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.     

Development of Specific hypothesis tests for estimated markov chains

We develop and evaluate the validity and power of two specific tests for the transition probabilities in a Markov chain estimated from aggregate frequency data. The two null hypotheses considered are (1) constancy of the diagonal elements of the one-step transition probability matrix and (2) an arbitrarily chosen transition probability’s being equal to a specific value. The formation of tests uses a general framework for statistical inference on estimated Markov processes; we also indicate how this framework can be used to form tests for a variety of other hypotheses. The validity and power performance of the two tests formed in this paper are examined in factorially designed Monte Carlo experiments. The results indicate that the proposed tests lead to type I error probabilities which are close to the desired levels and to high power against even small deviations from the null hypotheses considered.     

Testing the rate ratio under inverse sampling based on gradient statistic

Inverse sampling is widely applied in studies with dichotomous outcomes, especially when the subjects arrive sequentially or the response of interest is difficult to obtain. In this paper, we investigate the rate ratio test problem under inverse sampling based on gradient statistic with the asymptotic method and parametric bootstrap technique. The gradient statistic has many advantages, for example, it is simple to calculate and competitive with Wald-type, score and likelihood ratio tests in terms of local power. Numerical studies are carried out to evaluate the performance of our gradient test and the existing tests, namely Wald-type, score and likelihood ratio tests. The simulation results suggest that the gradient test based on the parametric bootstrap method has excellent type I error control and large powers even in small sample design. Two real examples, from a heart disease study and a drug comparison study, are applied to illustrate our methods.     

On testing for multivariate ARCH effects in vector time series models

Using a spectral approach, the authors propose tests to detect multivariate ARCH effects in the residuals from a multivariate regression model. The tests are based on a comparison, via a quadratic norm, between the uniform density and a kernel‐based spectral density estimator of the squared residuals and cross products of residuals. The proposed tests are consistent under an arbitrary fixed alternative. The authors present a new application of the test due to Hosking (1980) which is seen to be a special case of their approach involving the truncated uniform kernel. However, they typically obtain more powerful procedures when using a different weighting. The authors consider especially the procedure of Robinson (1991) for choosing the smoothing parameter of the spectral density estimator. They also introduce a generalized version of the test for ARCH effects due to Ling & Li (1997). They investigate the finite‐sample performance of their tests and compare them to existing tests including those of Ling & Li (1997) and the residual‐based diagnostics of Tse (2002).Finally, they present a financial application.     

ADAPTIVE GROUP SEQUENTIAL PROCEDURES FOR MULTIPLE COMPARISON OF TREATMENTS

ABSTRACT

We present a flexible group sequential procedure for comparing several treatments to a control. Though longitudinal data corresponding to a two stage mixed effects model are considered, ranges of application include any process with independent increments. The procedure allows the experimenter to drop the inferior treatments from the trial as soon as they are detected. It control strongly the familywise error rate. We also discuss a new error spending function (ESF) and study the performance of the procedure using various ESFs and time scales. Finally, the procedure is illustrated on a real example and implementation considerations are discussed.     

Approximate inference in heteroskedastic regressions: A numerical evaluation

The commonly made assumption that all stochastic error terms in the linear regression model share the same variance (homoskedasticity) is oftentimes violated in practical applications, especially when they are based on cross-sectional data. As a precaution, a number of practitioners choose to base inference on the parameters that index the model on tests whose statistics employ asymptotically correct standard errors, i.e. standard errors that are asymptotically valid whether or not the errors are homoskedastic. In this paper, we use numerical integration methods to evaluate the finite-sample performance of tests based on different (alternative) heteroskedasticity-consistent standard errors. Emphasis is placed on a few recently proposed heteroskedasticity-consistent covariance matrix estimators. Overall, the results favor the HC4 and HC5 heteroskedasticity-robust standard errors. We also consider the use of restricted residuals when constructing asymptotically valid standard errors. Our results show that the only test that clearly benefits from such a strategy is the HC0 test.     

Generalized likelihood ratio tests for the structure of semiparametric additive models

Semiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.     

