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Dose selection is critical in pharmaceutical drug development, as it directly impacts therapeutic efficacy and patient safety of a drug. The Generalized Multiple Comparison Procedures and Modeling (MCP-Mod) approach is commonly used in Phase II trials for testing and estimation of dose-response relationships. However, its effectiveness in small sample sizes, particularly with binary endpoints, is hindered by issues like complete separation in logistic regression, leading to non-existence of estimates. Motivated by an actual clinical trial using the MCP-Mod approach, this paper introduces penalized maximum likelihood estimation (MLE) and randomization-based inference techniques to address these challenges. Randomization-based inference allows for exact finite sample inference, while population-based inference for MCP-Mod typically relies on asymptotic approximations. Simulation studies demonstrate that randomization-based tests can enhance statistical power in small to medium-sized samples while maintaining control over type-I error rates, even in the presence of time trends. Our results show that residual-based randomization tests using penalized MLEs not only improve computational efficiency but also outperform standard randomization-based methods, making them an adequate choice for dose-finding analyses within the MCP-Mod framework. Additionally, we apply these methods to pharmacometric settings, demonstrating their effectiveness in such scenarios. The results in this paper underscore the potential of randomization-based inference for the analysis of dose-finding trials, particularly in small sample contexts.

This paper addresses the problem of deciding whether the dose response relationships between subgroups and the full population in a multi-regional trial are similar to each other. Similarity is measured in terms of the maximal deviation between the dose response curves. We consider a parametric framework and develop two powerful bootstrap tests for the similarity between the dose response curves of one subgroup and the full population, and for the similarity between the dose response curves of several subgroups and the full population. We prove the validity of the tests, investigate the finite sample properties by means of a simulation study and finally illustrate the methodology in a case study.

A common problem in clinical trials is to test whether the effect of an explanatory variable on a response of interest is similar between two groups, e.g. patient or treatment groups. In this regard, similarity is defined as equivalence up to a pre-specified threshold that denotes an acceptable deviation between the two groups. This issue is typically tackled by assessing if the explanatory variable's effect on the response is similar. This assessment is based on, for example, confidence intervals of differences or a suitable distance between two parametric regression models. Typically, these approaches build on the assumption of a univariate continuous or binary outcome variable. However, multivariate outcomes, especially beyond the case of bivariate binary response, remain underexplored. This paper introduces an approach based on a generalised joint regression framework exploiting the Gaussian copula. Compared to existing methods, our approach accommodates various outcome variable scales, such as continuous, binary, categorical, and ordinal, including mixed outcomes in multi-dimensional spaces. We demonstrate the validity of this approach through a simulation study and an efficacy-toxicity case study, hence highlighting its practical relevance., Comment: 20 pages, 5 figures

Indolent cancers are characterized by long overall survival (OS) times. Therefore, powering a clinical trial to provide definitive assessment of the effects of an experimental intervention on OS in a reasonable timeframe is generally infeasible. Instead, the primary outcome in many pivotal trials is an intermediate clinical response such as progression-free survival (PFS). In several recently reported pivotal trials of interventions for indolent cancers that yielded promising results on an intermediate outcome, however, more mature data or post-approval trials showed concerning OS trends. These problematic results have prompted a keen interest in quantitative approaches for monitoring OS that can support regulatory decision-making related to the risk of an unacceptably large detrimental effect on OS. For example, the US Food and Drug Administration, the American Association for Cancer Research, and the American Statistical Association recently organized a one-day multi-stakeholder workshop entitled 'Overall Survival in Oncology Clinical Trials'. In this paper, we propose OS monitoring guidelines tailored for the setting of indolent cancers. Our pragmatic approach is modeled, in part, on the monitoring guidelines the FDA has used in cardiovascular safety trials conducted in Type 2 Diabetes Mellitus. We illustrate proposals through application to several examples informed by actual case studies., Comment: 28 pages, 5 tables

