efficiencies of experimental designs for an illness death model disease development is described by the progressive markov illness death model with one disease the efficiencies of four experimental designs are discussed the simple survival experiment serial sacrifice periodic diagnosis and complete observation the discussion is based on the semiparametric assumption of piecewise constant intensities simple survival experiments are shown to have very low efficiency serial sacrifice experiments to be moderately efficient and periodic diagnosis to be almost as efficient as complete observation an application to the analysis of the development of leukaemia in laboratory mice is given

generalized logistic models abstract a class of models indexed by two shape parameters is introduced both to extend the scope of the standard logistic model to asymmetric probability curves and improve the fit in the noncentral probability regions one parameter subclasses can be used to examine symmetric or asymmetric deviations from the logistic model the delta algorithm is adapted to obtain maximum likelihood estimates of the parameters a review is made of other proposed generalizations the standard linear logistic model is widely used for modeling the dependence of binary data on explanatory variables its success is due to its broad applicability simplicity of form and ease of interpretation this model works well for many common applications however it assumes that the expected probability curve is skew symmetric about and that the shape of is the cumulative distribution function of the logistic distribution symmetric data with a shallower or steeper slope of ascent may not be fitted well by this model

e optimal designs for linear and nonlinear models with two parameters summary a method for constructing e optimal designs for a broad class of two parameter models is presented the procedure is based on the two dimensional geometry of the induced design space and of the first elfving set and guarantees to find e optimal designs which have exactly two points of support for e optimal designs based on three support points the procedure is less clear cut efficient designs based on essentially two points can be constructed using the geometry of the first elfving set but globally e optimal designs must at least in general be found by other strategies such as those based on symmetry considerations the geometry of the second elfving set systematic searches or numerical methods the methodology is illustrated by means of selected examples involving generalised linear models

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

elfving s theorem for d optimality we consider a model robust version of the c optimality criterion minimizing a weighted product with factors corresponding to the variances of the least squares estimates for linear combinations of the parameters in different models a generalization of elfving s theorem is proved for the optimal designs with respect to this criterion by an application of an equivalence theorem for mixtures of optimality criteria as a special case an elfving theorem for the d optimal design problem is obtained in the case of identical models the connection between the a optimality criterion and the model robust criterion is investigated the geometric characterizations of the optimal designs are illustrated by a couple of examples

calculation of carcinogenic potency from long term animal carcinogenesis experiments an index of carcinogenic potency for chemicals tested in chronic animal experiments is described by analogy with the well known lethal dose 50 ld50 of quantal bioassay a tumorigenic dose 50 td50 may be defined in the absence both of tumors in the control group and of intercurrent deaths as that daily dose of chemical which gives 50 of the test animals tumors by some fixed age tumors in the control zero dose group are handled exactly as for the ld50 and intercurrent deaths are handled by life table methods nonparametric procedures are developed for estimating the td50 and for constructing confidence intervals these are based on likelihoods which assume that the tumor hazard is linear in dose

group transformations and the nonlinear heat diffusion equation a nonlinear diffusion equation with a power law diffusion coefficient is studied self similar and partially invariant solutions are shown to be identical the theory of homology is introduced and used to generate classes of equations the solutions of which are related through backlund transformations an example of an application is given which generalizes the solution of storm j appl phys 22 1951 pp 940 951 on heat conduction in metals

analytical model of infiltration under constant concentration boundary conditions 1 known integrable models for 1d flow in unsaturated soil have a rescaled soil water diffusivity that is either constant or proportional to c c 1 c 2 where is the degree of saturation and c 1 is constant with a wider more realistic range of hydraulic conductivity functions than has been used in this context before a formal series solution is developed for infiltration subject to constant concentration boundary conditions a readily programmed iteration algorithm applicable for any value of c is used to construct many coefficients of the infiltration series without requiring any numerical integration in particular for either c 1 small or 1 c small several infiltration series coefficients are constructed as formal power series in c 1 or in 1 c for which we construct a number of terms explicitly in the limit as the diffusivity approaches a delta function the infiltration coefficients are obtained in simpler closed form all but the sorptivity depend on the form of the conductivity function

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

nonparametric prevalence and mortality estimators for animal experiments with incomplete cause of death data abstract age specific prevalence and mortality estimators are important descriptors of disease development and the subsequent effects of a disease on longevity nonparametric maximum likelihood estimators for the prevalence and mortality functions are available for the case in which incidental and fatal occurrences of a disease can always be distinguished this article generalizes these methods to allow the role of a disease in causing death to be uncertain for a subset of the animals dying with the disease the proposed analysis makes no assumptions about the degree of disease lethality data from sacrificed animals can be incorporated easily although sacrifices are not necessary if a certain representativeness assumption holds no restrictions are imposed on the distribution of survival times but the prevalence function is held constant over time intervals for stabilization purposes variances are estimated by inverting the observed information matrix an em algorithm simplifies the analysis when the

order restricted inference for multivariate binary data with application to toxicology in many applications researchers collect multivariate binary response data under two or more naturally ordered experimental conditions in such situations one is often interested in using all binary outcomes simultaneously to detect an ordering among the experimental conditions to make such comparisons we develop a general methodology for testing for the multivariate stochastic order between k 2 multivariate binary distributions our proposed test uses order restricted estimators which according to our simulation study are more efficient than the unrestricted estimators in terms of their mean squared error we compared the power of the proposed test with that of several alternative tests including procedures that combine individual univariate tests for order such as union intersection tests and a bonferroni based test we also compared the proposed test with an unrestricted hotelling t2 type test our simulations suggest that the proposed method competes well with these alternatives the gain i

