Modelling Claims Run-off with Reversible Jump Markov Chain Monte Carlo Methods

In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual adhoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied

Organic solvents and MS susceptibility Interaction with MS risk HLA genes

Objective We hypothesize that different sources of lung irritation may contribute to elicit an immune reaction in the lungs and subsequently lead to multiple sclerosis (MS) in people with a genetic susceptibility to the disease. We aimed to investigate the influence of exposure to organic solvents on MS risk, and a potential interaction between organic solvents and MS risk human leukocyte antigen (HLA) genes. Methods Using a Swedish population-based case-control study (2,042 incident cases of MS and 2,947 controls), participants with different genotypes, smoking habits, and exposures to organic solvents were compared regarding occurrence of MS, by calculating odds ratios with 95% confidence intervals using logistic regression. A potential interaction between exposure to organic solvents and MS risk HLA genes was evaluated by calculating the attributable proportion due to interaction. Results Overall, exposure to organic solvents increased the risk of MS (odds ratio 1.5, 95% confidence interval 1.2–1.8, p = 0.0004). Among both ever and never smokers, an interaction between organic solvents, carriage of HLA-DRB1*15, and absence of HLA-A*02 was observed with regard to MS risk, similar to the previously reported gene-environment interaction involving the same MS risk HLA genes and smoke exposure. Conclusion The mechanism linking both smoking and exposure to organic solvents to MS risk may involve lung inflammation with a proinflammatory profile. Their interaction with MS risk HLA genes argues for an action of these environmental factors on adaptive immunity, perhaps through activation of autoaggressive cells resident in the lungs subsequently attacking the CNS

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system's other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.Comment: 33 pages, 9 figure

Relative risks and effective number of meioses: A unified approach for general genetic models and phenotypes

Many common diseases are known to have genetic components, but since they are non-Mendelian, i.e. a large number of genetic factors affect the phenotype, these components are difficult to localize. These traits are often called complex and analysis of siblings is a valuable tool for mapping them. It has been shown that the power of the affected relative pairs method to detect linkage of a disease susceptibility locus depends on the locus contribution to increased risk of relatives compared with population prevalence Risch, 1990a,b). In this paper we generalize calculation of relative risk to arbitrary phenotypes and genetic models, but also show that the relative risk can be split into the relative risk at the main locus and the relative risk due to interaction between the main locus and loci at other chromosomes. We demonstrate how the main locus contribution to the relative risk is related to probabilities of allele sharing identical by descent at the main locus, as well as power to detect linkage. To this end we use the effective number of meioses, introduced by Hossjer (2005a) as a convenient tool. Relative risks and effective number of meioses are computed for several genetic models with binary or quantitative phenotypes, with or without polygenic effects

On computation of p-values in parametric linkage analysis

Parametric linkage analysis is usually used to find chromosomal regions linked to a disease (phenotype) that is described with a specific genetic model. This is done by investigating the relations between the disease and genetic markers, that is, well-characterized loci of known position with a clear Mendelian mode of inheritance. Assume we have found an interesting region on a chromosome that we suspect is linked to the disease. Then we want to test the hypothesis of no linkage versus the alternative one of linkage. As a measure we use the maximal lod score Z(max). It is well known that the maximal lod score has asymptotically a (2 ln 10)(-1) x (1/2 chi(2)(0) + 1/2 chi(2)(1)) distribution under the null hypothesis of no linkage when only one point ( one marker) on the chromosome is studied. In this paper, we show, both by simulations and theoretical arguments, that the null hypothesis distribution of Z(max) has no simple form when more than one marker is used ( multipoint analysis). In fact, the distribution of Z(max) depends on the number of families, their structure, the assumed genetic model, marker denseness, and marker informativity. This means that a constant critical limit of Z(max) leads to tests associated with different significance levels. Because of the above-mentioned problems, from the statistical point of view the maximal lod score should be supplemented by a p-value when results are reported. Copyright (C) 2004 S. Karger AG, Basel

Improving the calculation of statistical significance in genome-wide scans

Calculations of the significance of results from linkage analysis can be performed by simulation or by theoretical approximation, with or without the assumption of perfect marker information. Here we concentrate on theoretical approximation. Our starting point is the asymptotic approximation formula presented by Lander and Kruglyak (1995, Nature Genetics, 11, 241-247), incorporating the effect of finite marker spacing as suggested by Feingold et al. (1993, American Journal of Human Genetics, 53, 234-251). We consider two distinct ways in which this formula can be improved. Firstly, we present a formula for calculating the crossover rate rho for a pedigree of general structure. For a pedigree set, these values may then be weighted into an overall crossover rate which can be used as input to the original approximation formula. Secondly, the unadjusted p-value formula is based on the assumption of a Normally distributed nonparametric linkage (NPL) score. This leads to conservative or anticonservative p-values of varying magnitude depending on the pedigree set structure. We adjust for non-Normality by calculating the marginal distribution of the NPL score under the null hypothesis of no linkage with an arbitrarily small error. The NPL score is then transformed to have a marginal standard Normal distribution and the transformed maximal NPL score, together with a slightly corrected value of the overall crossover rate, is inserted into the original formula in order to calculate the p-value. We use pedigrees of seven different structures to compare the performance of our suggested approximation formula to the original approximation formula, with and without skewness correction, and to results found by simulation. We also apply the suggested formula to two real pedigree set structure examples. Our method generally seems to provide improved behavior, especially for pedigree sets which show clear departure from Normality, in relation to the competing approximations

Local Polynomial Variance Function Estimation

Local Polynomial Variance Function Estimatio

Asymptotics of Generalized S-Estimators

An S-estimator of regression is obtained by minimizing an M-estimator of scale applied to the residuals ri. On the other hand, a generalized S-estimator (or GS-estimator) minimizes an M-estimator of scale based on all pairwise differences ri - rj. Generalized S-estimators have similar robustness properties as S-estimators, including a high breakdown point. In this paper we prove asymptotic normality for the GS-esimator of the regression parameters, as well as for the accompanying scale estimator defined by the minimal value of the objective function. It turns out that the asymptotic efficiency can be much higher than that of S-estimators. For instance, by using a biweight [rho]-function we obtain a GS-estimator with 50% breakdown point and 68.4% efficiency.

The Repeated Median Intercept Estimator: Influence Function and Asymptotic Normality

Given the simple linear regression model Yi = [alpha] + [beta]Xi + ei for i = 1, ..., n, we consider the repeated median estimator of the intercept [alpha], defined as [alpha]n = medi medj,j [not equal to] i (XjYi - XiYj)/(Xj - Xi). We determine the influence function and prove asymptotic normality for [alpha]n when the carriers Xi and error terms ei are random. The resulting influence function is bounded, and is the same as if the intercept is estimated by the median of the residuals from a preliminary slope estimator. With bivariate gaussian data the efficiency becomes 2/[pi] [approximate] 63.7%. The asymptotic results are compared with sensitivity functions and finite-sample efficiencies.