Simultaneous confidence bands (SCBs) for percentiles in linear regression are valuable tools with many applications. In this paper, we propose a novel criterion for comparing SCBs for percentiles, termed the Minimum Area Confidence Set (MACS) criterion. This criterion utilizes the area of the confidence set for the pivotal quantities, which are generated from the confidence set of the unknown parameters. Subsequently, we employ the MACS criterion to construct exact SCBs over any finite covariate intervals and to compare multiple SCBs of different forms. This approach can be used to determine the optimal SCBs. It is discovered that the area of the confidence set for the pivotal quantities of an asymmetric SCB is uniformly and can be very substantially smaller than that of the corresponding symmetric SCB. Therefore, under the MACS criterion, exact asymmetric SCBs should always be preferred. Furthermore, a new computationally efficient method is proposed to calculate the critical constants of exact SCBs for percentiles. A real data example on drug stability study is provided for illustration., Comment: 26 pages, 6 figures

There has been a growing interest in covariate adjustment in the analysis of randomized controlled trials in past years. For instance, the U.S. Food and Drug Administration recently issued guidance that emphasizes the importance of distinguishing between conditional and marginal treatment effects. Although these effects coincide in linear models, this is not typically the case in other settings, and this distinction is often overlooked in clinical trial practice. Considering these developments, this paper provides a review of when and how to utilize covariate adjustment to enhance precision in randomized controlled trials. We describe the differences between conditional and marginal estimands and stress the necessity of aligning statistical analysis methods with the chosen estimand. Additionally, we highlight the potential misalignment of current practices in estimating marginal treatment effects. Instead, we advocate for the utilization of standardization, which can improve efficiency by leveraging the information contained in baseline covariates while remaining robust to model misspecification. Finally, we present practical considerations that have arisen in our respective consultations to further clarify the advantages and limitations of covariate adjustment.

The past few years have seen an increasing number of initiatives aimed at integrating information generated outside of confirmatory randomised clinical trials (RCTs) into drug development. However, data generated non-concurrently and through observational studies can provide results that are difficult to compare with randomised trial data. Moreover, the scientific questions these data can serve to answer often remain vague. Our starting point is to use clearly defined objectives for evidence generation, which are formulated towards early discussion with health technology assessment (HTA) bodies and are additional to regulatory requirements for authorisation of a new treatment. We propose FACTIVE (Flexible Augmented Clinical Trial for Improved eVidencE generation), a new class of study designs enabling flexible augmentation of confirmatory randomised controlled trials with concurrent and close-to-real-world elements. These enabling designs facilitate estimation of certain treatment effects in the confirmatory part and other, complementary treatment effects in a concurrent real-world part. Each stakeholder should use the evidence that is relevant within their own decision-making framework. High quality data are generated under one single protocol and the use of randomisation ensures rigorous statistical inference and interpretation within and between the different parts of the experiment. Evidence for the decision-making of HTA bodies could be available earlier than is currently the case.

Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It is realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this paper, the construction of confidence sets for the level set of linear regression is considered. In particular, $1-\alpha$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding $1-\alpha$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore the method proposed in this paper is widely applicable. Examples are given to illustrate the method.

The COVID-19 pandemic continues to affect the conduct of clinical trials globally. Complications may arise from pandemic-related operational challenges such as site closures, travel limitations and interruptions to the supply chain for the investigational product, or from health-related challenges such as COVID-19 infections. Some of these complications lead to unforeseen intercurrent events in the sense that they affect either the interpretation or the existence of the measurements associated with the clinical question of interest. In this article, we demonstrate how the ICH E9(R1) Addendum on estimands and sensitivity analyses provides a rigorous basis to discuss potential pandemic-related trial disruptions and to embed these disruptions in the context of study objectives and design elements. We introduce several hypothetical estimand strategies and review various causal inference and missing data methods, as well as a statistical method that combines unbiased and possibly biased estimators for estimation. To illustrate, we describe the features of a stylized trial, and how it may have been impacted by the pandemic. This stylized trial will then be re-visited by discussing the changes to the estimand and the estimator to account for pandemic disruptions. Finally, we outline considerations for designing future trials in the context of unforeseen disruptions.