tests for order restrictions in binary data in this article a general procedure is presented for testing for equality of k independent binary response probabilities against any given ordered alternative the proposed methodology is based on an estimation procedure developed in hwang and peddada 1994 annals of statistics 22 67 93 and can be used for a very broad class of order restrictions the procedure is illustrated through application to two data sets that correspond to three commonly encountered order restrictions simple tree order simple order and down turn order

immiscible two phase flow in a porous medium utilization of a laplace transform boost a boost operator is defined for the laplace transform it is used in conjunction with the laplace transform to solve the diffusion equation on a domain with multiple moving boundaries this problem arises in the study of immiscible multiphase flow in a finite medium including end effects extensions of this method to the solution of other linear differential equations are conceivable

effects of mismodeling on tests of association based on logistic regression models we consider the effects of misspecifying the covariate vector on tests of association between a particular covariate and the response variable in logistic regression models misspecification can include random measurement error discretizing a continuous explanatory variable and mismodeling the functional form of an explanatory variable the effects of misspecification are examined by deriving the asymptotic distribution of the test statistic under a sequence of models applications of this result are discussed

on the equivalence of constrained and compound optimal designs abstract constrained and compound optimal designs represent two well known methods for dealing with multiple objectives in optimal design as reflected by two functionals 1 and 2 on the space of information matrices a constrained optimal design is constructed by optimizing 2 subject to a constraint on 1 and a compound design is found by optimizing a weighted average of the functionals 1 1 2 0 1 we show that these two approaches to handling multiple objectives are equivalent

a geometric characterization of c optimal designs for heteroscedastic regression we consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables the c optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest a geometric characterization of c optimal designs in this context is presented which generalizes the classical result of elfving 1952 for c optimal designs as in elfving s famous characterization c optimal designs can be described as representations of boundary points of a convex set however in the case where there appear parameters of interest in the variance the structure of the elfving set is dierent roughly speaking the elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space the c optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new elfving set the theory is illustrated in several examples including pharmacokinetic models with random eects

an index of tumorigenic potency a new index of tumorigenic potency is derived using a survival adjusted quantal response estimator given by bailer and portier 1988 biometrics 44 417 431 the small sample bias and mean squared error properties of this estimator are explored using computer simulations of animal carcinogenicity experiments and are compared to the characteristics of the td50 an estimator proposed by sawyer et al 1984 biometrics 40 27 40 it is observed that this new index tends to have smaller bias and mean squared error relative to the td50 over varying shapes of the tumor onset distribution levels of tumor lethality and levels of treatment lethality

estimating the tumour onset distribution in animal carcinogenesis experiments summary in most survival sacrifice experiments for the detection and quantification of risks from chronic exposure to chemical agents the onset of the condition of interest is not clinically observable cancer is typically such an event in this paper an approximate maximum likelihood method is proposed for parametric estimation of the distribution of unobservable tumour onset times in the presence of competing risks when cause ofdeath information is unavailable

the use of a canonical form in the construction of locally optimal designs for non linear problems optimal experimental designs for non linear problems depend on the values of the underlying undknown parameters in the model for various reasons there is interest in providing explicit formulae for the optimal designs as a function of the unknown parameters this paper shows that for a certain class of generalized linear models the problem can be reduced to a canonical form this simplifies the underlying problem and designs are constructed for several contexts with a single variable using geometric and other arguments

exact transient solutions to nonlinear diffusion convection equations in higher dimensions the complete lie algebra of classical infinitesimal symmetries of the nonlinear diffusion convection equation in two and three dimensions is presented except for some cases involving constant diffusivity a complete reduction to an ordinary differential equation is not possible however closed form solutions are obtained for special forms of both the 2d and 3d nonlinear diffusion convection equations using a symmetry reduction and an additional physical constraint this extends the small list of closed form transient solutions already known

nonparametric methods for survival sacrifice experiments in many carcinogenicity studies the time to disease occurrence is not clinically observable a survival sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence a multistate model for disease development and death is considered and an algorithm of the em type for maximum likelihood estimation is obtained questions of identifiability and estimability are addressed under the model interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times a score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned

multiple solutions for discrete periodic nonlinear schrodinger equations in this paper we study the existence of multibump solutions for discrete nonlinear schrodinger equations with periodic potentials we first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group then we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear schrodinger equations and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case as an application we get without any symmetry assumptions infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity

bayesian models for multivariate current status data with informative censoring des donnees multivariees d etat consistent en un ensemble d indicatrices associees a la realisation de plusieurs evenements au moment d un examen nous nous interessons aux inferences sur les distributions jointes des temps d evenements les methodes conventionnelles d analyse des donnees de durees multiples ne peuvent etre utilisees parce que tous les temps d evenements sont censures et la censure peut etre informative nous prenons en compte la correlation des temps d evenements chez un sujet au moyen d une variable latente propre au sujet conditionnellement a laquelle les divers temps d evenements sont supposes independants nous supposons egalement que chaque evenement contribue independamment au risque de censure des fonctions en escalier non parametriques sont utilisees pour caracteriser les distributions de base des differents temps d evenements et des temps d examen les covariables et les effets specifiques du sujet sont incorpores grace au modele lineaire generalise un algorithme de monte carlo par chaines de markov est decrit pour l estimation des distributions a posteriori des inconnues les methodes sont illustrees par une application sur des donnees de sites multiples de tumeurs dans une etude de carcinogenicite animale

