A fast score test for generalized mixture models

In biomedical studies, testing for homogeneity between two groups, where one group is modeled by mixture models, is often of great interest. This paper considers the semiparametric exponential family mixture model proposed by Hong et al. (2017) and studies the score test for homogeneity under this model. The score test is nonregular in the sense that nuisance parameters disappear under the null hypothesis. To address this difficulty, we propose a modification of the score test, so that the resulting test enjoys the Wilks phenomenon. In finite samples, we show that with fixed nuisance parameters the score test is locally most powerful. In large samples, we establish the asymptotic power functions under two types of local alternative hypotheses. Our simulation studies illustrate that the proposed score test is powerful and computationally fast. We apply the proposed score test to an UK ovarian cancer DNA methylation data for identification of differentially methylated CpG sites.     

The size of the chi-square test for the Hardy-Weinberg law

Many scientific problems can be formulated in terms of a statistical model indexed by parameters, only some of which are of scientific interest and the other parameters, called nuisance parameters, are not of interest in themselves. For testing the Hardy-Weinberg law, a relation among genotype and allele probabilities is of interest and allele probabilities are of no interest and now nuisance parameters. In this paper we investigate how the size (the maximum of the type I error rate over the nuisance parameter space) of the chi-square test for the Hardy-Weinberg law is affected by the nuisance parameters. Whether the size is well controlled or not under the nominal level has been frequently investigated as basic components of statistical tests. The size represents the type I error rate at the worst case. We prove that the size is always greater than the nominal level as the sample size increases. Extensive computations show that the size of the chi-squared test (worst type I error rate over the nuisance parameter space) deviates more upwardly from the nominal level as the sample size gets larger. The value at which the maximum of the type I error rate was found moves closer to the edges of the the nuisance parameter space with increasing sample size. An exact test is recommended as an alternative when the type I error is inflated.     

Confidence interval for rate ratio in a 2 x 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests

In this article, we consider problems with correlated data that can be summarized in a 2 x 2 table with structural zero in one of the off-diagonal cells. Data of this kind sometimes appear in infectious disease studies and two-step procedure studies. Lui (1998, Biometrics54, 706-711) considered confidence interval estimation of rate ratio based on Fieller-type, Wald-type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false-negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false-negative rate ratio. Score test-based confidence interval construction for false-negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test-based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre-assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

On Some Nuisance Parameter Free Uniformly Most Powerful Tests

TAKEUCHI (1969) provides a uniformly most powerful (UMP) one side test for testing the location parameter of the two parameters exponential model when the scale parameter is unknown. The power of his similar size α test depends, however, on the unknown scale parameter. In this case and in more general situations when there exists a sufficient statistic for the nuisance parameter, the theory of generalized THOMPSON's distributions, more specifically, the Thompsonization of a test statistic, LAURENT (1959, 1972) provides a UMP test whose power does not depend on the nuisance parameter. Examples of application of the general nuisance parameter free test procedure include here the truncated exponential, the inverse Gaussian, and the geometric distributions.     

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

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

On Analysis of the Difference of Two Exposure-Adjusted Poisson Rates with Stratification: From Asymptotic to Exact Approaches

Long-term studies are frequently conducted to assess the treatment effect on rare diseases or the safety of a new treatment. To account for differential follow-up often encountered in long-term studies, exposure-adjusted incidence rates are used in evaluating the treatment effect. The difference of rates is sometimes used to quantify the treatment’s public health impact because the reciprocal of this difference can be interpreted as “the number needed to treat (or number needed to vaccinate) in order to cure (or prevent) 1 case of disease.” In this paper we focus on the stratified analysis of the difference of two exposure-adjusted rates in the setting of superiority, noninferiority and super-superiority hypothesis testing. After a brief review of asymptotic methods, we derive an exact method that guarantees control of the type I error. But it is conservative for noninferiority and super-superiority testing because of the search of the maximum tail probability over a multidimensional nuisance parameter space. Then, we present an approximate exact method where the p-value is estimated at the maximum likelihood estimates of the nuisance parameter. This method is identical to exact method for superiority testing and reduces the conservatism for noninferiority and super-superiority testing. In addition, a quasi-exact method is discussed. A real-life vaccine clinical trial example is used to illustrate these methods. Finally, we compare the performance of these methods via empirical studies and make some general practical recommendations.     

