The two-sample problem with multivariate censored data: a new rank test family

A new rank test family is proposed to test the equality of two multivariate failure times distributions with censored observations. The tests are very simple: they are based on a transformation of the multivariate rank vectors to a univariate rank score and the resulting statistics belong to the familiar class of the weighted logrank test statistics. The new procedure is also applicable to multivariate observations in general, such as repeated measures, some of which may be missing. To investigate the performance of the proposed tests, a simulation study was conducted with bivariate exponential models for various censoring rates. The size and power of these tests against Lehmann alternatives were compared to the size and power of two other tests (Wei and Lachin, 1984 and Wei and Knuiman, 1987). In all simulations the new procedures provide a relatively good power and an accurate control over the size of the test. A real example from the National Cooperative Gallstone Study is given     

A COMPARISON OF THE IMPRECISE BETA CLASS, THE RANDOMIZED PLAY-THE-WINNER RULE AND THE TRIANGULAR TEST FOR CLINICAL TRIALS WITH BINARY RESPONSES

This paper develops clinical trial designs that compare two treatments with a binary outcome. The imprecise beta class (IBC), a class of beta probability distributions, is used in a robust Bayesian framework to calculate posterior upper and lower expectations for treatment success rates using accumulating data. The posterior expectation for the difference in success rates can be used to decide when there is sufficient evidence for randomized treatment allocation to cease. This design is formally related to the randomized play‐the‐winner (RPW) design, an adaptive allocation scheme where randomization probabilities are updated sequentially to favour the treatment with the higher observed success rate. A connection is also made between the IBC and the sequential clinical trial design based on the triangular test. Theoretical and simulation results are presented to show that the expected sample sizes on the truly inferior arm are lower using the IBC compared with either the triangular test or the RPW design, and that the IBC performs well against established criteria involving error rates and the expected number of treatment failures.     

Double samble tests— a distribution free procedure

A method for constructing two-stage (double samble) tests is presented which does not require the evaluation of complicated bivariate distribution function. The procedure results from a modification of Fisher's method for combining independent tests of significance and is distribution free in the way it combines the test results from the two sampies. However, the one sample test statistics for the two samples are assumed to have continuous distributions and may be parametric. A rule is also given or the selection of a particular test out of a family of possible two-stage tests which can be generated by this method. Specific examples are given and comparisons are made with two double sample tests which have previously been presented in the literature.     

Performance of Preliminary Test Estimator Under Linex Loss Function

This article describes two bivariate geometric distributions. We investigate characterizations of bivariate geometric distributions using conditional failure rates and study properties of the bivariate geometric distributions. The bivariate models are fitted to real-life data using the Method of Moments, Maximum Likelihood, and Bayes Estimators. Two methods of moments estimators, in each bivariate geometric model, are compared and evaluated for their performance in terms of bias vector and covariance matrix. This comparison is done through a Monte Carlo simulation. Chi-square goodness-of-fit tests are used to evaluate model performance.     

Tests for bivariate exponentiality against BHNBUE alternatives

We present statistical procedures to test that a life distribution is bivariate exponential (BVE) against the alternative that it is bivariate harmonic new better than used in expectation (BHNBUE). We present a simulation study to compare the power the proposed test with tests proposed by Basu and Ebrahimi (1984) and Sen and Jain (1990) and we observe that the proposed test performs better than the other two tests.     

Bivariate zero-inflated negative binomial regression model with applications

Count data often display excessive number of zero outcomes than are expected in the Poisson regression model. The zero-inflated Poisson regression model has been suggested to handle zero-inflated data, whereas the zero-inflated negative binomial (ZINB) regression model has been fitted for zero-inflated data with additional overdispersion. For bivariate and zero-inflated cases, several regression models such as the bivariate zero-inflated Poisson (BZIP) and bivariate zero-inflated negative binomial (BZINB) have been considered. This paper introduces several forms of nested BZINB regression model which can be fitted to bivariate and zero-inflated count data. The mean–variance approach is used for comparing the BZIP and our forms of BZINB regression model in this study. A similar approach was also used by past researchers for defining several negative binomial and zero-inflated negative binomial regression models based on the appearance of linear and quadratic terms of the variance function. The nested BZINB regression models proposed in this study have several advantages; the likelihood ratio tests can be performed for choosing the best model, the models have flexible forms of marginal mean–variance relationship, the models can be fitted to bivariate zero-inflated count data with positive or negative correlations, and the models allow additional overdispersion of the two dependent variables.     