multi bump solutions for a strongly indefinite semilinear schrodinger equation without symmetry or convexity assumptions abstract in this paper we study the following semilinear schrodinger equation with periodic coefficient u v x u f x u u h 1 r n the functional corresponding to this equation possesses strongly indefinite structure the nonlinear term f x t satisfies some superlinear growth conditions and need not be odd or increasing in t using a new variational reduction method and a generalized morse theory we proved that this equation has infinitely many geometrically different solutions furthermore if the solutions of this equation under some energy level are isolated then we can show that this equation has infinitely many m bump solutions for any positive integer m 2

optimal weights for experimental designs on linearly independent support points an explicit formula is derived to compute the a optimal design weights on linearly independent regression vectors for the mean parameters in a linear model with homoscedastic variances the formula emerges as a special case of a general result which holds for a wide class of optimality criteria there are close links to iterative algorithms for computing optimal weights

effects of treatment induced mortality and tumor induced mortality on tests for carcinogenicity in small samples statistical tests of carcinogenicity are shown to have varying degrees of robustness to the effects of mortality mortality induced by two different mechanisms is studied mortality due to the tumor of interest and mortality due to treatment independent of the tumor the two most commonly used tests the life table test and the cochran armitage linear trend test are seen to be highly sensitive to increases in treatment lethality using small sample simulations increases in tumor lethality are seen to affect the performance of commonly used prevalence tests such as logistic regression a simple survival adjusted quantal response test appears to be the most robust of all the procedures considered

estimation of k for the poly k test with application to animal carcinogenicity studies this paper extends the survival adjusted cochran armitage test in order to achieve improved robustness to a variety of tumour onset distributions the cochran armitage test is routinely applied for detecting a linear trend in the incidence of a tumour of interest across dose groups to improve the robustness to the effects of differential mortality across groups bailer and portier introduced the poly 3 test by a survival adjustment using a fractional weighting scheme for subjects not at full risk of tumour development the performance of the poly 3 test depends on how closely it represents the correct specification of the time at risk weight in the data bailer and portier further suggested that this test can be improved by using a general k reflecting the shape of the tumour onset distribution in this paper we propose a method to estimate k by equating the empirical lifetime tumour incidence rate obtained from the data based on the fractional weighting scheme to a separately estimated cumulative lifetime tumour incidence rate this poly k test with the statistically estimated k appears to perform better than the poly 3 test which is conducted without prior knowledge of the tumour onset distribution our simulation shows that the proposed method improves the robustness to various tumour onset distributions in addition to the robustness to the effects of mortality achieved by the poly 3 test large sample properties are shown via simulations to illustrate the consistency of the proposed method the proposed methods are applied to analyse two real data sets one is to find a dose related linear trend on animal carcinogenicity and the other is to test an effect of calorie restriction on experimental animals

tumour incidence rate alternatives and the cause of death test for carcinogenicity summary lagakos louis 1988 and burnett krewski bleuer 1989 show that the cause ofdeath test of peto et al 1980 for carcinogenicity studies can be viewed as the approximate score test of a one dimensional joint hypothesis about tumour prevalences and the rates of death from tumour we relate this joint prevalence death rate hypothesis to hypotheses about differences in tumour incidence rates for two sample tests we show that when the control group tumour incidence is low when tumours are highly lethal or when the alternative is local the prevalence death rate family of alternatives is approximately equivalent to a proportional tumour incidence rate family more generally the prevalencedeath rate null hypothesis is always equivalent to a hypothesis of equal tumour incidence rates and each member of the family of prevalence death rate alternatives requires the tumour incidence rates not to cross using computer simulation we show that when the model does not hold the test of peto et al can be anti conservative and have poor power we discuss the model s implications for the interpretation of tests of regression hypotheses

bayesian optimal designs for estimating a set of symmetrical quantiles we propose multiple objective bayesian optimal designs for the logit model as an example we consider the design problem for estimating several percentiles with possibly unequal interest in each of the percentiles characteristics of these designs are studied and illustrated for the case when the interest lies in estimating the three quartiles we compare these optimal designs with the sequential designs generated via a generalized polya urn model and found the latter to be highly efficient in addition comparisons are made between locally optimal designs and bayesian optimal designs

a comparison of several methods to test for the existence of a monotonic dose response relationship in clinical and epidemiological studies the mantel extension chi square test for overall trend and an asymptotically equivalent test based on logistic regression are commonly used to test for a monotonic dose response relationship between exposure and disease in epidemiological and clinical studies however these tests present two important disadvantages as they i make the restrictive assumption of a parametric model of linear form on the logit scale and ii impose the a priori choice of scores to code for the exposure categories indeed the linear assumption if made incorrectly can lead to an invalid conclusion and the choice of scores lends arbitrariness to the test results some alternative tests have been proposed in the literature we have considered several of these tests namely one based on isotonic regression the t test based on contrasts and a recently published test based on adjacent contrasts dosemeci benichou test the aim of our study was to compare the statistical properties type i error and power of these tests and of the commonly used mantel extension test for overall trend we generated cohort and case control data and considered one and two sided versions of the tests moreover we studied the tests under the null hypothesis of no relationship between exposure and disease and under various alternative patterns of monotonic or non monotonic dose response relationships this study confirms that the commonly used trend tests can lead to erroneous conclusion of a monotonic dose response relationship the test based on isotonic regression does not represent a favourable alternative as it tends to be too powerful in case of non monotonic dose response relationship patterns the tests based on contrasts seem to possess more favourable properties by combining close to nominal type i error high power for monotonic alternatives and low power for non monotonic alternatives

