Wald-type Statistic Research Articles

Comparisons of the Satterthwaite Approaches for Fixed Effects in Linear Mixed Models

Four approximate F- tests derived by Fai and Cornelious in 1996 to make inference for fixed effects in mixed linear models of rank greater than one. Two of these approaches derived by introducing a Wald-type statistic distributed approximately as an F distribution, and the denominator degrees of freedom computed by matching the approximated one moment of the Wald-type statistic with the exact one moment of the F distribution. The other two approaches were derived by introducing a scaled Wald-type statistic to be distributed approximately as an F distribution, and the denominator degrees of freedom and the scale factor computed by matching the two moments of the statistic with the moments of the F distribution. This paper proposes two more approximate F-tests analogous to the four approaches where an adjusted estimator of the variance of the estimate of fixed effects used. In addition, the paper evaluates and compares the performance of the six approaches analytically, and some useful results are presented. Also, a simulation study for block designs was run to assess and compare the performance of the approaches based on their observed test levels. The simulation study shows that the approaches usually perform reasonably based on their test levels, and in some cases some approaches found to more adequately than other approaches.

A Naturalistic Intervention to Promote Executive Functions in Primary School Children: A Pilot Study.

Executive functions are related to the control of cognition, emotion, and behavior. They are essential to lifelong outcomes, including school performance. Naturalistic interventions embedded in children's daily activities and environments have greater effects. Therefore, this pilot study aimed to develop a naturalistic program suitable for schools, based on Goal Management Training (GMT), and to analyze its effects on executive functions and behavior. The participants consisted of 35 students from 2nd to 5th grade with executive dysfunction complaints. They underwent neuropsychological assessments of working memory, inhibition, cognitive flexibility, and intellectual capacity. Teachers and parents answered questionnaires on executive functions and behavior. Students were randomly assigned to an active control group, who participated in sessions on citizenship, and an experimental group (EG), stimulated through the executive function program, both with 16 sessions conducted by psychologists. After the intervention, all participants were reevaluated. The two-way Wald-type statistic (WTS) revealed greater improvement in executive functions for the EG, including working memory and inhibition. Additionally, parents and teachers, blind to the experimental conditions, reported improvements in some measures of executive functions and behavior. The results are encouraging, but further studies should test the intervention when implemented with larger samples and by teachers.

Improved inference for MCP-Mod approach using time-to-event endpoints with small sample sizes.

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

Non-Parametric Non-Inferiority Assessment in a Three-Arm Trial with Non-Ignorable Missing Data

A three-arm non-inferiority trial including a placebo is usually utilized to assess the non-inferiority of an experimental treatment to a reference treatment. Existing methods for assessing non-inferiority mainly focus on the fully observed endpoints. However, in some clinical trials, treatment endpoints may be subject to missingness for various reasons, such as the refusal of subjects or their migration. To address this issue, this paper aims to develop a non-parametric approach to assess the non-inferiority of an experimental treatment to a reference treatment in a three-arm trial with non-ignorable missing endpoints. A logistic regression is adopted to specify a non-ignorable missingness data mechanism. A semi-parametric imputation method is proposed to estimate parameters in the considered logistic regression. Inverse probability weighting, augmented inverse probability weighting and non-parametric methods are developed to estimate treatment efficacy for known and unknown parameters in the considered logistic regression. Under some regularity conditions, we show asymptotic normality of the constructed estimators for treatment efficacy. A bootstrap resampling method is presented to estimate asymptotic variances of the estimated treatment efficacy. Three Wald-type statistics are constructed to test the non-inferiority based on the asymptotic properties of the estimated treatment efficacy. Empirical studies show that the proposed Wald-type test procedure is robust to the misspecified missingness data mechanism, and behaves better than the complete-case method in the sense that the type I error rates for the former are closer to the pre-given significance level than those for the latter.