Bivariate Longitudinal Model for Detecting Prescribing Change in Two Drugs Simultaneously with Correlated Errors

Bivariate responses of repeated measures data are usually analysed as two separate responses in the literature by several authors. The two responses usually tend to be related in some way and analysing this data jointly presents an opportunity to account for the joint movement, which may impact on the conclusions reached compared to analysing the responses separately. In this paper, a bivariate regression model with random effects (linear mixed model) is used to detect a change if any in the prescribing habits in the UK at the general practice (family medicine) level due to an educational intervention given repeated measures data before and after the intervention and a control group. The message was to increase the prescribing of one drug while simultaneously decreasing the prescribing of another. The effects of modelling a bivariate auto-regressive process are evaluated.     

Two-sample post-stratified or subgroup analysis with restricted randomization rules

In a clinical trial to compare two treatments, subjects may be allocated sequentially to treatment groups by a restricted randomization rule. Suppose that at the end of the trial, the investigator is interested in a post-stratified or subgroup analysis with respect to a particular demographic or clinical factor which was not selected prior to the trial for stratified randomization. Under a randomization model, large sample theory of two-sample post-stratified permutational tests is developed with a broad class of restricted randomization treatment allocation rules. The test procedures proposed here are illustrated with a real-life example. The results of this example indicate that it is not always possible to ignore the treatment rule used in the trial in the design-based analysis.     

SMOOTH TESTS FOR THE BIVARIATE POISSON DISTRIBUTION

A theorem of Rayner & Best (1989) is generalised to permit the construction of smooth tests of goodness of fit without requiring a set of orthonormal functions on the hypothesised distribution. This result is used to construct smooth tests for the bivariate Poisson distribution. The test due to Crockett (1979) is similar to a smooth test that assesses the variance structure under the bivariate Poisson model; the test due to Loukas & Kemp (1986) is related to a smooth test that seeks to detect a particular linear relationship between the variances and covariance under the bivariate Poisson model. Using focused smooth tests may be more informative than using previously suggested tests. The distribution of the Loukas & Kemp (1986) statistic is not well approximated by the x²distribution for larger correlations, and a revised statistic is suggested.     

Tests for bivariate mean residual life

In this paper we have developed tests for bivariate exponentiaIity against the ‘bivariate decreasing mean residual life (BDMRL)’ and ‘bivariate new better than used in expectation (BNBUE)’ classes of non-exponentia1 probability distributions. We have also obtained a large-sample approximation to make the test readily applicable.     

Report of the ASA Technical Panel on the Census Undercount

A new model is proposed for the joint distribution of paired survival times generated from clinical trials and certain reliability settings. The new model can be considered an extension to the bivariate exponential models studied in the literature. Here, a more flexible bivariate Weibull model will be derived, and two exact parametric tests for testing the equality of marginal survival distributions are developed.     

Cumulative Incidence Association Models for Bivariate Competing Risks Data

Association models, like frailty and copula models, are frequently used to analyze clustered survival data and evaluate within-cluster associations. The assumption of noninformative censoring is commonly applied to these models, though it may not be true in many situations. In this paper, we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function (CIF), a nonparametrically identifiable quantity. Copula models are proposed which relate the bivariate CIF to its corresponding univariate CIFs, similarly to independently right censored data, and accommodate frailty models for the bivariate CIF. Two estimating equations are developed to estimate the association parameter, permitting the univariate CIFs to be estimated either parametrically or nonparametrically. Goodness-of-fit tests are presented for formally evaluating the parametric models. Both estimators perform well with moderate sample sizes in simulation studies. The practical use of the methodology is illustrated in an analysis of dementia associations.     

