Wald-type Statistic Research Articles

Test for linearity against STAR models with deterministic trends

This paper studies the linearity test of STAR models with deterministic trends. The results show that when the data generation process includes a deterministic trend, the Wald-type statistic proposed by Teräsvirta (1994) does not follow the standard χ2 distribution, but degenerates at the speed of T. Moreover, when the test for linearity is performed based on the residual of ordinary least squares detrending, the statistical power of the test is very low even when the Wald statistic follows the χ2 distribution.

Testing equality of risk ratios in multiple [formula omitted] tables with structural zero

This paper investigates the equality test of risk ratios in multiple 2 × 2 tables with structural zero, which is formed by a confounding factor such as age, gender, severity of disease, region or other variables of interest. Score statistic, Wald statistic and likelihood ratio statistic for testing equality of risk ratios in multiple 2 × 2 tables with structural zero are proposed. Three procedures for calculating p -values of the above three statistics based on the MaxT, MinP and Bonferroni methods are developed. An algorithm for determining sample size required for attaining the pre-specified power is presented. Empirical results show that (1) likelihood ratio statistic and score statistic via the MaxT and MinP methods behave satisfactorily in the sense that their type I error rates are close to the pre-given significance level; (2) the Wald-type statistic with logarithmic transformation behaves better than the Wald-type statistic. Simulation studies and a real example are used to illustrate the proposed methodologies.

Estimation and Testing in Elliptical Functional Measurement Error Models

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.

Testing and estimation of purely nonparametric effects in repeated measures designs

The several sample case of the so-called nonparametric Behrens–Fisher problem in repeated measures designs is considered. That is, even under the null hypothesis, the marginal distribution functions in the different groups may have different shapes, and are not assumed to be equal. Moreover, the continuity of the marginal distribution functions is not required so that data with ties and, particularly, ordered categorical data are covered by this model. A multiple relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within pairwise samples. The asymptotic distribution of this estimator is derived, along with a consistent estimator of its asymptotic covariance matrix. In addition, a multiple contrast test and related simultaneous confidence intervals for the relative marginal effects are derived and compared to rank-based Wald-type and ANOVA-type statistics. Simulations show that the ANOVA-type statistic and the multiple contrast test appear to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads, as expected, to somewhat liberal decisions. Regarding the power, none of the statistics is uniformly superior. A real data set illustrates the application.

Asymptotic confidence interval construction for proportion difference in medical studies with bilateral data

Bilateral dichotomous data are very common in modern medical comparative studies (e.g. comparison of two treatments in ophthalmologic, orthopaedic and otolaryngologic studies) in which information involving paired organs (e.g. eyes, ears and hips) is available from each subject. In this article, we study various confidence interval estimators for proportion difference based on Wald-type statistics, Fieller theorem, likelihood ratio statistic, score statistics and bootstrap resampling method under the dependence or/and independence models for bilateral binary data. Performance is evaluated with respect to the coverage probability and expected width via simulation studies. Our empirical results show that (1) ignoring the dependence feature of bilateral data could lead to severely incorrect coverage probabilities; and (2) Wald-type, score-type and bootstrap confidence intervals based on the dependence model perform satisfactorily for small to large sample sizes in the sense that their empirical coverage probabilities are close to the pre-specified nominal confidence level and are hence recommended. A real data from an otolaryngologic study is used to illustrate the proposed methods.

Simultaneous confidence intervals using ordinal effect measures for ordered categorical outcomes

When outcomes are ordered categorical, a model using an ordinal effect size measure is a good alternative of the cumulative logit model to compare several independent group differences. We present a method of constructing simultaneous confidence intervals for the ordinal effect size measures, using the studentized range distribution with the score test statistic. A simulation study shows that the proposed method performs well in terms of coverage probability, and it seems better than the method using a Bonferroni correction for Wald-type statistics and methods that account for the dependencies among pairwise ordinal effect size measures using the multivariate normal distribution (or the multivariate t-distribution for small samples).

The impact of the AIS 2005 revision on injury severity scores and clinical outcome measures