Approximate Confidence Intervals for a Binomial p—Once Again

The problem of constructing a reasonably simple yet well-behaved confidence interval for a binomial parameter p is old but still fascinating and surprisingly complex. During the last century, many alternatives to the poorly behaved standard Wald interval have been suggested. It seems though that the Wald interval is still much in use in spite of many efforts over the years through publications to point out its deficiencies. This paper constitutes yet another attempt to provide an alternative and it builds on a special case of a general technique for adjusted intervals primarily based on Wald type statistics. The main idea is to construct an approximate pivot with uncorrelated, or nearly uncorrelated, components. The resulting AN (Andersson–Nerman) interval, as well as a modification thereof, is compared with the well-renowned Wilson and AC (Agresti–Coull) intervals and the subsequent discussion will in itself hopefully shed some new light on this seemingly elementary interval estimation situation. Generally, an alternative to the Wald interval is to be judged not only by performance, its expression should also indicate why we will obtain a better behaved interval. It is argued that the well-behaved AN interval meets this requirement.

A general procedure for change-point detection in multivariate time series

We consider the change-point detection in a general class of time series models, including multivariate continuous and integer- valued time series. We propose a Wald-type statistic based on the estimator performed by a general contrast function, which can be constructed from the likelihood, a quasi-likelihood, a least squares method, etc. Sufficient conditions are provided to ensure that the test statistic convergences to a well-known distribution under the null hypothesis (of no change) and diverges to infinity under the alternative, which establishes the consistency of the procedure. Some examples of models are detailed to illustrate the scope of application of the proposed change-point detection tool. The procedure is applied to simulated and real data examples for numerical illustration.

Testing conditional multivariate rank correlations: the effect of institutional quality on factors influencing competitiveness

Joint distribution between two or more variables could be influenced by the outcome of a conditioning variable. In this paper, we propose a flexible Wald-type statistic to test for such influence. The test is based on a conditioned multivariate Kendall’s tau nonparametric estimator. The asymptotic properties of the test statistic are established under different null hypotheses to be tested for, such as conditional independence or testing for constant conditional dependence. Two simulation studies are presented: The first shows that the estimator proposed and the bandwidth selection procedure perform well. The second presents different bivariate and multivariate models to check the size and power of the test and runs comparisons with previous proposals when appropriate. The results support the contention that the test is accurate even in complex situations and that its computational cost is low. As an empirical application, we study the dependence between some pillars of European Regional Competitiveness when conditioned on the quality of regional institutions. We find interesting results, such as weaker links between innovation and higher education in regions with lower institutional quality.

Interval Estimation Under The Uniform Distribution U(a,b)

In this short note, we consider interval estimation for the parameters under the uniform distribution U(a; b). We study two approaches: (1) based on a Wald-type statistic, (2) based on a pivotal statistic. We show that the first approach in its common form is not valid and we propose a modified version of the first approach. It turns out it is equivalent to the confidence interval with the shortest length.

Identification-Robust Inference With Simulation-Based Pseudo-Matching

We develop a general simulation-based inference procedure for partially specified models. Our procedure is based on matching auxiliary statistics to simulated counterparts where nuisance parameters are calibrated neither assuming identification of parameters of interest nor a one-to-one binding function. The conditions underlying the asymptotic validity of our (pseudo-)simulators in conjunction with appropriate bootstraps are characterized beyond the strict and exact calibration of the parameters of the simulator. Our procedure is illustrated through impulse-response (IR) matching in a simulation study of a stylized dynamic stochastic equilibrium model, and two empirical applications on the New Keynesian Phillips curve and on the Industrial Production index. In addition to usual Wald-type statistics that combine structural or reduced form IRs, we analyze local projections IRs through a factor-analytic measure of distance which eschews the need to define a weighting matrix.

Posterior-based Wald-type statistics for hypothesis testing

A new Wald-type statistic is proposed for hypothesis testing based on Bayesian posterior distributions under the correct model specification. The new statistic can be explained as a posterior version of the Wald statistic and has several nice properties. First, it is well-defined under improper prior distributions. Second, it avoids Jeffreys–Lindley–Bartlett’s paradox. Third, under the null hypothesis and repeated sampling, it follows a χ2 distribution asymptotically, offering an asymptotically pivotal test. Fourth, it only requires inverting the posterior covariance for parameters of interest. Fifth and perhaps most importantly, when a random sample from the posterior distribution (such as MCMC output) is available, the proposed statistic can be easily obtained as a by-product of posterior simulation. In addition, the numerical standard error of the estimated proposed statistic can be computed based on random samples. A robust version of the test statistic is developed under model misspecification and inherits many nice properties of the new posterior statistic. The finite sample performance of the statistics is examined in Monte Carlo studies. The method is applied to two latent variable models used in microeconometrics and financial econometrics.

