Comparison of disease prevalence in two populations under double-sampling scheme with two fallible classifiers

A disease prevalence can be estimated by classifying subjects according to whether they have the disease. When gold-standard tests are too expensive to be applied to all subjects, partially validated data can be obtained by double-sampling in which all individuals are classified by a fallible classifier, and some of individuals are validated by the gold-standard classifier. However, it could happen in practice that such infallible classifier does not available. In this article, we consider two models in which both classifiers are fallible and propose four asymptotic test procedures for comparing disease prevalence in two groups. Corresponding sample size formulae and validated ratio given the total sample sizes are also derived and evaluated. Simulation results show that (i) Score test performs well and the corresponding sample size formula is also accurate in terms of the empirical power and size in two models; (ii) the Wald test based on the variance estimator with parameters estimated under the null hypothesis outperforms the others even under small sample sizes in Model II, and the sample size estimated by this test is also accurate; (iii) the estimated validated ratios based on all tests are accurate. The malarial data are used to illustrate the proposed methodologies.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.     

On triple sampling scremes for estimating from binomial data with misclassification errors ∗

Previous work has been carried out on the use of double sampling schemes for inference from binomial data which are subject to misclassification. The double sampling scheme utilizes a sample of n units which are classified by both a fallible and a true device and another sample of n₂ units which are classified only by a fallible device. A triple sampljng scheme incorporates an additional sample of n_l units which are classified only by the true device. In this paper we apply this triple sampling to estimation from binomialdata. First estimation of a binomial proportion is discussed under different misclassification structures. Then, the problem of optimal allocation of sample sizes is discussed.     

Optimal testing in a three way anova model

We consider likelihood ratio, score and Wald tests for a three-way random effects ANOVA model. Competitor tests are compared using criteria such as small sample power, asymptotic relative efficiency, and convenient null distribution. The final choice is between a new test and two tests long used in practice.     

Robust Small Sample Inference for Generalised Estimating Equations: An Application of the Anova‐type Test

Asymptotically, the Wald‐type test for generalised estimating equations (GEE) models can control the type I error rate at the nominal level. However in small sample studies, it may lead to inflated type I error rates. Even with currently available small sample corrections for the GEE Wald‐type test, the type I error rate inflation is still serious when the tested contrast is multidimensional. This paper extends the ANOVA‐type test for heteroscedastic factorial designs to GEE and shows that the proposed ANOVA‐type test can also control the type I error rate at the nominal level in small sample studies while still maintaining robustness with respect to mis‐specification of the working correlation matrix. Differences of inference between the Wald‐type test and the proposed test are observed in a two‐way repeated measures ANOVA model for a diet‐induced obesity study and a two‐way repeated measures logistic regression for a collagen‐induced arthritis study. Simulation studies confirm that the proposed test has better control of the type I error rate than the Wald‐type test in small sample repeated measures models. Additional simulation studies further show that the proposed test can even achieve larger power than the Wald‐type test in some cases of the large sample repeated measures ANOVA models that were investigated.     

An alternative method for evaluating stationarity in transition models

Transition models are an important framework that can be used to model longitudinal categorical data. A relevant issue in applying these models is the condition of stationarity, or homogeneity of transition probabilities over time. We propose two tests to assess stationarity in transition models: Wald and likelihood-ratio tests, which do not make use of transition probabilities, using only the estimated parameters of the models in contrast to the classical test available in the literature. In this paper, we present two motivating studies, with ordinal longitudinal data, to which proportional odds transition models are fitted and the two proposed tests are applied as well as the classical test. Additionally, their performances are assessed through simulation studies. The results show that the proposed tests have good performance, being better for control of type-I error and they present equivalent power functions asymptotically. Also, the correlations between the Wald, likelihood-ratio and the classical test statistics are positive and large, an indicator of general concordance. Additionally, both of the proposed tests are more flexible and can be applied in studies with qualitative and quantitative covariates.     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Analysis of multivariate categorical data with misclassification errors by triple sampling schemes

Previous work has been carried out on the use of double-sampling schemes for inference from categorical data subject to misclassification. The double-sampling schemes utilize a sample of n units classified by both a fallible and true device and another sample of n₂ units classified only by a fallible device. In actual applications, one often hasavailable a third sample of n₁ units, which is classified only by the true device. In this article we develop techniques of fitting log-linear models under various misclassification structures for a general triple-sampling scheme. The estimation is by maximum likelihood and the fitted models are hierarchical. The methodology is illustrated by applying it to data in traffic safety research from a study on the effectiveness of belts in reducing injuries.     