Background The abbreviated injury scale (AIS) was updated in 2005 from the AIS 1998 version. The purpose of this study is to describe the effects of this change on injury severity scoring and outcome measures. Materials and methods Analyses were performed on all trauma patients consecutively admitted over a 6-month period at two geographically separate Level I trauma centers. Injuries were manually double-coded according to the AIS 05 and the AIS 98. Changes in AIS, ISS, and new ISS (NISS) were analysed using paired t-tests. Apparent differences in outcome by ISS strata (<16, 16–24, >24) were compared for AIS 05 versus AIS 98 using the Wald-type statistic. Lastly, the percent of patients with a change in ISS strata are reported. Results There were 2250 patients included in the study. Nearly half (46.4%) of AIS codes changed, resulting in a different AIS score for 18.9% of all codes. The mean ISS was significantly lower using the AIS 05 (11.7) versus the AIS 98 (13.3, p < 0.001). Similarly, the mean NISS was significantly lower (16.3 versus 18.7, p < 0.001). In the ISS strata 16–24 an apparent increase in mortality, length of stay, and percent of patients not discharged home was observed for the AIS 05 versus AIS 98. Changes in outcome measures for this stratum were as follows (AIS 98 versus AIS 05): mortality, 4.3% versus 7.7% ( p = 0.002); hospital length of stay, 5.2 days versus 7.3 days ( p < 0.001); percent of patients not discharged home, 39.2% versus 49.3% ( p < 0.001). Finally, there was a 20.5% reduction in patients with an ISS ≥ 16 and a 26.2% reduction in patients with an ISS ≥ 25 using the AIS 05. Conclusions The AIS revision had a significant impact on overall injury severity measures, clinical outcome measures, and percent of patients in each ISS strata. Therefore, the AIS revision affects the ability to directly compare data generated using AIS 05 and AIS 98 which has implications in trauma research, reimbursement and ACS accreditation.

A flexible approximate likelihood ratio test for detecting differential expression in microarray data

Identifying differentially expressed genes in microarray data has been studied extensively and several methods have been proposed. Most popular methods in the study of gene expression microarray data analysis rely on normal distribution assumption and are based on a Wald statistic. These methods may be inefficient when expression levels follow a skewed distribution. To deal with possible violations of the normality assumption, we propose a method based on Generalized Logistic Distribution of Type II (GLDII). The motivation behind this distributional assumption is to allow longer tails than normal distribution. This is important in analyzing gene expression data since extreme values are common in such experiments. The shape parameter for GLDII allows flexibility in modeling a wide range of distributions. To simplify the computational complexity involved in carrying out Likelihood Ratio (LR) tests for several thousands of genes, an Approximate LR Test (ALRT) is proposed. We also generalize the two-class ALRT method to multi-class microarray data. The performance of the ALRT method under the GLDII assumption is compared to methods based on Wald-type statistics using simulation. The results from the simulations show that our method performs quite well compared to the significance analysis of microarrays (SAM) approach using standardized Wilcoxon rank statistics and the empirical Bayes (E-B) t -statistics. Our method is also less sensitive to extreme values. We illustrate our method using two publicly available gene expression data sets.

A heteroscedastic structural errors-in-variables model with equation error

It is not uncommon with astrophysical and epidemiological data sets that the variances of the observations are accessible from an analytical treatment of the data collection process. Moreover, in a regression model, heteroscedastic measurement errors and equation errors are common situations when modelling such data. This article deals with the limiting distribution of the maximum-likelihood and method-of-moments estimators for the line parameters of the regression model. We use the delta method to achieve it, making it possible to build joint confidence regions and hypothesis testing. This technique produces closed expressions for the asymptotic covariance matrix of those estimators. In the moment approach we do not assign any distribution for the unobservable covariate while with the maximum-likelihood approach, we assume a normal distribution. We also conduct simulation studies of rejection rates for Wald-type statistics in order to verify the test size and power. Practical applications are reported for a data set produced by the Chandra observatory and also from the WHO MONICA Project on cardiovascular disease.

Asymptotic theory of weighted F-statistics based on ranks

Using subspaces to describe the nonparametric null hypotheses introduced in Akritas and Arnold [Fully nonparametric hypotheses for factorial designs I: multivariate repeated measures designs, J. Amer. Statist. Assoc. 89 (1994), pp. 336–343.], leads to a natural extension of the models and the class of nonparametric hypotheses considered there. Nonparametric versions of all saturated or unsaturated parametric models for factorial designs, as well as nonparametric versions of all parametric hypotheses considered in such contexts, are included in the new formulation. To test these new hypotheses we introduce a new family of (mid-)rank statistics. The new statistics are modelled after the weighted F-statistics and are appropriate for (possibly) unbalanced designs with independent observations that can be heteroscedastic. Being rank versions of likelihood ratio statistics, the proposed statistics apply in situations where the Wald-type rank statistics of Akritas, Arnold and Brunner [Nonparametric hypotheses and rank statistics for unbalanced factorial designs, J. Amer. Statist. Assoc. 92 (1997), pp. 258–265.] have not been extended and are at least as efficient in the cases where both apply. We show that the new rank statistics converge in distribution to central chi-squared distributions under their respective null hypotheses. Finding the asymptotic distribution of statistics without a closed form expression requires a novel approach, which we introduce. A simulation study compares the achieved level and power of the new statistics with a number of competing procedures.

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2 x 2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2 x 2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald-type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre-specified power or a pre-determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald-type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies.

