Inference in group sequential designs with causal mechanisms: implications for power and mediation analysis

In cluster-specific studies, ordinary logistic regression and conditional logistic regression for binary outcomes provide maximum likelihood estimator (MLE) and conditional maximum likelihood estimator (CMLE), respectively. In this paper, we show that CMLE is approaching to MLE asymptotically when each individual data point is replicated infinitely many times. Our theoretical derivation is based on the observation that a term appearing in the conditional average log-likelihood function is the coefficient of a polynomial, and hence can be transformed to a complex integral by Cauchy’s differentiation formula. The asymptotic analysis of the complex integral can then be performed using the classical method of steepest descent. Our result implies that CMLE can be biased if individual weights are multiplied with a constant, and that we should be cautious when assigning weights to cluster-specific studies.

IntroductionAdaptive designs allow changes to an ongoing trial based on prespecified early examinations of accrued data. Opportunities are potentially being missed to incorporate health economic considerations into the design of these studies.MethodsWe describe how to estimate the expected value of sample information for group sequential design adaptive trials. We operationalize this approach in a hypothetical case study using data from a pilot trial. We report the expected value of sample information and expected net benefit of sampling results for 5 design options for the future full-scale trial including the fixed-sample-size design and the group sequential design using either the Pocock stopping rule or the O’Brien-Fleming stopping rule with 2 or 5 analyses. We considered 2 scenarios relating to 1) using the cost-effectiveness model with a traditional approach to the health economic analysis and 2) adjusting the cost-effectiveness analysis to incorporate the bias-adjusted maximum likelihood estimates of trial outcomes to account for the bias that can be generated in adaptive trials.ResultsThe case study demonstrated that the methods developed could be successfully applied in practice. The results showed that the O’Brien-Fleming stopping rule with 2 analyses was the most efficient design with the highest expected net benefit of sampling in the case study.ConclusionsCost-effectiveness considerations are unavoidable in budget-constrained, publicly funded health care systems, and adaptive designs can provide an alternative to costly fixed-sample-size designs. We recommend that when planning a clinical trial, expected value of sample information methods be used to compare possible adaptive and nonadaptive trial designs, with appropriate adjustment, to help justify the choice of design characteristics and ensure the cost-effective use of research funding.HighlightsOpportunities are potentially being missed to incorporate health economic considerations into the design of adaptive clinical trials.Existing expected value of sample information analysis methods can be extended to compare possible group sequential and nonadaptive trial designs when planning a clinical trial.We recommend that adjusted analyses be presented to control for the potential impact of the adaptive designs and to maintain the accuracy of the calculations.This approach can help to justify the choice of design characteristics and ensure the cost-effective use of limited research funding.

The Rasch model is one of the most prominent item response models. In this article, different item parameter estimation methods for the Rasch model are systematically compared through a comprehensive simulation study: Different alternatives of joint maximum likelihood (JML) estimation, different alternatives of marginal maximum likelihood (MML) estimation, conditional maximum likelihood (CML) estimation, and several limited information methods (LIM). The type of ability distribution (i.e., nonnormality), the number of items, sample size, and the distribution of item difficulties were systematically varied. Across different simulation conditions, MML methods with flexible distributional specifications can be at least as efficient as CML. Moreover, in many situations (i.e., for long tests), penalized JML and JML with ε adjustment resulted in very efficient estimates and might be considered alternatives to JML implementations currently used in statistical software. Moreover, minimum chi-square (MINCHI) estimation was the best-performing LIM method. These findings demonstrate that JML estimation and LIM can still prove helpful in applied research.

ABSTRACTSmall samples are common in growth models due to financial and logistical difficulties of following people longitudinally. For similar reasons, longitudinal studies often contain missing data. Though full information maximum likelihood (FIML) is popular to accommodate missing data, the limited number of studies in this area have found that FIML tends to perform poorly with small-sample growth models. This report demonstrates that the fault lies not with how FIML accommodates missingness but rather with maximum likelihood estimation itself. We discuss how the less popular restricted likelihood form of FIML, along with small-sample-appropriate methods, yields trustworthy estimates for growth models with small samples and missing data. That is, previously reported small sample issues with FIML are attributable to finite sample bias of maximum likelihood estimation not direct likelihood. Estimation issues pertinent to joint multiple imputation and predictive mean matching are also included and discussed.