On Small Sample Properties of the Wald,LR and LM Tests in a Linear Model with AR(1) Errors

When the error terms are autocorrelated, the conventional t-tests for individual regression coefficients mislead us to over-rejection of the null hypothesis. We examine, by Monte Carlo experiments, the small sample properties of the unrestricted estimator of ρ and of the estimator of ρ restricted by the null hypothesis. We compare the small sample properties of the Wald, likelihood ratio and Lagrange multiplier test statistics for individual regression coefficients. It is shown that when the null hypothesis is true, the unrestricted estimator of ρ is biased. It is also shown that the Lagrange multiplier test using the maximum likelihood estimator of ρ performs better than the Wald and likelihood ratio tests.     

The effect of number of clusters and cluster size on statistical power and Type I error rates when testing random effects variance components in multilevel linear and logistic regression models

When using multilevel regression models that incorporate cluster-specific random effects, the Wald and the likelihood ratio (LR) tests are used for testing the null hypothesis that the variance of the random effects distribution is equal to zero. We conducted a series of Monte Carlo simulations to examine the effect of the number of clusters and the number of subjects per cluster on the statistical power to detect a non-null random effects variance and to compare the empirical type I error rates of the Wald and LR tests. Statistical power increased with increasing number of clusters and number of subjects per cluster. Statistical power was greater for the LR test than for the Wald test. These results applied to both the linear and logistic regressions, but were more pronounced for the latter. The use of the LR test is preferable to the use of the Wald test.     

Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions

For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.     

Two-sample Non Randomized Response Techniques for Sensitive Questions

The problem facing most researchers is how to encourage participants to respond, and then to provide truthful response in surveys with sensitive questions. In this article, we consider a non randomized triangular model for testing the equality of the proportions of people with a sensitive characteristic between two independent populations. We derive the Wald, score, and likelihood ratio tests. Their respective sample size formulae are developed as well. Simulation studies are conducted to evaluate the performance of the three proposed test procedures, which show that the score test outperforms the other two.     

Vector autoregression and causality: a theoretical overview and simulation study

This paper provides a theoretical overview of Wald tests for Granger causality in levels vector autoregressions (VAR's) and Johansen-type error correction models (ECM's). The theory is based on results in Toda and Phillips (1991a) and allows for stochastic and deterministic trends as well as arbitrary degrees of cointegration. We recommend some operational procedures for conducting Granger causality tests that are based on the Gaussian maximum likelihood estimation of ECM's. These procedures are applicable in the important practical case of testing the causal effects of one variable on another group of variables and vice versa. This paper also investigates the sampling properties of these testing procedures through simulation exercises. Three sequential causality tests in ECM's are compared with conventional causality tests in levels and differences VAR's.     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Making wald tests work for cointegrated VAR systems

Wald tests of restrictions on the coefficients of vector autoregressive (VAR) processes are known to have nonstandard asymptotic properties for 1(1) and cointegrated systems of variables. A simple device is proposed which guarantees that Wald tests have asymptotic X2-distributions under general conditions. If the true generation process is a VAR(p) it is proposed to fit a VAR(p+l) to the data and perform a Wald test on the coefficients of the first p lags only. The power properties of the modified tests are studied both analytically and numerically by means of simple illustrative examples.     

The effect of spillover on the Granger causality test

In this paper, we investigate the properties of the Granger causality test in stationary and stable vector autoregressive models under the presence of spillover effects, that is, causality in variance. The Wald test and the WW test (the Wald test with White's proposed heteroskedasticity-consistent covariance matrix estimator imposed) are analyzed. The investigation is undertaken by using Monte Carlo simulation in which two different sample sizes and six different kinds of data-generating processes are used. The results show that the Wald test over-rejects the null hypothesis both with and without the spillover effect, and that the over-rejection in the latter case is more severe in larger samples. The size properties of the WW test are satisfactory when there is spillover between the variables. Only when there is feedback in the variance is the size of the WW test slightly affected. The Wald test is shown to have higher power than the WW test when the errors follow a GARCH(1,1) process without a spillover effect. When there is a spillover, the power of both tests deteriorates, which implies that the spillover has a negative effect on the causality tests.     