Performance of nonparametric multiple comparison tests under heteroscedasticity,dependency, and skewed error distribution

In this article, an extensive Monte Carlo simulation study is conducted to evaluate and compare nonparametric multiple comparison tests under violations of classical analysis of variance assumptions. Simulation space of the Monte Carlo study is composed of 288 different combinations of balanced and unbalanced sample sizes, number of groups, treatment effects, various levels of heterogeneity of variances, dependence between subgroup levels, and skewed error distributions under the single factor experimental design. By this large simulation space, we present a detailed analysis of effects of the violations of assumptions on the performance of nonparametric multiple comparison tests in terms of three error and four power measures. Observations of this study are beneficial to decide the optimal nonparametric test according to requirements and conditions of undertaken experiments. When some of the assumptions of analysis of variance are violated and number of groups is small, use of stepwise Steel-Dwass procedure with Holm's approach is appropriate to control type I error at a desired level. Dunn's method should be employed for greater number of groups. When subgroups are unbalanced and number of groups is small, Nemenyi's procedure with Duncan's approach produces high power values. Conover's procedure successfully provides high power values with a small number of unbalanced groups or with a greater number of balanced or unbalanced groups. At the same time, Conover's procedure is unable to control type I error rates.     

Comparing diagnostic tests: test of hypothesis for likelihood ratios

Likelihood ratios (LRs) are used to characterize the efficiency of diagnostic tests. In this paper, we use the classical weighted least squares (CWLS) test procedure, which was originally used for testing the homogeneity of relative risks, for comparing the LRs of two or more binary diagnostic tests. We compare the performance of this method with the relative diagnostic likelihood ratio (rDLR) method and the diagnostic likelihood ratio regression (DLRReg) approach in terms of size and power, and we observe that the performances of CWLS and rDLR are the same when used to compare two diagnostic tests, while DLRReg method has higher type I error rates and powers. We also examine the performances of the CWLS and DLRReg methods for comparing three diagnostic tests in various sample size and prevalence combinations. On the basis of Monte Carlo simulations, we conclude that all of the tests are generally conservative and have low power, especially in settings of small sample size and low prevalence.     

Properties of two tests for outliers in multivariate data

Mardia's multivariate kurtosis and the generalized distance have desirable properties as multivariate outlier tests. However, extensive critical values have not been published heretofore. A published approximation formula for critical values of the kurtosis is shown to inadequately control the type I error rate, with observed error rates often differing from their intended values by a factor of two or more. Critical values derived from simulations for both tests for up to 25 dimensions and 500 observations are presented. The power curves of both tests are discussed. The generalized distance is the more powerful test when exactly one outlier is present and the contaminant is substantially mean-shifted. However, as the number of outliers increases, the kurtosis becomes the more powerful test. The two tests are compared with respect to power and vulnerability to masking. Recommendations for the use of these tests and interpretation of results are given.     

Testing the equality of several multivariate normal mean vectors under heteroscedasticity: A fiducial approach and an approximate test

We consider the problem of testing the equality of several multivariate normal mean vectors under heteroscedasticity. We first construct a fiducial confidence region (FCR) for the differences between normal mean vectors and we then propose a fiducial test for comparing mean vectors by inverting the FCR. We also propose a simple approximate test that is based on a modification of the χ² approximation. This simple test avoids the complications of simulation-based inference methods. We show that the proposed fiducial test has correct type one error rate asymptotically. We compare the proposed fiducial and approximate tests with the parametric bootstrap test in terms of controlling the type one error rate via an extensive simulation study. Our simulation results show that the proposed fiducial and approximate tests control the type one error rate, while there are cases that the parametric bootstrap test is out of control. We also discuss the power performance of the tests. Finally, we illustrate with a real example how our proposed methods are applicable in analyzing repeated measure designs including a single grouping variable.     

Choosing appropriate covariance matrices in a nonparametric analysis of factorials in block designs

The standard nonparametric, rank-based approach to the analysis of dependent data from factorial designs is based on an estimated unstructured (UN) variance–covariance matrix, but the large number of variance–covariance terms in many designs can seriously affect test performance. In a simulation study for a factorial arranged in blocks, we compared estimates of type-I error probability and power based on the UN structure with the estimates obtained with a more parsimonious heterogeneous-compound-symmetry structure (CSH). Although tests based on the UN structure were anti-conservative with small number of factor levels, especially with four or six blocks, they became conservative at higher number of factor levels. Tests based on the CSH structure were anti-conservative, and results did not depend on the number of factor levels. When both tests were anti-conservative, tests based on the CSH structure were less so. Although use of the CSH structure is concluded to be more suitable than use of the UN structure for the small number of blocks typical in agricultural experiments, results suggest that further improvement of test statistics is needed for such situations.     

Testing purchasing power parity: some evidence of the effects of transaction costs

Empirical tests of purchasing power parity often recognize the problems created by simultaneous equations, but seldom recognize the effects of measurement error or transaction costs. Presumably because most researchers believe that they are unimportant. We present evidence that shows that measurement error and transaction costs and create serious econometric problems for testing purchasing power parity. One effect of these problems is that conventional tests of purchasing power parity can accept PPP when predictive errors are relatively large and reject it when predictive errors are relatively small. Another effect is to bias test of cointegration toward accepting the null of no cointegration between exchange rates and relative price indexes. We also construct a simple model of the determination of exchange rates that shows how transaction costs lead to regression switiching.     