Multipoint linkage detection in the presence of heterogeneity

Linkage heterogeneity is common for complex diseases. It is well known that loss of statistical power for detecting linkage will result if one assumes complete homogeneity in the presence of linkage heterogeneity. To this end, Smith (1963, Annals of Human Genetics 27, 175-182) proposed an admixture model to account for linkage heterogeneity. It is well known that for this model, the conventional chi-squared approximation to the likelihood ratio test for no linkage does not apply even when the sample size is large. By dealing with nuclear families and one marker at a time for genetic diseases with simple modes of inheritance, score-based test statistics (Liang and Rathouz, 1999, Biometrics 55, 65-74) and likelihood-ratio-based test statistics (Lemdani and Pons, 1995, Biometrics 51, 1033-1041) have been proposed which have a simple large-sample distribution under the null hypothesis of linkage. In this paper, we extend their work to more practical situations that include information from multiple markers and multi-generational pedigrees while allowing for a class of general genetic models. Three different approaches are proposed to eliminate the nuisance parameters in these test statistics. We show that all three approaches lead to the same asymptotic distribution under the null hypothesis of no linkage. Simulation results show that the proposed test statistics have adequate power to detect linkage and that the performances of these two classes of test statistics are quite comparable. We have applied the proposed method to a family study of asthma (Barnes et al., 1996), in which the score-based test shows evidence of linkage with p-value <0.0001 in the region of interest on chromosome 12. Additionally, we have implemented this score-based test within the frequently used computer package GENEHUNTER.     

Homogeneity score test for the intraclass version of the kappa statistics and sample-size determination in multiple or stratified studies

When the intraclass correlation coefficient or the equivalent version of the kappa agreement coefficient have been estimated from several independent studies or from a stratified study, we have the problem of comparing the kappa statistics and combining the information regarding the kappa statistics in a common kappa when the assumption of homogeneity of kappa coefficients holds. In this article, using the likelihood score theory extended to nuisance parameters (Tarone, 1988, Communications in Statistics-Theory and Methods 17(5), 1549-1556) we present an efficient homogeneity test for comparing several independent kappa statistics and, also, give a modified homogeneity score method using a noniterative and consistent estimator as an alternative. We provide the sample size using the modified homogeneity score method and compare it with that using the goodness-of-fit method (GOF) (Donner, Eliasziw, and Klar, 1996, Biometrics 52, 176-183). A simulation study for small and moderate sample sizes showed that the actual level of the homogeneity score test using the maximum likelihood estimators (MLEs) of parameters is satisfactorily close to the nominal and it is smaller than those of the modified homogeneity score and the goodness-of-fit tests. We investigated statistical properties of several noniterative estimators of a common kappa. The estimator (Donner et al., 1996) is essentially efficient and can be used as an alternative to the iterative MLE. An efficient interval estimation of a common kappa using the likelihood score method is presented.     

Non-inferiority of new procedure to standard procedure in stratified matched-pair design

We consider the statistical testing for non-inferiority of a new treatment compared with the standard one under matched-pair setting in a stratified study or in several trials. A non-inferiority test based on the efficient scores and a Mantel-Haenszel (M-H) like procedure with restricted maximum likelihood estimators (RMLEs) of nuisance parameters and their corresponding sample size formulae are presented. We evaluate the above tests and the M-H type Wald test in level and power. The stratified score test is conservative and provides the best power. The M-H like procedure with RMLEs gives an accurate level. However, the Wald test is anti-conservative and we suggest caution when it is used. The unstratified score test is not biased but it is less powerful than the stratified score test when base-line probabilities related to strata are not the same. This investigation shows that the stratified score test possesses optimum statistical properties in testing non-inferiority. A common difference between two proportions across strata is the basic assumption of the stratified tests, we present appropriate tests to validate the assumption and related remarks.     

Optimal Testing for Serial Correlation in a Large Number of Small Samples

In recent years, there has been an increased awareness of the potential one-sided nature of many testing problems in applied sciences. Usually, these testing problems can be reduced, either by conditioning on sufficient statistics or by invariant techniques. COX and SOLOMON (1988) considered testing the serial correlation coefficient of a stationary first order autoregressive process and concentrated on four independent samples, with each of size three. We outline a general method for testing the serial correlation coefficient, using locally best invariant, point optimal invariant and locally most mean powerful invariant test procedures. The first procedure optimizes power near the null hypothesis, the second optimizes it at a pre-determined point away from the null while the third optimizes the average curvature of the power hypersurface in the neighbourhood of the null hypothesis.     

