Adjustment of the Profile Likelihood for a Class of Normal Regression Models

We apply the method of McCullagh & Tibshirani (1990) to a generalization of the model for variance components in which the parameter of interest can appear in both the mean and variance. We obtain the exact adjusted profile log-likelihood score function. For the variance components model, we obtain the adjusted profile log-likelihood and show that it equals the restricted log-likelihood of Patterson & Thompson (1971). We discuss an example due to Kempton (1982) of a regression model with autoregressive terms in which the parameter of interest appears in both the mean and variance.     

Statistical analysis of right-censored failure-time data with partially specified hazard rates

This paper discusses the analysis of right-censored failure-time data in which the failure rate may have different forms in different time intervals. Such data occur naturally, for example, in demography studies and leukemia research, and a number of methods for the analysis have been proposed in the literature. However, most methods are purely parametric or nonparametric. Matthews and Farewell (1982), for example, discussed this problem and proposed a method for testing a constant failure rate against a failure rate involving a change point. To estimate an absolute limit on the attainable human life span, Zelterman (1992) discussed a hazard function that has different parametric forms over different time intervals. We consider a different situation in which the hazard function may follow a parametric form before a change point and is completely unknown after the change point. To test the existence of the change point, a modified maximal-censored-likelihood-ratio test is proposed and its asymptotic properties are studied. A bootstrap method is described for finding critical values of the proposed test. Simulation results indicate that the test performs well.     

Geometry of the log-likelihood ratio statistic in misspecified models

We show that the asymptotic mean of the log-likelihood ratio in a misspecified model is a differential geometric quantity that is related to the exponential curvature of Efron (1978), Amari (1982), and the preferred point geometry of [Critchley et al., 1993] and [Critchley et al., 1994]. The mean is invariant with respect to reparameterization, which leads to the differential geometrical approach where coordinate-system invariant quantities like statistical curvatures play an important role. When models are misspecified, the likelihood ratios do not have the chi-squared asymptotic limit, and the asymptotic mean of the likelihood ratio depends on two geometric factors, the departure of models from exponential families (i.e. the exponential curvature) and the departure of embedding spaces from being totally flat in the sense of Critchley et al. (1994). As a special case, the mean becomes the mean of the usual chi-squared limit (i.e. the half of the degrees of freedom) when these two curvatures vanish. The effect of curvatures is shown in the non-nested hypothesis testing approach of Vuong (1989), and we correct the numerator of the test statistic with an estimated asymptotic mean of the log-likelihood ratio to improve the asymptotic approximation to the sampling distribution of the test statistic.     

Likelihood Based Inference in Non-linear Regression Models Using the p* and R* Approach

We develop second order asymptotic results for likelihood-based inference in Gaussian non-linear regression models. We provide an approximation to the conditional density of the maximum likelihood estimator given an approximate ancillary statistic (the affine ancillary). From this approximation, we derive a statistic to test an hypothesis on one component of the parameter. This test statistic is an adjustment of the signed log-likelihood ratio statistic. The distributional approximations (for the maximum likelihood estimator and for the test statistic) are of second order in large deviation regions.     

Estimating changes in a multi-parameter exponential family

A two-stage procedure is studied for estimating changes in the parameters of the multi-parameter exponential family, given a sample X ₁,…,X _n. The first step is a likelihood ratio test of the hypothesis H_oof no change. Upon rejection of this hypothesis, the change point index and pre- and post-change parameters are estimated by maximum likelihood. The asymptotic (n → ∞) distribution of the log-likelihood ratio statistic is obtained under both H_oand local alternatives. The m.l.e.fs o of the pre- and post-change parameters are shown to be asymptotically jointly normal. The distribution of the change point estimate is obtained under local alternatives. Performance of the procedure for moderate samples is studied by Monte Carlo methods.     

Interval estimation for misclassification rate parameters in a complementary Poisson model

We investigate three interval estimators for binomial misclassification rates in a complementary Poisson model where the data are possibly misclassified: a Wald-based interval, a score-based interval, and an interval based on the profile log-likelihood statistic. We investigate the coverage and average width properties of these intervals via a simulation study. For small Poisson counts and small misclassification rates, the intervals can perform poorly in terms of coverage. The profile log-likelihood confidence interval (CI) is often proved to outperform the other intervals with good coverage and width properties. Lastly, we apply the CIs to a real data set involving traffic accident data that contain misclassified counts.     

