Inference with three-level prior distributions in quantile regression problems

In this paper, we propose a three level hierarchical Bayesian model for variable selection and estimation in quantile regression problems. Specifically, at the first level we consider a zero mean normal priors for the coefficients with unknown variance parameters. At the second level, we specify two different priors for the unknown variance parameters which introduce two different models producing different levels of sparsity. Then, at the third level we suggest joint improper priors for the unknown hyperparameters assuming they are independent. Simulations and Boston Housing data are utilized to compare the performance of our models with six existing models. The results indicate that our models perform good in the simulations and Boston Housing data.     

Non-parametric quantile estimate for length-biased and right-censored data with competing risks

In this article, we propose a non-parametric quantile inference procedure for cause-specific failure probabilities to estimate the lifetime distribution of length-biased and right-censored data with competing risks. We also derive the asymptotic properties of the proposed estimators of the quantile function. Furthermore, the results are used to construct confidence intervals and bands for the quantile function. Simulation studies are conducted to illustrate the method and theory, and an application to an unemployment data is presented.     

Conditional tests in a competing risks model

Testing for equality of competing risks based on their cumulative incidence functions (CIFs) or their cause specific hazard rates (CSHRs) has been considered by many authors. The finite sample distributions of the existing test statistics are in general complicated and the use of their asymptotic distributions can lead to conservative tests. In this paper we show how to perform some of these tests using the conditional distributions of their corresponding test statistics instead (conditional on the observed data). The resulting conditional tests are initially developed for the case of k = 2 and are then extended to k > 2 by performing a sequence of two sample tests and by combining several risks into one. A simulation study to compare the powers of several tests based on their conditional and asymptotic distributions shows that using conditional tests leads to a gain in power. A real life example is also discussed to show how to implement such conditional tests.     

Approximate tests for the equality of two cumulative incidence functions of a competing risk

In the context of a competing risks set-up, we discuss different inference procedures for testing equality of two cumulative incidence functions, where the data may be subject to independent right-censoring or left-truncation. To this end, we compare two-sample Kolmogorov–Smirnov- and Cramér–von Mises-type test statistics. Since, in general, their corresponding asymptotic limit distributions depend on unknown quantities, we utilize wild bootstrap resampling as well as approximation techniques to construct adequate test decisions. Here, the latter procedures are motivated from tests for heteroscedastic factorial designs but have not yet been proposed in the survival context. A simulation study shows the performance of all considered tests under various settings and finally a real data example about bloodstream infection during neutropenia is used to illustrate their application.     

Simulating competing cointegration tests in a bivariate system

In this paper, we consider the size and power of a set of cointegration tests in a number of Monte Carlo simulations. The behaviour of the competing methods is investigated in diff erent situations, including diff erent levels of variance and correlation in the error processes. The impact of violations of the common factor restriction (CFR) implied by the Engle-Granger framework is studied in these situations. The reactions to changes in the CFR condition depend on the error correlation. When the correlation is non-positive, the power increases with increasing CFR violations for the error correction model (ECM) test, while the other tests react in the opposite direction. We also note the reaction to diff erences in the error variances in the data-generating process. For positive correlation and equal variances, the reaction to changes in the CFR violations diff ers somewhat between the tests. We conclude that the ECM and the Z-tests show the best performance over diff erent parameter combinations. In most situations the ECM is best. Therefore, if we had to recommend a unit root test, it would be the ECM, especially for small samples. However, we do not think that one should use just one test, but two or more. Of course, the portfolio of tests we have considered here only represents a subset of the possible tests.     

Progressively censored tests for clinical experiments and life testing problems based on weighted empirical distributions

In the context of time-sequential studies, progressively censored tests for a simple regression model based on weighted empirical distributions are considered for ungrouped as well as grouped data situations. Early decision rules based on such tests are formulated. The asymptotic theory of the proposed tests rests on a construction of suitable empirical processes and their convergence (in distribution) to appropriate Gaussian functions. Critical values of the proposed test statistics are obtained by simulation, For a hypothetical example (of practical interest), a comparative study is made for the empirical powers and stopping times for some rival tests.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

Score tests for independence in semiparametric competing risks models

A popular model for competing risks postulates the existence of a latent unobserved failure time for each risk. Assuming that these underlying failure times are independent is attractive since it allows standard statistical tools for right-censored lifetime data to be used in the analysis. This paper proposes simple independence score tests for the validity of this assumption when the individual risks are modeled using semiparametric proportional hazards regressions. It assumes that covariates are available, making the model identifiable. The score tests are derived for alternatives that specify that copulas are responsible for a possible dependency between the competing risks. The test statistics are constructed by adding to the partial likelihoods for the individual risks an explanatory variable for the dependency between the risks. A variance estimator is derived by writing the score function and the Fisher information matrix for the marginal models as stochastic integrals. Pitman efficiencies are used to compare test statistics. A simulation study and a numerical example illustrate the methodology proposed in this paper.     

