Survival analysis is concerned with finding models to predict the survival of patients or to assess the efficacy of a clinical treatment. A key part of the model-building process is the selection of the predictor variables. It is standard to use a stepwise procedure guided by a series of significance tests to select a single model, and then to make inference conditionally on the selected model. However, this ignores model uncertainty, which can be substantial. We review the standard Bayesian model averaging solution to this problem and extend it to survival analysis, introducing partial Bayes factors to do so for the Cox proportional hazards model. In two examples, taking account of model uncertainty enhances predictive performance, to an extent that could be clinically useful. 1 Introduction From 1974 to 1984 the Mayo Clinic conducted a double-blinded randomized clinical trial involving 312 patients to compare the drug DPCA with a placebo in the treatment of primary biliary cirrhosis...

We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995) showed that BIC provides a close approximation to the Bayes factor when a unit-information prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is de ned in terms of the number of uncensored events instead of the number of observations. For the simplest censored data model, that of exponential distributions of survival times (i.e. a constant hazard rate), this revision results in a better approximation to the exact Bayes factor based on a conjugate unit-information prior. In the Cox proportional hazards regression model, we propose de ning BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space, and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.

. A minimax regret test is proposed for deciding whether one of N populations has slipped to the right of the rest, under the null hypothesis that all populations are identical. The problem is formulated as a multiple decision problem under uncertainty relative to a priori distribution of hypotheses and incomplete knowledge of probability distributions of observations. For the problem with nuisance parameters we use the combination of a minimax regret approach relative to the prior distribution and an invariance principle for nuisance parameters. The resulting minimax--invariant tests can be successfully applied to a variety of particular practical problems such as detection, identification, classification, and pattern recognition. The general results are illustrated by particular examples. Key words: slippage problem, minimax invariant test, minimax regret test, testing multiple hypotheses, least favorable distribution, prior uncertainty. AMS 1991 Subject Classification: Primary 62F05...

We show the combinatorics behind the Wilcoxon-Mann-Whitney two-sample test. This yields new combinatorial proofs of recurrences for its null distribution given recently by Brus and Chang, as well as new recurrences. It is shown how to convert these recurrences into generating functions. These generating functions are used to obtain closed expressions for the null distribution when one of the sample sizes is fixed and to compute moments. We also show how to perform these calculations with the aid of the computer algebra system Mathematica. Keywords Wilcoxon-Mann-Whitney two-sample test, partitions, Gaussian binomial coefficients, computer algebra, generating functions, recurrences. AMS classification 05A15, 05A17, 62-04, 62E15, 62E30, 62G10, 65U05 1 Introduction Let X 1 ; : : : ; Xm and Y 1 ; : : : ; Y n be independent random samples with continuous distribution functions F and G, respectively. In order to test whether X 1 is stochastically larger than Y 1 , Wilcoxon introduced in [1...

Summary.-Comparisons between oestrogen-receptor (RE)-positive or negative patients were made on a continuous series of 735 patients with primary breast tumours seen at the major treatment centre in British Columbia between 1975 and 1978. RE positivity was commoner in older patients, and was not associated with menopausal status independently of age. The concentration of receptor protein also increased with increasing age, but was not affected by menopausal status. Neither RE status nor quantity was associated with any of the epidemiological risk factors studied, which included parity, age at first birth, weight, family history and exposure to oestrogenic drugs and oral contraceptives. Patients with RE- tumours were more likely to present with symptoms other than a breast lump, pain or nipple inversion, and had less-differentiated tumours; they did not differ from RE+ patients in terms of stage, size of tumour, or interval from first symptom. These findings are discussed in terms of the biological origin and determinants of oestrogen receptors. BREAST-CANCER PATIENTS with oestrogen-receptor-positive (RE+) tumours differ from those with RE- tumours in their response to hormonal therapy (McGuire, 1978), survival from diagnosis (Bishop et al., 1979), time to first recurrence (Cooke et al., 1979; Knight et al., 1977) and probably in their response to chemotherapy (Lippman et al., 1978; Kiang et al., 1978). Oestrogen positivity has also been reported to vary with menopausal status (Allegra et al., 1979), age (Fisher et al., 1980), ethnic origin (Nomura et al., 1977) and features of tumour such as histological grade (Maynard et al., 1978), degree of elastosis (Masters et al., 1978), pathological type (Rosen et al., 1975; Kern, 1979) and cell-doubling time (McGuire, 1978). A report based on 45 patients suggests that RE- tumours are more common in women who have had an oophorectomy and have used oestrogenic drugs (Wallace et al., 1978). These many studies, each examining only a few factors, led us to

Abstract: We consider concepts and models for measuring inequality in the distribution of resources with a focus on how inequality varies as a function of covariates. Lorenz introduced a device for measuring inequality in the distribution of income that indicates how much the incomes below the u th quantile fall short of the egalitarian situation where everyone has the same income. Gini introduced a summary measure of inequality that is the average over u of the difference between the Lorenz curve and its values in the egalitarian case. More generally, measures of inequality are useful for other response variables in addition to income, e.g. wealth, sales, dividends, taxes, market share and test scores. In this paper we show that a generalized van Zwet type dispersion ordering for distributions of positive random variables induces an ordering on the Lorenz curve, the Gini coefficient and other measures of inequality. We use this result and distributional orderings based on transformations of distributions to motivate parametric and semiparametric models whose regression coefficients measure effects of covariates on inequality. In particular, we extend a parametric Pareto regression model to a flexible semiparametric regression model and give partial likelihood estimates of the regression coefficients and a baseline distribution that can be used to construct estimates of the various conditional measures of inequality. 1.

Several important multivariate probability inequalities can be formulated in terms of multivariate convolutions of the form J!I(x)h(x- (})dx, where usually!I = Ie is the indicator of a region C ~ R", h is a probability density on R n, and (} is a translation parameter. Often fl and f2 possess convexity, monotonicity, and/or symmetry properties. More general multivariate compositions of the form J h(x)f(xjO)J.l ( dx) also arise. Here several important convolution and composition theorems will be reviewed; these provide comparisons of Prob( differing multivariate distributions. The convolution theorems are to obtain concentration. or eurotrcau

For the analysis of data from matched pairs survival or reliability experiments the locally most powerful signed rank test under a bivariate exponential model is based on the signed rank exponential scores statistic S. In the present paper the null distribution of S is studied. Complete tables of its null distribution for sample sizes 2 through 10 are given as well as critical values for usual test levels for sample sizes 4 through 15. The accuracy of the Normal and Edgeworth approximations to the null distribution of S is investigated. A formula is given for the k t h moment of a general signed rank sum of scores statistic.-1-1.

g01dhc nag ranks and scores (g01dhc)

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