A simple two-stage design for quantitative responses with application to a study in diabetic neuropathic pain

Two-stage designs offer substantial advantages for early phase II studies. The interim analysis following the first stage allows the study to be stopped for futility, or more positively, it might lead to early progression to the trials needed for late phase II and phase III. If the study is to continue to its second stage, then there is an opportunity for a revision of the total sample size. Two-stage designs have been implemented widely in oncology studies in which there is a single treatment arm and patient responses are binary. In this paper the case of two-arm comparative studies in which responses are quantitative is considered. This setting is common in therapeutic areas other than oncology. It will be assumed that observations are normally distributed, but that there is some doubt concerning their standard deviation, motivating the need for sample size review. The work reported has been motivated by a study in diabetic neuropathic pain, and the development of the design for that trial is described in detail.     

Models for the analysis of C-reactive protein in statin trials

Elevation in C-reactive protein (CRP) is an independent risk factor for cardiovascular disease progression and levels are reduced by treatment with statins. However, on-treatment CRP, given baseline CRP and treatment, is not normally distributed and outliers exist even when transformations are applied. Although classical non-parametric tests address some of these issues, they do not enable straightforward inclusion of covariate information. The aims of this study were to produce a model that improved efficiency and accuracy of analysis of CRP data. Estimation of treatment effects and identification of outliers were addressed using controlled trials of rosuvastatin. The robust statistical technique of MM-estimation was used to fit models to data in the presence of outliers and was compared with least-squares estimation. To develop the model, appropriate transformations of the response and baseline variables were selected. The model was used to investigate how on-treatment CRP related to baseline CRP and estimated treatment effects with rosuvastatin. On comparing least-squares and MM-estimation, MM-estimation was superior to least-squares estimation in that parameter estimates were more efficient and outliers were clearly identified. Relative reductions in CRP were higher at higher baseline CRP levels. There was also evidence of a dose-response relationship between CRP reductions from baseline and rosuvastatin. Several large outliers were identified, although there did not appear to be any relationships between the incidence of outliers and treatments. In conclusion, using robust estimation to model CRP data is superior to least-squares estimation and non-parametric tests in terms of efficiency, outlier identification and the ability to include covariate information.     

Missing item imputation for quality-of-life instruments with application to asthma quality-of-life questionnaires

There has been increasing use of quality-of-life (QoL) instruments in drug development. Missing item values often occur in QoL data. A common approach to solve this problem is to impute the missing values before scoring. Several imputation procedures, such as imputing with the most correlated item and imputing with a row/column model or an item response model, have been proposed. We examine these procedures using data from two clinical trials, in which the original asthma quality-of-life questionnaire (AQLQ) and the miniAQLQ were used. We propose two modifications to existing procedures: truncating the imputed values to eliminate outliers and using the proportional odds model as the item response model for imputation. We also propose a novel imputation method based on a semi-parametric beta regression so that the imputed value is always in the correct range and illustrate how this approach can easily be implemented in commonly used statistical software. To compare these approaches, we deleted 5% of item values in the data according to three different missingness mechanisms, imputed them using these approaches and compared the imputed values with the true values. Our comparison showed that the row/column-model-based imputation with truncation generally performed better, whereas our new approach had better performance under a number scenarios.     

Nonparametric estimation of the shape function in a Gamma process for degradation data

The author considers estimation under a Gamma process model for degradation data. The setting for degradation data is one in which n independent units, each with a Gamma process with a common shape function and scale parameter, are observed at several possibly different times. Covariates can be incorporated into the model by taking the scale parameter as a function of the covariates. The author proposes using the maximum pseudo‐likelihood method to estimate the unknown parameters. The method requires usage of the Pool Adjacent Violators Algorithm. Asymptotic properties, including consistency, convergence rate and asymptotic distribution, are established. Simulation studies are conducted to validate the method and its application is illustrated by using bridge beams data and carbon‐film resistors data. The Canadian Journal of Statistics 37: 102‐118; 2009 © 2009 Statistical Society of Canada     

Variance estimation when donor imputation is used to fill in missing values

Donor imputation is frequently used in surveys. However, very few variance estimation methods that take into account donor imputation have been developed in the literature. This is particularly true for surveys with high sampling fractions using nearest donor imputation, often called nearest‐neighbour imputation. In this paper, the authors develop a variance estimator for donor imputation based on the assumption that the imputed estimator of a domain total is approximately unbiased under an imputation model; that is, a model for the variable requiring imputation. Their variance estimator is valid, irrespective of the magnitude of the sampling fractions and the complexity of the donor imputation method, provided that the imputation model mean and variance are accurately estimated. They evaluate its performance in a simulation study and show that nonparametric estimation of the model mean and variance via smoothing splines brings robustness with respect to imputation model misspecifications. They also apply their variance estimator to real survey data when nearest‐neighbour imputation has been used to fill in the missing values. The Canadian Journal of Statistics 37: 400–416; 2009 © 2009 Statistical Society of Canada     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.     

Multivariate Kendall's tau for change‐point detection in copulas

Statistical procedures for the detection of a change in the dependence structure of a series of multivariate observations are studied in this work. The test statistics that are proposed are $L_1$ , $L_2$ , and $L_{\infty }$ distances computed from vectors of differences of Kendall's tau; two multivariate extensions of Kendall's measure of association are used. Since the distributions of these statistics under the null hypothesis of no change depend on the unknown underlying copula of the vectors, a procedure based on the multiplier central limit theorem is used for the computation of p‐values; the method is shown to be valid both asymptotically and for moderate sample sizes. Alternative versions of the tests that take into account possible breakpoints in the marginal distributions are also investigated. Monte Carlo simulations show that the tests are powerful under many scenarios of change‐point. In addition, two estimators of the time of change are proposed and their efficiency is carefully studied. The methodologies are illustrated on simulated series from the Canadian Regional Climate Model. The Canadian Journal of Statistics 41: 65–82; 2013 © 2012 Statistical Society of Canada     

Robust location estimation with missing data

In a missing data setting, we have a sample in which a vector of explanatory variables ${\bf x}_i$ is observed for every subject i, while scalar responses $y_i$ are missing by happenstance on some individuals. In this work we propose robust estimators of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response's distribution. In particular, strongly consistent estimators of any continuous location functional, such as the median, L‐functionals and M‐functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimators is also derived. The proofs of the theorems are presented in Supplementary Material available online. The Canadian Journal of Statistics 41: 111–132; 2013 © 2012 Statistical Society of Canada     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada