The randomized concentration-controlled trial:mathematical definitions,a dose-adjusting algorithm,and sample size efficiency

A randomized concentration-controlled trial (RCCT) (Peck 1990) is distinguished from a randomized dose-controlled trial (RDCT) by controlling the plasma concentration and allowing the dose to vary, as opposed to controlling the dose and allowing the concentration to vary. Monte Carlo studies have suggested that the RCCT is more efficient (in terms of sample size) than the more traditional dose-controlled trials. This paper presents a general mathematical defminition and investigates the properties of an RCCT. A Bayesian adjustment algorithm used in an RCCT is presented and discussed. Expressions for the sample-size efficiency of the RCCT relative to the RDCT is found for both Gaussian and Bernoulli efficacy variables. The RCCT is found to be at least as efficient in terms of sample size. The RCCT can be considerably more efficient than the RDCT.     

Logistic regression analysis of non‐randomized response data collected by the parallel model in sensitive surveys

To study the relationship between a sensitive binary response variable and a set of non‐sensitive covariates, this paper develops a hidden logistic regression to analyse non‐randomized response data collected via the parallel model originally proposed by Tian (2014). This is the first paper to employ the logistic regression analysis in the field of non‐randomized response techniques. Both the Newton–Raphson algorithm and a monotone quadratic lower bound algorithm are developed to derive the maximum likelihood estimates of the parameters of interest. In particular, the proposed logistic parallel model can be used to study the association between a sensitive binary variable and another non‐sensitive binary variable via the measure of odds ratio. Simulations are performed and a study on people's sexual practice data in the United States is used to illustrate the proposed methods.     

Statistical profiling methods with hierarchical logistic regression for healthcare providers with binary outcomes

Within the context of California's public report of coronary artery bypass graft (CABG) surgery outcomes, we first thoroughly review popular statistical methods for profiling healthcare providers. Extensive simulation studies are then conducted to compare profiling schemes based on hierarchical logistic regression (LR) modeling under various conditions. Both Bayesian and frequentist's methods are evaluated in classifying hospitals into ‘better’, ‘normal’ or ‘worse’ service providers. The simulation results suggest that no single method would dominate others on all accounts. Traditional schemes based on LR tend to identify too many false outliers, while those based on hierarchical modeling are relatively conservative. The issue of over shrinkage in hierarchical modeling is also investigated using the 2005–2006 California CABG data set. The article provides theoretical and empirical evidence in choosing the right methodology for provider profiling.     

Design and analysis considerations for comparing dynamic treatment regimens with binary outcomes from sequential multiple assignment randomized trials

In behavioral, educational and medical practice, interventions are often personalized over time using strategies that are based on individual behaviors and characteristics and changes in symptoms, severity, or adherence that are a result of one's treatment. Such strategies that more closely mimic real practice, are known as dynamic treatment regimens (DTRs). A sequential multiple assignment randomized trial (SMART) is a multi-stage trial design that can be used to construct effective DTRs. This article reviews a simple to use ‘weighted and replicated’ estimation technique for comparing DTRs embedded in a SMART design using logistic regression for a binary, end-of-study outcome variable. Based on a Wald test that compares two embedded DTRs of interest from the ‘weighted and replicated’ regression model, a sample size calculation is presented with a corresponding user-friendly applet to aid in the process of designing a SMART. The analytic models and sample size calculations are presented for three of the more commonly used two-stage SMART designs. Simulations for the sample size calculation show the empirical power reaches expected levels. A data analysis example with corresponding code is presented in the appendix using data from a SMART developing an effective DTR in autism.     

Deterministic Evolution of Strength in Multiple Comparisons Models: Who is the Greatest Golfer?

