Modified Likelihood Ratio Test Research Articles

The Application of the Likelihood Ratio Test and the Cochran-Mantel-Haenszel Test to Discrimination Cases

In practice, the ultimate outcome of many important discrimination cases, for example, the Wal-Mart, Nike and Goldman-Sachs equal pay cases, is determined at the stage when the plaintiffs request that the case be certified as a class action. The primary statistical issue at this time is whether the employment practice in question leads to a common pattern of outcomes disadvantaging most plaintiffs. However, there are no formal procedures or government guidelines for checking whether an employment practice results in a common pattern of disparity. This article proposes using the slightly modified likelihood ratio test and the one-sided Cochran-Mantel-Haenszel (CMH) test to examine data relevant to deciding whether this commonality requirement is satisfied. Data considered at the class certification stage from several actual cases are analyzed by the proposed procedures. The results often show that the employment practice at issue created a common pattern of disparity, however, based on the evidence presented to the courts, the class action requests were denied.

Flexible-Elliptical Spatial Scan Method

The detection of disease clusters in spatial data analysis plays a crucial role in public health, while the circular scan method is widely utilized for this purpose, accurately identifying non-circular (irregular) clusters remains challenging and reduces detection accuracy. To overcome this limitation, various extensions have been proposed to effectively detect arbitrarily shaped clusters. In this paper, we combine the strengths of two well-known methods, the flexible and elliptic scan methods, which are specifically designed for detecting irregularly shaped clusters. We leverage the unique characteristics of these methods to create candidate zones capable of accurately detecting irregularly shaped clusters, along with a modified likelihood ratio test statistic. By inheriting the advantages of the flexible and elliptic methods, our proposed approach represents a practical addition to the existing repertoire of spatial data analysis techniques.

On hypothesis testing in latent class and finite mixture stochastic frontier models, with application to a contaminated normal-half normal model

Latent class and finite mixture stochastic frontier models have been proposed as a means of allowing either for technological heterogeneity or more flexible distributions of noise and inefficiency. As in the wider literature on latent class and finite mixture models, we are interested in class enumeration, particularly testing against homogeneity. We apply a modified likelihood ratio test for homogeneity in a stochastic frontier setting, based on established results for non-Gaussian latent class and finite mixture models, and provide evidence from Monte Carlo experiments which suggest the applicability of results regarding a modified likelihood ratio test to the stochastic frontier setting. We demonstrate an application to testing a model with a contaminated normal noise term against a model with a normally distributed noise term, finding that the former is preferred, with significant implications for efficiency prediction.

A new confidence interval for the ratio of two normal means and comparisons

The problem of estimating the ratio of means of two independent normal distributions is considered. Interval estimation method similar to Welch's approximate degrees of freedom solution for the Behrens–Fisher problem is proposed. The test and confidence interval (CI) for the ratio of means are compared with the available CIs including the one based on the modified likelihood ratio test. Extensive simulation studies indicate that the proposed method is accurate even for very small samples and simpler than other available likelihood-based tests and CIs. The methods are illustrated using an example.

Modified likelihood ratio tests for extreme value distributions

The modified Anderson-Darling test statistics have been suggested in testing the highly skewed distributions to detect more possible departures in the right (or left) tail. In this paper, we propose the corresponding modified likelihood ratio test statistics, and compare their performances for the Gumbel distribution with the modified Anderson-Darling test statistics. We also provide the approximate critical values for the generalized extreme value distribution and generalized Pareto distribution where the unknown shape parameter is present.

Modified Likelihood Ratio Test for Sub-mean Vectors with Two-step Monotone Missing Data in Two-sample Problem

This article deals with the problem of testing for two normal sub-mean vectors when the data set have two-step monotone missing observations. Under the assumptions that the population covariance matrices are equal, we obtain the likelihood ratio test (LRT) statistic. Furthermore, an asymptotic expansion for the null distribution of the LRT statistic is derived under the two-step monotone missing data by the perturbation method. Using the result, we propose two improved statistics with good chi-squared approximation. One is the modified LRT statistic by Bartlett correction,and the other is the modified LRT statistic using the modification coefficient by linear interpolation. The accuracy of the approximations are investigated by using a Monte Carlo simulation. The proposed methods are illustrated using an example.