Approximate variance formulas and asymptotic joint sampling distributions of standardized risk ratios in the presence of competing risks in cohort studies

The present paper is concerned with some results in cohort studies, in which the individuals in two study population are exposed simultaneously to several risks of death, which compete for their lives.

The morality experience of individuals in the two study populations is compared with respect to the morality experience of individuals in a well-defined and fixed population called the standard population.

Under some reasonable assumptions, not only simple variance formulas are-developed for the standardized risk ratio statistics (S[Rcirc]R_i) but also their joint asymptotic sampling distribution. It is demonstrated that these SRcirc;R_i's have asymptotically a multivariate normal distribtion corresponding to any given number of competing risks of death, These results are utilized to construct Scheffé-type and Sidak-type simultaneous confidence intervals for the SRR_i parameters which hold regardless of any covariance structure among the competing risks of death. The corresponding results for the cause-specific SMR and the externally standardized risk ratio parameters follow as special cases.

The present paper generalizes the available results in the literature in two directions, namely, to obtain simple variance formulas for the S[Rcirc]R_i, statistics and to treat the situation in the presence of competing risks to which individuals in a study are simultaneously exposed.

An empirical evaluation of these results is discussed in the last section utilizing some real cohort data from two recent occupational epidemiologic cohort studies.     

Proportionality of hazards in competing risk analysis

The proportional cause-specific hazards (CSHs) model and the proportional subdistribution (cumulative incidence function (CIF)) hazards model are widely used in competing risk analysis. In this paper, we prove that these two kinds of proportionalities cannot hold simultaneously for all CSH functions and all CIFs.     

Some linear and nonlinear rank tests for competing risks models

Using the methods of asymptotic decision theory asymptotically optimal for translation and scale families as well as for certian nonparmetric families. Moreover, two new classes of nonlinear rank tests are introduced. These tests are designed for detecting either “ omnibus alternatives ” or “ one sided alternatives of trend ”. Under the null hypothesis of randomness all tests are distribution - free. The asymptotic distributions of the test statistics are derived under contiguous alternatives.     

Measures and tests of heaping in discrete quantitative distributions

Heaping is often found in discrete quantitative data based on subject responses to open-ended interview questions or observer assessments. Heaping occurs when subjects or observers prefer some set of numbers as responses (e.g. multiples of 5) simply because of the features of this set. Although heaping represents a common type of measurement error, apparently no prior general measure of heaping exists. We present simple measures and tests of heaping in discrete quantitative data, illustrate them with data from an epidemiologic study, and evaluate the bias of these statistics. These techniques permit formal measurement of heaping and facilitate comparisons of the degree of heaping in data from different samples, substantive domains, and data collection methods.     

Implied distributions in multiple change point problems

A method for efficiently calculating exact marginal, conditional and joint distributions for change points defined by general finite state Hidden Markov Models is proposed. The distributions are not subject to any approximation or sampling error once parameters of the model have been estimated. It is shown that, in contrast to sampling methods, very little computation is needed. The method provides probabilities associated with change points within an interval, as well as at specific points.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

Power and sample size considerations in clinical trials with competing risk endpoints

In clinical trials with a time-to-event endpoint, subjects are often at risk for events other than the one of interest. When the occurrence of one type of event precludes observation of any later events or alters the probably of subsequent events, the situation is one of competing risks. During the planning stage of a clinical trial with competing risks, it is important to take all possible events into account. This paper gives expressions for the power and sample size for competing risks based on a flexible parametric Weibull model. Nonuniform accrual to the study is considered and an allocation ratio other than one may be used. Results are also provided for the case where two or more of the competing risks are of primary interest.     

Bivariate symmetry tests for complete and competing risks data: a saddlepoint approach

A class of bivariate symmetry tests for complete data and competing risks data is considered. Saddlepoint approximation for the exact p-values of the underlying permutation distribution of these tests is derived. Several simulation studies are conducted to evaluate the performance of the saddlepoint approximation and the asymptotic approximation. The saddlepoint approximation was found to be highly accurate and superior to the asymptotic approximations in replicating the exact permutation significance.     

Asymptotic distributions of quadratic forms occurring in changepoint problems

A stochastic-process approach is used to derive the asymptotic distributions of quadratic forms occurring in the analysis of changepoint data.