BackgroundIn investigating differentially expressed genes or other selected features, researchers conduct hypothesis tests to determine which biological categories, such as those of the Gene Ontology (GO), are enriched for the selected features. Multiple comparison procedures (MCPs) are commonly used to prevent excessive false positive rates. Traditional MCPs, e.g., the Bonferroni method, go to the opposite extreme: strictly controlling a family-wise error rate, resulting in excessive false negative rates. Researchers generally prefer the more balanced approach of instead controlling the false discovery rate (FDR). However, the q-values that methods of FDR control assign to biological categories tend to be too low to reliably estimate the probability that a biological category is not enriched for the preselected features. Thus, we study an application of the other estimators of that probability, which is called the local FDR (LFDR).ResultsWe considered five LFDR estimators for detecting enriched GO terms: a binomial-based estimator (BBE), a maximum likelihood estimator (MLE), a normalized MLE (NMLE), a histogram-based estimator assuming a theoretical null hypothesis (HBE), and a histogram-based estimator assuming an empirical null hypothesis (HBE-EN). Since NMLE depends not only on the data but also on the specified value of Π0, the proportion of non-enriched GO terms, it is only advantageous when either Π0 is already known with sufficient accuracy or there are data for only 1 GO term. By contrast, the other estimators work without specifying Π0 but require data for at least 2 GO terms. Our simulation studies yielded the following summaries of the relative performance of each of those four estimators. HBE and HBE-EN produced larger biases for 2, 4, 8, 32, and 100 GO terms than BBE and MLE. BBE has the lowest bias if Π0 is 1 and if the number of GO terms is between 2 and 32. The bias of MLE is no worse than that of BBE for 100 GO terms even when the ideal number of components in its underlying mixture model is unknown, but has high bias when the number of GO terms is small compared to the number of estimated parameters. For unknown values of Π0, BBE has the lowest bias for a small number of GO terms (2-32 GO terms), and MLE has the lowest bias for a medium number of GO terms (100 GO terms).ConclusionsFor enrichment detection, we recommend estimating the LFDR by MLE given at least a medium number of GO terms, by BBE given a small number of GO terms, and by NMLE given either only 1 GO term or precise knowledge of Π0.

ABSTRACTIn Rheumatoid Arthritis studies, scientists usually need to focus on multiple endpoints simultaneously. Regulatory agencies require ACR20 as the primary endpoint for approval and making decisions. However, the proportional measurement for a binary variable is less reliable with a small sample size. DAS28 is an increasingly popular co-primary or key secondary endpoint for decision making in interim analyses because it is a continuous variable as well as a linear combination of multiple measurements that are also contained in ACR20. In a group sequential (GS) design, one of the most important parts is to compute the correlation among test statistics in interim and final analyses. We perform a logistic regression to link DAS28 and ACR20 and compute the correlation numerically. We also prove that the covariance matrix between binary and continuous variables with underlying logistic regression is a consistent estimator. Three methods including “GS design with univariate variable (ACR20),” “GS design with change of endpoints (DAS28 and ACR20),” and “Bonferroni tests with change of endpoints” are compared in three cases—“early stopping for efficacy,” “early stopping for futility,” and “early stopping for both” through simulation. The results show that in the first and third cases, the power of tests with a change of endpoint is much higher than tests that only use ACR20. In the case of “early stopping for futility,” the methods with a change of endpoint also provide a higher probability of correctly stopping early for futility.

To demonstrate average bioequivalence, the ninety-percent confidence intervals (CI) on the ratio of geometric means for area under the concentration-time curve (AUC) and maximum observed plasma concentration (Cmax) must lie within 0.80-1.25. Demonstration of average bioequivalence (ABE) for highly variable drug products requires large numbers of subjects in a standard, adequately powered, two-period crossover. Application of non-traditional study designs can help to meet this hurdle. Study design and analysis for replicate and group sequential-replicate study designs are presented and illustrated using examples. It is demonstrated how to use such approaches to meet the difficult regulatory hurdle of average bioequivalence for a highly variable drug product. To illustrate, data are provided from three separate ABE studies for a highly variable drug product at three dosage strengths. In all three studies, a replicate study design was used to compensate for high intrasubject variation. Additionally, for the last study, a group sequential study design was imposed to provide early evidence of conclusive results. Replicate designs and group-sequential designs in bioequivalence should be used to demonstrate average bioequivalence for highly variable drug products or when uncertain of true intrasubject variability in order to ensure conclusive study results.

Sequential analysis is frequently employed to address ethical and financial issues in clinical trials. Sequential analysis may be performed using standard group sequential designs, or, more recently, with adaptive designs that use estimates of treatment effect to modify the maximal statistical information to be collected. In the general setting in which statistical information and clinical trial costs are functions of the number of subjects used, it has yet to be established whether there is any major efficiency advantage to adaptive designs over traditional group sequential designs. In survival analysis, however, statistical information (and hence efficiency) is most closely related to the observed number of events, while trial costs still depend on the number of patients accrued. As the number of subjects may dominate the cost of a trial, an adaptive design that specifies a reduced maximal possible sample size when an extreme treatment effect has been observed may allow early termination of accrual and therefore a more cost-efficient trial. We investigate and compare the tradeoffs between efficiency (as measured by average number of observed events required), power, and cost (a function of the number of subjects accrued and length of observation) for standard group sequential methods and an adaptive design that allows for early termination of accrual. We find that when certain trial design parameters are constrained, an adaptive approach to terminating subject accrual may improve upon the cost efficiency of a group sequential clinical trial investigating time-to-event endpoints. However, when the spectrum of group sequential designs considered is broadened, the advantage of the adaptive designs is less clear.