Estimation in the Three-Parameter Gamma Distribution Based on the Profile Log-Likelihood Function

In this article, we express the profile log-likelihood function for the three-parameter gamma distribution in terms of the location parameter only and we study its properties. The behavior of the profile function is examined as the location parameter tends to the boundary values, i.e., to ? ∞ and to the minimum value of the sample. As a result, we obtain that if the log-likelihood function has a local maximum then it has another stationary value which is a saddle point. The results are supported with the use of simulation results.     

Ratio test for variance change point in linear process with long memory

In this paper we consider the detection problem of variance change point in linear process with long memory. We propose the ratio test to detect the variance change point. The limiting distribution for test statistics under H ₀ is derived and the consistency of the test is also established. In comparison with the existing CUSUM of squares (SCUSUM) test, the ratio test does not need to estimate the long memory parameter in practical situation and therefore it can be used more conveniently.     

A simultaneous variable selection methodology for linear mixed models

Selecting an appropriate structure for a linear mixed model serves as an appealing problem in a number of applications such as in the modelling of longitudinal or clustered data. In this paper, we propose a variable selection procedure for simultaneously selecting and estimating the fixed and random effects. More specifically, a profile log-likelihood function, along with an adaptive penalty, is utilized for sparse selection. The Newton-Raphson optimization algorithm is performed to complete the parameter estimation. By jointly selecting the fixed and random effects, the proposed approach increases selection accuracy compared with two-stage procedures, and the usage of the profile log-likelihood can improve computational efficiency in one-stage procedures. We prove that the proposed procedure enjoys the model selection consistency. A simulation study and a real data application are conducted for demonstrating the effectiveness of the proposed method.     

Robust tests for the common principal components model

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.     

Consistent parametric estimation of the intensity of a spatial–temporal point process

We consider conditions under which parametric estimates of the intensity of a spatial–temporal point process are consistent. Although the actual point process being estimated may not be Poisson, an estimate involving maximizing a function that corresponds exactly to the log-likelihood if the process is Poisson is consistent under certain simple conditions. A second estimate based on weighted least squares is also shown to be consistent under quite similar assumptions. The conditions for consistency are simple and easily verified, and examples are provided to illustrate the extent to which consistent estimation may be achieved. An important special case is when the point processes being estimated are in fact Poisson, though other important examples are explored as well.     

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.     

Computation of Two- and Three-Dimensional Confidence Regions With the Likelihood Ratio

The asymptotic results pertaining to the distribution of the log-likelihood ratio allow for the creation of a confidence region, which is a general extension of the confidence interval. Two- and three-dimensional regions can be displayed visually to describe the plausible region of the parameters of interest simultaneously. While most advanced statistical textbooks on inference discuss these asymptotic confidence regions, there is no exploration of how to numerically compute these regions for graphical purposes. This article demonstrates the application of a simple trigonometric transformation to compute two- and three-dimensional confidence regions; we transform the Cartesian coordinates of the parameters to create what we call the radial profile log-likelihood. The method is applicable to any distribution with a defined likelihood function, so it is not limited to specific data distributions or model paradigms. We describe the method along with the algorithm, follow with an example of our method, and end with an examination of computation time. Supplementary materials for this article are available online.     

Monte carlo sampling approach to testing nonnested hypothesis: monte carlo results

Alternative ways of using Monte Carlo methods to implement a Cox-type test for separate families of hypotheses are considered. Monte Carlo experiments are designed to compare the finite sample performances of Pesaran and Pesaran's test, a RESET test, and two Monte Carlo hypothesis test procedures. One of the Monte Carlo tests is based on the distribution of the log-likelihood ratio and the other is based on an asymptotically pivotal statistic. The Monte Carlo results provide strong evidence that the size of the Pesaran and Pesaran test is generally incorrect, except for very large sample sizes. The RESET test has lower power than the other tests. The two Monte Carlo tests perform equally well for all sample sizes and are both clearly preferred to the Pesaran and Pesaran test, even in large samples. Since the Monte Carlo test based on the log-likelihood ratio is the simplest to calculate, we recommend using it.     

