Log-likelihood Ratio Test Statistic Research Articles

Decorrelated empirical likelihood for generalized linear models with high-dimensional longitudinal data

This paper focuses on decorrelated empirical likelihood-based inference for longitudinal data with ultrahigh-dimensional covariates. The primary issues we aim to address involve parameter estimation and hypothesis testing for a low-dimensional parameter of interest. Under the framework of the generalized linear model, we initially consider the within-subject correlation by linearizing the precision matrix with certain known matrices, which retains optimality even if the working correlated structure is misspecified. Coupled with the decorrelated matrix, we then eliminate the influence of nuisance parameters on the estimation procedure. The proposed approach not only yields more efficient estimators compared to generalized decorrelated estimating equations but also shares the same asymptotic variance as quadratic decorrelated inference function based methods. Furthermore, we define the decorrelated empirical log-likelihood ratio test statistic to assess the significance of regression coefficients. Finally, to evaluate the performance of the proposed procedure, we conduct simulation studies and apply it to a real data example.

A longitudinal complex likelihood ratio test for pleiotropy

. To test pleiotropy based on longitudinal data, two main difficulties are the construction of test statistics and the formulation of decision-making criteria. In particular, the commonly used log-likelihood ratio test is not available since there does not exist a likelihood function and the estimation is based on estimating equations in general. To address this and construct a statistic similar to the -2 log-likelihood ratio test statistic, we propose a new concept, a complex log-likelihood function, such that its first derivative is equal to a given estimating function. In consequence, the -2 complex log-likelihood ratio test is constructed for testing genetic pleiotropy, and the validity of the test procedure is established. An extensive simulation study is performed, which suggests that the proposed method works well for practical situations. Furthermore, it is applied to a set of longitudinal data on tropical storms.

Asymptotic distributions for likelihood ratio tests for the equality of covariance matrices

Consider k independent random samples from p-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of k covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when k and p are fixed integers. Jiang and Qi (Scand J Stat 42:988–1009, 2015) and Jiang and Yang (Ann Stat 41(4):2029–2074, 2013) have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality p goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either p or k goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for p and k. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.

Asymptotics for Markov chain mixture detection

Sufficient conditions are provided under which the log-likelihood ratio test statistic fails to have a limiting chi-squared distribution under the null hypothesis when testing between one and two components under a general two-component mixture model, but rather tends to infinity in probability. These conditions are verified when the component densities describe continuous-time, discrete-state-space Markov chains and the results are illustrated via a parametric bootstrap simulation on an analysis of the migrations over time of a set of corporate bonds ratings. The precise limiting distribution is derived in a simple case with two states, one of which is absorbing which leads to a right-censored exponential scale mixture model. In that case, when centred by a function growing logarithmically in the sample size, the statistic has a limiting distribution of Gumbel extreme-value type rather than chi-squared.

Modeling Z-valued time series based on new versions of the Skellam INGARCH model

Recently, there has been a growing interest in integer-valued time series models, including integer-valued autoregressive (INAR) models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, but only a few of them can deal with data on the full set of integers, that is, Z={…,−2,−1,0,1,2,…}. Although some attempts have been made to deal with Z-valued time series, these models do not provide enough flexibility in modeling some specific integers (e.g., 0, ±1). A symmetric Skellam INGARCH(1,1) model was proposed in the literature, but it only considered zero-mean processes, which limits its application. We first extend the symmetric Skellam INGARCH model to an asymmetric version, which can deal with non-zero-mean processes. Then we propose a modified Skellam model which adopts a careful treatment on integers 0 and ±1 to satisfy a special feature of the data. Our models are easy-to-use and flexible. The maximum likelihood method is used to estimate unknown parameters and the log-likelihood ratio test statistic is provided for testing the asymmetric model against the modified one. Simulation studies are given to evaluate performances of the parametric estimation and log-likelihood ratio test. A real data example is also presented to demonstrate good performances of newly proposed models.

