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A robust test for linear contrast using modified maximum likelihood estimators based on symmetrically censored samples proposed by Tiku (1973, 1982a) is studied in this paper from the Bayesian point of view. The effects of asymmetric censoring on this testing procedure is investigated and a good approximation to its posterior distribution in this case is worked out. We also present an example which illustrates the method of obtaining the highest posterior density interval for the linear combination of the unknown location parameters.

In this paper, I have shown that under some mild conditions, the number of initiated cells in an extended two-stage model of carcinogenesis can be approximated by a diffusion process. By using this approximation, I have derived the probability distribution for the number of initiated cells in terms of Laguerre polynomials under normal prevention conditions. This follows from the fact that many of the dietary components are antioxidants which would neutralize the hydroxyl free radicals and hence, reduce the proliferation rates of initiated cells to interrupt or slow down the promotion stage in carcinogenesis.

In this paper we have developed a state space model for carcinogenesis. By using this state space model we have also developed statistical procedures to estimate the unknown parameters via multi-level Gibbs sampling method. We have applied this model and the methods to the British physician data on lung cancer with smoking. Our results indicate that the tobacco nicotine is an initiator. If $t > 60$ years old, then the tobacco nicotine is also a promoter.

In this paper, we have developed some state space models for carcinogenesis involving multievent models and multiple pathways models. In these state space models, the stochastic system models are stochastic models of carcinogenesis expressed in terms of stochastic differential equations, whereas the observation models are statistical models based on the observed number of detectable preneoplastic lesions per individual over time and the observed number of detectable cancer tumors per individual over time. In this paper, we have applied some of the theories to some animal papillomas data from some initiation-promotion experiments on skin cancer in mice to estimate some unknown parameters. For this data set we have obtained excellent fit by a model with three piece-wise intervals.

By surveying recent studies by molecular biologists and cancer geneticists, in this paper we have proposed some stochastic models of carcinogenesis and provided some biological evidences for these models. Because most of these models are quite complicated far beyond the scope of the MVK two-stage model, the traditional Markov theory approach becomes too complicated to be of much use. In this paper, we thus propose an alternative approach through stochastic differential equations. For validating the model and for estimating unknown parameters, we further use these stochastic differential equations to develop state space models (Kalman filter models) for carcinogenesis. In this paper, we have used the multievent model as an example to illustrate our modeling approach and some basic theories. These theories will be used by these authors to analyze data from experiments by scientists at BPNNL (Battelle Pacific Northwest National Laboratory) in Richland, WA.

Assuming some general models for the HIV epidemic, in this paper this paper I derive the HIV incubation distributions under AZT treatment. It is shown that under some conditions, these probability distributions are mixtures of some generalized Gamma distributions and products of generalized Gamma distributions

In this paper, I derive some first passage probability distributions for some continuous time Markov chains. By using these results, I then derive some exact HIV incubation distributions under treatment by anti-viral drugs such as AZT under some general conditions. It is shown that under some special conditions, these distributions are mixtures of generalized Gamma distributions.

In this paper we develop a stochastic model for the HIV epidemic involving both sexual contact and IV drug use by taking into account the dynamic of the HIV epidemic. The model is formulated in terms of the chain multi-nomial model which provides a mechanism by means of which one may derive the moments of the numbers of S people, L people, I people and A people. By using the model we show by Monte Carlo studies that the randomness of the number of different partners per unit time seem to have little effects on the HIV epidemic if the expected number of different partners per unit time is not very small (say >= 0.5). However, the numerical results indicate clearly that in almost all the cases the results of the deterministic model provide poor approximation to the corresponding expected numbers of the stochastic model.

By assuming the Longini et al. staging system in this paper, we derive the probability distributions of times to pre-AIDS symptoms, to clinical AIDS and to death from AIDS under various conditions. Specific conditions involve random test for HIV antibodies, left censoring for HIV transmission and central interval censoring for seroconversion. Using these distribution results, we then derive the means and the variances of times to pre-AIDS symptoms, to clinical AIDS and to death from AIDS as well as the average incubation periods under various conditions. Some numerical results show clearly that these distributions were affected significantly by the transition parameters, the random test, and the censoring.

