Orthogonalization of vectors with minimal adjustment

Two transformations are proposed that give orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim is that each component should be close to the vector with which it is paired, orthogonality imposing a constraint. The transformations lead to a variety of new statistical methods, including a unified approach to the identification and diagnosis of collinearities, a method of setting prior weights for Bayesian model averaging, and a means of calculating an upper bound for a multivariate Chebychev inequality. One transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors, and is determined using a new algorithm that also provides the decomposition of a positive-definite matrix in terms of a diagonal matrix and a correlation matrix. The algorithm is shown to converge to a global optimum

Incidence and survival of childhood bone cancer in northern England and the West Midlands, 1981–2002

There is a paucity of population-based studies examining incidence and survival trends in childhood bone tumours. We used high quality data from four population-based registries in England. Incidence patterns and trends were described using Poisson regression. Survival trends were analysed using Cox regression. There were 374 cases of childhood (ages 0–14 years) bone tumours (206 osteosarcomas, 144 Ewing sarcomas, 16 chondrosarcomas, 8 other bone tumours) registered in the period 1981–2002. Overall incidence (per million person years) rates were 2.63 (95% confidence interval (CI) 2.27–2.99) for osteosarcoma, 1.90 (1.58–2.21) for Ewing sarcoma and 0.21 (0.11–0.31) for chondrosarcoma. Incidence of Ewing sarcoma declined at an average rate of 3.1% (95% CI 0.6–5.6) per annum (P=0.04), which may be due to tumour reclassification, but there was no change in osteosarcoma incidence. Survival showed marked improvement over the 20 years (1981–2000) for Ewing sarcoma (hazard ratio (HR) per annum=0.95 95% CI 0.91–0.99; P=0.02). However, no improvement was seen for osteosarcoma patients (HR per annum=1.02 95% CI 0.98–1.05; P=0.35) over this time period. Reasons for failure to improve survival including potential delays in diagnosis, accrual to trials, adherence to therapy and lack of improvement in treatment strategies all need to be considered

The geometric combination of Bayesian forecasting models

A nonlinear geometric combination of statistical models is proposed as an alternative approach to the usual linear combination or mixture. Contrary to the linear, the geometric model is closed under the regular exponential family of distributions, as we show. As a consequence, the distribution which results from the combination is unimodal and a single location parameter can be chosen for decision making. In the case of Student t-distributions (of particular interest in forecasting) the geometric combination can be unimodal under a sufficient condition we have established. A comparative analysis between the geometric and linear combinations of predictive distributions from three Bayesian regression dynamic linear models, in a case of beer sales forecasting in Zimbabwe, shows the geometric model to consistently outperform its linear counterpart as well as its component models. Copyright © 2008 John Wiley & Sons, Ltd.