a measure of tumorigenic potency incorporating dose response shape many researchers have considered the problem of ranking chemical agents based on their carcinogenic potency sawyer et al 1984 biometrics 40 27 40 proposed a carcinogenic potency estimate that incorporates both intercurrent mortality and background tumor rates since then many authors have either generalized the method outlined by sawyer et al or developed their own method based on a slightly different adjustment for treatment related changes in survival none of these methods however has estimated the shape of the dose response curve and incorporated such an estimate in potency estimation in this manuscript a measure of tumorigenic potency is proposed that utilizes the estimated shape of the dose response relationship in addition to estimated dose effects in order to rank chemicals on the basis of carcinogenic risk comparison of this new measure to that of sawyer et al is done using a large database of animal carcinogenicity experiments from the national cancer institute and the national toxicology program

nonparametric joint estimators for disease resistance and survival functions in survival sacrifice experiments the nonparametric estimators proposed in this paper simultaneously estimate the disease resistance function s t for the distribution of time to occurrence of disease and the survival function p t for the distribution of time to death caused by the disease the estimators are applicable to animal survival data when the disease of interest is irreversible and the cause of death of each animal can be determined diseases need not be classified as universally lethal or universally nonlethal and time to occurrence of the disease of interest need not be observable directly it is shown that the proposed estimators are maximum likelihood estimators under the condition that s t p t is monotonically decreasing application of the derived estimators is illustrated with experimental survival sacrifice data and results of a simulation study of the estimators performance are reported

age adjusted exact trend tests in the event of rare occurrences summary preclinical animal carcinogenicity studies are usually concerned with testing the statistical significance of a dose response relationship when the response consists of a rare event such as the development of a certain type of tumor exact statistical methods are often employed the exact randomization trend test based on the multivariate hypergeometric distribution is less powerful in the presence of treatment related risks other than the specified response particularly the loss of power becomes more pronounced when competing risks cause progressively higher mortality rates with increasing dose which is usual in practice an age adjusted form of the randomization test is proposed to adjust for this effect permutational distribution for peto s cause of death cod test is also explored and compared with its asymptotic counterpart by simulation the use of cod information has been a controversial issue due to the subjectivity in the pathologists determinations as well as for economic reasons the proposed age adjusted exact test does not require cod and it is shown to compare favorably to the cod tests via an extensive monte carlo simulation applications of the methods to two real data sets are included

ratio estimates the delta method and quantal response tests for increased carcinogenicity this paper demonstrates the use of the delta method for estimating the variance of ratio statistics derived from animal carcinogenicity experiments the cochran armitage test cochran 1954 biometrika 10 417 451 and armitage 1955 biometrics 11 375 386 is routinely applied to carcinogenicity data as a test for linear trend in lifetime tumor incidence rates the computing formula for this test derives from the assumption that the denominators of the quantal response rates are fixed however when time at risk weights are introduced to correct for treatment related differences in survival the denominators of the quantal response rates are subject to random variation the delta method and weighted least squares techniques are applied here to approximate the variance of such ratio statistics and test for a linear dose response relationship among treatments this technique is compared to that of bailer and portier 1988 biometrics 44 417 431 who introduced a survival adjusted quantal response test for trend in lifetime tumor incidence rates their test modifies the usual cochran armitage computing formula by weighting the denominators of the response rates to reflect less than whole animal contributions to risk within the framework of a weighted least squares linear regression model that underlies the cochran armitage test the time at risk weights of bailer and portier are incorporated using the delta method although the delta method approach is slightly more computationally intensive small sample simulations indicate that it has superior operating characteristics over the poly 3 trend test of bailer and portier when background tumor incidence rates are low under 3 and survival patterns differ markedly across treatments abstract truncated at 250 words

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

locally ϕp optimal designs for generalized linear models with a single variable quadratic polynomial predictor finding optimal designs for generalized linear models is a challenging problem recent research has identified the structure of optimal designs for generalized linear models with single or multiple unrelated explanatory variables that appear as first order terms in the predictor we consider generalized linear models with a single variable quadratic polynomial as the predictor under a popular family of optimality criteria when the design region is unrestricted our results establish that optimal designs can be found within a subclass of designs based on a small support with symmetric structure we show that the same conclusion holds with certain restrictions on the design region but in other cases a larger subclass may have to be considered in addition we derive explicit expressions for some d optimal designs

first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

nonparametric estimation of lifetime and disease onset distributions from incomplete observations in this paper we derive and investigate nonparametric estimators of the distributions of lifetime and time until onset associated with an irreversible disease that is detectable only at death the nonparametric maximum likelihood solution requires an iterative algorithm an alternative though closely related pair of estimators for the lifetime and onset distributions exists in closed form these estimators are the familiar kaplan meier estimator and an isotonic regression estimator respectively first order approximations provide variance estimators the proposed methods generalize and shed additional light on the constrained estimators presented by kodell shaw and johnson 1982 biometrics 38 43 58 data from an animal experiment illustrate the techniques

potential conservation laws we prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws therefore characteristics of pure potential conservation laws have to essentially depend on potential variables this statement provides a significant generalization of results of the recent paper by bluman et al j math phys 47 113505 2006 moreover we present extensions to gauged potential systems abelian and general coverings and general foliated systems of differential equations an example illustrating possible applications of these results is given a special version of the hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

