On Goodness-of-Fit Tests for Aalen's Additive Risk Model

Abstract.  In this paper we propose goodness-of-fit tests for Aalen's additive risk model. They are based on test statistics the asymptotic distributions of which are determined under both the null and alternative hypotheses. The results are derived using martingale techniques for counting processes. An important feature of these tests is that they can be adjusted to particular alternatives. One of the alternatives we consider is Cox's multiplicative risk model. It is perhaps remarkable that such a test needs no estimate of the baseline hazard in the Cox model. We present simulation studies which give an impression of the performance of the proposed tests. In addition, the tests are applied to real data sets.     

Optimal goodness-of-fit tests for recurrent event data

A class of tests for the hypothesis that the baseline intensity belongs to a parametric class of intensities is given in the recurrent event setting. Asymptotic properties of a weighted general class of processes that compare the non-parametric versus parametric estimators for the cumulative intensity are presented. These results are given for a sequence of Pitman alternatives. Test statistics are proposed and methods of obtaining critical values are examined. Optimal choices for the weight function are given for a class of chi-squared tests. Based on Khmaladze’s transformation we propose distributional free tests. These include the types of Kolmogorov–Smirnov and Cramér–von Mises. The tests are used to analyze two different data sets.     

A unified approach of testing for discrete and continuous Pareto laws

New tests are proposed for the Pareto distribution as well as its discrete version, the so called Zipf’s law. In both cases the discrepancy between the empirical moment of arbitrary negative order and its theoretical counterpart is utilized in a weighted integral test statistic. If the weight function is of exponential rate of decay interesting limit statistics are obtained. The tests are shown to be consistent under fixed alternatives and a Monte Carlo study is drawn to investigate the performance of the proposed procedures in small samples. Furthermore a bootstrap procedure is proposed to cope with the case of unknown shape parameter. We conclude with applications to real data.     

Bayesian analysis of paired survival data using a bivariate exponential distribution

We consider a Bayesian analysis method of paired survival data using a bivariate exponential model proposed by Moran (1967, Biometrika 54:385–394). Important features of Moran’s model include that the marginal distributions are exponential and the range of the correlation coefficient is between 0 and 1. These contrast with the popular exponential model with gamma frailty. Despite these nice properties, statistical analysis with Moran’s model has been hampered by lack of a closed form likelihood function. In this paper, we introduce a latent variable to circumvent the difficulty in the Bayesian computation. We also consider a model checking procedure using the predictive Bayesian P-value.     

Modeling stock markets’ volatility using GARCH models with Normal,Student’s <Emphasis Type="Italic">t</Emphasis> and stable Paretian distributions

As GARCH models and stable Paretian distributions have been revisited in the recent past with the papers of Hansen and Lunde (J Appl Econom 20: 873–889, 2005) and Bidarkota and McCulloch (Quant Finance 4: 256–265, 2004), respectively, in this paper we discuss alternative conditional distributional models for the daily returns of the US, German and Portuguese main stock market indexes, considering ARMA-GARCH models driven by Normal, Student’s t and stable Paretian distributed innovations. We find that a GARCH model with stable Paretian innovations fits returns clearly better than the more popular Normal distribution and slightly better than the Student’s t distribution. However, the Student’s t outperforms the Normal and stable Paretian distributions when the out-of-sample density forecasts are considered.     

A new functional statistic for multivariate normality

The test statistics of assessing multivariate normality based on Roy’s union-intersection principle (Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1953) are generalizations of univariate normality, and are formed as the optimal value of a nonlinear multivariate function. Due to the difficulty of solving multivariate optimization problems, researchers have proposed various approximations. However, this paper shows that the (nearly) global solution contrarily results in unsatisfactory power performance in Monte Carlo simulations. Thus, instead of searching for a true optimal solution, this study proposes a functional statistic constructed by the q% quantile of the objective function values. A comparative Monte Carlo analysis shows that the proposed method is superior to two highly recommended tests when detecting widely-selected alternatives that characterize the various properties of multivariate normality.     

