On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

A self-adjusted weighted likelihood ratio test for global clustering of disease

Compared to tests for localized clusters, the tests for global clustering only collect evidence for clustering throughout the study region without evaluating the statistical significance of the individual clusters. The weighted likelihood ratio (WLR) test based on the weighted sum of likelihood ratios represents an important class of tests for global clustering. Song and Kulldorff (Likelihood based tests for spatial randomness. Stat Med. 2006;25(5):825–839) developed a wide variety of weight functions with the WLR test for global clustering. However, these weight functions are often defined based on the cell population size or the geographic information such as area size and distance between cells. They do not make use of the information from the observed count, although the likelihood ratio of a potential cluster depends on both the observed count and its population size. In this paper, we develop a self-adjusted weight function to directly allocate weights onto the likelihood ratios according to their values. The power of the test was evaluated and compared with existing methods based on a benchmark data set. The comparison results favour the suggested test especially under global chain clustering models.     

Tests of independence in incomplete multi-way tables using likelihood functions

Kang (2006) and Kang and Larsen (in press) used the log likelihood function with Lagrangian multipliers for estimation of cell probabilities in two-way incomplete contingency tables. This paper extends results and simulations to three-way and multi-way tables. Numerous studies cross-classify subjects by three or more categorical factors. Constraints on cell probabilities are incorporated through Lagrangian multipliers. Variances of the MLEs are derived from the matrix of second derivatives of the log likelihood with respect to cell probabilities and the Lagrange multiplier. Wald and likelihood ratio tests of independence are derived using the estimates and estimated variances. In simulation results in Kang and Larsen (in press), for data missing at random, maximum likelihood estimation (MLE) produced more efficient estimates of population proportions than either multiple imputation (MI) based on data augmentation or complete case (CC) analysis. Neither MLE nor MI, however, lead to an improvement over CC analysis with respect to power of tests for independence in two-way tables. Results are extended to multidimensional tables with arbitrary patterns of missing data when the variables are recorded on individual subjects. In three-way and higher-way tables, however, there is information relevant for judging independence in partially classified information, as long as two or more variables are jointly observed. Simulations study three-dimensional tables with three patterns of association and two levels of missing information.     

Evidence From Marginally Significant t Statistics

ABSTRACT

This article examines the evidence contained in t statistics that are marginally significant in 5% tests. The bases for evaluating evidence are likelihood ratios and integrated likelihood ratios, computed under a variety of assumptions regarding the alternative hypotheses in null hypothesis significance tests. Likelihood ratios and integrated likelihood ratios provide a useful measure of the evidence in favor of competing hypotheses because they can be interpreted as representing the ratio of the probabilities that each hypothesis assigns to observed data. When they are either very large or very small, they suggest that one hypothesis is much better than the other in predicting observed data. If they are close to 1.0, then both hypotheses provide approximately equally valid explanations for observed data. I find that p-values that are close to 0.05 (i.e., that are “marginally significant”) correspond to integrated likelihood ratios that are bounded by approximately 7 in two-sided tests, and by approximately 4 in one-sided tests.

The modest magnitude of integrated likelihood ratios corresponding to p-values close to 0.05 clearly suggests that higher standards of evidence are needed to support claims of novel discoveries and new effects.     

A parametric multistate model for the analysis of carcinogenicity experiments

A fully parametric multistate model is explored for the analysis of animal carcinogenicity experiments in which the time of tumour onset is not known. This model does not require assumptions about tumour lethality or cause of death judgements and can be fitted in the absence of sacrifice data. The model is constructed as a three-state model with simple parametric forms for the transition rates. Maximum likelihood methods are used to estimate the transition rates and different treatment groups are compared using likelihood ratio tests. Selection of an appropriate model and methods to assess the fit of the model are illustrated with data from animal experiments. Comparisons with standard methods are made.     

Weighted Logrank Permutation Tests for Randomly Right Censored Life Science Data

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.     

