Group testing in graphs

This paper studies the group testing problem in graphs as follows. Given a graph G=(V,E), determine the minimum number t(G) such that t(G) tests are sufficient to identify an unknown edge e with each test specifies a subset X⊆V and answers whether the unknown edge e is in G[X] or not. Damaschke proved that ⌈log ₂ e(G)⌉≤t(G)≤⌈log ₂ e(G)⌉+1 for any graph G, where e(G) is the number of edges of G. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. Later, they proved that the conjecture is true for complete bipartite graphs. Chang and Juan verified the conjecture for bipartite graphs G with e(G)≤2⁴ or for k≥5. This paper proves the conjecture for bipartite graphs G with e(G)≤2⁵ or for k≥6. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. J.S.-t.J. is supported in part by the National Science Council under grant NSC89-2218-E-260-013. G.J.C. is supported in part by the National Science Council under grant NSC93-2213-E002-28. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.     

Coherent Hypothesis Testing

Multiple hypothesis testing, an important quantitative tool to report the results of scientific inquiries, frequently leads to contradictory conclusions. For instance, in an analysis of variance (ANOVA) setting, the same dataset can lead one to reject the equality of two means, say μ₁ = μ₂, but at the same time to not reject the hypothesis that μ₁ = μ₂ = 0. These two conclusions violate the coherence principle introduced by Gabriel in 1969 Gabriel, K. R. (1969), “Simultaneous Test Procedures - Some Theory of Multiple Comparisons,” The Annals of Mathematical Statistics, 40, 224–250. Available at http://www.jstor.org/stable/2239213[Crossref] , [Google Scholar], and lead to results that are difficult to communicate, and, many times, embarrassing for practitioners of statistical methods. Although this situation is common in the daily life of statisticians, it is usually not discussed in courses of statistics. In this work, we enrich the teaching and discussion of this important topic by investigating through a few examples whether several standard test procedures are coherent or not. We also discuss the relationship between coherent tests and measures of support. Finally, we show how a Bayesian decision-theoretical framework can be used to build coherent tests. These approaches to coherence enlighten when such property is appealing in multiple testing and provide means of obtaining it.     

Testing of homogeneity of variance and autocorrelation coefficients of nonlinear mixed models with AR(1) errors based on M-estimation

Homogeneity of between-individual variance and autocorrelation coefficients is one of assumptions in the study of longitudinal data. However, the assumption could be challenging due to the complexity of the dataset. In the paper we propose and analyze nonlinear mixed models with AR(1) errors for longitudinal data, intend to introduce Huber's function in the log-likelihood function and get robust estimation, which may help to reduce the influence of outliers, by Fisher scoring method. Testing of homogeneity of variance among individuals and autocorrelation coefficients on the basis of Huber's M-estimation is studied later in the paper. Simulation studies are carried to assess performance of score test we proposed. Results obtained from plasma concentrations data are reported as an illustrative example.     

Maximum Likelihood Estimators in Regression Models for Error‐prone Group Testing Data

Since Dorfman's seminal work on the subject, group testing has been widely adopted in epidemiological studies. In Dorfman's context of detecting syphilis, group testing entails pooling blood samples and testing the pools, as opposed to testing individual samples. A negative pool indicates all individuals in the pool free of syphilis antigen, whereas a positive pool suggests one or more individuals carry the antigen. With covariate information collected, researchers have considered regression models that allow one to estimate covariate‐adjusted disease probability. We study maximum likelihood estimators of covariate effects in these regression models when the group testing response is prone to error. We show that, when compared with inference drawn from individual testing data, inference based on group testing data can be more resilient to response misclassification in terms of bias and efficiency. We provide valuable guidance on designing the group composition to alleviate adverse effects of misclassification on statistical inference.     