On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H(0) : γ = γ(0), based on the pseudolikelihood L(θ, ??), where ?? is a consistent estimator of ?, the nuisance parameter. We show that the asymptotic distribution of T under H(0) is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest, θ(0), or the true value of the nuisance parameter, ?(0), lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.     

Testing for Absence of Qualitative Interactions Between Risk Factors and Treatment Effects

The problem of investigating qualitative interactions (QI's) between subsets of patients defined by means of some risk factors, and treatment effects in clinical trials is considered from a viewpoint which leads to interchanging the hypotheses of the testing problem dealed with in the existing literature on QI's. A natural way of approaching this reverse problem is to apply one of the tests available for the original problem of detecting QI's at level 1 — α and to reject the null hypothesis of the new problem if and only if this test accepts. Unfortunately, this would require unbiasedness of the level 1 — α test for existence of QI's to start with, a property which exhibits neither the likelihood ratio procedure derived in the seminal paper of GAIL and SIMON (1985), nor the test based on the extreme order statistics which was introduced by several authors in 1993. Nevertheless we show that there is a valid test for absence of QI's which depends on the extreme values only and coincides with the maximum likelihood ratio procedure for the same problem. Furthermore, the procedure is generalized to the problem of testing for absence of relevant QI's, i.e. of qualitative interactions exceeding some specified tolerance ε > 0.     

Fisher Lecture: the 2002 R. A. Fisher lecture: dedicated to the memory of Shanti S. Gupta. Variances are not always nuisance parameters

In classical problems, e.g., comparing two populations, fitting a regression surface, etc., variability is a nuisance parameter. The term "nuisance parameter" is meant here in both the technical and the practical sense. However, there are many instances where understanding the structure of variability is just as central as understanding the mean structure. The purpose of this article is to review a few of these problems. I focus in particular on two issues: (a) the determination of the validity of an assay; and (b) the issue of the power for detecting health effects from nutrient intakes when the latter are measured by food frequency questionnaires. I will also briefly mention the problems of variance structure in generalized linear mixed models, robust parameter design in quality technology, and the signal in microarrays. In these and other problems, treating variance structure as a nuisance instead of a central part of the modeling effort not only leads to inefficient estimation of means, but also to misleading conclusions.     

Statistical analysis of microarray data: a Bayesian approach

The potential of microarray data is enormous. It allows us to monitor the expression of thousands of genes simultaneously. A common task with microarray is to determine which genes are differentially expressed between two samples obtained under two different conditions. Recently, several statistical methods have been proposed to perform such a task when there are replicate samples under each condition. Two major problems arise with microarray data. The first one is that the number of replicates is very small (usually 2-10), leading to noisy point estimates. As a consequence, traditional statistics that are based on the means and standard deviations, e.g. t-statistic, are not suitable. The second problem is that the number of genes is usually very large (approximately 10,000), and one is faced with an extreme multiple testing problem. Most multiple testing adjustments are relatively conservative, especially when the number of replicates is small. In this paper we present an empirical Bayes analysis that handles both problems very well. Using different parametrizations, we develop four statistics that can be used to test hypotheses about the means and/or variances of the gene expression levels in both one- and two-sample problems. The methods are illustrated using experimental data with prior knowledge. In addition, we present the result of a simulation comparing our methods to well-known statistics and multiple testing adjustments.     

Score Tests for Exploring Complex Models: Application to HIV Dynamics Models

In biostatistics, more and more complex models are being developed. This is particularly the case in system biology. Fitting complex models can be very time‐consuming, since many models often have to be explored. Among the possibilities are the introduction of explanatory variables and the determination of random effects. The particularity of this use of the score test is that the null hypothesis is not itself very simple; typically, some random effects may be present under the null hypothesis. Moreover, the information matrix cannot be computed, but only an approximation based on the score. This article examines this situation with the specific example of HIV dynamics models. We examine the score test statistics for testing the effect of explanatory variables and the variance of random effect in this complex situation. We study type I errors and the statistical powers of this score test statistics and we apply the score test approach to a real data set of HIV‐infected patients.     