analytical and numerical investigations of spontaneous imbibition in porous media we present semi analytical solutions for co current displacements with some degree of counter current flow the solution assumes a one dimensional horizontal displacement of two immiscible incompressible fluids with arbitrary viscosities and saturation dependent relative permeability and capillary pressures we address the impact of the system length on the degree of counter current flow when there is no pressure drop in the non wetting phase across the system assuming negligible capillary back pressure at the inlet boundary of the system it is shown that in such displacements the fractional flow can be used to determine a critical water saturation from which regions of both co current and counter current flow are identified this critical saturation changes with time as the saturation front moves into the porous medium furthermore the saturation profile in the approach presented here is not necessarily a function of distance divided by the square root of time we also present approximate solutions using a perturbative approach which is valid for a wide range of flow conditions this approach requires less computational power and is much easier to implement than the implicit integral solutions used in previous work finally a comprehensive comparison between analytical and numerical solutions is presented numerical computations are performed using traditional finite difference formulations and convergence analysis shows a generally slow convergence rate for water imbibition rates and saturation profiles this suggests that most coarsely gridded simulations give a poor estimate of imbibition rates while demonstrating the value of these analytical solutions as benchmarks for numerical studies complementing buckley leverett analysis this article is protected by copyright all rights reserved

efficiency robust score tests for rodent tumourigenicity experiments summary the standard statistical test for assessing adverse effects in rodent tumourigenicity experiments is based on a time adjusted analysis of lethal and incidental tumours this test is shown to arise as the sum of two score statistics for fatal and incidental lesions respectively the former optimal against extreme value hazard alternatives and the latter optimal against logistic prevalence alternatives in terms of asymptotic relative efficiency in the pitman sense an efficiency robust test is developed which retains near full efficiency in the extreme value logistic situation as well as in other situations where the standard test may be only 70 efficient rodent tumourigenicity experiments are used to determine whether or not a particular agent results in an increase in tumour occurrence in animals exposed to this substance throughout their lifetimes as compared to unexposed controls the tumour status of an individual animal is normally determined at the time of death since many animals will die either from natural mortality or as a result of tumour development prior to the end of the study period information on tumourigenicity will accrue throughout the course of the experiment typically lasting about two years animals surviving to the end of the experiment are sacrificed and subjected to the same pathological evaluation as those dying prior to the termination time peto et al 1980 have proposed life table analysis of the lethal or fatal tumours and a prevalence analysis of the nonlethal or incidental tumours in the present paper we show that when the substance under study affects the occurrence of fatal and incidental lesions in a similar fashion their test can be formally derived as the sum of two score statistics for fatal and incidental tumours respectively the former statistic is the score test under extreme value hazard alternatives whereas the latter statistic is optimal against logistic prevalence alternatives because the true form of the underlying hazard and prevalence functions is unknown we develop in 3 an efficiency robust version of this test this is done by maximizing the minimum asymptotic relative efficiency within a finite class of hazard and prevalence functions applications of the maximin efficiency robust test are discussed in 4

semiparametric bayes multiple testing applications to tumor data in national toxicology program ntp studies investigators want to assess whether a test agent is carcinogenic overall and specific to certain tumor types while estimating the dose response profiles because there are potentially correlations among the tumors a joint inference is preferred to separate univariate analyses for each tumor type in this regard we propose a random effect logistic model with a matrix of coefficients representing log odds ratios for the adjacent dose groups for tumors at different sites we propose appropriate nonparametric priors for these coefficients to characterize the correlations and to allow borrowing of information across different dose groups and tumor types global and local hypotheses can be easily evaluated by summarizing the output of a single monte carlo markov chain mcmc two multiple testing procedures are applied for testing local hypotheses based on the posterior probabilities of local alternatives simulation studies are conducted and an ntp tumor data set is analyzed illustrating the proposed approach

a dose response test via closed form solutions for constrained mles in survival sacrifice experiments in most survival sacrifice experiments in animal carcinogenicity studies the onset of the tumour of interest is not clinically observable due to the complexity of constraints for a biological justification recently developed methods for estimating the tumour onset function and tumour specific survival function employ computer intensive numerical solutions in this paper closed form solutions for nonparametric maximum likelihood estimators are derived under explicit and implicit inequality constraints obtained from the monotonicity of the survival functions our methods do not require cause of death information the proposed methods can be used to estimate the tumour onset function and the survival function of the tumour of interest we use the proposed estimators for the development of our new dose response trend test a modification of the poly k test is made by replacing the time at risk weight to a function of the tumour onset survival function the weighted least square regression method is applied to the estimated survival functions in order to construct a dose response trend test a simulation study is conducted to evaluate the performance of the proposed test and compare it with existing trend tests a real example is used to illustrate the methods

estimation of the proportional hazard in two treatment group clinical trials two easily computed approximate methods of obtaining an estimate of the ratio of the hazard rates for two treatments in survival analysis are compared by simulation with the partial maximum likelihood cml estimator which assumes the cox model with no covariates the two methods studied are the mantel haenszel estimator and the o e method which employs the ratio of the relative death rates for the two treatments uncensored and censored cases are examined for both continuous and tied data with the assumption of either an underlying exponential distribution or a weibull distribution with shape parameter 2 the o e method is biased but within the range of values of the ratio of the hazard rates of interest in clinical trials it is more efficient in terms of mean square error than either cml or the mantel haenszel method for all but the largest trials the mantel haenszel method is minimally biased gives answers very close to those obtained using cml and may be used to provide satisfactory approximate confidence intervals