To describe the dynamics of the AIDS epidemic, the purpose of this paper is to develop a stochastic model for the AIDS epidemic in complex situations. To take into account different risk groups, social factors and levels of sexual activities, in this paper we consider l risk populations with each risk population being further classified into n subpopulations. To take into account some important biological aspects of AIDS epidemic, in each population we postulate an L person and divide the infectious persons (I persons) into @k stages and the AIDS cases (A persons) into m stages. Given these specifications, the probability generating function (PGF) of the numbers of L persons, infectious persons and AIDS cases is then derived under some general conditions. By using this PGF, it is shown that the expected values, the variances and the covariances of the latent persons, infective persons and AIDS cases satisfy some ordinary differential equations. These equations have been solved numerically to assess effects of various factors on the AIDS epidemic.

Extending my previous work [1-3], in this paper I proceed to develop a stochastic model for HIV epidemic in a homosexual population under general conditions. Through computer generated data, I assess various deterministic models as compared with the expected numbers of the stochastic models. It is shown that different mixing patterns have significant impacts on the HIV epidemic except possibly restricted mixing. Thus, in populations with preferred mixing (mixing proportion less than 1) and with proportional mixing, the numbers of S people, L people, I people and A people differ significantly from the corresponding expected numbers of the stochastic models. For the L people, I people and A people, the numbers of the deterministic models first appear to be smaller and later appear to be larger than the corresponding mean numbers of the stochastic models, indicating that while in the short run the deterministic models would underestimate the true numbers, in the long run the deterministic models would overestimate the true numbers.

In this paper I develop a stochastic model for the AIDS epidemic involving several risk populations. The probability generating function (PGF) of the latent persons, infective persons and AIDS cases is derived. By using this PGF, it is shown that the expected values, the variances and the covariances of the latent persons, infective persons and AIDS cases satisfy some ordinary differential equations. These equations have been solved numerically to assess effects of various factors on the AIDS epidemic.

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n1, …, nk)th degree, then all the mixed moments up to the order (n1, …, nk) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.

In this article we consider the unbalanced random-effects model as given in (1.1) with random variables from nonormal universes. Assuming that the αi 's are independently distributed of the eij 's, an approximation to the joint distribution of the mean squares (and hence the F ratio) by using Laguerre polynomials is derived. These results are then used to examine the effects of departure from normality on the traditional F-ratio distribution.

This article deals with the exact non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix in a multinormal distribution is diagonal. The exact non-null moments as well as the exact non-null distribution are derived. The distribution is also expressed in computable form with the help of inverse Mellin transform and calculus of residues. The results obtained in this article are useful in studying the power of testing several correlation coefficients simultaneously.

SUMMARY We propose a test of the null hypothesis of no treatment effect in a randomized clinical trial that is based on the randomization distribution of residuals. These residuals result from regressing the response on covariates, but not treatment. In contrast to model-based score tests, this procedure maintains nominal size when the model is misspecified, and, in particular, when relevant covariates are omitted from the regression. The efficiency of the procedure is evaluated for regressions with some, but not all, required covariates. For many generalized linear models and survival models, conventional model-based score tests are shown to have supranominal size when relevant covariates are omitted, but logistic regression and the proportional hazards model are robust.

Consider a two organs system for which one of the two organs must function for the individual to survive. This paper derives the survival probability distributions under more general situations by adopting [the] Markov process approach thus providing extensions of results given in Gross Clark and Liu (1971) and Kodlin (1967). The results obtained are then extended to a k-organs system. (EXCERPT)

This paper provides a Monte Carlo study of approximating the non-central Wishart distribution by Central Wishart distribution by mean of 1. Multivariate Gram-Chalier expansion and 2. Laguerre polynomial expansion. For assessing the closeness of these approximations, 1,000 independent 2×2 non-central Wishart matrices are generated by computer. The numerical results indicate that the multivariate Gram-Chalier expansion provides a close approximation to the non-central Wishart distribution as long as the correlation coefficient is less than 0.8. Also, it appears that the Gram-Chalier expansion approximation is better than the Laguerre polynomial expansion approximation when the probability values are large.