Testing the Number of Components in a Birnbaum-Saunders Mixture Model under a Random Censoring Scheme

This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under randomly right censored data. We focus on two methods, one based on the modified likelihood ratio test and the other based on the shortcut of bootstrap test. Based on extensive Monte Carlo simulation studies, we evaluate and compare the performance of the proposed tests through their size and power. A power analysis provides guidance for researchers to examine the factors that affect the power of the proposed tests used in detecting the correct number of components in a Birnbaum-Saunders mixture model. Finally an example of aircraft Windshield data is used to illustrate the testing procedure.

Improved testing inferences for beta regressions with parametric mean link function

Beta regressions are widely used for modeling random variables that assume values in the standard unit interval, (0, 1), such as rates, proportions, and income concentration indices. Parameter estimation is typically performed via maximum likelihood, and hypothesis testing inferences on the model parameters are commonly performed using the likelihood ratio test. Such a test, however, may deliver inaccurate inferences when the sample size is small. It is thus important to develop alternative testing procedures that are more accurate when the sample contains only few observations. In this paper, we consider the beta regression model with parametric mean link function and derive two modified likelihood ratio test statistics for that class of models. We provide simulation evidence that shows that the new tests usually outperform the standard likelihood ratio test in samples of small to moderate sizes. We also present and discuss two empirical applications.

Homogeneity testing under finite location‐scale mixtures

The testing problem for the order of finite mixture models has a long history and remains an active research topic. Since Ghosh & Sen (1985) revealed the hard‐to‐manage asymptotic properties of the likelihood ratio test, many successful alternative approaches have been developed. The most successful attempts include the modified likelihood ratio test and the EM‐test, which lead to neat solutions for finite mixtures of univariate normal distributions, finite mixtures of single‐parameter distributions, and several mixture‐like models. The problem remains challenging, and there is still no generic solution for location‐scale mixtures. In this article, we provide an EM‐test solution for homogeneity for finite mixtures of location‐scale family distributions. This EM‐test has nonstandard limiting distributions, but we are able to find the critical values numerically. We use computer experiments to obtain appropriate values for the tuning parameters. A simulation study shows that the fine‐tuned EM‐test has close to nominal type I errors and very good power properties. Two application examples are included to demonstrate the performance of the EM‐test.

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model's parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.

A New Test for the Homogeneity of Inverse Gaussian Scale Parameters Based on Computational Approach Test

In this paper, we focused on testing homogeneity of scale parameters of k Inverse Gaussian distributions (IGDs) since this distribution is one of the most common distribution for analyzing nonnegative right-skewed data. We have proposed a new test statistic based on the Computational Approach Test (CAT), which is a type of parametric bootstrap method, for testing homogeneity of scale parameters of k IGDs. Simulation results have been presented to compare the performances of the proposed method and existing methods such as the likelihood ratio test, modified likelihood ratio test and generalized likelihood ratio test in terms of type I error rate and power. The results showed that the proposed CAT is better than the others in terms of the type I error rates and powers in some cases.

ModL: exploring and restoring regularity when testing for positive selection.

Likelihood ratio tests are commonly used to test for positive selection acting on proteins. They are usually applied with thresholds for declaring a protein under positive selection determined from a chi-square or mixture of chi-square distributions. Although it is known that such distributions are not strictly justified due to the statistical irregularity of the problem, the hope has been that the resulting tests are conservative and do not lose much power in comparison with the same test using the unknown, correct threshold. We show that commonly used thresholds need not yield conservative tests, but instead give larger than expected Type I error rates. Statistical regularity can be restored by using a modified likelihood ratio test. We give theoretical results to prove that, if the number of sites is not too small, the modified likelihood ratio test gives approximately correct Type I error probabilities regardless of the parameter settings of the underlying null hypothesis. Simulations show that modification gives Type I error rates closer to those stated without a loss of power. The simulations also show that parameter estimation for mixture models of codon evolution can be challenging in certain data-generation settings with very different mixing distributions giving nearly identical site pattern distributions unless the number of taxa and tree length are large. Because mixture models are widely used for a variety of problems in molecular evolution, the challenges and general approaches to solving them presented here are applicable in a broader context. https://github.com/jehops/codeml_modl. Supplementary data are available at Bioinformatics online.

