A formula for a missing plot in a general incomplete block design,when recovery of interblock information is used

In this paper, Yate's missing plot technique is used to derive the formula for substitution in a missing plot in a general incomplete block design, where blocks are assumed to be independent normal. The use of penalized normal equations, using BLUPS, makes this task simpler.     

Andrews plots for multivariate data: some new suggestions and applications

Andrews plots (Biometrics 28 (1972) 125-136), as a tool to graphically interpret multivariate data, have recently gained considerable recognition. In this article, we first review the previous literature and then suggest a modification to the traditional Andrews plots. Finally, we illustrate a few new applications of these plots in robust design studies and in correspondence analysis, using real data.     

Analysis of an experiment aimed at improving the reliability of transmission centre shafts

Smith (1991) presents a paper proposing the use of Weibull regression models to establish dependence of failure data (usually times) on covariates related to the design of the test specimens and test procedures. In his article Smith made the point that good experimental design was as important in reliability applications as elsewhere, and in view of the current interest in design inspired by Taguchi and others, we pay some attention in this article to that topic. A real case study from the Ford Motor Company is presented.Our main approach is to utilize suggestions in the literature for applying standard least squares techniques of experimental analysis even when there is likely to be nonnormal error, and censoring. This approach lacks theoretical justification, but its appeal is its simplicity and flexibility. For completeness we also include some analysis based on the proportional hazards model, and in an attempt to link back to Smith (1991), look at a Weibull regression model.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Diagnostics for repeated measurements in nonlinear mixed effects models

Abstract

This paper introduces a new non-deletion method which identifies discordant subjects in nonlinear mixed effects models under a self-modeling framework. The new method decomposes the population level residuals into two parts and suggests two-dimensional plots to identify discordant subjects. An observation-wise investigation for alleged discordant subjects is also presented. The performances of the new methods are illustrated with simulation data and two real data examples. The new methods successfully identify the intended or important discordant subjects and observations. In a comparison with the local influence method, the new method reaches a consistent conclusion in a simpler and more efficient manner.     

Minimal–maximal correlation-type goodness-of-fit tests

ABSTRACT

In this paper we consider correlation-type tests based on plotting points which are modifications to the simultaneous closeness probability plotting points as recently introduced in the literature. In particular, we consider a maximal correlation test and a minimal correlation test. Furthermore, we provide two methods to carry out each test, where one method uses plotting points which are data dependent and the other test uses plotting points which are not. Some numerical properties on the associated correlation statistics are provided for various distributions, as well as a comprehensive power study to assess their performance in comparison to correlation-type tests based on more traditional plotting points. Two illustrative examples are also provided to demonstrate the tests. Finally, we make some observations and provide ideas for future work.     

On the strengths of the self-updating process clustering algorithm

The self-updating process (SUP) is a clustering algorithm that stands from the viewpoint of data points and simulates the process how data points move and perform self-clustering. It is an iterative process on the sample space and allows for both time-varying and time-invariant operators. By simulations and comparisons, this paper shows that SUP is particularly competitive in clustering (i) data with noise, (ii) data with a large number of clusters, and (iii) unbalanced data. When noise is present in the data, SUP is able to isolate the noise data points while performing clustering simultaneously. The property of the local updating enables SUP to handle data with a large number of clusters and data of various structures. In this paper, we showed that the blurring mean-shift is a static SUP. Therefore, our discussions on the strengths of SUP also apply to the blurring mean-shift.     

Applications of bivariate and univariate local lyapunov exponents

Chaotic systems are characterized by sensitivity to initial conditions, and this property can be measured by global Lyapunov exponents, which are measures of the average divergence rate of initially close trajectories. Wolff (1992) introduced local Lyapunov exponents and used them to obtain two diagnostic plots for differentiating between stochastic and deterministic time series. We extend the definition of the local Lyapunov exponent and the diagnostic plots to accommodate time series that arise from bivariate maps and investigate the behaviour of the local Lyapunov exponents and the corresponding diagnostic plots for some dynamical systems and stochastic time series. We consider the application of these diagnostic plots to some heart rate variability data.     