Permutation test for the multivariate coefficient of variation in factorial designs

New inference methods for the multivariate coefficient of variation and its reciprocal, the standardized mean, are presented. While there are various testing procedures for both parameters in the univariate case, it is less known how to do inference in the multivariate setting appropriately. There are some existing procedures but they rely on restrictive assumptions on the underlying distributions. We tackle this problem by applying Wald-type statistics in the context of general, potentially heteroscedastic factorial designs. In addition to the k-sample case, higher-way layouts can be incorporated into this framework allowing the discussion of main and interaction effects. The resulting procedures are shown to be asymptotically valid under the null hypothesis and consistent under general alternatives. To improve the finite sample performance, we suggest permutation versions of the tests and show that the tests’ asymptotic properties can be transferred to them. An exhaustive simulation study compares the new tests, their permutation counterparts and existing methods. To further analyze the differences between the tests, we conduct two illustrative real data examples.

The Effects of the Parenting Education Performed through Whatsapp on the Level of Maternal-Paternal and Infant Attachment in Turkey

PurposeThis study aimed to identify the effects of parenting education with online WhatsApp messages on the level of maternal-paternal and infant attachment. Design and methodsThe study adopted a randomized-controlled experimental design. The sample of the study was 32 couples. The experimental group was provided with online parenting education in the WhatsApp groups. Data were collected through the Socio-Demographic Form, the Maternal Attachment Inventory, and the Paternal Postnatal Attachment Questionnaire. Data were analyzed using the Chi-square test, Wilcoxon test, Paired t-test, Independent sample t-test, Mann Whitney U test, and Wald type statistics test. ResultsIntra-group comparisons showed that there was a statistical significance between the Maternal Attachment Inventory total scores (p < 0.001). There was a significant difference between group time interactions (p < 0.001), and these differences had a high effect size. When the intra-group comparisons were performed in terms of the Paternal Postnatal Attachment Questionnaire, there were significant differences between the total score differences (p = 0.001). An analysis of the group time interactions indicated no statistically significant differences (p = 0.653). ConclusionsThe results of this study showed that the parenting education intervention increased maternal attachment levels in a significant way.

Homogeneity Test of Many-to-One Risk Differences for Correlated Binary Data under Optimal Algorithms

In clinical studies, it is important to investigate the effectiveness of different therapeutic designs, especially, multiple treatment groups to one control group. The paper mainly studies homogeneity test of many-to-one risk differences from correlated binary data under optimal algorithms. Under Donner’s model, several algorithms are compared in order to obtain global and constrained MLEs in terms of accuracy and efficiency. Further, likelihood ratio, score, and Wald-type statistics are proposed to test whether many-to-one risk differences are equal based on optimal algorithms. Monte Carlo simulations show the performance of these algorithms through the total averaged estimation error, SD, MSE, and convergence rate. Score statistic is more robust and has satisfactory power. Two real examples are given to illustrate our proposed methods.

Meta-analysis of published excess relative risk estimates.

A meta-analytic summary effect estimate often is calculated as an inverse-variance-weighted average of study-specific estimates of association. The variances of published estimates of association often are derived from their associated confidence intervals under assumptions typical of Wald-type statistics, such as normality of the parameter. However, in some research areas, such as radiation epidemiology, epidemiological results typically are obtained by fitting linear relative risk models, and associated likelihood-based confidence intervals are often asymmetric; consequently, reasonable estimates of variances associated with study-specific estimates of association may be difficult to infer from the standard approach based on the assumption of a Wald-type interval. Here, a novel method is described for meta-analysis of published results from linear relative risk models that uses a parametric transformation of published results to improve on the normal approximation used to assess confidence intervals. Using simulations, it is illustrated that the meta-analytic summary obtained using the proposed approach yields less biased summary estimates, with better confidence interval coverage, than the summary obtained using the more classical approach to meta-analysis. The proposed approach is illustrated using a previously published example of meta-analysis of epidemiological findings regarding circulatory disease following exposure to low-level ionizing radiation.

