Log-linear models for correlated marginal totals of a contingency table

The marginal totals of a contingency table can be rearranged to form a new table. If at least twoof these totals include the same cell of the original table, the new table cannot be treated as anordinary contingency table. An iterative method is proposed to calculate maximum likelihood estimators for the expected cell frequencies of the original table under the assumption that some marginal totals (or more generally, some linear functions) of these expected frequencies satisfy a log-linear model.In some cases, a table of correlated marginal totals is treated as if it was an ordinary contingency table. The effects of ignoring the special structure of the marginal table on thedistributionof the goodness-of-fit test statistics are discussed and illustrated, with special reference to stationary Markov chains.     

Test procedures in coyariamce analysis assuming a general correlation pattern

The analysis of covariance procedure is considered when the observations in each cell are equicorrelated. A correction procedure is given, A computationally easier conservative test statistic is also given. The conservative test statistic allows one to more readily determine the consequences of ignoring correlations, even slight correlations, in the analysis of covariance procedure. The difference of the corrected test and the conservative test is shown to converge in probability to zero. This conservative test is easy to implement on statistical computer packages, It is shown, that for the general correlation pattern, any test involving the regression coefficients of the covariables is an exact test. An example illustrates the procedure     

Letters to the Editor

Although several authors have indicated that the median test has low power in small samples, it continues to be presented in many statistical textbooks, included in a number of popular statistical software packages, and used in a variety of application areas. We present results of a power simulation study that shows that the median test has noticeably lower power, even for the double exponential distribution for which it is asymptotically most powerful, than other readily available rank tests. We suggest that the median test be “retired” from routine use and recommend alternative rank tests that have superior power over a relatively large family of symmetric distributions.     

The rainbow test for lack of fit in regression

A test for lack of fit in regression is presented. Unlike other methods, this one doesn't require replicates or a prior estimate of variance. It can be used for linear or multiple regression, and would be easy to add to existing computer packages. It is based on comparing a fit over low leverage points with a fit over the entire set of data. Distribution theory results are pre¬sented, with examples of power. A discussion of its use for de¬tecting violations of other regression assumptions is also given.     

A revisit to test the equality of variances of several populations

We revisit the problem of testing homoscedasticity (or, equality of variances) of several normal populations which has applications in many statistical analyses, including design of experiments. The standard text books and widely used statistical packages propose a few popular tests including Bartlett's test, Levene's test and a few adjustments of the latter. Apparently, the popularity of these tests have been based on limited simulation study carried out a few decades ago. The traditional tests, including the classical likelihood ratio test (LRT), are asymptotic in nature, and hence do not perform well for small sample sizes. In this paper we propose a simple parametric bootstrap (PB) modification of the LRT, and compare it against the other popular tests as well as their PB versions in terms of size and power. Our comprehensive simulation study bursts some popularly held myths about the commonly used tests and sheds some new light on this important problem. Though most popular statistical software/packages suggest using Bartlette's test, Levene's test, or modified Levene's test among a few others, our extensive simulation study, carried out under both the normal model as well as several non-normal models clearly shows that a PB version of the modified Levene's test (which does not use the F-distribution cut-off point as its critical value), and Loh's exact test are the “best” performers in terms of overall size as well as power.     

Pearson-type goodness-of-fit tests for regression

A procedure for testing the goodness of fit of linear regression models is introduced. For a given partition of the real line into cells, the proposed test is a quadratic form based on the vector of observed minus expected frequencies of the residuals obtained by maximum-likelihood estimation of the regression parameters. The quadratic form is of the same computational difficulty as the traditional Pearson-type tests with uncensored data. A statistic based on only one cell is particularly easy to apply and is used for testing the normality assumption in a real data set from astronomy. A simulation study examines the finite-sample properties of the proposed tests.     

Testing the Significance of Categorical Predictor Variables in Nonparametric Regression Models

In this paper we propose a test for the significance of categorical predictors in nonparametric regression models. The test is fully data-driven and employs cross-validated smoothing parameter selection while the null distribution of the test is obtained via bootstrapping. The proposed approach allows applied researchers to test hypotheses concerning categorical variables in a fully nonparametric and robust framework, thereby deflecting potential criticism that a particular finding is driven by an arbitrary parametric specification. Simulations reveal that the test performs well, having significantly better power than a conventional frequency-based nonparametric test. The test is applied to determine whether OECD and non-OECD countries follow the same growth rate model or not. Our test suggests that OECD and non-OECD countries follow different growth rate models, while the tests based on a popular parametric specification and the conventional frequency-based nonparametric estimation method fail to detect any significant difference.     

Goodness of fit tests for the multiple logistic regression model

Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions.

The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

An R Package for Value at Risk and Expected Shortfall

Value at risk and expected shortfall are the two most popular measures of financial risk. But the available R packages for their computation are limited. Here, we introduce an R contributed package written by the authors. It computes the two measures for over 100 parametric distributions, including all commonly known distributions. We expect that the R package could be useful to researchers and to the financial community.     

Testing Periodicity in Short Series and Application to Gene Expression Data

Testing for periodicity in microarray time series encounters the challenges of short series length, missing values and presence of non-Fourier frequencies. In this article, a test method for such series has been proposed. The method is completely simulation based and finds p-values for test of periodicity through fitting Pearson Type VI distribution. The simulation results compare and reveal the excellence of this method over Fisher's g test for varying series length, frequencies, and error variance. This approach is applied to Caulobacter crescentus cell cycle data in order to demonstrate the practical performance.     