Estimating a sensitive proportion through randomized response procedures based on auxiliary information

Randomized response techniques are widely employed in surveys dealing with sensitive questions to ensure interviewee anonymity and reduce nonrespondents rates and biased responses. Since Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, many ingenious devices have been suggested to increase respondent’s privacy protection and to better estimate the proportion of people, π _A , bearing a sensitive attribute. In spite of the massive use of auxiliary information in the estimation of non-sensitive parameters, very few attempts have been made to improve randomization strategy performance when auxiliary variables are available. Moving from Zaizai’s (Model Assist Stat Appl 1:125–130, 2006) recent work, in this paper we provide a class of estimators for π _A , for a generic randomization scheme, when the mean of a supplementary non-sensitive variable is known. The minimum attainable variance bound of the class is obtained and the best estimator is also identified. We prove that the best estimator acts as a regression-type estimator which is at least as efficient as the corresponding estimator evaluated without allowing for the auxiliary variable. The general results are then applied to Warner and Simmons’ model.     

On locally optimal invariant unbiased tests for the variance components ratio in mixed linear models

In the paper the problem of testing of two-sided hypotheses for variance components in mixed linear models is considered. When the uniformly most powerful invariant test does not exist (see e.g. Das and Sinha, in Proceedings of the second international Tampere conference in statistics, 1987; Gnot and Michalski, in Statistics 25:213–223, 1994; Michalski and Zmyślony, in Statistics 27:297–310, 1996) then to conduct the optimal statistical inference on model parameters a construction of a test with locally best properties is desirable, cf. Michalski (in Tatra Mountains Mathematical Publications 26:1–21, 2003). The main goal of this article is the construction of the locally best invariant unbiased test for a single variance component (or for a ratio of variance components). The result has been obtained utilizing Andersson’s and Wijsman’s approach connected with a representation of density function of maximal invariant (Andersson, in Ann Stat 10:955–961, 1982; Wijsman, in Proceedings of fifth Berk Symp Math Statist Prob 1:389–400, 1967; Wijsman, in Sankhyā A 48:1–42, 1986; Khuri et al., in Statistical tests for mixed linear models, 1998) and from generalized Neyman–Pearson Lemma (Dantzig and Wald, in Ann Math Stat 22:87–93, 1951; Rao, in Linear statistical inference and its applications, 1973). One selected real example of an unbalanced mixed linear model is given, for which the power functions of the LBIU test and Wald’s test (the F-test in ANOVA model) are computed, and compared with the attainable upper bound of power obtained by using Neyman–Pearson Lemma.     

Data-driven smooth tests of the proportional hazards assumption

A new test of the proportional hazards assumption in the Cox model is proposed. The idea is based on Neyman’s smooth tests. The Cox model with proportional hazards (i.e. time-constant covariate effects) is embedded in a model with a smoothly time-varying covariate effect that is expressed as a combination of some basis functions (e.g., Legendre polynomials, cosines). Then the smooth test is the score test for significance of these artificial covariates. Furthermore, we apply a modification of Schwarz’s selection rule to choosing the dimension of the smooth model (the number of the basis functions). The score test is then used in the selected model. In a simulation study, we compare the proposed tests with standard tests based on the score process.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

Tests for Equality of Survival Distributions Against Non-Location Alternatives

We propose new two andk-sample tests for evaluating the equality of survival distributions against alternatives that include crossing of survival functions, and proportional and monotone hazard ratios. The tests allow for right censored data. The asymptotic power against local alternatives is investigated. Simulation results demonstrate that the new tests are more powerful than known tests when survival functions cross. We apply the tests to a well known study of chemo- and radio-therapy conducted by the Gastrointestinal Tumor Study Group. TheP-values for both proposed tests are much smaller than for other known tests.     

Testing Independence of Covariates and Errors in Non‐parametric Regression

Consider a non‐parametric regression model Y =m (X )+ϵ , where m is an unknown regression function, Y is a real‐valued response variable, X is a real covariate, and ϵ is the error term. In this article, we extend the usual tests for homoscedasticity by developing consistent tests for independence between X and ϵ . Further, we investigate the local power of the proposed tests using Le Cam's contiguous alternatives. An asymptotic power study under local alternatives along with extensive finite sample simulation study shows that the performance of the new tests is competitive with existing ones. Furthermore, the practicality of the new tests is shown using two real data sets.     