Causal inference methods are gaining increasing prominence in pharmaceutical drug development in light of the recently published addendum on estimands and sensitivity analysis in clinical trials to the E9 guideline of the International Council for Harmonisation. The E9 addendum emphasises the need to account for post-randomization or `intercurrent' events that can potentially influence the interpretation of a treatment effect estimate at a trial's conclusion. Instrumental Variables (IV) methods have been used extensively in economics, epidemiology and academic clinical studies for `causal inference', but less so in the pharmaceutical industry setting until now. In this tutorial paper we review the basic tools for causal inference, including graphical diagrams and potential outcomes, as well as several conceptual frameworks that an IV analysis can sit within. We discuss in detail how to map these approaches to the Treatment Policy, Principal Stratum and Hypothetical `estimand strategies' introduced in the E9 addendum, and provide details of their implementation using standard regression models. Specific attention is given to discussing the assumptions each estimation strategy relies on in order to be consistent, the extent to which they can be empirically tested and sensitivity analyses in which specific assumptions can be relaxed. We finish by applying the methods described to simulated data closely matching two recent pharmaceutical trials to further motivate and clarify the ideas, Comment: 29 pages, 9 Figures, 1 Table

Very recently the new pathogen severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was identified and the coronavirus disease 2019 (COVID-19) declared a pandemic by the World Health Organization. The pandemic has a number of consequences for the ongoing clinical trials in non-COVID-19 conditions. Motivated by four currently ongoing clinical trials in a variety of disease areas we illustrate the challenges faced by the pandemic and sketch out possible solutions including adaptive designs. Guidance is provided on (i) where blinded adaptations can help; (ii) how to achieve type I error rate control, if required; (iii) how to deal with potential treatment effect heterogeneity; (iv) how to utilize early readouts; and (v) how to utilize Bayesian techniques. In more detail approaches to resizing a trial affected by the pandemic are developed including considerations to stop a trial early, the use of group-sequential designs or sample size adjustment. All methods considered are implemented in a freely available R shiny app. Furthermore, regulatory and operational issues including the role of data monitoring committees are discussed.

Much of the research on multiple comparison and simultaneous inference in the past sixty years or so has been for the comparisons of several population means. Spurrier (1999) seems to be the first to study the multiple comparison of several simple linear regression lines by using simultaneous confidence bands. In this paper, the work of Liu et al. (2004) for finite comparisons of several univariate linear regression models by using simultaneous confidence bands has been extended to finite comparison of several multivariate linear regression models by using simultaneous confidence tubes. We show how simultaneous confidence tubes can be constructed to allow more informative inferences for the comparison of several multivariate linear regression models than the current approach of hypotheses testing. The methodologies are illustrated with examples.

When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al. (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. However, the FWER can be a very stringent criterion that leads to procedures with low power, and may not be appropriate in exploratory trial settings. This motivates controlling generalised error rates, particularly when the number of hypotheses tested is no longer small. We consider the generalised familywise error rate (k-FWER), which is the probability of making k or more false rejections, as well as the tail probability of the false discovery proportion (FDP), which is the probability that the proportion of false rejections is greater than some threshold. We also consider asymptotic control of the false discovery rate (FDR), which is the expectation of the FDP. In this paper, we show how to control these generalised error rates when using the graphical approach and its extensions. We demonstrate the utility of the resulting graphical procedures on three clinical trial case studies., Comment: Accepted for publication in Statistics in Medicine

Clinical trials often aim to compare a new drug with a reference treatment in terms of efficacy and/or toxicity depending on covariates such as, for example, the dose level of the drug. Equivalence of these treatments can be claimed if the difference in average outcome is below a certain threshold over the covariate range. In this paper we assume that the efficacy and toxicity of the treatments are measured as binary outcome variables and we address two problems. First, we develop a new test procedure for the assessment of equivalence of two treatments over the entire covariate range for a single binary endpoint. Our approach is based on a parametric bootstrap, which generates data under the constraint that the distance between the curves is equal to the pre-specified equivalence threshold. Second, we address equivalence for bivariate binary (correlated) outcomes by extending the previous approach for a univariate response. For this purpose we use a 2-dimensional Gumbel model for binary efficacy-toxicity responses. We investigate the operating characteristics of the proposed approaches by means of a simulation study and present a case study as an illustration.

