Abstract: For many years actuaries and demographers have been doing curve fitting of age-specific mortality data. We use the eight-parameter Heligman Pollard (HP) empirical law to fit the mortality curve. It consists of three nonlinear curves, child mortality, mid-life mortality and adult mortality. It is now well-known that the eight unknown parameters in the HP law are difficult to estimate because numerical algorithms generally do not converge when model fitting is done. We consider a novel idea to fit the three curves (nonlinear splines) separately, and then connect them smoothly at the two knots. To connect the curves smoothly, we express uncertainty about the knots because these curves do not have turning points. We have important prior information about the location of the knots, and this helps in the es timation convergence problem. Thus, the Bayesian paradigm is particularly attractive. We show the theory, method and application of our approach. We discuss estimation of the curve for English and Welsh mortality data. We also make comparisons with the recent Bayesian method.
Abstract: In Bayesian analysis of mortality rates it is standard practice to present the posterior mean rates in a choropleth map, a stepped statistical surface identified by colored or shaded areas. A natural objection against the posterior mean map is that it may not be the “best” representation of the mortality rates. One should really present the map that has the highest posterior density over the ensemble of areas in the map (i.e., the coordinates that maximize the joint posterior density of the mortality rates). Thus, the posterior modal map maximizes the joint posterior density of the mortality rates. We apply a Poisson regression model, a Bayesian hierarchical model, that has been used to study mortality data and other rare events when there are occurrences from many areas. The model provides convenient Rao-Blackwellized estimators of the mortality rates. Our method enables us to construct the posterior modal map of mortality data from chronic obstructive pulmonary diseases (COPD) in the continental United States. We show how to fit the Poisson regression model using Markov chain Monte Carlo methods (i.e., the Metropolis-Hastings sampler), and obtain both the posterior modal map and posterior mean map are obtained by an output analysis from the Metropolis-Hastings sampler. The COPD data are used to provide an empirical comparison of these two maps. As expected, we have found important differences between the two maps, and recommended that the posterior modal map should be used.
Abstract: The chi-squared test for independence in two-way categorical tables depends on the assumptions that the data follow the multinomial distribution. Thus, we suggest alternatives when the assumptions of multi nomial distribution do not hold. First, we consider the Bayes factor which is used for hypothesis testing in Bayesian statistics. Unfortunately, this has the problem that it is sensitive to the choice of prior distributions. We note here that the intrinsic Bayes factor is not appropriate because the prior distribu tions under consideration are all proper. Thus, we propose using Bayesian estimation which is generally not as sensitive to prior specifications as the Bayes factor. Our approach is to construct a 95% simultaneous credible re gion (i.e., a hyper-rectangle) for the interactions. A test that all interactions are zero is equivalent to a test of independence in two-way categorical tables. Thus, a 95% simultaneous credible region of the interactions provides a test of independence by inversion.
Abstract: When there is a rare disease in a population, it is inefficient to take a random sample to estimate a parameter. Instead one takes a random sample of all nuclear families with the disease by ascertaining at least one affected sibling (proband) of each family. In these studies, an estimate of the proportion of siblings with the disease will be inflated. For example, studies of the issue of whether a rare disease shows an autosomal recessive pattern of inheritance, where the Mendelian segregation ratios are of interest, have been investigated for several decades. How do we correct for this ascertainment bias? Methods, primarily based on maximum likelihood estimation, are available to correct for the ascertainment bias. We show that for ascertainment bias, although maximum likelihood estimation is optimal under asymptotic theory, it can perform badly. The problem is exasperated in the situation where the proband probabilities are allowed to vary with the number of affected siblings. We use two data sets to illustrate the difficulties of maximum likelihood estimation procedure, and we use a simulation study to assess the quality of the maximum likelihood estimators.