Investigation of disease-free survival as a surrogate endpoint for survival in cancer clinical trials

A stochastic model wiuh exponential components is used to describe our data collected from a phase III cancer clinical trial. Criteria which guarantee that disease-free survival (DFS) can be used as a surrogate for overall survival are explored under this model. We examine several colorectal adjuvant clinical trials and find that these conditions are not satisfied. The relationship between the hazard ratio of DFS for an active treatment versus a control treatment and the cumulative hazard ratio of survival for the same two treatments is then explored. An almost linear relationship is found such that a hazard ratio for DFS of less than a threshold R corresponds to a non-null treatment effect on survival The threshold value R is determined for our colorectal adjuvant trial data. Based on this relationship, a one-sided test of equal hazard rate of survival is equivalent to a test of hazard ratio of DFS small than R This approach assumes that recurrence information is unbiasedly and accurately assessed; an assumpion which is sometimes difficult to ensure for multicenter clinical trials, particularly for interim analyses.     

Mixture Distributions Based Methods of Calibration for the Empirical Log-Likelihood Ratio

Empirical likelihood ratio confidence regions based on the chi-square calibration suffer from an undercoverage problem in that their actual coverage levels tend to be lower than the nominal levels. The finite sample distribution of the empirical log-likelihood ratio is recognized to have a mixture structure with a continuous component on [0, + ∞) and a point mass at + ∞. The undercoverage problem of the Chi-square calibration is partly due to its use of the continuous Chi-square distribution to approximate the mixture distribution of the empirical log-likelihood ratio. In this article, we propose two new methods of calibration which will take advantage of the mixture structure; we construct two new mixture distributions by using the F and chi-square distributions and use these to approximate the mixture distributions of the empirical log-likelihood ratio. The new methods of calibration are asymptotically equivalent to the chi-square calibration. But the new methods, in particular the F mixture based method, can be substantially more accurate than the chi-square calibration for small and moderately large sample sizes. The new methods are also as easy to use as the chi-square calibration.     

Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

In this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.     

Maximum log-likelihood ratio test for a change in three parameter Weibull distribution

Asymptotic behavior of a log-likelihood ratio statistic for testing a change in a three parameter Weibull distribution is studied. It is shown that if a shape parameter α>2$\alpha >2$ the law of iterated logarithm for maximum-likelihood estimators is still valid and the log-likelihood testing statistic is asymptotically distributed (after an appropriate normalization) according to a Gumbel distribution.     

Test for multiple upper outliers in an exponential sample irrespective of origin

An approach for testing multiple upper outliers with slippage alternative in an exponential sample, irrespective of origin, is discussed. The outlier detection procedure is based on a ratio of two estimates, obtained by the maximization of the two log-likelihood functions. One is the complete data log-likelihood and the other is its conditional expectation, given the regular observations. The exact null distribution of the test statistic is derived and no new table for critical values is required. A simulation study is also carried out to compare the performance of the test with the earlier work.     

Comparison of the restricted mean survival time with the hazard ratio in superiority trials with a time‐to‐event end point

With the emergence of novel therapies exhibiting distinct mechanisms of action compared to traditional treatments, departure from the proportional hazard (PH) assumption in clinical trials with a time‐to‐event end point is increasingly common. In these situations, the hazard ratio may not be a valid statistical measurement of treatment effect, and the log‐rank test may no longer be the most powerful statistical test. The restricted mean survival time (RMST) is an alternative robust and clinically interpretable summary measure that does not rely on the PH assumption. We conduct extensive simulations to evaluate the performance and operating characteristics of the RMST‐based inference and against the hazard ratio–based inference, under various scenarios and design parameter setups. The log‐rank test is generally a powerful test when there is evident separation favoring 1 treatment arm at most of the time points across the Kaplan‐Meier survival curves, but the performance of the RMST test is similar. Under non‐PH scenarios where late separation of survival curves is observed, the RMST‐based test has better performance than the log‐rank test when the truncation time is reasonably close to the tail of the observed curves. Furthermore, when flat survival tail (or low event rate) in the experimental arm is expected, selecting the minimum of the maximum observed event time as the truncation timepoint for the RMST is not recommended. In addition, we recommend the inclusion of analysis based on the RMST curve over the truncation time in clinical settings where there is suspicion of substantial departure from the PH assumption.