Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.     

Simultaneous prediction intervals for regression models with an intercept

Consider the usual linear regression model y = x’β+?, relating a response y to a vector of predictors x. Suppose that n observations on y together with the corresponding values of x are available , and it is desired to construct simultaneous prediction intervals for k future values of y at values of x which can not be ascertained beforehand. In most applications the regression model contains an intercept. This paper presents two sets of prediction intervals appropriate to this case. The proposed intervals are compared with those of Carlstein (1986), and the improvements are illustrated in the case of simple linear regression.     

Maximum likelihood estimators for extended growth curve model with orthogonal between-individual design matrices

The extended growth curve model is discussed in this paper. There are two versions of the model studied in the literature, which differ in the way how the column spaces of the design matrices are nested. The nesting is applied either to the between-individual or to the within-individual design matrices. Although both versions are equivalent via reparametrization, the properties of estimators cannot be transferred directly because of non-linearity of estimators. Since in many applications the between-individual matrices are one-way ANOVA matrices, it is reasonable to assume orthogonality of the column spaces of between-individual design matrices along with nestedness of the column spaces of within-individual design matrices. We present the maximum likelihood estimators and their basic moments for the model with such orthogonality condition.     

Simultaneous choice of time points and the block design in the growth curve model

The aim of this paper is to consider the optimality in the growth curve model with respect to two aspects: time and the block design and to show some relations between information functions for different designs. The A-, D- and E- optimality are studied.     

Probability matching priors for an extended statistical calibration model

Statistical calibration or inverse prediction involves data collected in two stages. In the first stage, several values of an endogenous variable are observed, each corresponding to a known value of an exogenous variable; in the second stage, one or more values of the endogenous variable are observed which correspond to an unknown value of the exogenous variable. When estimating the value of the latter, it has been suggested that the variability about the regression relationship should not be assumed to be equal for the two stages of data collection. In this paper, the authors present a Bayesian method of analysis based on noninformative priors that takes this heteroscedasticity into account.     

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.     

Confidence intervals and bands for the binormal ROC curve revisited

Two types of confidence intervals (CIs) and confidence bands (CBs) for the receiver operating characteristic (ROC) curve are studied: pointwise CIs and simultaneous CBs. An optimized version of the pointwise CI with the shortest width is developed. A new ellipse-envelope simultaneous CB for the ROC curve is suggested as an adaptation of the Working-Hotelling-type CB implemented in a paper by Ma and Hall (1993). Statistical simulations show that our ellipse-envelope CB covers the true ROC curve with a probability close to nominal while the coverage probability of the Ma and Hall CB is significantly smaller. Simulations also show that our CI for the area under the ROC curve is close to nominal while the coverage probability of the CI suggested by Hanley and McNail (1982) uniformly overestimates the nominal value. Two examples illustrate our simultaneous ROC bands: radiation dose estimation from time to vomiting and discrimination of breast cancer from benign abnormalities using electrical impedance measurements.     

Constructing quantile confidence intervals using extended simple random sample in finite populations

This paper constructs quantile confidence intervals based on extended simple random sample (SRS) from a finite population, where ranks of population units are all known. Extended simple random sample borrows additional information from unmeasured observations in the population by conditioning on the population ranks of the measured units in SRS. The confidence intervals are improved using Rao-Blackwell theorem over the conditional distribution of sample ranks given the measured sample units. Empirical evidence shows that the proposed confidence intervals have shorter lengths than confidence intervals constructed from an SRS sample.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

Simultaneous prediction intervals for multiple comparisons with a standard

k normal populations having common variance are used to construct two-sided and one-sided simultaneous prediction intervals for the differences between the future means of independent random sample from each of these populations compared to a standard. These prediction intervals are particularly useful if one has sampled the performance of several products and wishes to simultaneously predict the differences between future sample mean performance of these products and a standard with a predetermined joint probability. Methods on sample size determination are also given. The procedures are illustrated with a numerical example. Received: February 25, 2000; revised version: February 6, 2001     

Estimating parameters in extended growth curve models with special covariance structures

In this paper the estimation of the unknown parameters is considered in standard growth curve model with special covariance structures. Based on the unbiased estimating equations, some new methods are proposed. The resulting estimators can be expressed in explicit forms. The statistical properties of the proposed estimators are investigated. Some simulation results are presented to compare the performance of the proposed estimator with that of the existing approaches. Finally, these methods are applied in general extended growth curve model with special covariance structures.     