Empirical likelihood inference for semi-parametric transformation models with length-biased sampling

The semi-parametric transformation models under length-biased sampling are considered. The well-known proportional hazards model and proportional odds model are special cases of the semi-parametric transformation models. Empirical likelihood and adjusted empirical likelihood inferences for semi-parametric transformation models with length-biased sampling are proposed, and the empirical log-likelihood ratio test statistic is shown to converge to a standard chi-squared distribution. In addition, statistical inferences for the regression parameters are made based on the results. Moreover, extensive simulation studies are carried out. Finally, a real data set is analyzed to illustrate the proposed empirical likelihood and adjusted empirical likelihood methods.

Practical Aspects of False Alarm Control for Change Point Detection: Beyond Average Run Length

A popular method for detecting changes in the probability distribution of a sequence of observations is CUSUM, which proceeds by sequentially evaluating a log-likelihood ratio test statistic and comparing it to a predefined threshold; a change point is detected as soon as the threshold is exceeded. It is desirable to choose the threshold such that the number of false alarms is kept to a specified level. Traditionally, the number of false alarms is measured by the average run length – the expected stopping time until the first false alarm. However, this is does not in general allow one to control the number of false alarms at every particular time instance. Thus, in this paper two stronger false alarm criteria are considered, for which approximation methods are investigated to facilitate the selection of a threshold.

Statistical inference in a growth curve quantile regression model for longitudinal data.

This article describes a polynomial growth curve quantile regression model that provides a comprehensive assessment about the treatment effects on the changes of the distribution of outcomes over time. The proposed model has the flexibility, as it allows the degree of a polynomial to vary across quantiles. A high degree polynomial model fits the data adequately, yet it is not desirable due to the complexity of the model. We propose the model selection criterion based on an empirical loglikelihood that consistently identifies the optimal degree of a polynomial at each quantile. After the parsimonious model is fitted to the data, the hypothesis test is further developed to evaluate the treatment effects by comparing the growth curves. It is shown that the proposed empirical loglikelihood ratio test statistic follows a chi-square distribution asymptotically under the null hypothesis. Various simulation studies confirm that the proposed test successfully detects the difference between the curves across quantiles. When the empirical loglikelihood is employed, we incorporate the within-subject correlation commonly existing in longitudinal data and gain estimation efficiency of the quantile regression parameters in the growth curve model. The proposed process is illustrated through the analysis of randomized controlled longitudinal depression data.

Bayesian penalized log-likelihood ratio approach for dose response clinical trial studies

ABSTRACTIn literature, there are a few unified approaches to test proof of concept and estimate a target dose, including the multiple comparison procedure using modeling approach, and the permutation approach proposed by Klingenberg. We discuss and compare the operating characteristics of these unified approaches and further develop an alternative approach in a Bayesian framework based on the posterior distribution of a penalized log-likelihood ratio test statistic. Our Bayesian approach is much more flexible to handle linear or nonlinear dose–response relationships and is more efficient than the permutation approach. The operating characteristics of our Bayesian approach are comparable to and sometimes better than both approaches in a wide range of dose–response relationships. It yields credible intervals as well as predictive distribution for the response rate at a specific dose level for the target dose estimation. Our Bayesian approach can be easily extended to continuous, categorical, and time-to-event responses. We illustrate the performance of our proposed method with extensive simulations and Phase II clinical trial data examples.

Inference and Sample Size Calculations Based on Statistical Tests in a Negative Binomial Distribution for Differential Gene Expression in RNAseq Data

The high throughput RNA sequencing (RNA-seq) technology has become the popular method of choice for transcriptomics and the detection of differentially expressed genes. Sample size calculations for RNA-seq experimental design are an important consideration in biological research and clinical trials. Currently, the sample size formulas derived from the Wald and the likelihood ratio statistical tests with a Poisson distribution to model RNA-seq data have been developed. However, since the mean read counts in the real RNA-seq data are not equal to the variance, an extended method to calculate sample sizes based on a negative binomial distribution using an exact test statistic was proposed by Li et al. in 2013. In this study, we alternatively derive five sample size calculation methods based on the negative binomial distribution using the Wald test, the log-transformed Wald test and the log-likelihood ratio test statistics. A comparison of our five methods and an existing method was performed by calculating the sample sizes and the simulated power in different scenarios. We first calculated the sample sizes for testing a single gene using the six methods given a nominal significance level α at 0.05 and 80% power. Then, we calculated the sample sizes for testing multiple genes given a false discovery rate (FDR) at 0.05 and 0.10. The empirical power and true prognostic genes for differential gene expression analysis corresponding to the estimated sample sizes from the six methods are also estimated via the simulation studies. Using the sample size formulas derived from log-transformed and Wald-based tests, we observed smaller sample properties while maintaining the nominal power close to or higher than 80% in all the settings compared to other methods. Moreover, the Wald test based sample size calculation method is easier to compute and faster in an RNA-seq experimental design.