Simulating competing cointegration tests in a bivariate system

In this paper, we consider the size and power of a set of cointegration tests in a number of Monte Carlo simulations. The behaviour of the competing methods is investigated in diff erent situations, including diff erent levels of variance and correlation in the error processes. The impact of violations of the common factor restriction (CFR) implied by the Engle-Granger framework is studied in these situations. The reactions to changes in the CFR condition depend on the error correlation. When the correlation is non-positive, the power increases with increasing CFR violations for the error correction model (ECM) test, while the other tests react in the opposite direction. We also note the reaction to diff erences in the error variances in the data-generating process. For positive correlation and equal variances, the reaction to changes in the CFR violations diff ers somewhat between the tests. We conclude that the ECM and the Z-tests show the best performance over diff erent parameter combinations. In most situations the ECM is best. Therefore, if we had to recommend a unit root test, it would be the ECM, especially for small samples. However, we do not think that one should use just one test, but two or more. Of course, the portfolio of tests we have considered here only represents a subset of the possible tests.     

A Simple Method for Computing the Arl of [ILM0001]-Charts with Zone Tests

In practice the [ILM0002]-chart has been augmented with one or more zone tests to improve the sensitivity of detecting small shifts in the process mean. The average run length (ARL) is one of the indices used to evaluate the performance of control chart procedures. Two unified patterns of transition probability matrices and four closed form expressions of the ARL are found based on the Markov Chain approach. These closed form expressions can be used for computing the ARL for both one-sided and two-sided [ILM0003]-charts with zone tests.     

DOUBLE LENGTH ARTIFICIAL REGRESSIONS FOR TESTING SPATIAL DEPENDENCE

This paper derives two simple artificial Double Length Regressions (DLR) to test for spatial dependence. The first DLR tests for spatial lag dependence while the second DLR tests for spatial error dependence. Both artificial regressions utilize only least squares residuals of the restricted model and are therefore easy to compute. These tests are illustrated using two simple examples. In addition, Monte Carlo experiments are performed to study the small sample performance of these tests. As expected, these DLR tests have similar performance to their corresponding LM counterparts.     

Testing methods for the one-way fixed effects ANOVA models of log-normal samples

For one-way fixed effects of log-normal data with unequal variance, the present study proposes a method to deal with heterogeneity. An appropriate hypothesis testing is demonstrated; and one of the approximate tests, such as the Alexander-Govern test, Welch test or James second-order test, is applied to control Type I error rate. Monte Carlo simulation is used to investigate the performance of the F test for log-scale, the F test for original scale, the James second-order test, the Welch test, and the Alexander-Govern test. The simulated results and real data analysis show that the proposed method is valid and powerful.     

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.     

Zone control charts with estimated parameters for detecting prespecified changes in linear profiles

ABSTRACT

In profile monitoring, control charts are proposed to detect unanticipated changes, and it is usually assumed that the in-control parameters are known. However, due to the characteristics of a system or process, the prespecified changes would appear in the process. Moreover, in most applications, the in-control parameters are usually unknown. To overcome these issues, we develop the zone control charts with estimated parameters to detect small shifts of these prespecified changes. The effects of estimation error have been investigated on the performance of the proposed charts. To account for the practitioner-to-practitioner variability, the expected average run length (ARL) and the standard deviation of the average run length (SDARL) is used as the performance metrics. Our results show that the estimation error results in the significant variation in the ARL distribution. Furthermore, in order to adequately reduce the variability, more phase I samples are required in terms of the SDARL metric than that in terms of the expected ARL metric. In addition, more observations on each sampled profile are suggested to improve the charts' performance, especially for small phase I sample sizes. Finally, an illustrative example is given to show the performance of the proposed zone control charts.     

A monte carlo study of the friedman and conover tests in the single-factor repeated measures design

A Monte Carlo study was used to compare the Type I error rates and power of two nonparametric tests against the F test for the single-factor repeated measures model. The performance of the nonparametric Friedman and Conover tests was investigated for different distributions, numbers of blocks and numbers of repeated measures. The results indicated that the type of the distribution has little effect on the ability of the Friedman and Conover tests to control Type error rates. For power, the Friedman and Conover tests tended to agree in rejecting the same false hyporhesis when the design consisted of three repeated measures. However, the Conover test was more powerful than the Friedman test when the number of repeated measures was 4 or 5. Still, the F test is recommended for the single-factor repeated measures model because of its robustness to non-normality and its good power across a range of conditions.