Graphical Tests for the Frailty Distribution in the Shared Frailty Model

We extend a diagnostic plot for the frailty distribution in proportional hazards models to the case of shared frailty. The plot is based on a closure property of exponential family failure distributions with canonical statistics z and g(z), namely that the frailty distribution among survivors at time t has the same form, with the same values of the parameters associated with g(z). We extend this property to shared frailty, considering various definitions of a “surviving” cluster at time t. We illustrate the effectiveness of the method in the case where the “death” of the cluster is defined by the first death among its members.     

A. S. A. Programs

Arranged for the 25th Session of the INTERNATIONAL STATISTICAL INSTITUTE     

A graphical evaluation of logistic ridge estimator in mixture experiments

In comparison to other experimental studies, multicollinearity appears frequently in mixture experiments, a special study area of response surface methodology, due to the constraints on the components composing the mixture. In the analysis of mixture experiments by using a special generalized linear model, logistic regression model, multicollinearity causes precision problems in the maximum-likelihood logistic regression estimate. Therefore, effects due to multicollinearity can be reduced to a certain extent by using alternative approaches. One of these approaches is to use biased estimators for the estimation of the coefficients. In this paper, we suggest the use of logistic ridge regression (RR) estimator in the cases where there is multicollinearity during the analysis of mixture experiments using logistic regression. Also, for the selection of the biasing parameter, we use fraction of design space plots for evaluating the effect of the logistic RR estimator with respect to the scaled mean squared error of prediction. The suggested graphical approaches are illustrated on the tumor incidence data set.     

Diagnostics for repeated measurements in generalized linear mixed effects models

ABSTRACT

As there is an extensive body of research on diagnostics in regression models, various outlier detection methods have been developed. These methods have been extended to mixed effects models and generalized linear models, but there exist intrinsic drawbacks and limitations. This paper presents two-dimensional plots to identify discordant subjects and observations in generalized linear mixed effects models, displaying discordance in two directions. The sTudentized Residual Sum of Squares is not an extension of any regression tools but a new approach designed to efficiently reflect the characteristics of repeated measures. And this noteworthy clustering of outliers is identified in the plot. Applications to real-life examples are presented to illustrate the favorable/beneficial performance of the new tool.     

Diagnostic Plots for Assessing the Frailty Distribution in Multivariate Survival Data

In biomedical studies, frailty models arecommonly used in analyzing multivariate survival data, wherethe objective of the study is to estimate both the covariateeffect and the dependence between the multivariate survival times.However, inference based on these models are dependent on thedistributional assumption of frailty. We propose a diagnosticplot for assessing the frailty assumption. The proposed methodis based on the cross-ratio function and the diagnostic plotsuggested by Oakes (1989). We use kernel regression smoothingwith bandwidth choice by cross-validation, to obtain the proposedplot. The resulting plot is capable of differentiating betweenthe gamma and positive stable frailty models when strong associationis present. We illustrate the feasibility of our method usingsimulation studies under known frailty distributions. The approachis applied to data on blindness for each eye of diabetic patientswith adult onset diabetes and a reasonable fit to the gamma frailtymodel is found.     

Comparison of Prediction Capabilities of Partially Replicated Central Composite Designs in Cuboidal Region

Three variations of partially replicated central composite designs have been compared. Their prediction capabilities with respect to their prediction variances are assessed using variance dispersion graphs in cuboidal region when the axial distance is α = 1. The results show that for n₀ = 1, 2, and 3 center points, the replicated star designs have better prediction potentials than the replicated cube designs when interest is in predicting responses in the face-centered cube. The face-centered cube is the second-order central composite design with axial distance, α = 1.     