We present a statistical methodology for fitting time‐varying rankings, by estimating the strength parameters of the Plackett–Luce multiple comparisons model at regularly spaced times for each ranked item. We use the little‐known method of barycentric rational interpolation to interpolate between the strength parameters so that a competitor's strength can be evaluated at any time. We chose the time‐varying strengths to evolve deterministically rather than stochastically, a preference that we reason often has merit. There are many statistical and computational problems to overcome on fitting anything beyond ‘toy’ data sets. The methodological innovations here include a method for maximizing a likelihood function for many parameters, approximations for modelling tied data and an approach to the elimination of secular drift of the estimated ‘strengths’. The methodology has obvious applications to fields such as marketing, although we demonstrate our approach by analysing a large data set of golf tournament results, in search of an answer to the question ‘who is the greatest golfer of all time?’     

Colours and Cocktails: Compositional Data Analysis 2013 Lancaster Lecture

The different constituents of physical mixtures such as coloured paint, cocktails, geological and other samples can be represented by d‐dimensional vectors called compositions with non‐negative components that sum to one. Data in which the observations are compositions are called compositional data. There are a number of different ways of thinking about and consequently analysing compositional data. The log‐ratio methods proposed by Aitchison in the 1980s have become the dominant methods in the field. One reason for this is the development of normative arguments converting the properties of log‐ratio methods to ‘essential requirements’ or Principles for any method of analysis to satisfy. We discuss different ways of thinking about compositional data and interpret the development of the Principles in terms of these different viewpoints. We illustrate the properties on which the Principles are based, focussing particularly on the key subcompositional coherence property. We show that this Principle is based on implicit assumptions and beliefs that do not always hold. Moreover, it is applied selectively because it is not actually satisfied by the log‐ratio methods it is intended to justify. This implies that a more open statistical approach to compositional data analysis should be adopted.     

Predictive Inference for Big,Spatial, Non‐Gaussian Data: MODIS Cloud Data and its Change‐of‐Support

Remote sensing of the earth with satellites yields datasets that can be massive in size, nonstationary in space, and non‐Gaussian in distribution. To overcome computational challenges, we use the reduced‐rank spatial random effects (SRE) model in a statistical analysis of cloud‐mask data from NASA's Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on board NASA's Terra satellite. Parameterisations of cloud processes are the biggest source of uncertainty and sensitivity in different climate models’ future projections of Earth's climate. An accurate quantification of the spatial distribution of clouds, as well as a rigorously estimated pixel‐scale clear‐sky‐probability process, is needed to establish reliable estimates of cloud‐distributional changes and trends caused by climate change. Here we give a hierarchical spatial‐statistical modelling approach for a very large spatial dataset of 2.75 million pixels, corresponding to a granule of MODIS cloud‐mask data, and we use spatial change‐of‐Support relationships to estimate cloud fraction at coarser resolutions. Our model is non‐Gaussian; it postulates a hidden process for the clear‐sky probability that makes use of the SRE model, EM‐estimation, and optimal (empirical Bayes) spatial prediction of the clear‐sky‐probability process. Measures of prediction uncertainty are also given.     

Assessing the treatment effect in a randomized controlled trial with extensive non‐adherence: the EVOLVE trial

Intention‐to‐treat (ITT) analysis is widely used to establish efficacy in randomized clinical trials. However, in a long‐term outcomes study where non‐adherence to study drug is substantial, the on‐treatment effect of the study drug may be underestimated using the ITT analysis. The analyses presented herein are from the EVOLVE trial, a double‐blind, placebo‐controlled, event‐driven cardiovascular outcomes study conducted to assess whether a treatment regimen including cinacalcet compared with placebo in addition to other conventional therapies reduces the risk of mortality and major cardiovascular events in patients receiving hemodialysis with secondary hyperparathyroidism. Pre‐specified sensitivity analyses were performed to assess the impact of non‐adherence on the estimated effect of cinacalcet. These analyses included lag‐censoring, inverse probability of censoring weights (IPCW), rank preserving structural failure time model (RPSFTM) and iterative parameter estimation (IPE). The relative hazard (cinacalcet versus placebo) of mortality and major cardiovascular events was 0.93 (95% confidence interval 0.85, 1.02) using the ITT analysis; 0.85 (0.76, 0.95) using lag‐censoring analysis; 0.81 (0.70, 0.92) using IPCW; 0.85 (0.66, 1.04) using RPSFTM and 0.85 (0.75, 0.96) using IPE. These analyses, while not providing definitive evidence, suggest that the intervention may have an effect while subjects are receiving treatment. The ITT method remains the established method to evaluate efficacy of a new treatment; however, additional analyses should be considered to assess the on‐treatment effect when substantial non‐adherence to study drug is expected or observed. Copyright © 2015 John Wiley & Sons, Ltd.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