Inference about the shape parameters of several inverse Gaussian distributions: testing equality and confidence interval for a common value

In this paper, we consider inference about the shape parameters of several inverse Gaussian distributions. At first, an approach is given to test the equality of these parameters based on modified likelihood ratio test. Then, five approaches are presented to construct confidence intervals for the common shape parameter. The performance of these approaches is studied using Monte Carlo simulation, and illustrated using a real data set.

Asymptotic null distribution of the modified likelihood ratio test for homogeneity in finite mixture models

Likelihood-based methods play a central role in statistical inference for parametric models. Among these, the modified likelihood ratio test is preferred in testing for homogeneity in finite mixture models. The test statistic is related to the maximum of a quadratic function under general regularity conditions. Re-parameterization is shown to have overcome the difficulty when linear independence is not satisfied. Models with parameter constraints are also considered. The asymptotic null distribution of the test statistic is shown to have a chi-bar-squared distribution in both constrained and unconstrained cases. We extend the result to linear models and demonstrate that the chi-bar-squared distribution is also applicable. The general asymptotic result provides a much simpler testing procedure with an exact form of the asymptotic distribution compared to re-sampling approach in the literature. It also offers accurate p-value as shown in simulation. The results are checked by extensive simulation and are supplemented by a breast cancer data example.

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

ABSTRACTIn this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.

Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions

Let x 1 , … , x n be a random sample from a Gaussian random vector of dimension p < n with mean μ and covariance matrix Σ . Based on this sample, we consider the moderate deviation principle of the modified likelihood ratio test (LRT) for the testing problem H 0 : Σ = λ I p versus H 1 : Σ ≠ λ I p in the high-dimensional setting, where λ is some unknown constant (Jiang and Yang (2013)). We assume that both the dimension p and sample size n go to infinity in such a way that p / n → y ∈ ( 0 , 1 ] . Under H 0 , our results give the exponential convergence rate of the LRT statistic to the corresponding asymptotic distribution.

On finite mixture models

ABSTRACTFinite mixture models are widely used in scientific investigations. Due to their non-regularity, there are many technical challenges concerning inference problems on various aspects of the finite mixture models. After decades of effort by statisticians, substantial progresses are recorded recently in characterising large sample properties of some classical inference methods when applied to finite mixture models, providing effective numerical solutions for mixture model-based data analysis, and the invention of novel inference approaches. This paper aims to provide a comprehensive summary on large sample properties of some classical statistical methods and recently developed modified likelihood ratio test and EM-test for the order of the finite mixture model. The presentation de-emphasises the rigour in order to gain some insights behind some complex technical issues. The paper wishes to recommend the EM-test as the most promising approach to data analysis problems from all models with mixture structures.

A modified likelihood ratio test for a mean vector with monotone missing data

A modified likelihood ratio test for a mean vector with monotone missing data

Testing equality of generalized variances of k multivariate normal populations

ABSTRACTGeneralized variance is a measure of dispersion of multivariate data. Comparison of dispersion of multivariate data is one of the favorite issues for multivariate quality control, generalized homogeneity of multidimensional scatter, etc. In this article, the problem of testing equality of generalized variances of k multivariate normal populations by using the Bartlett's modified likelihood ratio test (BMLRT) is proposed. Simulations to compare the Type I error rate and power of the BMLRT and the likelihood ratio test (LRT) methods are performed. These simulations show that the BMLRT method has a better chi-square approximation under the null hypothesis. Finally, a practical example is given.

On the likelihood ratio test for the equality of multivariate normal populations with two-step monotone missing data

ABSTRACTIn this article, we consider the problem of testing the equality of multivariate normal populations when the data set has missing observations with a two-step monotone pattern. The likelihood ratio test (LRT) statistic for the simultaneous testing of the mean vectors and the covariance matrices is given under the condition of two-step monotone missing data. An approximate modified likelihood ratio test (MLRT) statistic is presented using linear interpolation based on the coefficients of the MLRT statistic in the case of complete data sets. As an alternative approach, we propose approximate MLRT statistics of two kinds with two-step monotone missing data using the decompositions of the likelihood ratio (LR). An approximate upper percentile of the LRT statistic with two-step monotone missing data is also derived based on an asymptotic expansion for the LRT statistic in the case of complete data sets. Finally, we investigate the accuracy of the approximations using Monte Carlo simulation.