Modelling udder infection data using copula models for quadruples

We study copula models for correlated infection times in the four udder quarters of dairy cows. Both a semi-parametric and a nonparametric approach are considered to estimate the marginal survival functions, taking into account the effect of a binary udder quarter level covariate. We use a two-stage estimation approach and we briefly discuss the asymptotic behaviour of the estimators obtained in the first and the second stage of the estimation. A pseudo-likelihood ratio test is used to select an appropriate copula from the power variance copula family that describes the association between the outcomes in a cluster. We propose a new bootstrap algorithm to obtain the p-value for this test. This bootstrap algorithm also provides estimates for the standard errors of the estimated parameters in the copula. The proposed methods are applied to the udder infection data. A small simulation study for a setting similar to the setting of the udder infection data gives evidence that the proposed method provides a valid approach to select an appropriate copula within the power variance copula family.     

Estimating standard error of intra-class correlation coefficients up to three level unbalanced nested clinical trials

ABSTRACT

Very often researchers plan a balanced design for cluster randomization clinical trials in conducting medical research, but unavoidable circumstances lead to unbalanced data. By adopting three or more levels of nested designs, they usually ignore the higher level of nesting and consider only two levels, this situation leads to underestimation of variance at higher levels. While calculating the sample size for three-level nested designs, in order to achieve desired power, intra-class correlation coefficients (ICCs) at individual level as well as higher levels need to be considered and must be provided along with respective standard errors. In the present paper, the standard errors of analysis of variance (ANOVA) estimates of ICCs for three-level unbalanced nested design are derived. To conquer the strong appeal of distributional assumptions, balanced design, equality of variances between clusters and large sample, general expressions for standard errors of ICCs which can be deployed in unbalanced cluster randomization trials are postulated. The expressions are evaluated on real data as well as highly unbalanced simulated data.     

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Adjusted homogeneity tests of odds ratios when data are clustered

Three modified tests for homogeneity of the odds ratio for a series of 2 × 2 tables are studied when the data are clustered. In the case of clustered data, the standard tests for homogeneity of odds ratios ignore the variance inflation caused by positive correlation among responses of subjects within the same cluster, and therefore have inflated Type I error. The modified tests adjust for the variance inflation in the three existing standard tests: Breslow–Day, Tarone and the conditional score test. The degree of clustering effect is measured by the intracluster correlation coefficient, ρ. A variance correction factor derived from ρ is then applied to the variance estimator in the standard tests of homogeneity of the odds ratio. The proposed tests are an application of the variance adjustment method commonly used in correlated data analysis and are shown to maintain the nominal significance level in a simulation study. Copyright © 2004 John Wiley & Sons, Ltd.     

Re-weighting estimation of the coefficients in the varying coefficient model with heteroscedastic errors

ABSTRACT

This article explores the estimation problem of the coefficients in the varying coefficient model with heteroscedastic errors. Specifically, we first present a method for estimating the variance function of the error term and the resulting estimator is proved to be consistent. Then, motivated by the generalized least-squares procedure for dealing with heteroscedasticity in the linear regression literature, we re-weight each squared residual term in the local linear smoother with the inverse of the corresponding estimated error variance to construct estimates of the coefficients. Simulation experiments and practical data analysis conducted demonstrate that the re-weighting approach can improve the accuracy of the coefficient estimates under a finite sample size, especially when the error heteroscedasticity is strong.     

Nonlinear IV panel unit root testing under structural breaks in the error variance

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$ -neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.     

Critical point and power calculations for the studentized range test for generally correlated means

The studentized range test is a widely applied statistical procedure to compare several normal means within the analysis of variance. However, up to now no general methodology is available to perform the all-pair comparisons precisely, such as the computation of p-values or quantiles in the simple unbalanced one-way layout. Instead, a variety of approximations have been proposed in the past. This article focuses on exact computations of simultaneous confidence intervals and exact sample size determinations for all-pair comparisons in the analysis of variance involving arbitrary variance-covariance matrices. General power expressions in closed form are developed and numerical issues concerning the arising multivariate central and noncentral t-distributions are discussed. An application to the usual fixed effects analysis of covariance illustrates the use of the obtained results.     

Analysis of Linear Models with One Factor Having Both Fixed and Random Levels

ABSTRACT

A general theory for a case where a factor has both fixed and random effect levels is developed under one-way treatment structure model. Estimation procedures for the fixed effects and variance components are consider for the model. The testing of fixed effects is considered when the variance–covariance matrix is known and unknown. Confidence intervals for estimable functions and prediction intervals for predictable functions are constructed. The computational procedures are illustrated using data from an on-farm trial.     