Intent‐to‐treat analysis in the presence of off‐treatment or missing data

Intent‐to‐treat (ITT) analysis is viewed as the analysis of a clinical trial that provides the least bias, but difficult issues can arise. Common analysis methods such as mixed‐effects and proportional hazards models are usually labeled as ITT analysis, but in practice they can often be inconsistent with a strict interpretation of the ITT principle. In trials where effective medications are available to patients withdrawing from treatment, ITT analysis can mask important therapeutic effects of the intervention studied in the trial. Analysis of on‐treatment data may be subject to bias, but can address efficacy objectives when combined with careful review of the pattern of withdrawals across treatments particularly for those patients withdrawing due to lack of efficacy and adverse events. Copyright © 2010 John Wiley & Sons, Ltd.     

A robust I-sample analysis of means type randomization test for variances for unbalanced designs

Two analysis of means type randomization tests for testing the equality of I variances for unbalanced designs are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples ‘in a bowl’ forms the combined sample. Drawing samples ‘from the bowl’ forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. A Monte Carlo study is used to verify that the permutation version of the tests exhibit excellent power when compared to other robust tests. The Monte Carlo study also identifies circumstances under which the popular Levene's test fails.     

Impact of Model Choice on LR Assessment in Case of Rare Haplotype Match (Frequentist Approach)

The likelihood ratio (LR) measures the relative weight of forensic data regarding two hypotheses. Several levels of uncertainty arise if frequentist methods are chosen for its assessment: the assumed population model only approximates the true one, and its parameters are estimated through a limited database. Moreover, it may be wise to discard part of data, especially that only indirectly related to the hypotheses. Different reductions define different LRs. Therefore, it is more sensible to talk about ‘a’ LR instead of ‘the’ LR, and the error involved in the estimation should be quantified. Two frequentist methods are proposed in the light of these points for the ‘rare type match problem’, that is, when a match between the perpetrator's and the suspect's DNA profile, never observed before in the database of reference, is to be evaluated.     

A New Approach to the Estimation of Inter-Variable Correlation

The use of different measures of similarity between observed vectors for the purposes of classifying or clustering them has been expanding dramatically in recent years. One result of this expansion has been the use of many new similarity measures, designed for the purpose of satisfying various criteria. A noteworthy application involves estimating the relationships between genes using microarray experimental data. We consider the class of ‘correlation-type’ similarity measures. The use of these new measures of similarity suggest that the whole problem needs to be formulated in statistical terms to clarify their relative benefits. Pursuant to this need, we define, for each given observed vector, a baseline representing the ‘true’ value common to each of the component observations. These ‘true’ values are taken to be parameters. We define the ‘true correlation’ between each two observed vectors as the average (over the distribution of the observations for given baseline parameters) of Pearson's correlation with sample means replaced by the corresponding baseline parameters. Estimators of this true correlation are assessed using their mean squared error (MSE). Proper Bayes estimators of this true correlation, being based on the predictive posterior distribution of the data, are both difficult to calculate/analyze and highly non robust. By constrast, empirical Bayes estimators are: (i) close to their Bayesian counterparts; (ii) easy to analyze; and (iii) strongly robust. For these reasons, we employ empirical Bayes estimators of correlation in place of their Bayesian counterparts. We show how to construct two different kinds of simultaneous Bayes correlation estimators: the first assumes no apriori correlation between baseline parameters; the second assumes a common unknown correlation between them. Estimators of the latter type frequently have significantly smaller MSE than those of the former type which, in turn, frequently have significantly smaller MSE than their Pearson estimator counterparts. For purposes of illustrating our results, we examine the problem of inferring the relationships between gene expression level vectors, in the context of observing microarray experimental data.     