Likelihood inference for Type I bivariate Pólya–Aeppli distribution

This article discusses likelihood inference for the Type I bivariate Pólya–Aeppli distribution. The Type I bivariate Pólya–Aeppli distribution was derived by Minkova and Balakrishnan by using compounding with geometric random variables and the trivariate reduction method. They also discussed the moment estimation of the parameters of the Type I bivariate Pólya–Aeppli distribution. Here, we carry out a simulation study to compare the performance of the developed Maximum Likelihood Estimation (MLE) method with the moment estimation. The obtained results show that, through the MLEs require more computational time compared to the moment estimates (MoM), the MLEs perform better, in most of the settings, than the MoM. Finally, we apply the Type I bivariate Pólya–Aeppli model to a real dataset containing the frequencies of railway accidents in two subsequent six-year periods for the purpose of illustration. We also carry out some hypothesis tests using the Wald test statistic. From these results, we conclude that the two variables belong to the same univariate Pólya–Aeppli distribution, but are correlated.     

Inferences on correlation coefficients of bivariate log-normal distributions

This article considers inference on correlation coefficients of bivariate log-normal distributions. We developed generalized confidence intervals and hypothesis tests for the correlation coefficients, and extended the results to compare two independent correlations. Simulation studies show that the suggested methods work well. Two practical examples are used to illustrate the application of the proposed methods.     

Bivariate frailty model for the analysis of multivariate survival time

Because of limitations of the univariate frailty model in analysis of multivariate survival data, a bivariate frailty model is introduced for the analysis of bivariate survival data. This provides tremendous flexibility especially in allowing negative associations between subjects within the same cluster. The approach involves incorporating into the model two possibly correlated frailties for each cluster. The bivariate lognormal distribution is used as the frailty distribution. The model is then generalized to multivariate survival data with two distinguished groups and also to alternating process data. A modified EM algorithm is developed with no requirement of specification of the baseline hazards. The estimators are generalized maximum likelihood estimators with subject-specific interpretation. The model is applied to a mental health study on evaluation of health policy effects for inpatient psychiatric care.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

A likelihood ratio test and its modifications for the homogeneity of the covariance matrices of dependent multivariate normals

For the problem of testing the homogeneity of the covariance matrices of several dependent multivariate normals, a likelihood ratio test is derived in this paper. The software SAS can be used to perform the test and the code is given and explained. Several modifications of the test that allow their distributions to be better approximated by the chi-square distribution are also considered. Formulae for calculating approximate sample size and power are derived. Small sample performances of these tests in the case of two dependent bivariate normals are compared to each other and to the competing tests by simulating levels of significance and powers, and recommendation is made of the ones that have good performance. The recommended tests are then applied to real data from a crossover bioequivalence trial.     

A maxmin linear test of normal means and its application to lachin’s data

For stochastic ordering tests for normal distributions there exist two well known types of tests. One of them is based on the maximum likelihood ratio principle, the other is the most stringent somewhere most powerful test of Schaafsma and Smid(for a comprehensive treatment see Robertson, Wright and Dykstra(1988), for the latter test also Shi and Kudo(1987)). All these tests are in general numerically tedious. Wei, Lachin(1984)and particularly Lachin(1992)formulate a simple and easily computable test. However, it is not known so far for which sort of ordered alternatives his test is optimal

In this paper it is shown that his procedure is a maxmin test for reasonable subalternatives, provided the covariance matrix has nonnegative row sums. If this property is violated then his procedure can be altered in such a manner that the resul ting test again is a maxmin test. An example is glven where the modified procedure even in the least favourable case leads to a nontrifling increase in power. The fact that Lachins test resp. the modified version are maxmin tests on appropriate subalternatives amounts to the property that they are maxmin tests on subhypotheses which are relevant in practical applications.     

Robust inference for bivariate point processes

In the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures.     

Some inference results in bivariate exponential distributions rased on censored samples

In this paper, two bivariate exponential distributions based on time(right) censored samples are presented. We assume that the censoring time is independent of the life-times of the two components. This paper obtains comparison of different tests for testing zero and non-zero values of the parameter λ₃ which measures the degree of

dependence between the two components and also testing symmetry of the two components or λ₁=λ₂ in

the bivariate exponential distribution (BVED) formulated by Marshall and Olkin (1967) based on the above censored sample. It is observed from simulated study that the test based on MLE's performs better in both tests of independence as well as symmetry. The above results have been extended also in Block and Basu (19874) model.