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

Multivariate data with missing observations

In this paper, multivariate data with missing observations, where missing values could be by chance or by design, are considered for various models including the growth curve model. The likelihood equations are derived and the consistency of the estimates established. The likelihood ratio tests are explicity derived.     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Repeated measurements,log likelihood ratio tests and small samples

Various models have previously been proposed for data comprising m repeated measurements on each of N subjects. Log likelihood ratio tests may be used to help choose between possible models, but these tests are based on distributions which in theory apply only asymptotically. With small N , the log likelihood ratio approximation is unreliable, tending to reject the simpler of two models more often than it should. This is shown by reference to three datasets and analogous simulated data. For two of the three datasets, subjects fall into two groups. Log likelihood ratio tests confirm that for each of these two datasets group means over time differ. Tests suggest that group covariance structures also differ.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

Robust multivariate transformations to normality: Constructed variables and likelihood ratio tests

The assumption of multivariate normality provides the customary powerful and convenient ways of analysing multivariate data: if the data are not normal, the analysis may often be simplified by an appropriate transformation. In this context, the most widely used test is the likelihood ratio, which requires the maximum likelihood estimate of the transformation parameter for each variable. Given that this estimate can only be found numerically, when the number of variables is large (> 20) it is impossible or infeasible to compute the test. In this paper we introduce alternative tests which do not require the maximum likelihood estimate of the transformation parameters and prove algebraically their relationships. We also give insights both using theoretical arguments and a robust simulation study, based on the forward search algorithm, about the distribution of the tests previously introduced.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

Semi-empirical likelihood inference for the ROC curve with missing data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performance of two or more laboratory or diagnostic tests. In this paper, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric and the other one is non-parametric and both have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It is shown that the log-semi-empirical likelihood ratio statistic is asymptotically scaled chi-squared. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. We conduct extensive simulation studies to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing probabilities.     

The limiting power of point optimal autocorrelation tests

This paper considers the point optimal tests for AR(1) errors in the linear regression model. It is shown that these tests have the same limiting power characteristics as the Durbin-Watson test. . The limiting power is zero or one when the regression has no intercept, but lies strictly between these values when an intercept is included.     

A comment on “testing the lucas critique: A review ”

We derive general distribution tests based on the method of maximum entropy (ME) density. The proposed tests are derived from maximizing the differential entropy subject to given moment constraints. By exploiting the equivalence between the ME and maximum likelihood (ML) estimates for the general exponential family, we can use the conventional likelihood ratio (LR), Wald, and Lagrange multiplier (LM) testing principles in the maximum entropy framework. In particular, we use the LM approach to derive tests for normality. Monte Carlo evidence suggests that the proposed tests are compatible with and sometimes outperform some commonly used normality tests. We show that the proposed tests can be extended to tests based on regression residuals and non-i.i.d. data in a straightforward manner. An empirical example on production function estimation is presented.     

TEST FOR TREATMENT EFFECT BASED ON BINARY DATA WITH RANDOM SAMPLE SIZES

The problem of testing for treatment effect based on binary response data is considered, assuming that the sample size for each experimental unit and treatment combination is random. It is assumed that the sample size follows a distribution that belongs to a parametric family. The uniformly most powerful unbiased tests, which are equivalent to the likelihood ratio tests, are obtained when the probability of the sample size being zero is positive. For the situation where the sample sizes are always positive, the likelihood ratio tests are derived. These test procedures, which are unconditional on the random sample sizes, are useful even when the random sample sizes are not observed. Some examples are presented as illustration.     

基于加权似然比检验的监控线性曲线的控制图

文章提出了一种基于加权似然比检验的阶段二监控线性曲线的控制图,称为WLRT图,并通过平均运行长度来衡量控制图的性能表现。模拟结果表明,WLRT图对于线性曲线的截距、斜率、标准差的变化及截距和斜率同时变化都具有很好的检测能力。通过与其他几种控制图的性能比较,得出WLRT图能较快地发现过程变化,而且设计简单、操作方便。     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Likelihood Asymptotics

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the usual asymptotic likelihood ratio tests, and for one-dimensional hypotheses the test statistic is known as r *, introduced by Barndorff-Nielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace-type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi-dimensional tests is developed. In the final part of the paper further problems and ideas are presented. Among these are linear models with non-normal error, non-parametric linear models obtained by estimation of the residual density in combination with the present results, and the generalization of the results to restricted maximum likelihood and similar structured models.