In clinical trials the comparison of two different populations is a frequently addressed problem. Non-linear (parametric) regression models are commonly used to describe the relationship between covariates as the dose and a response variable in the two groups. In some situations it is reasonable to assume some model parameters to be the same, for instance the placebo effect or the maximum treatment effect. In this paper we develop a (parametric) bootstrap test to establish the similarity of two regression curves sharing some common parameters. We show by theoretical arguments and by means of a simulation study that the new test controls its level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters a considerable more powerful test can be constructed compared to the test which does not use this assumption. Finally, we illustrate potential applications of the new methodology by a clinical trial example.

We consider two problems that are attracting increasing attention in clinical dose finding studies. First, we assess the similarity of two non-linear regression models for two non-overlapping subgroups of patients over a restricted covariate space. To this end, we derive a confidence interval for the maximum difference between the two given models. If this confidence interval excludes the equivalence margins, similarity of dose response can be claimed. Second, we address the problem of demonstrating the similarity of two target doses for two non-overlapping subgroups, using again a confidence interval based approach. We illustrate the proposed methods with a real case study and investigate their operating characteristics (coverage probabilities, Type I error rates, power) via simulation., Comment: Keywords and Phrases: equivalence testing, multiregional trial, target dose estimation, subgroup analyses

We describe a general framework for weighted parametric multiple test procedures based on the closure principle. We utilize general weighting strategies that can reflect complex study objectives and include many procedures in the literature as special cases. The proposed weighted parametric tests bridge the gap between rejection rules using either adjusted significance levels or adjusted $p$-values. This connection is possible by allowing intersection hypotheses to be tested at level smaller than $\alpha$, which may be needed for certain study considerations. For such cases we introduce a subclass of exact $\alpha$-level parametric tests which satisfy the consonance property. When only subsets of test statistics are correlated, a new procedure is proposed to fully utilize the parametric assumptions within each subset. We illustrate the proposed weighted parametric tests using a clinical trial example., Comment: 10 pages, 1 figures

Nonlinear regression models addressing both efficacy and toxicity outcomes are increasingly used in dose-finding trials, such as in pharmaceutical drug development. However, research on related experimental design problems for corresponding active controlled trials is still scarce. In this paper we derive optimal designs to estimate efficacy and toxicity in an active controlled clinical dose finding trial when the bivariate continuous outcomes are modeled either by polynomials up to degree 2, the Michaelis- Menten model, the Emax model, or a combination thereof. We determine upper bounds on the number of different doses levels required for the optimal design and provide conditions under which the boundary points of the design space are included in the optimal design. We also provide an analytical description of the minimally supported $D$-optimal designs and show that they do not depend on the correlation between the bivariate outcomes. We illustrate the proposed methods with numerical examples and demonstrate the advantages of the $D$-optimal design for a trial, which has recently been considered in the literature., Comment: Keywords and Phrases: Active controlled trials, dose finding, optimal design, admissible design, Emax model, Equivalence theorem, Particle swarm optimization, Tchebycheff system

Phase II dose finding studies in clinical drug development are typically conducted to adequately characterize the dose response relationship of a new drug. An important decision is then on the choice of a suitable dose response function to support dose selection for the subsequent Phase III studies. In this paper we compare different approaches for model selection and model averaging using mathematical properties as well as simulations. Accordingly, we review and illustrate asymptotic properties of model selection criteria and investigate their behavior when changing the sample size but keeping the effect size constant. In a large scale simulation study we investigate how the various approaches perform in realistically chosen settings. Finally, the different methods are illustrated with a recently conducted Phase II dosefinding study in patients with chronic obstructive pulmonary disease., Comment: Keywords and Phrases: Model selection; model averaging; clinical trials; simulation study

