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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.

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.

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.

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.

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.

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.

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.

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 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

Assuming common variance, the moment estimators of a mixture of two normal densities are obtained. The asymptotic covariance matrix of the moment estimators and the information matrix are then derived and hence the efficiency of the moment method computed.

In this paper some distribution theories of the segregating populations in self-fertilized populations are derived, assuming that the random disturbance (or environmental effect) follows the normal distribution. Utilizing these distribution results, some efficient statistical procedures such as the maximum likelihood (ML) method are introduced for estimating the number of genes and the effects of genes. The methods and the techniques developed are then applied to analyze Shakudo's data of panicle length of rice and Goodwin's data of flowering time of Solidag sempervirens L. These numerical examples seem to indicate that the present approach is useful and desirable; it extracts more information from limited data than by using the classical approaches.

In this paper, we apply some finite Markov chain theories to study the two locus selfing model with selection. Specifically, the effects of selection and linkage on the ultimate absorption probabilities, the first time absorption probabilities, the mean absorption times and the variances of first absorption times, are assessed. Also discussed are the effects of selection on the absolute probability distributions, the variances of the nth generation and the covariances between the nith and the (n, + n2)th generations. Some formulas using the spectral expansion of matrix are given for the absorption probabilities, the mean absorption times and the variances of the first time absorptions.