This paper derives the probabilities of obtaining negative estimates of additive and dominance genetic variances when one uses the traditional weighted least square method for estimating genetic variances as given in MATHER and JINKS (1971). The model considered involves P1, P2, F2, B1 (Backcross to P1) and B2 (Backcross to P2). The results are derived under the ordinary assumptions as made in the genetic literatures. It is shown that unless the genetic effects are very large and environmental effects small, the probabilities of obtaining negative estimates of additive and dominance variances are in general quite large.

In this paper we consider a cell population such as bacteria consisting of two types of cells, mutant and nonmutant. Under the mutation and homogeneous pure birth processes, this paper derives a maximum likelihood estimation procedure for estimating mutation rate and birth rate. The method is applied to Newcombe's data; further some Monte Carlo studies are generated. The numerical results indicate that the method is quite efficient for estimating genetic parameters in cell populations.

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber's M-estimators with the Brown-Forsythe test , several robust procedures were developed; these procedures were compared through computer simulation studies with-the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test . The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions

A robust test (to be referred to as M* test) is proposed for testing equality of several group means without assuming normality and equality of variances. This test statistic is obtained by combining Tiku's MML robust procedure with the James statistic. Monte Carlo simulation studies indicate that the M* test is more powerful than the Welch test, the James test, and the tests based on Huber's M-estimators over a wide range of nonnormal universes. It is also more powerful than the Brown and Forsythe test under most of nonnormal distributions and has substantially the same power as the Brown and Forsythe test under normal distribution. Comparing with Tan-Tabatabai test, M* is almost as powerful as Tan-Tabatabai test.

In this paper we extend the Moolgavkar-Venzon-Knudson two-stage model into nonhomogeneous cases. The probability generating functions (PGF) of initiated cells and tumors are derived under very general conditions. Using these PGFs we then obtain expected incidence functions of tumor and provide an iterative procedure for computing probability distributions of tumors. These results can be used to take account of time-varying exposures to help identify etiologic agents and to assess their mechanisms of action in epidemiologic studies.

Using a quantitative genetic model, this paper compares four different methods for estimating genetic variance components. Given various genetic parameters, data were generated and estimates computed. The number of negative estimates, the sample mean, the sample variance, and the sample mean squared error were computed for each method. It is shown that, if the genetic values are not very small, the traditional MATHER-JINKS method is at least as good as any other method. The ML method might be preferable only if the genetic values are very small and the number of loci large.

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.

In this paper, by combining the harmonic mean approach with the Welch and the James procedure (see WELCH 1951, JAMES, 1951), we develop some robust procedures for testing parallelism in several straight lines under heteroscedasticity and nonnormality. Through Monte Carlo simulations it is shown that these new tests are quite robust with respect to departure from normality. For small sample sizes, however, the TAN-TABATABAI (1984) Fβ and F*β tests appear to be more powerful than the new tests, although when sample sizes are not small, there are hardly any differences between the Tan-Tabatabai Fβ and F*β tests and the new tests.

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber‘s M-estimators with the Brown-Forsythe test, several robust procedures were developed; these procedures were compared through computer simulation studies with the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test. The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions.

In this paper we analyze a quantitative genetic character which is controlled by both major genes and polygenes. Assuming that there are no epistatic effects, no linkage and no genetic-environmental interactions, we follow TAN and CHANG (1972) to derive the probability distributions for segregating populations. The numbers of major genes and polygenes, and the additive and dominance effects of major genes and polygenes are then estimated by using the procedures developed in TAN and CHANG (1972) and the POWELL-FLETCHER search procedure for maximum values. In this paper, we consider the case involving data from P1, P2, F2, B1 (Backcross to P1) and B2 (Backcross to P2) as this type of experiment is common in practical applications. The analyses are applied to a simulated model generated by using binomial, multinomial and normal variables and to the data of an experiment on kernel weight of sorghum plant provided to the authors by Professor GEORGE H. L. LIANG of Kansas State University. The analysis of these data indicate clearly that the method derived in this paper is useful and desirable.

