Frequentist-Bayes Lack-of-Fit Tests Based on Laplace Approximations

A Nonparametric Lack-of-Fit Test of Constant Regression in the Presence of Heteroscedastic Variances

We consider a k-nearest neighbor-based nonparametric lack-of-fit test of constant regression in presence of heteroscedastic variances. The asymptotic distribution of the test statistic is derived under the null and local alternatives for a fixed number of nearest neighbors. Advantages of our test compared to classical methods include: (1) The response variable can be discrete or continuous regardless of whether the conditional distribution is symmetric or not and can have variations depending on the predictor. This allows our test to have broad applicability to data from many practical fields; (2) this approach does not need nonlinear regression function estimation that often affects the power for moderate sample sizes; (3) our test statistic achieves the parametric standardizing rate, which gives more power than smoothing-based nonparametric methods for moderate sample sizes. Our numerical simulation shows that the proposed test is powerful and has noticeably better performance than some well known tests when the data were generated from high frequency alternatives or binary data. The test is illustrated with an application to gene expression data and an assessment of Richards growth curve fit to COVID-19 data.

Flexible omnibus test in 1:M matched case-crossover study with measurement error in covariate

The matched case-crossover study design is used in public health, biomedical, and epidemiological research with clustered binary outcomes. Conditional logistic regression is commonly used for analysis because any effects associated with the matching covariates by stratum can be removed. However, some matching covariates often play an important role as effect modifications, causing incorrect statistical testing. The covariates in such studies are often measured with error, so that not accounting for this error can also lead to incorrect inferences for all covariates in the model. However, the methods available for simultaneously evaluating effect modification by matching covariates as well as assessing and characterizing error-in-covariates are limited. In this paper, we propose a flexible omnibus test for testing (1) the significance of a functional association between the clustered binary outcomes and covariates with the measurement error, (2) the existence of effect modifications by matching covariates, and (3) the significance of an interaction effect between the measurement error covariate and other covariates, without specifying the functional forms for these testings. The proposed omnibus test has the flexibility to allow inferences on various hypothesis settings. The advantages of the proposed flexible omnibus test are demonstrated through simulation studies and 1:4 bidirectional matched data analyses from an epidemiology study.

A hybrid omnibus test for generalized semiparametric single-index models with high-dimensional covariate sets

Numerous statistical methods have been developed for analyzing high-dimensional data. These methods often focus on variable selection approaches but are limited for the purpose of testing with high-dimensional data. They are often required to have explicit-likelihood functions. In this article, we propose a "hybrid omnibus test" for high-dicmensional data testing purpose with much weaker requirements. Our hybrid omnibus test is developed under a semiparametric framework where a likelihood function is no longer necessary. Our test is a version of a frequentist-Bayesian hybrid score-type test for a generalized partially linear single-index model, which has a link function being a function of a set of variables through a generalized partially linear single index. We propose an efficient score based on estimating equations, define local tests, and then construct our hybrid omnibus test using local tests. We compare our approach with an empirical-likelihood ratio test and Bayesian inference based on Bayes factors, using simulation studies. Our simulation results suggest that our approach outperforms the others, in terms of type I error, power, and computational cost in both the low- and high-dimensional cases. The advantage of our approach is demonstrated by applying it to genetic pathway data for type II diabetes mellitus.

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.