Elja Arjas

A Graphical Method for Assessing Goodness of Fit in Cox's Proportional Hazards Model

Abstract Suggested here is a simple graphical method for studying the goodness of fit in Coxs regression model for survival data. The method is easy to use, as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation. The rationale behind the graphs is very intuitive: They make a direct comparison between observed and expected failure frequencies, as estimated from the model. In a correctly specified model one anticipates an approximate balance between such frequencies; otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data, and this shows in the graphs introduced here. In the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data. There is always one graph for each stratum. S...

BAYESIAN METHODS FOR ANALYZING MOVEMENTS IN HETEROGENEOUS LANDSCAPES FROM MARK-RECAPTURE DATA

Spatially referenced mark-recapture data are becoming increasingly available, but the analysis of such data has remained difficult for a variety of reasons. One of the fundamental problems is that it is difficult to disentangle inherent movement behavior from sampling artifacts. For example, in a typical study design, short distances are sampled more frequently than long distances. Here we present a modeling-based alternative that combines a diffusion-based process model with an observation model to infer the inherent movement behavior of the species from the data. The movement model is based on classifying the landscape into a number of habitat types, and assuming habitat-specific diffusion and mortality parameters, and habitat selection at edges between the habitat types. As the problem is computationally highly intensive, we provide software that implements adaptive Bayesian methods for effective sampling of the posterior distribution. We illustrate the modeling framework by analyzing individual mark-recapture data on the Glanville fritillary butterfly (Melitaea cinxia), and by comparing our results with earlier ones derived from the same data using a purely statistical approach. We use simulated data to perform an analysis of statistical power, examining how accuracy in parameter estimates depends on the amount of data and on the study design. Obtaining precise estimates for movement rates and habitat preferences turns out to be especially challenging, as these parameters can be highly correlated in the posterior density. We show that the parameter estimates can be considerably improved by alternative study designs, such as releasing some of the individuals into the unsuitable matrix, or spending part of the recapture effort in the matrix.

The Failure and Hazard Processes in Multivariate Reliability Systems

The component failures in a complex multi-component system are treated in the paper as a multivariate point process, and the system failure as a univariate point process derived from this. Our main concern is to extend the well-known concept of a hazard function to this general case. It is suggested that, e.g., the system failure hazard be defined as a stochastic process, viz., the compensator of the system failure counting process relative to the σ-fields describing the systems past. The connections between such a hazard process and the lifetime distribution classes of Arjas Arjas, E. 1981. A stochastic process approach to multivariate reliability systems: Notions based on conditional stochastic order. Math. Oper. Res.6 263--276. are also discussed.

Non‐parametric Bayesian Estimation of a Spatial Poisson Intensity

A method introduced by Arjas & Gasbarra (1994) and later modified by Arjas & Heikkinen (1997) for the non-parametric Bayesian estimation of an intensity on the real line is generalized to cover spatial processes. The method is based on a model approximation where the approximating intensities have the structure of a piecewise constant function. Random step functions on the plane are generated using Voronoi tessellations of random point patterns. Smoothing between nearby intensity values is applied by means of a Markov random field prior in the spirit of Bayesian image analysis. The performance of the method is illustrated in examples with both real and simulated data.

BAYESIAN ANALYSIS OF METAPOPULATION DATA

A Bayesian approach is used to develop a method for fitting a metapopulation model (the incidence function model) to data on habitat patch occupancy, providing esti- mates of the five model parameters. Parameter estimation is carried out using a Markov chain Monte Carlo sampler, and data augmentation is used to include the effect of missing data in the analysis. The Bayesian approach allows us to take into account uncertainty about the parameter estimates when making predictions with the model. We demonstrate the methods of parameter estimation and prediction with simulated data. We first simulated metapopulation dynamics in real habitat patch networks with given parameter values and sampled the simulated data. Parameters were estimated both from full data sets, and from data sets with data for many years treated as missing. These estimates were then used to predict the distribution of time to extinction in modified networks, where patch areas had been reduced so that the real parameter values led to metapopulation extinction within ;30 yr. We were successfully able to fit the model and found that, in some cases, the predictions can be sensitive to one of the parameters.