Decomposition of cumulative chi-squared statistics,with some new tools for their interpretation

It is well known that the Pearson statistic \(\chi ^{2}\) can perform poorly in studying the association between ordinal categorical variables. Taguchi’s and Hirotsu’s statistics have been introduced in the literature as simple alternatives to Pearson’s chi-squared test for contingency tables with ordered categorical variables. The aim of this paper is to shed new light on these statistics, stressing their interpretations and characteristics, providing in this way new and different interpretations of these statistics. Moreover, a theoretical scheme is developed showing the links between the different proposals and classes of cumulative chi-squared statistical tests, starting from a unifying index of heterogeneity, unalikeability and variability measures. Users of statistics may find it attractive to understand well the different proposals. Some decompositions of both statistics are also highlighted. This paper presents a case study of optimizing the polysilicon deposition process in a very large-scale integrated circuit, to identify the optimal combination of factor levels. It is obtained by means of the information coming from a correspondence analysis based on Taguchi’s statistic and regression models for binary dependent variables. A new optimal combination of factor levels is obtained, different from many others proposed in the literature for this data.     

Notes on odds ratio estimation for a randomized clinical trial with noncompliance and missing outcomes

The odds ratio (OR) has been recommended elsewhere to measure the relative treatment efficacy in a randomized clinical trial (RCT), because it possesses a few desirable statistical properties. In practice, it is not uncommon to come across an RCT in which there are patients who do not comply with their assigned treatments and patients whose outcomes are missing. Under the compound exclusion restriction, latent ignorable and monotonicity assumptions, we derive the maximum likelihood estimator (MLE) of the OR and apply Monte Carlo simulation to compare its performance with those of the other two commonly used estimators for missing completely at random (MCAR) and for the intention-to-treat (ITT) analysis based on patients with known outcomes, respectively. We note that both estimators for MCAR and the ITT analysis may produce a misleading inference of the OR even when the relative treatment effect is equal. We further derive three asymptotic interval estimators for the OR, including the interval estimator using Wald’s statistic, the interval estimator using the logarithmic transformation, and the interval estimator using an ad hoc procedure of combining the above two interval estimators. On the basis of a Monte Carlo simulation, we evaluate the finite-sample performance of these interval estimators in a variety of situations. Finally, we use the data taken from a randomized encouragement design studying the effect of flu shots on the flu-related hospitalization rate to illustrate the use of the MLE and the asymptotic interval estimators for the OR developed here.     

On the Breakdown Properties of Some Multivariate M‐Functionals*

Abstract. For probability distributions on ? ^q, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/( q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.     

Coefficients of lee-gurland two-sample test on normal means

The Behrens-Fisher problem in comparing means of two normal populations is revisited Lee and Gurland (1975) suggested a solution to the problem and provided the set of coefficients required in computing critical values for the case α=005, where α is the nominal level of significance This solution, called the Lee-Guiland Test in this article, has proven to be practical as far as calculation is involved, and more importantly, it maintains the actual size very close to α= 0.05 for possible values of the ratio of population variances This merit has not been attained by most of the Behrens-Fisher solutions in the literature. In this article, the coefficients for other values of α, namely 0 025, 0 01 and 0.005 are provided for wider applications of the test Moreover, careful and detailed comparisons are made in terms of size and power with the other practical solution:the Welch's Approximate t I est Due to a possible drawback of the Welch's Approximate t I est in controlling the actual size, especially for small a and small sample sizes, the Lee-Gurland lest presents itself as a slightly better' alternative in testing equality of two normal population means I he coefficients mentioned above are also fitted by the functions of the reciprocals of the degrees of freedom, so that the substantial amount of table-looking can be avoided Some discussions are also made in regarding the recent “Welch vs Gosset” argument: Should the Student's t Test be dispensed off’from the routine use in testing the equality of two normal means?.     