A new distribution-free k-sample test: Analysis of kernel density functionals

A novel distribution-free k-sample test of differences in location shifts based on the analysis of kernel density functional estimation is introduced and studied. The proposed test parallels one-way analysis of variance and the Kruskal–Wallis (KW) test aiming at testing locations of unknown distributions. In contrast to the rank (score)-transformed non-parametric approach, such as the KW test, the proposed F-test uses the measurement responses along with well-known kernel density estimation (KDE) to estimate the locations and construct the test statistic. A practical optimal bandwidth selection procedure is also provided. Our simulation studies and real data example indicate that the proposed analysis of kernel density functional estimate (ANDFE) test is superior to existing competitors for fat-tailed or heavy-tailed distributions when the k groups differ mainly in location rather than shape, especially with unbalanced data. ANDFE is also highly recommended when it is unclear whether test groups differ mainly in shape or location. The Canadian Journal of Statistics 48: 167–186; 2020 © 2019 Statistical Society of Canada     

Nonparametric Applications of Shaffer's Extension of Dennett's Procedure

Shaffer's extensions and generalization of Dunnett's procedure are shown to be applicable in several nonparametric data analyses. Applications are considered within the context of the Kruskal-Wallis one-way analysis of variance (ANOVA) test for ranked data, Friedman's two-way ANOVA test for ranked data, and Cochran's test of change for dichotomous data.     

Tests for the umbrella alternative under normality

Consider a one-way layout of the analysis of variance assuming independence, normality, and homogeneity of variance. Test the null hypothesis H_o that the means, j., of each of Amp; columns, i = 1,…, k are equal versus the alternative that they follow an umbrella pattern. That is, the alternative is H₁-H₀ where H₁: μ₁> μ₂>… > μk, and m is known. We derive a class of tests which are unbiased and lie in a nontrivial complete class. We recommend specific tests within the class. A simulation of the power functions of some tests is contrasted with the simulated power function of the likelihood ratio test.     

Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

A comparison of bayes and maximum likelihood estimation of the intraclass correlation coefficient

Two methods of estimating the intraclass correlation coefficient (p) for the one-way random effects model were compared in several simulation experiments using balanced and unbalanced designs. Estimates based on a Bayes approach and a maximum likelihood approach were compared on the basis of their biases (differences between estimates and true values of p) and mean square errors (mean square errors of estimates of p) in each of the simulation experiments. The Bayes approach used the median of a conditional posterior density as its estimator.     

A Note on Fitting a Regression without an Intercept Term

By entering the data (y _i ,x _i ) followed by (–y _i ,–x _i ), one can obtain an intercept-free regression Y = Xβ + ε from a program package that normally uses an intercept term. There is no bias in the resultant regression coefficients, but a minor postanalysis adjustment is needed to the residual variance and standard errors.     

A finite population quantile estimation by unequal probability sampling

Abstract

Using a model-assisted approach, this paper studies asymptotically design-unbiased (ADU) estimation of a population “distribution function” and extends to deriving an asymptotic and approximate unbiased estimator for a population quantile from a sample chosen with varying probabilities. The respective asymptotic standard errors and confidence intervals are then worked out. Numerical findings based on an actual data support the theory with efficient results.     

Resampling method to estimate intra-cluster correlation for clustered binary data

ABSTRACT

Various methods have been proposed to estimate intra-cluster correlation coefficients (ICCs) for correlated binary data, and many are very sensitive to the type of design and underlying distributional assumptions. We proposed a new method to estimate ICC and its 95% confidence intervals based on resampling principles and U-statistics, where we resampled with replacement pairs of individuals from within and between clusters. We concluded from our simulation study that the resampling-based estimates approximate the population ICC more precisely than the analysis of variance and method of moments techniques for different event rates, varying number of clusters, and cluster sizes.     

A new computational approach test for one-way ANOVA under heteroscedasticity

This paper proposes a new test statistic based on the computational approach test (CAT) for one-way analysis of variance (ANOVA) under heteroscedasticity. The proposed test was compared with other popular tests according to type I error and power of tests under different combinations of variances, means, number of groups and sample sizes. As a result, it was observed that the proposed test yields better results than other tests in many cases.     

Tests of heteroscedasticity and correlation in multivariate t regression models with AR and ARMA errors

Heteroscedasticity checking in regression analysis plays an important role in modelling. It is of great interest when random errors are correlated, including autocorrelated and partial autocorrelated errors. In this paper, we consider multivariate t linear regression models, and construct the score test for the case of AR(1) errors, and ARMA(s,d) errors. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. Based on modified profile likelihood, the adjusted score test is also developed. The finite sample performance of the tests is investigated through Monte Carlo simulations, and also the tests are illustrated with two real data sets.     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

The impact of integrated measurement errors on modeling long-run macroeconomic time series

ABSTRACT

Data spanning long time periods, such as that over 1860–2012 for the UK, seem likely to have substantial errors of measurement that may even be integrated of order one, but which are probably cointegrated for cognate variables. We analyze and simulate the impacts of such measurement errors on parameter estimates and tests in a bivariate cointegrated system with trends and location shifts which reflect the many major turbulent events that have occurred historically. When trends or shifts therein are large, cointegration analysis is not much affected by such measurement errors, leading to conventional stationary attenuation biases dependent on the measurement error variance, unlike the outcome when there are no offsetting shifts or trends.