Rao's Score Research Articles

Differential Geometric Least Angle Regression: A Differential Geometric Approach to Sparse Generalized Linear Models

SummarySparsity is an essential feature of many contemporary data problems. Remote sensing, various forms of automated screening and other high throughput measurement devices collect a large amount of information, typically about few independent statistical subjects or units. In certain cases it is reasonable to assume that the underlying process generating the data is itself sparse, in the sense that only a few of the measured variables are involved in the process. We propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family. In our approach we generalize the equiangular condition in a generalized linear model. Although the geometry involves the Fisher information in a way that is not obvious in the simple regression setting, the equiangular condition turns out to be equivalent with an intuitive condition imposed on the Rao score test statistics. In certain special cases the method can be tweaked to obtain L1-penalized generalized linear model solution paths, but the method itself defines sparsity more directly. Although the computation of the solution paths is not trivial, the method compares favourably with other path following algorithms.

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Rao's score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test's statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.

Identifying genetic marker sets associated with phenotypes via an efficient adaptive score test

In recent years, genome-wide association studies (GWAS) and gene-expression profiling have generated a large number of valuable datasets for assessing how genetic variations are related to disease outcomes. With such datasets, it is often of interest to assess the overall effect of a set of genetic markers, assembled based on biological knowledge. Genetic marker-set analyses have been advocated as more reliable and powerful approaches compared with the traditional marginal approaches (Curtis and others, 2005. Pathways to the analysis of microarray data. TRENDS in Biotechnology 23, 429-435; Efroni and others, 2007. Identification of key processes underlying cancer phenotypes using biologic pathway analysis. PLoS One 2, 425). Procedures for testing the overall effect of a marker-set have been actively studied in recent years. For example, score tests derived under an Empirical Bayes (EB) framework (Liu and others, 2007. Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models. Biometrics 63, 1079-1088; Liu and others, 2008. Estimation and testing for the effect of a genetic pathway on a disease outcome using logistic kernel machine regression via logistic mixed models. BMC bioinformatics 9, 292-2; Wu and others, 2010. Powerful SNP-set analysis for case-control genome-wide association studies. American Journal of Human Genetics 86, 929) have been proposed as powerful alternatives to the standard Rao score test (Rao, 1948. Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 50-57). The advantages of these EB-based tests are most apparent when the markers are correlated, due to the reduction in the degrees of freedom. In this paper, we propose an adaptive score test which up- or down-weights the contributions from each member of the marker-set based on the Z-scores of their effects. Such an adaptive procedure gains power over the existing procedures when the signal is sparse and the correlation among the markers is weak. By combining evidence from both the EB-based score test and the adaptive test, we further construct an omnibus test that attains good power in most settings. The null distributions of the proposed test statistics can be approximated well either via simple perturbation procedures or via distributional approximations. Through extensive simulation studies, we demonstrate that the proposed procedures perform well in finite samples. We apply the tests to a breast cancer genetic study to assess the overall effect of the FGFR2 gene on breast cancer risk.

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.

Robust Logistic Regression for Binomial Responses

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed analytically by means of the Influence Function (IF) and empirically by means of simulations. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification errors in the responses or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts and they are applied to the analysis of binary data from a study on breastfeeding.

New results on the likelihood ratio and score tests for the von Mises distribution

In this paper, we derive Bartlett and Bartlett-type corrections [G.M. Cordeiro and S.L.P. Ferrari 1991, A modified score test statistic having chi-squared distribution to order n −1 , Biometrika 78 (1991), pp. 573–582] to improve the likelihood ratio and Rao's score statistics for testing the mean parameter and the concentration parameter in the von Mises distribution. Simple formulae are suggested for the corrections valid for small and large values of the concentration parameter that do not depend on the modified Bessel functions and can be useful in practical applications.

GENERAL SPECIFICATION TESTING WITH LOCALLY MISSPECIFIED MODELS

A well known result is that many of the tests used in econometrics, such as the Rao score (RS) test, may not be robust to misspecified alternatives, that is, when the alternative model does not correspond to the underlying data generating process. Under this scenario, these tests spuriously reject the null hypothesis too often. We generalize this result to generalized method of moments–based (GMM-based) tests. We also extend the method proposed in Bera and Yoon (1993, Econometric Theory 9, 649–658) for constructing RS tests that are robust to local misspecification to GMM-based tests. Finally, a further generalization for general estimating and testing functions is developed. This framework encompasses both likelihood and GMM-based results.

Rao score test

Rao score test

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.

Correcting Four Test Statistics for One-Parameter Distributions Using Mathematica

Likelihood ratio, Rao score, Wald, and modified Wald statistics are commonly used to test composite hypotheses on the parameters of distributions. We present source code for a Mathematica program (script) that can be used for generating analytic expressions for Bartlett and Bartlett-type corrections to improve all these four statistics for testing the parameter of an arbitrary regular one-parameter distribution. The script is tested on various discrete and continuous distributions, and the results are compared with those published in earlier works, confirming all of the previously reported expressions. The presented script may in principle be applied to calculate Bartlett and Bartlett-type corrections to improve the accuracy of the asymptotic distributions of these four statistics for any regular one-parameter distribution. In particular, new results are derived for distributions outside the exponential family, which have not been previously published. We apply our results on the t distribution in a simulation study to compare the performance of the four classical tests and their corrected versions.

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n −3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n −1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.

On testing equality of intraclass correlations under unequal family sizes

The three likelihood-based tests, namely, likelihood ratio test, Rao score test, and Wald test and two more asymptotic tests which use Srivastava's estimator of intraclass correlation coefficient are considered to test the null hypothesis of equality of intraclass correlation coefficients when the families have unequal number of children. Methods are illustrated on Galton's data set. Using simulation experiment we compute the sizes and powers of these tests and compare. It is found that our proposed test using Srivastava's estimator and the score test perform the best among all tests.