A pragmatic dose‐finding approach using short‐term surrogate efficacy outcomes to evaluate binary efficacy and toxicity outcomes in phase I cancer clinical trials

There is a growing need for study designs that can evaluate efficacy and toxicity outcomes simultaneously in phase I or phase I/II cancer clinical trials. Many dose‐finding approaches have been proposed; however, most of these approaches assume binary efficacy and toxicity outcomes, such as dose‐limiting toxicity (DLT), and objective responses. DLTs are often defined for short time periods. In contrast, objective responses are often defined for longer periods because of practical limitations on confirmation and the criteria used to define ‘confirmation’. This means that studies have to be carried out for unacceptably long periods of time. Previous studies have not proposed a satisfactory solution to this specific problem. Furthermore, this problem may be a barrier for practitioners who want to implement notable previous dose‐finding approaches. To cope with this problem, we propose an approach using unconfirmed early responses as the surrogate efficacy outcome for the confirmed outcome. Because it is reasonable to expect moderate positive correlation between the two outcomes and the method replaces the surrogate outcome with the confirmed outcome once it becomes available, the proposed approach can reduce irrelevant dose selection and accumulation of bias. Moreover, it is also expected that it can significantly shorten study duration. Using simulation studies, we demonstrate the positive utility of the proposed approach and provide three variations of it, all of which can be easily implemented with modified likelihood functions and outcome variable definitions. Copyright © 2013 John Wiley & Sons, Ltd.     

On generalised estimating equations for vector regression

Generalised estimating equations (GEE) for regression problems with vector‐valued responses are examined. When the response vectors are of mixed type (e.g. continuous–binary response pairs), the GEE approach is a semiparametric alternative to full‐likelihood copula methods, and is closely related to Prentice & Zhao's mean‐covariance estimation equations approach. When the response vectors are of the same type (e.g. measurements on left and right eyes), the GEE approach can be viewed as a ‘plug‐in’ to existing methods, such as the vglm function from the state‐of‐the‐art VGAM package in R. In either scenario, the GEE approach offers asymptotically correct inferences on model parameters regardless of whether the working variance–covariance model is correctly or incorrectly specified. The finite‐sample performance of the method is assessed using simulation studies based on a burn injury dataset and a sorbinil eye trial dataset. The method is applied to data analysis examples using the same two datasets, as well as to a trivariate binary dataset on three plant species in the Hunua ranges of Auckland.     

Misspecification of at‐risk periods and distributional assumptions in estimating COPD exacerbation rates: The resultant bias in treatment effect estimation

In trials comparing the rate of chronic obstructive pulmonary disease exacerbation between treatment arms, the rate is typically calculated on the basis of the whole of each patient's follow‐up period. However, the true time a patient is at risk should exclude periods in which an exacerbation episode is occurring, because a patient cannot be at risk of another exacerbation episode until recovered. We used data from two chronic obstructive pulmonary disease randomized controlled trials and compared treatment effect estimates and confidence intervals when using two different definitions of the at‐risk period. Using a simulation study we examined the bias in the estimated treatment effect and the coverage of the confidence interval, using these two definitions of the at‐risk period. We investigated how the sample size required for a given power changes on the basis of the definition of at‐risk period used. Our results showed that treatment efficacy is underestimated when the at‐risk period does not take account of exacerbation duration, and the power to detect a statistically significant result is slightly diminished. Correspondingly, using the correct at‐risk period, some modest savings in required sample size can be achieved. Using the proposed at‐risk period that excludes recovery times requires formal definitions of the beginning and end of an exacerbation episode, and we recommend these be always predefined in a trial protocol.