Goodness-of-fit tests in semi-linear models

Specification tests for the error distribution are proposed in semi-linear models, including the partial linear model and additive models. The tests utilize an integrated distance involving the empirical characteristic function of properly estimated residuals. These residuals are obtained from an initial estimation step involving a combination of penalized least squares and smoothing techniques. A bootstrap version of the tests is utilized in order to study the small sample behavior of the procedures in comparison with more classical approaches. As an example, the tests are applied on some real data sets.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

Rank repeated measures analysis of covariance

When analyzing a response variable at the presence of both factors and covariates, with potentially correlated responses and violated assumptions of the normal residual or the linear relationship between the response and the covariates, rank-based tests can be an option for inferential procedures instead of the parametric repeated measures analysis of covariance (ANCOVA) models. This article derives a rank-based method for multi-way ANCOVA models with correlated responses. The generalized estimating equations (GEE) technique is employed to construct the proposed rank tests. Asymptotic properties of the proposed tests are derived. Simulation studies confirmed the performance of the proposed tests.     

Tests for compound periodicities and estimating a non linear function

We propose two tests for testing compound periodicities which are the uniformly most powerful invariant decision procedures against simple periodicities. The second test can provide an excellent estimation of a compound periodic non linear function from observed data. These tests were compared with the tests proposed by Fisher and Siegel by Monte Carlo studies and we found that all the tests showed high power and high probability of a correct decision when all the amplitudes of underlying periods were the same. However, if there are at least several different periods with unequal amplitudes, then the second test proposed always showed high power and high probability of a correct decision, whereas the tests proposed by Fisher and Siegel gave 0 for the power and 0 for the probability of a correct decision, whatever the standard deviation of pseudo normal random numbers. Overall, the second test proposed is the best of all in view of the probability of a correct decision and power.     

Comparing distributions of time to onset of disease in animal tumorigenicity experiments

The cause-of-death test of Peto et al.(1980)pools information from a Hoel-Walburg test on incidental tumors with information from a logrank test on fatal tumors in order to compare the tumor rate of a group of rodents exposed to a carcinogen against the tumor rate of a group of unexposed animals. The cause-of-death test, which can arise as a partial likelihood score test from a model that assumes proportional odds for tumor prevalence and proportional hazards for tumor mortality, is not, in general, a direct test for equality of tumor onset distributions for occult tumors that are observed in both fatal and incidental contexts. This paper develops a direct cause-of-death test for comparing distributions of time to onset of occultumors. The test is derived as a partial likelihood score test under an assumed proportional hazards model for tumor onset distributions. The size and power of the proposed test are compared in a Monte Carlo simulation study to the size and power of competitive procedures, including procedures that do not require cause-of-death information.     

Exact testing in multivariate regression

An F statistic due to Rao (1951,1973) tests uniform mixed linear restrictions in the multivariateregression model. In combination with a generalization of the Bera-Evans-Savin exact functional relationship between the W, LR, and LM statistics, Rao's F serves to unify a number of exact test procedures commonly applied in disparate empirical literatures. Examples in demand analysis and asset pricing are provided. The availability of exact tests of restrictions in certain nonlinear models when the model is linear under the null, originally explored by Milliken-Graybill (1970), is extended to multivariate regression. Generalized RESET, J-, and Hausman-Wu tests are resented. As an extension of Dufour (1989), bounds tests exist for nonlinear and inequality restrictions. Applications include conservative bound tests for symmetry or negativity of the substitution matrix in demand systems.     

Testing linearity against nonlinear moving average models

Lagrange multiplier (LM) test statistics are derived for testing a linear moving average model against an asymmetric moving average model and an LM type test against an additive smooth transition moving average model. The latter model is introduced in the paper. The small sample performance of the proposed tests are evaluated in a Monte Carlo study and compared to Wald and likelihood ratio statistics. The size properties of the Lagrange multiplier test are better than those of other 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.     

Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.     

Rank-based ridge estimation in multiple linear regression

Multicollinearity and model misspecification are frequently encountered problems in practice that produce undesirable effects on classical ordinary least squares (OLS) regression estimator. The ridge regression estimator is an important tool to reduce the effects of multicollinearity, but it is still sensitive to a model misspecification of error distribution. Although rank-based statistical inference has desirable robustness properties compared to the OLS procedures, it can be unstable in the presence of multicollinearity. This paper introduces a rank regression estimator for regression parameters and develops tests for general linear hypotheses in a multiple linear regression model. The proposed estimator and the tests have desirable robustness features against the multicollinearity and model misspecification of error distribution. Asymptotic behaviours of the proposed estimator and the test statistics are investigated. Real and simulated data sets are used to demonstrate the feasibility and the performance of the estimator and the tests.     

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.     

Tests for a simple tree order restriction with application to dose–response studies

Summary.  We propose 'Dunnett-type' test procedures to test for simple tree order restrictions on the means of p independent normal populations. The new tests are based on the estimation procedures that were introduced by Hwang and Peddada and later by Dunbar, Conaway and Peddada. The procedures proposed are also extended to test for 'two-sided' simple tree order restrictions. For non-normal data, nonparametric versions based on ranked data are also suggested. Using computer simulations, we compare the proposed test procedures with some existing test procedures in terms of size and power. Our simulation study suggests that the procedures compete well with the existing procedures for both one-sided and two-sided simple tree alternatives. In some instances, especially in the case of two-sided alternatives or for non-normally distributed data, the gains in power due to the procedures proposed can be substantial.     

A CONTINUOUSLY ADAPTIVE RANK TEST FOR SHIFT IN LOCATION

This paper considers the problem of testing for shift in location when the symmetry of the underlying distribution is in doubt. Various adaptive test procedures have been suggested in the literature; they are mainly based on a preliminary test or measure of asymmetry, and then choosing between the sign or the Wilcoxon tests accordingly. However, as this paper demonstrates, there are some disadvantages with such procedures. This paper develops a test that does not suffer from such disadvantages. The proposed test is based on modifying the Wilcoxon scores according to the evidence of asymmetry of the distribution present in the data as indicated by the magnitude of the P‐value from a preliminary test of symmetry. A simulation study investigates and compares the performance of the proposed test and other known adaptive procedures in terms of power and attainment of the nominal size. The performance of a suitable bootstrap procedure for the situation under consideration is also studied. In most cases under consideration, the proposed test is found to be superior to the other tests.     

Invariant tests based on M-estimators,estimating functions,and the generalized method of moments

We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)?type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)?type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.     

Dimension test approach of heteroscedasticity in the linear model

Heteroscedasticity testing has a long history and is still an important matter in the linear model. There exist many types of tests, but they are limited in use to their own specific cases and sensitive to normality. Here, we propose a dimension test approach to heteroscedasticity. The proposed test overcomes the shortcomings of the existing methods, so that it is robust to normality and is unified in sense that it is applicable in the linear model with multi-dimensional response. Numerical studies confirm that the proposed test is favorable over the existing tests with moderate sample sizes, and real data analysis is presented.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

Re-examining two tests for bivariate normality

Many goodness of fit tests for bivariate normality are not rigorous procedures because the distributions of the proposed statistics are unknown or too difficult to manipulate. Two familiar examples are the ring test and the line test. In both tests the statistic utilized generally is approximated by a chi-square distribution rather than compared to its known beta distribution. These two procedures are re-examined and re-evaluated in this paper. It is shown that the chi-square approximation can be too conservative and can lead to unnecessary

rejection of normality.     

Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.