ground state and multiple solutions via generalized nehari manifold abstract by using variational methods and the generalized nehari manifold due to szulkin and weth the existence of the ground states and the multiplicity of solutions for a wide class of superlinear elliptic equations is studied

inverse transformations and the reduction of nonlinear dirichlet problems involutory transformations are introduced which allow the solution of a class on nonlinear dirichlet problems the application of coupled involutory and backlund transformations is also described

sequential experimental designs for generalized linear models we consider the problem of experimental design when the response is modeled by a generalized linear model glm and the experimental plan can be determined sequentially most previous research on this problem has been limited either to one factor binary response experiments or to augmenting the design when there are already sufficient data to compute parameter estimates we suggest a new procedure for the sequential choice of observations that offers five important advantages 1 it can be applied to multifactor experiments and is not limited to the one factor setting 2 it can be used with any glm not just binary responses 3 both fully sequential and group sequential settings are treated 4 it enables efficient design from the outset of the experiment and 5 the experimenter is not constrained to specify a single model and can use the prior to reflect uncertainty as to the link function and the form of the linear predictor our procedure is based on a d optimality criterion and on a bayesian

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

a comparison of tumour incidence analyses applicable in single sacrifice animal experiments monte carlo methods are used to evaluate the operating characteristics of several trend tests for comparing incidence rates of occult tumours the candidate tests are applicable in long term animal experiments that have just one sacrifice time and no information on cause of death when survival decreases with dose and the tumour is lethal the tests that do not adjust for survival or that assume tumours are non lethal behave conservatively whereas the test that assumes tumours are rapidly lethal rejects the hypothesis of equal incidence rates too frequently the only test which consistently operates at the correct level is one that specifies a constant difference between the death rates for animals with and without the tumour surprisingly the test that specifies a constant ratio of these death rates often is conservative finally a test based on a simple modification of the unadjusted test which accounts for differential mortality by scaling down the size of the risk set performs reasonably well in many cases among the tests that operate at the proper level the constant risk difference test consistently exhibits high power across a wide range of situations

a penalized likelihood approach for a progressive three state model with censored and truncated data application to aids summary we consider the estimation of the intensity and survival functions for a continuous time progressive three state semi markov model with intermittently observed data the estimator of the intensity function is defined nonparametrically as the maximum of a penalized likelihood we thus obtain smooth estimates of the intensity and survival functions this approach can accommodate complex observation schemes such as truncation and interval censoring the method is illustrated with a study of hemophiliacs infected by hiv the intensity functions and the cumulative distribution functions for the time to infection and for the time to aids are estimated covariates can easily be incorporated into the model

optimal designs for dose response models with restricted design spaces in dose response studies the dose range is often restricted due to concerns over drug toxicity and or efficacy we present restricted and unrestricted interval locally optimal designs with respect to a very general class of optimality criteria for estimating the underlying dose response curve the underlying curve belongs to a diversified set of link functions suitable for the dose response studies and having a common canonical form these include the fundamental binary response models the logit and the probit as well as the skewed versions of these models the results are illustrated through the re design of a dose ranging trial conducted at the merck research laboratories zeng and zhu 1997 this work is a generalization of the results of dai and zhu 2002 in terms of the design interval the underlying dose response curve and the optimality criterion

nonparametric estimation of the distribution of time to onset for specific diseases in survival sacrifice experiments this paper concerns the analysis of an animal survival sacrifice experiment designed to investigate the incidence of a particular disease of interest the disease is assumed to be irreversible and detectable only at death for example by a necropsy each observation can be of one of three types i death caused by the disease ii death from a competing cause such as sacrifice with the disease present or iii death with the disease absent a two dimensional em algorithm is proposed for the nonparametric maximum likelihood estimation of the distributions of the time to onset and of the time to death from the disease these can be compared with nonparametric estimators recently proposed by kodell shaw and johnson 1982 biometrics 38 43 58 and by dinse and lagakos 1982 biometrics 38 921 932 a slight modification of the algorithm permits the construction of likelihood based interval estimates of quantiles of the distributions some extensions and generalizations are indicated

weighted p value adjustments for animal carcinogenicity trend test summary a typical animal carcinogenicity experiment routinely analyzes approximately 10 30 tumor sites comparisons of tumor responses between dosed and control groups and dose related trend tests are often evaluated for each individual tumor site type separately p value adjustment approaches have been proposed for controlling the overall type i error rate or familywise error rate fwe however these adjustments often result in reducing the power to detect a dose effect this paper proposes using weighted adjustments by assuming that each tumor can be classified as either class a or class b based on prior considerations the tumors in class a which are considered as more critical endpoints are given less adjustment two weighted methods of adjustments are presented the weighted p adjustment and weighted oa adjustment a monte carlo simulation shows that both weighted adjustments control the fwe well furthermore the power increases if a treatment dependent tumor is analyzed as in class a tumors and the power decreases if it is analyzed as in class b tumors a data set from a national toxicology program ntp 2 year animal carcinogenicity experiment with 13 tumor types sites observed in male mice was analyzed using the proposed methods the modified poly 3 test was used to test for increased carcinogenicity since it has been adopted by the ntp as a standard test for a dose related trend the unweighted adjustment analysis concluded that there was no statistically significant dose related trend using the food and drug administration classification scheme for the weighted adjustment analyses two rare tumors with background rates of 1 or less were analyzed as class a tumors and 11 common tumors with background rates higher than 1 as class b both weighted analyses showed a significant dose related trend for one rare tumor