Exact unconditional testing procedures for comparing two independent Poisson rates

We consider seven exact unconditional testing procedures for comparing adjusted incidence rates between two groups from a Poisson process. Exact tests are always preferable due to the guarantee of test size in small to medium sample settings. Han [Comparing two independent incidence rates using conditional and unconditional exact tests. Pharm Stat. 2008;7(3):195–201] compared the performance of partial maximization p-values based on the Wald test statistic, the likelihood ratio test statistic, the score test statistic, and the conditional p-value. These four testing procedures do not perform consistently, as the results depend on the choice of test statistics for general alternatives. We consider the approach based on estimation and partial maximization, and compare these to the ones studied by Han (2008) for testing superiority. The procedures are compared with regard to the actual type I error rate and power under various conditions. An example from a biomedical research study is provided to illustrate the testing procedures. The approach based on partial maximization using the score test is recommended due to the comparable performance and computational advantage in large sample settings. Additionally, the approach based on estimation and partial maximization performs consistently for all the three test statistics, and is also recommended for use in practice.     

Comparison of k independent,zero‐heavy lognormal distributions

Lachenbruch ( 1976 , 2001 ) introduced two‐part tests for comparison of two means in zero‐inflated continuous data. We are extending this approach and compare k independent distributions (by comparing their means, either overall or the departure from equal proportion of zeros and equal means of nonzero values) by introducing two tests: a two‐part Wald test and a two‐part likelihood ratio test. If the continuous part of the distributions is lognormal then the proposed two test statistics have asymptotically chi‐square distribution with $2(k-1)$ degrees of freedom. A simulation study was conducted to compare the performance of the proposed tests with several well‐known tests such as ANOVA, Welch ( 1951 ), Brown & Forsythe ( 1974 ), Kruskal–Wallis, and one‐part Wald test proposed by Tu & Zhou ( 1999 ). Results indicate that the proposed tests keep the nominal type I error and have consistently best power among all tests being compared. An application to rainfall data is provided as an example. The Canadian Journal of Statistics 39: 690–702; 2011. © 2011 Statistical Society of Canada     

Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions

We evaluated the properties of six statistical methods for testing equality among populations with zero-inflated continuous distributions. These tests are based on likelihood ratio (LR), Wald, central limit theorem (CLT), modified CLT (MCLT), parametric jackknife (PJ), and nonparametric jackknife (NPJ) statistics. We investigated their statistical properties using simulated data from mixed distributions with an unknown portion of non zero observations that have an underlying gamma, exponential, or log-normal density function and the remaining portion that are excessive zeros. The 6 statistical tests are compared in terms of their empirical Type I errors and powers estimated through 10,000 repeated simulated samples for carefully selected configurations of parameters. The LR, Wald, and PJ tests are preferred tests since their empirical Type I errors were close to the preset nominal 0.05 level and each demonstrated good power for rejecting null hypotheses when the sample sizes are at least 125 in each group. The NPJ test had unacceptable empirical Type I errors because it rejected far too often while the CLT and MCLT tests had low testing powers in some cases. Therefore, these three tests are not recommended for general use but the LR, Wald, and PJ tests all performed well in large sample applications.     

Grouping Tests for Misspecification: An Application to Housing Demand

We consider several grouping tests for regression misspecification, with reference to housing-demand function estimation. We compare three existing test procedures, demonstrate modifications necessary in most applications, and propose a fourth test to distinguish between two categories of potential specification error. The test procedures are evaluated in artificial simulations of alternative errors. Finally, we apply the tests to FHA home purchase data. We reject the hypothesis that household and grouped regressions differ only by sampling error or random mismeasurement of household income or price. Our results have implications for choices among test procedures and interpretations of previous housing-demand analysis.