Transmission of Pneumococcal Carriage in Families: A Latent Markov Process Model for Binary Longitudinal Data

Abstract We present a Bayesian data augmentation model to estimate acquisition and clearance rates of carriage of Streptococcus pneumoniae (Pnc) bacteria. The panel observation data comprise 10 measurements of Pnc carriage (carrier/noncarrier of the bacteria) in all members of 97 families with young children over a period of 2 years. Using natural conditional independence assumptions, a transmission model is constructed for the unobserved dependent binary processes of the augmented data. The model explicitly considers carriage transmission within the family and carriage acquisition from the surrounding community. The joint posterior of the model parameters and the augmented data is explored by Markov chain Monte Carlo sampling. The analysis shows that in young children the rate of acquiring carriage of three common Pnc serotypes increases with age. In children less than 2 years old, the duration of carriage is longer than in older family members. Asymptomatic Pnc carriage is found highly transmittable between members of the same family. In young children, the estimated rate of acquiring carriage from a family member carrying Pnc is more than 20-fold to that from acquiring it from the community.

A note on random intensities and conditional survival functions

One of the interesting directions of research in IIASAs Population Program deals with the methodological aspects of population heterogeneity dynamics. The crucial notion in this analysis is the stochastic intensity which is widely used in the stochastic processes models of human morbidity and mortality or technical failure. This paper provides the probabilistic specification of this notion which gives an opportunity to use the results of modern general theory of processes in analyzing factors that influence demographic characteristics.

Approximating Many Server Queues by Means of Single Server Queues

Obtaining time dependent results for many server queues is, under general structural assumptions, a hard problem. This paper makes an attempt to approximate stochastically the behaviour of a general many server queue by using single server queues as stochastic bounds. We propose three alternative ways of constructing approximating single server queues. The first technique utilizes special classes of service time distributions new better than used, new worse than used, the second is via dividing the service times by the number of servers, and the third is based on a grouping idea of the customers. The first and third techniques yield in fact two bounding queues each, one of which is faster and one slower than the original s-server queue.

Increasing incidence of Type 1 diabetes – role for genes?

BackgroundThe incidence of Type 1 diabetes (T1DM) is increasing fast in many populations. The reasons for this are not known, although an increase in the penetrance of the diabetes-associated alleles, through changes in the environment, might be the most plausible mechanism. After the introduction of insulin treatment in 1930s, an increase in the pool of genetically susceptible individuals has been suggested to contribute to the increase in the incidence of Type 1 diabetes.ResultsTo explore this hypothesis, the authors formulate a simple population genetic model for the incidence change driven by non-Mendelian transmission of a single susceptibility factor, either allele(s) or haplotype(s). A Poisson mixture model is used to model the observed number of cases. Model parameters were estimated by maximizing the log-likelihood function. Based on the Finnish incidence data 1965–1996 the point estimate of the transmission probability was 0.998. Given our current knowledge of the penetrance of the most diabetic gene variants in the HLA region and their transmission probabilities, this value is exceedingly unrealistic.ConclusionsAs a consequence, non-Mendelian transmission of diabetic allele(s)/haplotype(s) if present, could explain only a small part of the increase in incidence in Finland. Hence, the importance of other, probably environmental factors modifying the disease incidence is emphasized.

Bayesian versus frequentist analysis of multiple quantitative trait loci with an application to an outbred apple cross

Abstract Two methods, following different statistical paradigms for mapping multiple quantitative trait loci (QTLs), were compared: the first is a frequentist, the second a Bayesian approach. Both methods were applied to previously published experimental data from an outbred progeny of a single cross between two apple cultivars (Malus pumila Mill.). These approaches were compared with respect to (1) the models used, (2) the number of putative QTLs, (3) their estimated map positions and accuracies thereof and (4) the choice of cofactor markers. In general, the strongest evidence for QTLs, provided by both methods, was for the same linkage groups and for similar map positions. However, some differences were found with respect to evidence for QTLs on other linkage groups. The effect of using cofactor markers which were selected differently was also somewhat different.