Selection of terms in random coefficient regression models

The selection of suitable terms in random coefficient regression models is a challenging problem to practitioners. Although many techniques, ranging from those with a theoretical flavour to those with an exploratory spirit, have been proposed for such purposes, no particular one may be considered as a paradigm. In fact, many authors advocate that they should be used in a complementary way. We consider exploratory methods based on fitting standard regression models to the individual response profiles or to the rows of the sample within-units covariance matrix (for balanced data) that may serve as additional tools in the process of selecting an appropriate model. We evaluate the performance of the proposal via a simulation study and consider applications to two examples in the field of Biostatistics.     

Simple graphical methods of displaying multiple clinical results

A common challenge in clinical research trials is for applied statistics to manage, analyse, summarize and report an enormous amount of data. Nowadays, due to advances in medical technology, situations frequently arise where it is difficult to display and interpret results. Consequently, a creative approach is required to summarize the main outcomes of the statistical analyses in a form which is easy to grasp, to interpret and possibly to remember. In this paper a number of clinical case studies are provided. Firstly, a topographical map of the brain summarizing P-values obtained from comparisons across different EEG sites; secondly, a bulls eye plot, showing the agreement between observers in different regions of the heart; thirdly, a pictorial table reporting inter- and intra-rater reliability scores of a speech assessment; fourthly a star-plot to deal with numerous questionnaire results and finally a correlogram to illustrate significant correlation values between two diagnostic tools. The intention of this paper is to encourage the effort of visual representations of multiple statistical outcomes. Such representations do not only embellish the report, but aid interpretation by conveying a specific statistical meaning.     

Covariance Analysis for Split-Plot and Split-Block Designs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

Proper methodology for the analysis of covariance for experiments designed in a split-plot or split-block design is not found in the statistical literature. Analyses for these designs are often performed incompletely or even incorrectly. This is especially true when popular statistical computer software packages are used for the analysis of these designs. This article provides several appropriate models, ANOVA tables, and standard errors for comparisons from experiments arranged in a standard split-plot, split–split-plot, or split-block design where a covariate has been measured on the smallest size experimental unit.     

Investigating the assumption of homogeneity of treatment effects in clinical studies with application to meta-analysis

An important part of the evaluation of a therapy is an investigation of the assumption of homogeneity of its effect across pre-defined subpopulations. In this paper we describe simple graphical presentations that could be used to assess the homogeneity of treatment effect and identify outliers. The emphasis in the paper is on meta-analysis but the methods described can be generalized to other investigations.     

Bayesian sample size determination for phase IIA clinical trials using historical data and semi‐parametric prior's elicitation

The Simon's two‐stage design is the most commonly applied among multi‐stage designs in phase IIA clinical trials. It combines the sample sizes at the two stages in order to minimize either the expected or the maximum sample size. When the uncertainty about pre‐trial beliefs on the expected or desired response rate is high, a Bayesian alternative should be considered since it allows to deal with the entire distribution of the parameter of interest in a more natural way. In this setting, a crucial issue is how to construct a distribution from the available summaries to use as a clinical prior in a Bayesian design. In this work, we explore the Bayesian counterparts of the Simon's two‐stage design based on the predictive version of the single threshold design. This design requires specifying two prior distributions: the analysis prior, which is used to compute the posterior probabilities, and the design prior, which is employed to obtain the prior predictive distribution. While the usual approach is to build beta priors for carrying out a conjugate analysis, we derived both the analysis and the design distributions through linear combinations of B‐splines. The motivating example is the planning of the phase IIA two‐stage trial on anti‐HER2 DNA vaccine in breast cancer, where initial beliefs formed from elicited experts' opinions and historical data showed a high level of uncertainty. In a sample size determination problem, the impact of different priors is evaluated.     

Save: a method for dimension reduction and graphics in regression

Sliced average variance estimation (SAVE) is a method for constructing sufficient summary plots in regressions with many predictors. The summary plots are designed to capture all the information about the response that is available from the predictors, and do not require a model for their construction. They can be particularly helpful for guiding the choice of a first model. Methodological aspects of SAVE are studied in this article.