By using Monte Carlo studies, this paper compares the Welch test, the James test and the Brown-Forsythe test for comparing several means under heteroscedasticity. It appears that all the tests are quite robust with respect to departure from normality. The Brown-Forsythe test is at least as good as the James test and the Welch test; but the differences are so small that the choice is immaterial for practical purposes.

This paper gives some matrix results and some necessary and sufficient conditions for a matrix quadratic form in normal variables to be distributed as a difference of two independent Wishart distributions. Some extensions of Cochran’s theorem are then derived concerning differences of independent Wishart distributions.

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.

In this paper we consider a two-way layout random-effects model with interaction proportional to the product of two random components and derive the distributions of the sums of squares and the F-ratio under the usual normality assumption for the random effects. Approximations to these distributions by finite series Laguerre polynomial expansions are then developed. Some numerical results are given to illustrate the applications of the theory.

In this paper we consider the following nested random effect model where the αi's, the βij's and the eijk are independent random variables. By assuming that these variables follow a mixture of two normal densities, we study the effects of departure from normality on the classical JT-tests for variance components. It is shown that the departure from normality has little effects on the type 1 error and the power function; this indicates that the classical F-tests are quite robust with respect to departure from normality.

Some justifications for the Roy-Tiku (1962) approximation to the distributions of sample variances by Laguerre polynomial series are provided. It is shown that if one approximates any nonnegative continuous random variable by a finite series of Laguerre polynomials up to the kth degree, then the first k moments of the approximated distribution are equal to the first k moments of the random variable. We also derive an alternative Laguerre series approximation, including the Box (1953) approximation as a special case.

By applying Tiku's MML robust procedure to Brown and Forsythe's (1974) statistic, this paper derives a robust and more powerful procedure for comparing several means under hetero-scedasticity and nonnormality. Some Monte Carlo studies indicate clearly that among five nonnormal distributions, except for the uniform distribution, the new test is more powerful than the Brown and Forsythe test under nonnormal distributions in all cases investigated and has substantially the same power as the Brown and Forsythe test under normal distribution.

Fitting the growth curves by polynomials, this paper is intended to test whether or not there is any correlation between two characters. The likelihood ratio statistic is derived and is shown to be distributed under the null hypothesis as the product of three independent U statistics as defined in Anderson (1958). Box's procedure is then applied to approximate the critical region. A numerical example is given to illustrate the procedure.

In this paper we consider several straight lines and compare the regression coefficients under heteroscedastic variances. A test Fβ is proposed for this purpose by assuming normality for the random disturbances. Some Monte Carlo studies indicate that the Fβ-test is quite powerful and robust with respect to departure from normality. By combining Tiku’s robust procedure with the Fβ-test, a new robust test is developed. Some Monte Carlo studies indicate clearly that except for the uniform distribution, the -test is more powerful than the Fβ-test under non-normal distributions but has substantially the same power as the Fβtest under a normal distribution.

In this paper we consider a simple linear regression model under heteroscedasticity and nonnormality. A statistical test for testing the regression coefficient is then derived by assuming normality for the random disturbances and by applying Welch's method. Some Monte Carlo studies are generated for assessing robustness of this test. By combining Tiku's robust procedure with the new test, a robust but more powerful test is developed.

Consider a population of cells such as bacteria consisting of mutant and nonmutant cells. By recognizing that the mutation-birth-death process may be modeled as a bivariate branching process, this paper derives the probability distribution of the number of mutants for any fixed time interval. The probability generating functions, the cumulants and an iterative formula for computing probabilities are obtained; a representation formula for the number of mutants is also derived. The results obtained iti this paper are useful for stochastic modelling of many mutagenicity and carcinogenesis testing assays.