Testing the Equality of Two Proportions for Combined Unilateral and Bilateral Data

In many medical comparative studies (e.g., comparison of two treatments in an otolaryngological study), subjects may produce either bilateral (e.g., responses from a pair of ears) or unilateral (response from only one ear) data. For bilateral cases, it is meaningful to assume that the information between the two ears from the same subject are generally highly correlated. In this article, we would like to test the equality of the successful cure rates between two treatments with the presence of combined unilateral and bilateral data. Based on the dependence and independence models, we study ten test statistics which utilize both the unilateral and bilateral data. The performance of these statistics will be evaluated with respect to their empirical Type I error rates and powers under different configurations. We find that both Rosner's and Wald-type statistics based on the dependence model and constrained maximum likelihood estimates (under the null hypothesis) perform satisfactorily for small to large samples and are hence recommended. We illustrate our methodologies with a real data set from an otolaryngology study.

Testing for structural change in regression quantiles

Abstract Most studies in the structural change literature focus solely on the conditional mean, while under various circumstances, structural change in the conditional distribution or in conditional quantiles is of key importance. This paper proposes several tests for structural change in regression quantiles. Two types of statistics are considered, namely, a fluctuation type statistic based on the subgradient and a Wald type statistic, based on comparing parameter estimates obtained from different subsamples. The former requires estimating the model under the null hypothesis, and the latter involves estimation under the alternative hypothesis. The tests proposed can be used to test for structural change occurring in a pre-specified quantile, or across quantiles, which can be viewed as testing for change in the conditional distribution with a linear specification of the conditional quantile function. Both single and multiple structural changes are considered. We derive the limiting distributions under the null hypothesis, and show they are nuisance parameter free and can be easily simulated. A simulation study is conducted to assess the size and power in finite samples.

Robust tests for the common principal components model

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.

Testing for Structural Change in Regression Quantiles

Most studies in the structural change literature focus solely on the conditional mean, while under various circumstances structural change in the conditional distribution or in conditional quantiles is of key importance. This paper proposes several tests for structural change in regression quantiles. Two types of statistics are considered, namely, a fluctuation type statistic based on the subgradient and a Wald type statistic based on comparing parameter estimates obtained from different subsamples. The former requires estimating the model under the null hypothesis and the latter involves estimation under the alternative hypothesis. The tests proposed can be used to test for structural change occurring in a pre-specified quantile, or across quantiles, which can be viewed as testing for change in the conditional distribution with a linear specification of the conditional quantile function. Both single and multiple structural changes are considered. We derive the limiting distributions under the null hypothesis and show they are nuisance parameter free and can be easily simulated. A simulation study is conducted to assess the size and power in finite samples.

Homogeneity Test of Risk Differences of Marginal and Conditional Probabilities in Several Incomplete Correlated 2 × 2 Tables

This article considers K pairs of incomplete correlated 2 × 2 tables in which the interesting measurement is the risk difference between marginal and conditional probabilities. A Wald-type statistic and a score-type statistic are presented to test the homogeneity hypothesis about risk differences across strata. Powers and sample size formulae based on the above two statistics are deduced. Figures about sample size against risk difference (or marginal probability) are given. A real example is used to illustrate the proposed methods.

Rank test statistics for unbalanced nested designs

We formulate rank statistics for testing hypotheses in unbalanced, and possibly heteroscedastic, two-factor nested designs with independent observations. These include Wald-type statistics based on the theory introduced by Akritas, Arnold and Brunner, as well as a Box-type approximation which is intended to improve the accuracy of approximation to asymptotic distributions. We also present statistics based on a recent theory of weighted F -statistics for ranks. The actual sizes of the statistics at various nominal levels are compared in a simulation study. Our main conclusion is that the Box-adjusted Wald-type statistic is the only statistic that is accurate across all the situations considered and therefore we recommend it for general use.

Hypothesis testing in the unrestricted and restricted parametric spaces of structural models

Hypothesis for testing the equality of slopes in measurement error models, incorporating the additional assumption that the slopes lie in a closed interval is considered. Wald type statistics based on the maximum likelihood estimators are considered under both, restricted and unrestricted parametric spaces. Their asymptotic behaviors are analyzed through different simulation studies. The statistics performance is analyzed in terms of power. Analysis of the results suggests that the statistics considered under the restricted parametric space has shown a better performance since the tests based on these statistics are more powerful. An example illustrates the proposal.

Hypothesis Testing in Functional Comparative Calibration Models

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.

Sample size determination for matched-pair equivalence trials using rate ratio

In this article, we compare Wald-type, logarithmic transformation, and Fieller-type statistics for the classical 2-sided equivalence testing of the rate ratio under matched-pair designs with a binary end point. These statistics can be implemented through sample-based, constrained least squares estimation and constrained maximum likelihood (CML) estimation methods. Sample size formulae based on the CML estimation method are developed. We consider formulae that control a prespecified power or confidence width. Our simulation studies show that statistics based on the CML estimation method generally outperform other statistics and methods with respect to actual type I error rate and average width of confidence intervals. Also, the corresponding sample size formulae are valid asymptotically in the sense that the exact power and actual coverage probability for the estimated sample size are generally close to their prespecified values. The methods are illustrated with a real example from a clinical laboratory study.