Testing the equality of several log-normal means based on a computational approach

In this study, we propose a new test based on a computational approach to test the equality of several log-normal means. We compare this test with some existing methods in terms of the type-I error rate and power using Monte Carlo simulations for varying values of number of groups and sample sizes. The simulation results indicate that the proposed test could be suggested as a good alternative for testing the equality of several log-normal means.     

Darling-Erdös-type test for change detection in parameters and variance for stationary VAR models

The aim of this article is the construction of the test statistic for the detection of changes in vector autoregressive (AR) models where both AR parameters and the variance matrix of the error term are the subjects of a change. The approximating distribution of the proposed statistic is the Gumbel distribution. The proof stands on the approximation of weakly dependent random vectors by independent ones and by application of Horváth’s extension of Darling-Erdös extremal result for random vectors, see Darling and Erdös (1956) Darling, D.A., Erdös, P. (1956). A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J. 23:143–155.[Crossref], [Web of Science ®] , [Google Scholar] and Horváth (1993) Horváth, L. (1993). The maximum likelihood method for testing changes in the parameters of normal observations. Ann. Stat. 21(2):671–680.[Crossref], [Web of Science ®] , [Google Scholar]. The test statistic is a modification of the likelihood ratio.     

Invariant tests based on M-estimators,estimating functions,and the generalized method of moments

We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)?type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)?type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.     

A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

Nonparametric Knn estimation with monotone constraints

ABSTRACT

The K-nearest-neighbor (Knn) method is known to be more suitable in fitting nonparametrically specified curves than the kernel method (with a globally fixed smoothing parameter) when data sets are highly unevenly distributed. In this paper, we propose to estimate a nonparametric regression function subject to a monotonicity restriction using the Knn method. We also propose using a new convergence criterion to measure the closeness between an unconstrained and the (monotone) constrained Knn-estimated curves. This method is an alternative to the monotone kernel methods proposed by Hall and Huang (2001 Hall, P., Huang, L.-S. (2001). Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics 29:624–647.[Crossref], [Web of Science ®] , [Google Scholar]), and Du et al. (2013 Du, P., Parmeter, C. F., Racine, J. S. (2013). Nonparametric kernel regression with multiple predictors and multiple shape constraints. Statistic Sinica 23:1347–1371.[Web of Science ®] , [Google Scholar]). We use a bootstrap procedure for testing the validity of the monotone restriction. We apply our method to the “Job Market Matching” data taken from Gan and Li (2016 Gan, L., Li, Q. (2016). Efficiency of thin and thick market. Journal of Econometrics 192(1):40–54.[Crossref], [Web of Science ®] , [Google Scholar]) and find that the unconstrained/constrained Knn estimators work better than kernel estimators for this type of highly unevenly distributed data.     

Analyzing multiple endpoints in a confirmatory randomized clinical trial—an approach that addresses stratification,missing values,baseline imbalance and multiplicity for strictly ordinal outcomes

Confirmatory randomized clinical trials with a stratified design may have ordinal response outcomes, ie, either ordered categories or continuous determinations that are not compatible with an interval scale. Also, multiple endpoints are often collected when 1 single endpoint does not represent the overall efficacy of the treatment. In addition, random baseline imbalances and missing values can add another layer of difficulty in the analysis plan. Therefore, the development of an approach that provides a consolidated strategy to all issues collectively is essential. For a real case example that is from a clinical trial comparing a test treatment and a control for the pain management for patients with osteoarthritis, which has all aforementioned issues, multivariate Mann‐Whitney estimators with stratification adjustment are applicable to the strictly ordinal responses with stratified design. Randomization based nonparametric analysis of covariance is applied to account for the possible baseline imbalances. Several approaches that handle missing values are provided. A global test followed by a closed testing procedure controls the family wise error rate in the strong sense for the analysis of multiple endpoints. Four outcomes indicating joint pain, stiffness, and functional status were analyzed collectively and also individually through the procedures. Treatment efficacy was observed in the combined endpoint as well as in the individual endpoints. The proposed approach is effective in addressing the aforementioned problems simultaneously and straightforward to implement.