This paper investigates the problem whether the difference between two parametric models $m_1,m_2$ describing the relation between a response variable and several covariates in two different groups is practically irrelevant, such that inference can be performed on the basis of the pooled sample. Statistical methodology is developed to test the hypotheses $H_0 : d(m_1,m_2)\geq \epsilon$ versus $H_1 : d(m_1,m_2) < \epsilon$ to demonstrate equivalence between the two regression curves $m_1,m_2$ for a pre-specified threshold $\epsilon$, where $d$ denotes a distance measuring the distance between $m_1$ and $m_2$. Our approach is based on the asymptotic properties of a suitable estimator $d(\hat{m}_1; \hat{m}_2)$ of this distance. In order to improve the approximation of the nominal level for small sample sizes a bootstrap test is developed, which addresses the specific form of the interval hypotheses. In particular, data has to be generated under the null hypothesis, which implicitly defines a manifold for the parameter vector. The results are illustrated by means of a simulation study and a data example. It is demonstrated that the new methods substantially improve currently available approaches with respect to power and approximation of the nominal level., Comment: 28 pages Keywords and Phrases: dose response studies; nonlinear regression; equivalence of curves; constrained parameter estimation; parametric bootstrap

In a recent paper Dette et al. (2014) introduced optimal design problems for dose fnding studies with an active control. These authors concentrated on regression models with normal distributed errors (with known variance) and the problem of determining optimal designs for estimating the smallest dose, which achieves the same treatment effect as the active control. This paper discusses the problem of designing active-controlled dose fnding studies from a broader perspective. In particular, we consider a general class of optimality criteria and models arising from an exponential family, which are frequently used analyzing count data. We investigate under which circumstances optimal designs for dose fnding studies including a placebo can be used to obtain optimal designs for studies with an active control. Optimal designs are constructed for several situations and the differences arising from different distributional assumptions are investigated in detail. In particular, our results are applicable for constructing optimal experimental designs to analyze active-controlled dose fnding studies with discrete data, and we illustrate the efficiency of the new optimal designs with two recent examples from our consulting projects., Comment: 24 pages 3 figures

Statistical methodology for the design and analysis of clinical Phase II dose response studies, with related software implementation, are well developed for the case of a normally distributed, homoscedastic response considered for a single timepoint in parallel group study designs. In practice, however, binary, count, or time-to-event endpoints are often used, typically measured repeatedly over time and sometimes in more complex settings like crossover study designs. In this paper we develop an overarching methodology to perform efficient multiple comparisons and modeling for dose finding, under uncertainty about the dose-response shape, using general parametric models. The framework described here is quite general and covers dose finding using generalized non-linear models, linear and non-linear mixed effects models, Cox proportional hazards (PH) models, etc. In addition to the core framework, we also develop a general purpose methodology to fit dose response data in a computationally and statistically efficient way. Several examples, using a variety of different statistical models, illustrate the breadth of applicability of the results. For the analyses we developed the R add-on package DoseFinding, which provides a convenient interface to the general approach adopted here., Comment: Now references were correctly compiled

Dose-finding studies are frequently conducted to evaluate the effect of different doses or concentration levels of a compound on a response of interest. Applications include the investigation of a new medicinal drug, a herbicide or fertilizer, a molecular entity, an environmental toxin, or an industrial chemical. In pharmaceutical drug development, dose-finding studies are of critical importance because of regulatory requirements that marketed doses are safe and provide clinically relevant efficacy. Motivated by a dose-finding study in moderate persistent asthma, we propose response-adaptive designs addressing two major challenges in dose-finding studies: uncertainty about the dose-response models and large variability in parameter estimates. To allocate new cohorts of patients in an ongoing study, we use optimal designs that are robust under model uncertainty. In addition, we use a Bayesian shrinkage approach to stabilize the parameter estimates over the successive interim analyses used in the adaptations. This approach allows us to calculate updated parameter estimates and model probabilities that can then be used to calculate the optimal design for subsequent cohorts. The resulting designs are hence robust with respect to model misspecification and additionally can efficiently adapt to the information accrued in an ongoing study. We focus on adaptive designs for estimating the minimum effective dose, although alternative optimality criteria or mixtures thereof could be used, enabling the design to address multiple objectives., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS445 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)