In this paper we propose a general mixed model of carcinogenesis involving one-stage and two-stage models. These models were proposed after careful examination of some cancers which are related to oncogenes, antioncogenes and accessory cancer genes. The probability generating functions, the expected incidence functions and the probabilities of number of tumors are derived. These mixed models are useful for deriving risk assessment procedures for evaluating risks of environmental agents.

In this paper we consider a covariance model under heteroscedastic variances and propose a procedure for testing treatment effects. A simultaneous procedure for comparing the treatment effects is also developed. Some Monte Carlo studies indicate that the effects of departure from normality may not be very serious although some effects do exist due to high level of heteroscedasticity.

By adopting the Bayesian approach, this paper proposes an alternative procedure for the estimation of linkage.Through simulation studies, the proposed method was then compared with the classical maximum likelihood method. The numerical results indicate clearly that in most of the cases, the Bayesian method is better than the maximum likelihood method, especially in the repulsion phase.

In this paper Bayesian approach is adopted to develop inferences about parameters in proportional odds models. Bayesian posterior intervals for coefficients in proportional odds models are derived by using approximation given in Pregibon (1981). The results are illustrated by using the lung cancer survival data reported by Prentice (1973).

In this paper we derive a stochastic model for the formation of metastatic foci at distant sites. The probability generation functions and hence the probabilities are derived under very general conditions. Some waiting time distributions for the metastatic foci are also derived. These results are useful for modelling cancer progression under quite general conditions.

In this paper we study the distribution of T= XS, where X with order p × q is normally distributed with zero expectations and covariance matrix Σ⊗V, (that is, Cov(, ) = σij, being the j-th column of X, Σ=(σij)) and where S1′S1 = S is distris uted as Wishart independently of X, with d.f.=f and parameter matrix Σ=(σij) This is then a multivariate extension of the stu-dentization process. As we shall see, however, the distribution of T depends on how S is partitioned. In this paper we shall consider the cases where S1 is lower triangular with positive diagonal elements and where S, is upper triangular with positive diagonal elements. The distributions of T corresponding to these two decompositions of S are called respectively the upper and lower disguised matrix-t distributions. Properties of these distributions are derived. Some applications of these distributions to multivariate regression problems are also pointed out. Cet article traite d'une generalisation a plusieurs dimensions du procede appele “studentization”. Nous etudions la loi de probabilite de la matrice T= XS ou la matrice X, d'ordre p par q, suit une loi normale avec moyenne zero et matrice des covariances Σ⊗V [c.-a-d., Cov(xi, xj)=σijV, xj etant la j'-ieme colonne de la matrice X et Σ=(σij)], et ou la matrice S=S1S1 est distribuee independamment de X selon la loi de Wishart avec f degres de liberte et matrice des parametres Σ=(σij). Nous considerons le cas ou la matrice S1 est triangulaire superieure (ou triangulaire inferieure) avec des elements positifs sur la diagonale. Nous etudions des proprietes des lois de T correspondantes et les illustrons a l'aide de quelques applications a des modeles de regression multivaries.

In this paper some properties of the multivariate Beta distributions are discussed. Specifically it is shown that the density of the multivariate Beta I distribution can be decomposed as a product of independent ordinary Beta I densities and the Dirichlet densities, while the density of the multivariate Beta II distribution as a product of independent ordinary Beta II densities and the inverted Dirichlet densities. Some extentions of the results to the generalized multivariate Beta distributions are indicated.

Thus, E(yij-_U)2 = o2 + oa2. The parameters or2 and oa2 are therefore also called variancecomponents. The problem of estimating variance-components has attracted the attention of many writers-see, for example, Bross (1950), Bulmer (1957), Bush & Anderson (1963), Crump (1946, 1951), Daniels (1939), Fisher (1935), Green (1954) and Healy (1963), etc. In most of these works, the problem has been analysed from the sampling theory point of view. A difficulty which has concerned many of these writers is the so-called 'negative estimated variance' problem. For example, using the model in (1.fi1) with the added assumption that the ai's and eij's are independent among themselves, the following unbiased estimator for