Impact of breast structure on lesion detection in breast tomosynthesis, a simulation study.

This study aims to characterize the effect of background tissue density and heterogeneity on the detection of irregular masses in breast tomosynthesis, while demonstrating the capability of the sophisticated tools that can be used in the design, implementation, and performance analysis of virtual clinical trials (VCTs). Twenty breast phantoms from the extended cardiac-torso (XCAT) family, generated based on dedicated breast computed tomography of human subjects, were used to extract a total of 2173 volumes of interest (VOIs) from simulated tomosynthesis images. Five different lesions, modeled after human subject tomosynthesis images, were embedded in the breasts and combined with the lesion absent condition yielded a total of [Formula: see text] VOIs. Effects of background tissue density and heterogeneity on the detection of the lesions were studied by implementing a composite hypothesis signal detection paradigm with location known exactly, lesion known exactly or statistically, and background known statistically. Using the area under the receiver operating characteristic curve, detection performance deteriorated as density was increased, yielding findings consistent with clinical studies. A human observer study was performed on a subset of the simulated tomosynthesis images, confirming the detection performance trends with respect to density and serving as a validation of the implemented detector. Performance of the implemented detector varied substantially across the 20 breasts. Furthermore, background tissue density and heterogeneity affected the log-likelihood ratio test statistic differently under lesion absent and lesion present conditions. Therefore, considering background tissue variability in tissue models can change the outcomes of a VCT and is hence of crucial importance. The XCAT breast phantoms have the potential to address this concern by offering realistic modeling of background tissue variability based on a wide range of human subjects, comprising various breast shapes, sizes, and densities.

Generalized Correlation Coefficient Based on Log Likelihood Ratio Test Statistic

In this paper, I point out that both Joe’s and Ding’s strength statistics can only be used for testing the pair-wise independence, and I propose a novel G-square based strength statistic, called Liu’s generalized correlation coefficient, it can be used to detect and compare the strength of not only the pair-wise independence but also the mutual independence of any multivariate variables. Furthermore, I proved that only Liu’s generalized correlation coefficient is strictly increasing on its number of variables, it is more sensitive and useful than Cramer’s V coefficient, in other words, Liu generalized correlation coefficient is not only the G-square based strength statistic, but also an improved statistic for detecting and comparing the strengths of deferent associations of any two or more sets of multivariate variables, moreover, this new strength statistic can also be tested by G 2 .

Resolved target detection in clutter using correlated, dual-band imagery

This paper develops a log-likelihood ratio test statistic for resolved target detection in dual-band imagery because the previous work indicates that most of the processing gains come from processing just two bands. Simple, closed-form equations for the closed-form probabilities of false alarm and detection are given. A computer simulation validates the theory. A constant false alarm rate version of the theory is applied to real available multiband data with quasi-resolved target sets and fixed clutter noise. The results show very reasonable performance in target detectability using three sets of correlated dual-band images. Finally, this paper shows that the resolved target detection problem depends on the weighted difference between the dual-band target contrasts. The theoretical development reaffirms that the signal-to-noise ratio or contrast-to-noise ratio is approximately the weighted difference squared, divided by the normalized total image noise variance.

Multiple Comparisons for a Psychophysical Test in Bootstrap Logistic Regression

We propose an algorithm of multiple comparisons with a control for a psychophysical test. Our algorithm is based on the step-down procedure and is applicable to the bootstrap test in logistic regression. We use logistic regression combined with guessing rate and log-likelihood ratio test statistics of multiple samples in order to test hypotheses by using non-parametric bootstrap resampling. We apply our algorithm to visual acuity measurement, and show that bootstrap resampling can be used to resolve problems with multiple comparisons especially when the numbers of observations among samples are not identical.