Asymptotic expansions of the distributions of some test statistics in generalized linear models

The asymptotic expansions of the distributions of three statistics, namely, the likelihood ratio, Wald and Rao's score statistics, are obtained for testing a subset of parameters in the generalized linear models. The asymptotic expansions are given in closed forms up to n −1 in the null case and up to n −1/2 in the nonnull case under a sequence of Pitman alternatives. It is found that the power of Wald and Rao's score tests are same up to n −1/2. Numerical examples are presented to illustrate the performance of the asymptotic null distributions.

A New Generalization of the Logarithmic Distribution Arising from the Inverse Trinomial Distribution

A new generalization of the logarithmic distribution (LD)—the generalized logarithmic distribution (GLD)—is introduced. The GLD is a cluster size distribution of the inverse trinomial distribution, which arises in a generalized random walk problem. Major properties of the GLD are derived and discussed. The method of first moment and frequency of one count and maximum likelihood estimation (MLE) are used for parameter estimation. Rao's score test statistic is derived as a model selection tool. Finally, we demonstrate an application of the GLD in fitting insect data from temperate and tropical regions.

Statistical inference for the risk ratio in [formula omitted] binomial trials with structural zero

In some statistical analyses, researchers may encounter the problem of analyzing a correlated 2 × 2 table with a structural zero in one of the off diagonal cells. Structural zeros arise in situation where it is theoretically impossible for a particular cell to be observed. For instance, Agresti [2002. Categorical Data Analysis, 2nd ed. Wiley, Hoboken, New Jersey] provided an example involving a sample of 156 calves born in Okeechobee County, Florida. Calves are first classified according to whether they get a pneumonia infection within certain time. They are then classified again according to whether they get a secondary infection within a period after the first infection clears up. Because subjects cannot, by definition, have a secondary infection without first having a primary infection, a structural void in the cell of the summary table that corresponds with no primary infection and has secondary infection is introduced. For discussion of this phenomenon, see Tang and Tang [2002. Exact unconditional inference for risk ratio in a correlated 2 × 2 table with structural zero. Biometrics 58, 972–980], and Lui [1998. Interval estimation of the risk ratio between a secondary infection, given a primary infection, and the primary infection. Biometrics 54, 706–711]. The risk ratio (RR) between the secondary infection, given the primary infection, and the primary infection may be a useful measure of change in the pneumonia infection rates of the primary infection and the secondary infection. In this paper, we first develop and evaluate the large sample confidence intervals of RR. In addition to the three confidence intervals in the literatures, we propose a confidence interval based on Rao's score test. The performance of these confidence intervals is studied by means of extensive simulation studies. We also investigate the tests of hypothesis for the RR and the power of these tests. Simulation studies are carried out to examine the performance of these tests in terms of their power. An example, from the literature, is also provided to illustrate these procedures. Finally, the four confidence intervals were compared with those obtained by corrected version of Tang et al. [2004. Confidence interval for rate ratio in a 2 × 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests. Biometrics 60, 550–555; 2006. Sample size determination for 2-step studies with dichotomous response. Journal of Statistical Planning and Inference 136, 1166–1180].

On Rao score and PearsonX 2 statistics in generalized linear models

The identity of the Rao score and PearsonX 2 statistics is well known in the areas where the latter was first introduced: goodness-of-fit in contingency tables and binary responses. We show in this paper that the same identity holds when the two statistics are used for testing goodness-of-fit of Generalized Linear Models. We also highlight the connections that exist between the two statistics when they are used for the comparison of nested models. Finally, we discuss some merits of these unifying results.

Testing panel data regression models with spatial error correlation

This paper derives several lagrange multiplier (LM) tests for the panel data regression model with spatial error correlation. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin (Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Dordrecht; Rao's score test in spatial econometrics, J. Statist. Plann. Inference 97 (2001) 113) and Anselin et al. (Regional Sci. Urban Econom. 26 (1996) 77), and the second is the LM tests for the error component panel data model discussed in Breusch and Pagan (Rev. Econom. Stud. 47(1980) 239) and Baltagi et al. (J. Econometrics 54 (1992) 95). The idea is to allow for both spatial error correlation as well as random region effects in the panel data regression model and to test for their joint significance. Additionally, this paper derives conditional LM tests, which test for random regional effects given the presence of spatial error correlation. Also, spatial error correlation given the presence of random regional effects. These conditional LM tests are an alternative to the one-directional LM tests that test for random regional effects ignoring the presence of spatial error correlation or the one-directional LM tests for spatial error correlation ignoring the presence of random regional effects. We argue that these joint and conditional LM tests guard against possible misspecification. Extensive Monte Carlo experiments are conducted to study the performance of these LM tests as well as the corresponding likelihood ratio tests.

Variable selection for structural models

Theory of variable selection for structural models that do not have clear dependent variables is developed. Theory is derived within the framework of the curved exponential family of distributions for observed variables. The idea of Rao's score test was taken to construct a test statistic for variable selection, and its statistical properties are examined. In particular, the test statistic is shown to have asymptotic central chi-square distribution under a kind of alternative hypothesis. This fact will provide an evidence for excellent performance of the score statistic for real data sets.

Robust inference with binary data

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.

STATISTICAL TESTS FOR NESTED DESIGNS UNDER PARTIAL BALANCE

ABSTRACT In this article, we derive locally optimum tests and Rao's score tests for a two-stage nested design under a certain partial balance. Specifically, it is assumed that for any given level of leading factor, the number of observations at every error stage below it are equal. Formulas are provided so as to efficiently compute the test statistics.