Goodness-of-Fit Tests for the Additive Risk Model with (p > 2)-Dimensional Time-Invariant Covariates

This paper presents methods for checking the goodness-of-fit of the additive risk model with p(> 2)-dimensional time-invariant covariates. The procedures are an extension of Kim and Lee (1996) who developed a test to assess the additive risk assumption for two-sample censored data. We apply the proposed tests to survival data from South Wales nikel refinery workers. Simulation studies are carried out to investigate the performance of the proposed tests for practical sample sizes.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

A comparative study of tests for paired lifetime data

In this paper, we investigate different procedures for testing the equality of two mean survival times in paired lifetime studies. We consider Owen’s M-test and Q-test, a likelihood ratio test, the paired t-test, the Wilcoxon signed rank test and a permutation test based on log-transformed survival times in the comparative study. We also consider the paired t-test, the Wilcoxon signed rank test and a permutation test based on original survival times for the sake of comparison. The size and power characteristics of these tests are studied by means of Monte Carlo simulations under a frailty Weibull model. For less skewed marginal distributions, the Wilcoxon signed rank test based on original survival times is found to be desirable. Otherwise, the M-test and the likelihood ratio test are the best choices in terms of power. In general, one can choose a test procedure based on information about the correlation between the two survival times and the skewness of the marginal survival distributions.     

Laplace’s approximation for relative risk frailty models

Relative risk frailty models are used extensively in analyzing clustered and/or recurrent time-to-event data. In this paper, Laplace’s approximation for integrals is applied to marginal distributions of data arising from parametric relative risk frailty models. Under regularity conditions, the approximate maximum likelihood estimators (MLE) are consistent with a rate of convergence that depends on both the number of subjects and number of members per subject. We compare the approximate MLE against alternative estimators using limited simulation and demonstrate the utility of Laplace’s approximation approach by analyzing U.S. patient waiting time to deceased kidney transplant data.     

Testing for Uncorrelated Residuals in Dynamic Count Models With an Application to Corporate Bankruptcy

This article proposes new model checks for dynamic count models. Both portmanteau and omnibus-type tests for lack of residual autocorrelation are considered. The resulting test statistics are asymptotically pivotal when innovations are uncorrelated but possibly exhibit higher order serial dependence. Moreover, the tests are able to detect local alternatives converging to the null at the parametric rate T^{? 1/2}, with T the sample size. The finite sample performance of the test statistics are examined by means of Monte Carlo experiments. Using a dataset on U.S. corporate bankruptcies, the proposed tests are applied to check if different risk models are correctly specified. Supplementary materials for this article are available online.     

An extended cure model and model selection

We propose a novel interpretation for a recently proposed Box–Cox transformation cure model, which leads to a natural extension of the cure model. Based on the extended model, we consider an important issue of model selection between the mixture cure model and the bounded cumulative hazard cure model via the likelihood ratio test, score test and Akaike’s Information Criterion (AIC). Our empirical study shows that AIC is informative and both the score test and the likelihood ratio test have adequate power to differentiate between the mixture cure model and the bounded cumulative hazard cure model when the sample size is large. We apply the tests and AIC methods to leukemia and colon cancer data to examine the appropriateness of the cure models considered for them in the literature.     

Comparison of goodness of fit tests for the cox proportional hazards model

The proportional hazards regression model of Cox(1972) is widely used in analyzing survival data. We examine several goodness of fit tests for checking the proportionality of hazards in the Cox model with two-sample censored data, and compare the performance of these tests by a simulation study. The strengths and weaknesses of the tests are pointed out. The effects of the extent of random censoring on the size and power are also examined. Results of a simulation study demonstrate that Gill and Schumacher's test is most powerful against a broad range of monotone departures from the proportional hazards assumption, but it may not perform as well fail for alternatives of nonmonotone hazard ratio. For the latter kind of alternatives, Andersen's test may detect patterns of irregular changes in hazards.     

Model checks for Cox-type regression models based on optimally weighted martingale residuals

We introduce directed goodness-of-fit tests for Cox-type regression models in survival analysis. “Directed” means that one may choose against which alternatives the tests are particularly powerful. The tests are based on sums of weighted martingale residuals and their asymptotic distributions. We derive optimal tests against certain competing models which include Cox-type regression models with different covariates and/or a different link function. We report results from several simulation studies and apply our test to a real dataset.