Distinguishing ‘Higgs’ spin hypotheses using γγ and WW ∗ decays

The new particle X recently discovered by the ATLAS and CMS Collaborations in searches for the Higgs boson has been observed to decay into , ZZ and WW , but its spin and parity, J P , remain a mystery, with J P = 0 + and 2 + being open possibilities. We use PYTHIA and Delphes to simulate an analysis of the angular distribution of gg! X! decays in a full 2012 data set, including realistic background levels. We show that this angular distribution should provide strong discrimination between the possibilities of spin zero and spin two with graviton-like couplings: 3 if a conservative symmetric interpretation of the log-likelihood ratio (LLR) test statistic is used, and 6 if a less conservative asymmetric interpretation is used. The WW and ZZ couplings of the Standard Model Higgs boson and of a 2 + particle with graviton-like couplings are both expected to exhibit custodial symmetry. We simulate the present ATLAS and CMS search strategies for X! WW using PYTHIA and Delphes, and show that their eciencies

A Test for an Abrupt Change in Weibull Hazard Functions with Staggered Entry and Type I Censoring

We consider a test for an abrupt change in a Weibull hazard function with unknown scale parameter when the data are subject to staggered entry and Type I censoring. We formulate the profile log-likelihood ratio test statistic as a function of the change point and show that it converges weakly to an Ornstein-Uhlenbeck process. We determine critical values from an approximation to the tail probability of the supremum of the process. We study the power of the test through simulation, and demonstrate its applicability using real data from a clinical study for the treatment of chronic granulomatous disease.

A test for abrupt change in hazard regression models with Weibull baselines

We develop a likelihood ratio test for an abrupt change point in Weibull hazard functions with covariates, including the two-piece constant hazard as a special case. We first define the log-likelihood ratio test statistic as the supremum of the profile log-likelihood ratio process over the interval which may contain an unknown change point. Using local asymptotic normality (LAN) and empirical measure, we show that the profile log-likelihood ratio process converges weakly to a quadratic form of Gaussian processes. We determine the critical values of the test and discuss how the test can be used for model selection. We also illustrate the method using the Chronic Granulomatous Disease (CGD) data.

Drug safety data mining with a tree‐based scan statistic

In post-marketing drug safety surveillance, data mining can potentially detect rare but serious adverse events. Assessing an entire collection of drug-event pairs is traditionally performed on a predefined level of granularity. It is unknown a priori whether a drug causes a very specific or a set of related adverse events, such as mitral valve disorders, all valve disorders, or different types of heart disease. This methodological paper evaluates the tree-based scan statistic data mining method to enhance drug safety surveillance. We use a three-million-member electronic health records database from the HMO Research Network. Using the tree-based scan statistic, we assess the safety of selected antifungal and diabetes drugs, simultaneously evaluating overlapping diagnosis groups at different granularity levels, adjusting for multiple testing. Expected and observed adverse event counts were adjusted for age, sex, and health plan, producing a log likelihood ratio test statistic. Out of 732 evaluated disease groupings, 24 were statistically significant, divided among 10 non-overlapping disease categories. Five of the 10 signals are known adverse effects, four are likely due to confounding by indication, while one may warrant further investigation. The tree-based scan statistic can be successfully applied as a data mining tool in drug safety surveillance using observational data. The total number of statistical signals was modest and does not imply a causal relationship. Rather, data mining results should be used to generate candidate drug-event pairs for rigorous epidemiological studies to evaluate the individual and comparative safety profiles of drugs.

Analysis of ϕ-divergence for loglinear models with constraints under product-multinomial sampling

The loglinear model under product-multinomial sampling with constraints is considered. The asymptotic expansion and normality of the restricted minimum ϕ-divergence estimator (RMϕDE) which is a generalization of the maximum likelihood estimator is presented. Then various statistics based on ϕ-divergence and RMϕDE are used to test various hypothesis test problems under the model considered. These statistics contain the classical loglikelihood ratio test statistics and Pearson chi-squared test statistics. In the last section, a simulation study is implemented.

Lack-of-Fit Tests for Generalized Linear Models via Splines

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.