An $\boldsymbol{L^{2}}$-norm based test for high-dimensional two-way MANOVA

In this paper, we propose and study an $L^{2}$-norm based test for high-dimensional two-way MANOVA where there are fewer observations than the dimension. The test statistic is constructed by removing the inverse sample covariance matrix in the Wald-type statistic for the general linear hypothesis.We propose to approximate the null distribution of our test statistic by using the well-known Welch-Satterthwaite chi-squared approximation and discuss the relationship between chi-squared approximation and the commonly-used normal approximation. The $L^{2}$-norm based test is also shown to admit several invariant properties under certain transformations. The asymptotic and approximate powers of the proposed test are investigated. Simulation studies and real data applications show that the proposed test performs well for high-dimensional data.

Partial Sum Processes of Residual-Based and Wald-type Break-Point Statistics in Time Series Regression Models

Partial Sum Processes of Residual-Based and Wald-type Break-Point Statistics in Time Series Regression Models

Homogeneity Test of Ratio of Two Proportions in Stratified Bilateral Data

ABSTRACTHomogeneity is a nonnegligible condition in the statistical analysis of the stratified bilateral data. The correlation between responses from two paired body parts should be taken into account to avoid biased or misleading results. In this article, we derive four test statistics (i.e., likelihood ratio statistic TL, Wald-type statistic TW, and score statistics and ) for testing common ratios of two proportions across strata under the assumption of equal correlation coefficient within each stratum. Monte Carlo simulations show that the score test () generally produces robust Type I error and satisfactory power. Two real examples are used to demonstrate the performance of the four proposed tests. Supplementary materials for this article are available online.

A Posterior-Based Wald-Type Statistic for Hypothesis Testing

A new Wald-type statistic is proposed for hypothesis testing based on Bayesian posterior distributions. The new statistic can be explained as a posterior version of Wald test and have several nice properties. First, it is well-defined under improper prior distributions. Second, it avoids Jeffreys-Lindley’s paradox. Third, under the null hypothesis and repeated sampling, it follows a χ2 distribution asymptotically, offering an asymptotically pivotal test. Fourth, it only requires inverting the posterior covariance for the parameters of interest. Fifth and perhaps most importantly, when a random sample from the posterior distribution (such as an MCMC output) is available, the proposed statistic can be easily obtained as a by-product of posterior simulation. In addition, the numerical standard error of the estimated proposed statistic can be computed based on the random sample. The finite sample performance of the statistic is examined in Monte Carlo studies. The method is applied to two latent variable models used in micro-econometrics and financial econometrics.

Why are there time-varying comovements in the European stock market?

ABSTRACTThis paper analyzes the dynamics of pair comovements between different domestic European stock market returns (Spain, France, Germany, Switzerland and the United Kingdom) seeking to check whether there is a unique source of risk driving those dynamics. Once it is shown that the comovements are time-varying, the question is to find whether a global index such as the Euro Stoxx can be considered the main source of risk. To that end we estimate and test for time-varying global pair covariances and for time-varying remaining pair covariances once the effect of the Euro Stoxx is removed. The empirical results are obtained considering locally stationary variables, a family that includes variables with first and second time-varying moments. Under that framework time-varying means and covariances can be estimated using a spline-based procedure and Wald-type statistics can be computed to test for time-variations. A simulation study shows that the role of the mean estimation part is crucial to the good performance of the tests for second moments. The empirical results evidence that all global pair covariances for the European countries analyzed are time-varying, but also that the Euro Stoxx can be considered as the driving source of risk for these time-varying dynamics. This conclusion is very useful for modeling purpose and financial strategies. Finally, we repeat the analysis considering the Nasdaq as an alternative global index and find that it explains only a small part of the dynamics in the European pair comovements.

Rank-based permutation approaches for non-parametric factorial designs.

Inference methods for null hypotheses formulated in terms of distribution functions in general non-parametric factorial designs are studied. The methods can be applied to continuous, ordinal or even ordered categorical data in a unified way, and are based only on ranks. In this set-up Wald-type statistics and ANOVA-type statistics are the current state of the art. The first method is asymptotically exact but a rather liberal statistical testing procedure for small to moderate sample size, while the latter is only an approximation which does not possess the correct asymptotic α level under the null. To bridge these gaps, a novel permutation approach is proposed which can be seen as a flexible generalization of the Kruskal-Wallis test to all kinds of factorial designs with independent observations. It is proven that the permutation principle is asymptotically correct while keeping its finite exactness property when data are exchangeable. The results of extensive simulation studies foster these theoretical findings. A real data set exemplifies its applicability.