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

estimating tumor incidence rates in animal carcinogenicity experiments tumor incidence is the primary measure of carcinogenesis this article focuses on estimating time dependent incidence rates in animal experiments with few sacrifices when the context of observation is known for none or all of the animals dying with the tumor of interest previous results are obtained under relaxed assumptions the link with existing semiparametric and nonparametric procedures based on latent failure times is exploited by using these methods to compute maximum likelihood estimates of the incidence rates without introducing latent random variables nonparametric estimators that are appropriate when all contexts of observation are known are generalized to the case in which the contexts of observation are unknown for a subset of the tumor bearing animals

the equivalence problem for the heat equation in this note we ask for the classes of equations of the second order which can be transformed into the heat equation ut uxx to give a partial answer to the question we express the heat equation by differential forms and prolong it by the estabrook wahlquist method this is motivated by the fact that our analysis is based upon conservation laws for which ideals of differential forms are a very suitable framework necessary conditions are derived for deciding whether a given equation can be transformed by some invertible point transformation into the heat equation or into its prolongation in particular the prolongation method enables us to understand the connection of various equations to the heat equation

potential equivalence transformations for nonlinear diffusion convection equations potential equivalence transformations pets are effectively applied to a class of nonlinear diffusion convection equations for this class all possible potential symmetries are classified and a theorem on their connection with point symmetries via pets is also proved it is shown that the known nonlocal transformations between equations under consideration are nothing but pets the action of pets on sets of exact solutions of a fast diffusion equation is investigated

efficient design for estimation of median lethal dose and quantal dose response curves summary the results of quantal dose response experiments are often summarized by an estimate of the median lethal dose denoted ld50 and many sequential designs have been proposed for efficient estimation of ld50 these designs strive to produce a sequence of trials at dose levels that get closer and closer to ld50 consequently they may not provide very good estimates of the overall shape of the dose response curve in this paper we propose guidelines for the design of experiments that estimate ld50 fairly efficiently and that also allow for efficient global estimation of the curve

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

modeling and identifying optimum designs for fitting dose response curves based on raw optical density data abstract for raw optical density rod data such as those generated in biological assays using a 96 well plate reader d optimal designs are identified for a family of homogeneous nonlinear models with two parameters developed for fitting and studying dose response curves further within the class of k point equally spaced and uniform designs d optimal designs are specified also the d efficiency of the d optimal equally spaced and uniform k point designs are compared to d optimal design for 3 k 7 the efficiency robustness of these optimal designs with respect to initial nominal values of the parameters are investigated in addition d optimal designs are derived for a family of heterogeneous nonlinear models

large sample tests for binary outcomes in fixed dose combination drug studies several test statistics are developed for testing the hypothesis that the combination of two drugs at a fixed dose regimen is more effective than both of the single drugs used alone with respect to a dichotomous response variable the response probability logit and arcsine root scales are considered the power function and the significance level are derived for large samples for the sample size per group of 20 or greater the power and type i error rate can be accurately calculated using the large sample power function when the response probability ranges from 0 2 to 0 8 these tests have similar power behaviors in small samples the large sample power functions of two of the tests can severely underestimate the type i error rate while overestimation can occur with one other test the utilities of these tests are extended to unbalanced sample size cases generally speaking there is a loss of power with unequal sample size allocation but the loss is not severe

extension of the fujita solution to air and water movement in soils for specific soil properties the fundamental solution of fujita 1952b is extended to include the effect of air movement on horizontal absorption this exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by parlange et al 1982 and sander and parlange 1984 the optimal sorptivity of parlange et al has very good accuracy while the surface water content s is near saturation but decreases in accuracy as s decreases the approximate sorptivity of sander and parlange however proves to be extremely accurate for all s with a maximum error of only 0 5 sander and parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0 5 the aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above alternatively the solution may be used for checking the accuracy of numerical models

using historical controls to adjust for covariates in trend tests for binary data abstract historical data often play an important role in helping interpret the results of a current study this article is motivated primarily by one specific application the analysis of data from rodent carcinogenicity studies by proposing a suitable informative prior distribution on the relationship between control outcome data and covariates we derive modified trend test statistics that incorporate historical control information to adjust for covariate effects frequentist and fully bayesian methods are presented and novel computational techniques are developed to compute the test statistics several attractive theoretical and computational properties of the proposed priors are derived in addition a semiautomatic elicitation scheme for the priors is developed our approach is used to modify a widely used prevalence test for carcinogenicity studies the proposed methodology is applied to data from a national toxicology program carcinogenicity experiment and is shown to provide helpful insight on t

explicit solutions for constrained maximum likelihood estimators in survival sacrifice experiments summary regulatory agencies routinely base conclusions regarding carcinogenicity of compounds tested in long term animal studies on tests which are known to be biased under conditions of treatment lethality and tumour lethality the recognition of these biases has led to a variety of proposed approaches which rely on survival and sacrifice data williams portier 1992 derived analytic expressions for maximum likelihood estimators of the tumour incidence rate and discrete death rates based on a discrete multistate model a disadvantage of the proposed estimators was that they sometimes resulted in negative estimates of the tumour incidence rate in this paper explicit solutions for constrained estimators are derived under the imposition of boundary conditions for a study design with one or two interim sacrifices and a terminal sacrifice for study designs with more than two interim sacrifices alternative estimators of the tumour incidence rate and discrete death rates are developed heuristically by pooling data together from adjacent

