceive a child) will be the probability of infertility, and the variation of the frailty variable Z expresses the varying fecund abilities among fertile couples. The model was applied by Hougaard et al. (1994) [4] to diabetic nephropathy onset data, a serious complication experienced by some diabetic patients. This model was also applied by Aalen and Tretli (1999) [3] to testicular cancer. Testicular cancer has two striking epidemiological features. First, its incidence has increased speedily over the past few decades and secondly, the incidence is greatest among younger men, and then declines from a certain age. The idea of the model is that a subgroup of men is particularly susceptible to testicular cancer, which results in selection over time. The model is fit to incidence data from the Norwegian Cancer Registry collected between 1953 and 1993. This work is extended by Moger et al. (2004a) [10] . Haukka et al. (2003) [8] applied the model to schizophrenia data from the Finnish population born 1950 -1968. They concluded that only a small part of the population is susceptible to schizophrenia and found increasing individual risk with higher age among the vulnerable part of the population.
Estimation of Compound Poisson distribution
After some simulation the p.d.f. of Compound Poisson variate is
Consider n observations , , … , from (3.1) so that the likelihood will be
After differentiating w.r.t. , and we have
The asymptotic equations with after some simplifications, we shall get following equations.
Where ̃ ∏ Taking n observations , , … , from (5.13.1), the likelihood L will be Limiting the summation of numerator for 1 and 2 that of denominator for 1, we get following equations after simplification.
Compound negative binomial distribution
Compound negative binomial distribution
References
8 in the original paper- Aalen OO. Heterogeneity in survival analysis. Statistics in Medicine 1988; 7:1121-1137.
- Aalen OO. Modelling heterogeneity in survival analysis by the compound Poisson distribution. Annals of Applied Probability 1992; 4:951-972
- Aalen OO, Tretli S. Analyzing incidence of testis cancer by means of a frailty model. Cancer Causes and Control 1999; 10:285-292
- Hougaard P. Life table methods for heterogeneous populations: Distributions describing the heterogeneity. Biometrika, 1984; 71(1):75-83.
- Hougaard P, Myglegaard P, Borch-Johnsen K. Heterogeneity models of disease susceptibility, with application to diabetic nephropathy. Biometrics 1994; 50:1178-1188.
- Haukka J, Suvisaari J, Lonnqvist J. Increasing age does not decrease risk of schizophrenia up to age 40. Schizophrenia Research. 2003; 61:105-110.
- Manton K, Stallard E, Vaupel J. Alternative models for heterogeneity of mortality risks among the aged. Journal of the American Statistical Association 1986; 81:635644.
- Vaupel JW, Yashin AI. Heterogeneity's ruses: Some surprising effects of selection on population dynamics. The American Statistician, 1985; 39:176-185.