In this article we introduce multiple stuttering-Poisson distributions and discuss their genesis, properties and applications. We consider a special case of triple stuttering-Poisson distribution with probability generating function G(t) = Exp [a(t − 1) + b(t 2 − 1) + c(t 3 − l)] and investigate (1) maximum likelihood estimation, (2) moment estimation, and (3) mixed moment estimation of the parameters a, b, c. First order terms in the expressions for the biases and covariances of moment estimators are presented. The joint asymptotic efficiencies of moment estimators are tabulated for a few selected parameter points. The first order terms in the biases of maximum likelihood estimators and moment estimators are also given for a few selected parameter points. As judged from the contribution of the first order term, the possible bias of maximum likelihood estimators can be substantial over a considerable region of the parameter space. The moment estimators have low joint asymptotic efficiency over a large par...

An order-preserving property of the maximum likelihood estimates for the Rasch model

Comparative diagnostic accuracy (CDA) studies are typically small retrospective studies supporting a higher accuracy for one modality over another for either staging a particular disease or assessing response to therapy, and they are used to generate hypotheses for larger prospective trials. The purpose of this article is to introduce the group sequential design (GSD) approach in planning these larger trials. Methodology needed for using GSD in the CDA studies is recently developed. In this article, GSD with the O'Brien and Fleming (OBF) stopping rule is described and guidelines for sample size calculation are provided. Simulated data is used to demonstrate the application of GSD in the design/analysis of a clinical trial in the CDA study setting. The expected sample size needed for planning a trial with GSD (under the OBF stopping rule) is slightly inflated but may ultimately result in greater savings of patient resources. GSD is a specialized statistical method that is helpful in balancing the ethical and financial advantages of stopping a study early against the risk of an incorrect conclusion and should be adopted for planning CDA studies.

BackgroundPhase III trials often require large sample sizes, leading to high costs and delays in clinical decision-making. Group sequential designs can improve trial efficiency by allowing for early stopping for efficacy and/or futility and thus may decrease the sample size, trial duration and associated costs. Bayesian approaches may offer additional benefits by incorporating previous information into the analyses and using decision criteria that are more practically relevant than those used in frequentist approaches. Frequentist group sequential designs have often been used for phase III studies, but the use of Bayesian group sequential designs is less common. The aim of this work was to explore how Bayesian group sequential designs could be constructed for phase III trials conducted in emergency medicine.MethodsThe PARAMEDIC2 trial was a phase III randomised controlled trial that compared the use of adrenaline to placebo in out-of-hospital cardiac arrest patients on 30-day survival rates. It used a frequentist group sequential design to allow early stopping for efficacy or harm. We constructed several alternative Bayesian group sequential designs and studied their operating characteristics via simulation. We then virtually re-executed the trial by applying the Bayesian designs to the PARAMEDIC2 data to demonstrate what might have happened if these designs had been used in practice.ResultsWe produced three alternative Bayesian group sequential designs, each of which had greater than 90% power to detect the target treatment effect. A Bayesian design which performed interim analyses every 500 patients recruited produced the lowest average sample size. Using the alternative designs, the PARAMEDIC2 trial could have declared adrenaline superior for 30-day survival with approximately 1500 fewer patients.ConclusionsUsing the PARAMEDIC2 trial as a case study, we demonstrated how Bayesian group sequential designs can be constructed for phase III emergency medicine trials. The Bayesian framework enabled us to obtain efficient designs using decision criteria based on the probability of benefit or harm. It also enabled us to incorporate information from previous studies on the treatment effect via the prior distributions. We recommend the wider use of Bayesian approaches in phase III clinical trials.Trial registrationPARAMEDIC2 Trial registration ISRCTN, ISRCTN73485024. Registered 13 March 2014, http://www.isrctn.com/ISRCTN73485024

The bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated. For the two sequential designs considered in detail, the sequential probability ratio test and the triangular test, this bias is appreciable. A method of calculating an adjusted estimate with reduced bias is described, and an approximation to the standard error of the new estimate is provided. Examples of the implementation of the method are given, and its advantages and disadvantages relative to alternative approaches are discussed.

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

The authors present polynomial-based Newton algorithms for maximum likelihood estimation (MLE) of the parameters of multiple exponential signals in noise. This formulation can be used in the estimation, for example, of the directions of arrival of multiple noise-corrupted narrowband plane waves using uniform linear arrays and the frequencies of multiple noise-corrupted complex sine waves. The algorithms offer rapid convergence and exhibit the computation efficiency associated with the polynomial approach. Compact, closed-form expressions are presented for the gradients and Hessians. Various model assumptions concerning the statistics of the underlying signals are considered. Numerical simulations are presented to demonstrate the algorithms' performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>