Exact Statistical Tests for Heterogeneity of Frequencies Based on Extreme Values

Sophisticated statistical analyses of incidence frequencies are often required for various epidemiologic and biomedical applications. Among the most commonly applied methods is the Pearson's χ² test, which is structured to detect non specific anomalous patterns of frequencies and is useful for testing the significance for incidence heterogeneity. However, the Pearson's χ² test is not efficient for assessing the significance of frequency in a particular cell (or class) to be attributed to chance alone. We recently developed statistical tests for detecting temporal anomalies of disease cases based on maximum and minimum frequencies; these tests are actually designed to test of significance for a particular high or low frequency. The purpose of this article is to demonstrate merits of these tests in epidemiologic and biomedical studies. We show that our proposed methods are more sensitive and powerful for testing extreme cell counts than is the Pearson's χ² test. This feature could provide important and valuable information in epidemiologic or biomeidcal studies. We elucidated and illustrated the differences in sensitivity among our tests and the Pearson's χ² test by analyzing a data set of Langerhans cell histiocytosis cases and its hypothetical sets. We also computed and compared the statistical power of these methods using various sets of cell numbers and alternative frequencies. The investigation of statistical sensitivity and power presented in this work will provide investigators with useful guidelines for selecting the appropriate tests for their studies.     

Conditional logistic regression in cluster-specific 1: m matched treatment–control designs

Conditional logistic regression is a popular method for estimating a treatment effect while eliminating cluster-specific nuisance parameters when they are not of interest. Under a cluster-specific 1: m matched treatment–control study design, we present a new closed-form relationship between the conditional logistic regression estimator and the ordinary logistic regression estimator. In addition, we prove an equivalence between the ordinary logistic regression and the conditional logistic regression estimators, when the clusters are replicated infinitely often, which indicates that potential bias concerns when applying conditional logistic regression to complex survey samples.     

A goodness-of-fit test using randomly selected cell boundaries

This paper describes a technique for testing for the goodness-of-fit of a set of observations on a random variable to a hypothesized distribution. The cell boundaries needed for partitioning the range of the random variable are randomly chosen using the computer, thus eliminating an arbitrary choice by the analyst. The power of this test procedure is compared to the test procedure using equal expected cell frequencies     

Expected estimating equations to accommodate covariate measurement error

Estimating equations which are not necessarily likelihood-based score equations are becoming increasingly popular for estimating regression model parameters. This paper is concerned with estimation based on general estimating equations when true covariate data are missing for all the study subjects, but surrogate or mismeasured covariates are available instead. The method is motivated by the covariate measurement error problem in marginal or partly conditional regression of longitudinal data. We propose to base estimation on the expectation of the complete data estimating equation conditioned on available data. The regression parameters and other nuisance parameters are estimated simultaneously by solving the resulting estimating equations. The expected estimating equation (EEE) estimator is equal to the maximum likelihood estimator if the complete data scores are likelihood scores and conditioning is with respect to all the available data. A pseudo-EEE estimator, which requires less computation, is also investigated. Asymptotic distribution theory is derived. Small sample simulations are conducted when the error process is an order 1 autoregressive model. Regression calibration is extended to this setting and compared with the EEE approach. We demonstrate the methods on data from a longitudinal study of the relationship between childhood growth and adult obesity.     

Regression-based estimation of the false negative fraction when multiple negatives are unverified

Summary.  The paper describes a method of estimating the false negative fraction of a multiple-screening test when individuals who test negatively on all K tests do not have their true disease status verified. The method proposed makes no explicit assumptions about the underlying heterogeneity of the population or about serial correlation of test results within an individual. Rather, it is based on estimating false negative fractions conditionally on observed diagnostic histories and extrapolating the observed patterns in these empirical frequencies by using logistic regression. The method is illustrated on, and motivated by, data on a multiple-screening test for bowel cancer.     

Pre-checking for non-monotonicity of the Wald statistic

The non-monotonic behaviour of the Wald test in some finite-sample applications leads to low power when the null hypothesis needs rejection most. This article proposes a simple check for discerning if the Wald statistic for testing significance of regression coefficients is non-monotonic in the neighbourhood of the parameter space from which the sample data are drawn. Monte Carlo simulations show that this method works rather well for detecting situations where the Wald test can be safely applied. An example is provided to illustrate the use of this check.     

The effects of additive outliers on the seasonal KPSS test: a Monte Carlo analysis

This article builds on the test proposed by Lyhagen [The seasonal KPSS statistic, Econom. Bull. 3 (2006), pp. 1–9] for seasonal time series and having the null hypothesis of level stationarity against the alternative of unit root behaviour at some or all of the zero and seasonal frequencies. This new test is qualified as seasonal-frequency Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and it is not originally supported by a regression framework.

The purpose of this paper is twofold. Firstly, we propose a model-based regression method and provide a clear illustration of Lyhagen's test and we establish its asymptotic theory in the time domain. Secondly, we use the Monte Carlo method to study the finite-sample performance of the seasonal KPSS test in the presence of additive outliers. Our simulation analysis shows that this test is robust to the magnitude and the number of outliers and the statistical results obtained cast an overall good performance of the test finite-sample properties.     

基于Lasso改进的一般因果关系检验

通常所说的Granger因果关系检验，实际上是对线性因果关系的检验，无法检验非线性因果关系。Peguin和Terasvirta（1999）进行了基于泰勒展式的一般性扩展，应用于非线性因果关系检验，并采用提取主成分的方法解决其中的多重共线性问题。但是，提取主成分对解决多重共线性的效果并不太好。Lasso回归是目前处理多重共线性的主要方法之一，相对于其他方法，更容易产生稀疏解，在参数估计的同时实现变量选择，因而可以用来解决检验中的多重共线性问题，以提高检验的效率。对检验程序的模拟结果表明，基于Lasso回归的检验取得较好的效果。