Plug‐in tests for nonequivalence of means of independent normal populations

In this paper, we consider the problem of testing nonequivalence of several independent normal population means. It is a well‐known problem to test the equality of several means using the analysis of variance (ANOVA). Instead of determining the equality, one may consider more flexible homogeneity, which allows a predetermined level of difference. This problem is known as testing nonequivalence of populations. We propose the plug‐in statistics for two different measures of variability: the sum of the absolute deviations and the maximum of the absolute deviations. For each test, the least favorable configuration (LFC) to ensure the maximum rejection probability under the null hypothesis is investigated. Furthermore, we demonstrate the numerical studies based on both simulation and real data to evaluate the plug‐in tests and compare these with the range test.     

Mapping quantitative trait loci using multiple phenotypes in general pedigrees

The use of correlated phenotypes may dramatically increase the power to detect the underlying quantitative trait loci (QTLs). Current approaches for multiple phenotypes include regression-based methods, the multivariate variance of components method, factor analysis and structural equations. Issues with these methods include: 1) They are computation intensive and are subject to problems of optimization algorithms; 2) Existing claims on the asymptotic distribution of the likelihood ratio statistic for the multivariate variance of components method are contradictory and erroneous; 3) The dimension reduction of the parameter space under the null hypothesis, a phenomenon that is unique to multivariate analyses, makes the asymptotic distribution of the likelihood ratio statistic more complicated than expected. In this article, three cases of varying complexity are considered. For each case, the efficient score statistic, which is asympotically equivalent to the likelihood ratio statistic, is derived, so is its asymptotic distribution [correction]. These methods are straightforward to calculate. Finite-sample properties of these score statistics are studied through extensive simulations. These score statistics are for use with general pedigrees.     

One‐Sided Test to Assess Correlation in Linear Logistic Models using Estimating Equations

A score‐type test is proposed for testing the hypothesis of independent binary random variables against positive correlation in linear logistic models with sparse data and cluster specific covariates. The test is developed for univariate and multivariate one‐sided alternatives. The main advantage of using score test is that it requires estimation of the model only under the null hypothesis, that in this case corresponds to the binomial maximum likelihood fit. The score‐type test is developed from a class of estimating equations with block‐diagonal structure in which the coefficients of the linear logistic model are estimated simultaneously with the correlation. The simplicity of the score test is illustrated in two particular examples.     

Approximate Bayesian computation in population genetics

We propose a new method for approximate Bayesian statistical inference on the basis of summary statistics. The method is suited to complex problems that arise in population genetics, extending ideas developed in this setting by earlier authors. Properties of the posterior distribution of a parameter, such as its mean or density curve, are approximated without explicit likelihood calculations. This is achieved by fitting a local-linear regression of simulated parameter values on simulated summary statistics, and then substituting the observed summary statistics into the regression equation. The method combines many of the advantages of Bayesian statistical inference with the computational efficiency of methods based on summary statistics. A key advantage of the method is that the nuisance parameters are automatically integrated out in the simulation step, so that the large numbers of nuisance parameters that arise in population genetics problems can be handled without difficulty. Simulation results indicate computational and statistical efficiency that compares favorably with those of alternative methods previously proposed in the literature. We also compare the relative efficiency of inferences obtained using methods based on summary statistics with those obtained directly from the data using MCMC.     

Testing the non-unity of rate ratio under inverse sampling

Inverse sampling is considered to be a more appropriate sampling scheme than the usual binomial sampling scheme when subjects arrive sequentially, when the underlying response of interest is acute, and when maximum likelihood estimators of some epidemiologic indices are undefined. In this article, we study various statistics for testing non-unity rate ratios in case-control studies under inverse sampling. These include the Wald, unconditional score, likelihood ratio and conditional score statistics. Three methods (the asymptotic, conditional exact, and Mid-P methods) are adopted for P-value calculation. We evaluate the performance of different combinations of test statistics and P-value calculation methods in terms of their empirical sizes and powers via Monte Carlo simulation. In general, asymptotic score and conditional score tests are preferable for their actual type I error rates are well controlled around the pre-chosen nominal level, and their powers are comparatively the largest. The exact version of Wald test is recommended if one wants to control the actual type I error rate at or below the pre-chosen nominal level. If larger power is expected and fluctuation of sizes around the pre-chosen nominal level are allowed, then the Mid-P version of Wald test is a desirable alternative. We illustrate the methodologies with a real example from a heart disease study.