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

a bayesian analysis of quantal bioassay experiments incorporating historical controls via bayes factors this paper addresses model based bayesian inference in the analysis of data arising from bioassay experiments in such experiments increasing doses of a chemical substance are given to treatment groups usually rats or mice for a fixed period of time usually 2 years the goal of such an experiment is to determine whether an increased dosage of the chemical is associated with increased probability of an adverse effect usually presence of adenoma or carcinoma the data consists of dosage survival time and the occurrence of the adverse event for each unit in the study to determine whether such relationship exists this paper proposes using bayes factors to compare two probit models the model that assumes increasing dose effects and the model that assumes no dose effect these models account for the survival time of each unit through a poly k type correction in order to increase statistical power the proposed approach allows the incorporation of information from control groups from previous studies the proposed method is able to handle data with very few occurrences of the adverse event the proposed method is compared with a variation of the peddada test via simulation and is shown to have higher power we demonstrate the method by applying it to the two bioassay experiment datasets previously analyzed by other authors copyright 2017 john wiley sons ltd

an infinite dimensional linking theorem without upper semi continuous assumption and its applications abstract we obtain a new infinite dimensional linking theorem this theorem replaces the upper semi continuous assumption in the classical infinite dimensional linking theorem of kryszewski and szulkin by a new assumption as an application of this theorem we obtain a nontrivial solution for a strongly indefinite periodic schrodinger equation with sign changing nonlinearity

tests for differences in tumor incidence based on animal carcinogenesis experiments abstract in animal carcinogenesis studies statistical methods that compare the production of occult tumors should compare tumor incidence rates in general the incidence rate cannot be identified even approximately without both survival and sacrifice information if information from frequent sacrifices is available however a function that approximates the incidence rate can be identified without assumptions about the relationship between tumor incidence and death some earlier proposals that based comparisons on tumor prevalence rates and other functions also compared tumor incidence rates under special conditions but these tests can be biased when the conditions are violated tests that compare functions approximating tumor incidence rates can be developed using survival sacrifice information but these tests may be seriously inefficient in special cases in which sacrifices are not necessary to identify the tumor incidence rate

order restricted dose related trend tests methods of isotonic regression are proposed to increase the power of common trend tests in situations where a monotonicity constraint is imposed upon the dose response function isotonic versions of cochran armitage type trend tests for binary response data are developed and the bootstrap method is used in finding the empirical distributions of the test statistics and their critical values the isotonic likelihood ratio test with a survival adjustment is also proposed this survival adjustment can be applied to the likelihood ratio test for either the order restricted or unrestricted parameter cases to achieve the isotonic modifications an amalgamation algorithm is applied when the observed dose response is non monotonic a monte carlo simulation study comparing these trend tests shows the advantages of the isotonic modifications and survival adjustment by applying the proposed methods to data from a toxicology and carcinogenesis study conducted as part of the national toxicology program the effect of ci pigment red 23 is investigated

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

on the periodic and asymptotically periodic nonlinear helmholtz equation abstract in the first part of this paper the existence of infinitely many l p standing wave solutions for the nonlinear helmholtz equation u u q x u p 2 u in r n is proven for n 2 and 0 under the assumption that q be a nonnegative periodic and bounded function and the exponent p lies in the helmholtz subcritical range in a second part the existence of a nontrivial solution is shown in the case where the coefficient q is only asymptotically periodic

an age adjusted bootstrap based poly k test the assumption of an asymptotic normal distribution of some test statistics may be invalid in certain dose response trend tests for instance the survival adjusted cochran armitage test known as the poly k test is asymptotically standard normal under the null hypothesis however the asymptotic normality is not valid if there is a deviation from the tumour onset distribution that is assumed in this test or if the competing risks survival rates differ across groups we develop an age adjusted bootstrap based method to assess the significance of assumed asymptotic normal tests for animal carcinogenicity data the proposed method differs from conventional bootstrap methods in the aspect of preserving the mortality rate in each dose group under the null hypothesis of equal tumour incidence rates among the groups we investigate an empirical distribution of the poly 3 p3 trend test statistic using the proposed age adjusted bootstrap based method and compare it with the p3 test statistic referenced to the assumed standard normal distribution a simulation study is conducted to evaluate the robustness of these tests to various weibull family tumour onset distributions the proposed method is applied to national toxicology program data sets to evaluate a dose related trend of a test substance on the incidence of neoplasms

a classification of pseudospherical surface equations of type ut uxxx g u ux uxx the equations of type ut uxxx g u ux uxx which describe pseudospherical surfaces are characterized a new class of such equations is obtained this class together with the korteweg de vries kdv equation the modified korteweg de vries mkdv equation and the linear equation completes the classification

a two phase free boundary problem for a nonlinear diffusion convection equation a two phase free boundary problem associated with a diffusion convection equation is considered the problem is reduced to a system of nonlinear integral equations which admits a unique solution for small times the system admits an explicit two component solution corresponding to a two component shock wave of the burgers equation the stability of such a solution is also discussed

use of historical controls in time adjusted trend tests for carcinogenicity we develop a method for incorporating historical control information into time adjusted tests for dose effects in carcinogenicity studies after discretizing the time scale we use a multinomial distribution to model the number of animals dying with tumor in each interval data from past studies are used to estimate the parameters characterizing the prior a score test derived from the resulting dirichlet multinomial generalizes the test of tarone 1982 biometrics 38 215 220 and reduces in the limit to the log rank test in the case of a diffuse prior the methodology is illustrated with data from a study of the fire retardant 2 2 bis bromomethyl 1 3 propanediol