Divergence-based confidence intervals in false-positive misclassification model

In this article, we introduce minimum divergence estimators of parameters of a binary response model when data are subject to false-positive misclassification and obtained using a double-sampling plan. Under this set up, the problem of goodness-of-fit is considered and divergence-based confidence intervals (CIs) for a population proportion parameter are derived. A simulation experiment is carried out to compare the coverage probabilities of the new CIs. An application to real data is also given.     

An extended cure model and model selection

We propose a novel interpretation for a recently proposed Box–Cox transformation cure model, which leads to a natural extension of the cure model. Based on the extended model, we consider an important issue of model selection between the mixture cure model and the bounded cumulative hazard cure model via the likelihood ratio test, score test and Akaike’s Information Criterion (AIC). Our empirical study shows that AIC is informative and both the score test and the likelihood ratio test have adequate power to differentiate between the mixture cure model and the bounded cumulative hazard cure model when the sample size is large. We apply the tests and AIC methods to leukemia and colon cancer data to examine the appropriateness of the cure models considered for them in the literature.     

Marginal correlation from an extended random-effects model for repeated and overdispersed counts

Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]] derived approximate correlation functions for longitudinal sequences of general data type, Gaussian and non-Gaussian, based on generalized linear mixed-effects models (GLMM). Their focus was on binary sequences, as well as on a combination of binary and Gaussian sequences. Here, we focus on the specific case of repeated count data, important in two respects. First, we employ the model proposed by Molenberghs et al. [13 Molenberghs, G., Verbeke, G. and Demétrio, C. G.B. 2007. An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Anal., 13: 513–531. [Crossref], [PubMed], [Web of Science ®] [Google Scholar]], which generalizes at the same time the Poisson-normal GLMM and the conventional overdispersion models, in particular the negative-binomial model. The model flexibly accommodates data hierarchies, intra-sequence correlation, and overdispersion. Second, means, variances, and joint probabilities can be expressed in closed form, allowing for exact intra-sequence correlation expressions. Next to the general situation, some important special cases such as exchangeable clustered outcomes are considered, producing insightful expressions. The closed-form expressions are contrasted with the generic approximate expressions of Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]]. Data from an epileptic-seizures trial are analyzed and correlation functions derived. It is shown that the proposed extension strongly outperforms the classical GLMM.     

Simultaneous one-sided tolerance intervals for sets of contrasts from normal populations

This paper considers the problem of constructing simultaneous prediction and tolerance intervals for sets of contrasts of normal variables in situations where simultaneous intervals are available. Tables are given with critical values used in simultaneous tolerance bounds for two classes of contrasts: pairwise many-one and profile type.     

Simultaneous confidence intervals for comparing several inverse Gaussian means under heteroscedasticity

Recently, Zhang [Simultaneous confidence intervals for several inverse Gaussian populations. Stat Probab Lett. 2014;92:125–131] proposed simultaneous pairwise confidence intervals (SPCIs) based on the fiducial generalized pivotal quantity concept to make inferences about the inverse Gaussian means under heteroscedasticity. In this paper, we propose three new methods for constructing SPCIs to make inferences on the means of several inverse Gaussian distributions when scale parameters and sample sizes are unequal. One of the methods results in a set of classic SPCIs (in the sense that it is not simulation-based inference) and the two others are based on a parametric bootstrap approach. The advantages of our proposed methods over Zhang’s (2014) method are: (i) the simulation results show that the coverage probability of the proposed parametric bootstrap approaches is fairly close to the nominal confidence coefficient while the coverage probability of Zhang’s method is smaller than the nominal confidence coefficient when the number of groups and the variance of groups are large and (ii) the proposed set of classic SPCIs is conservative in contrast to Zhang’s method.     

Simultaneous confidence intervals on partial means of classes in the two-stage stratified sampling

Let’s consider a finite population of P units, each of them assumes a specific amount of the quantitative variable X. Moreover we assume that the range of values of X is subdivided into k classes and the sampling data come out from a two stage stratified sampling. The main purpose of the work is to determine the estimators, as well as their asymptotic distribution, of the partial means of classes, each of them is defined as a non linear function of the other parameters. Particularly, we are interested in determining the linear approximation estimators and, under convergence theorems, the asymptotic distribution. Afterwards we define the estimator of the vector of the partial means of classes and its asymptotic convergence to multivariate normal distribution is determined. These results are useful to develop simultaneous inferential procedures.