Confidence sets for a level set in linear regression.

Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It has been realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, ie, the set of covariate values for which the regression function exceeds a predefined level, 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 article, the construction of confidence sets for the level set of linear regression is considered. In particular, 1 - α $$ 1-\alpha $$ 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 - α $$ 1-\alpha $$ 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 article is widely applicable. Simulation studies with both linear and generalized linear models are conducted to assess the method and real examples are used to illustrate the method.

Distribution-free hyperrectangular tolerance regions for setting multivariate reference regions in laboratory medicine.

Reference regions are important in laboratory medicine to interpret the test results of patients, and usually given by tolerance regions. Tolerance regions of p ( ≥ 2 ) $$ p\;\left(\ge 2\right) $$ dimensions are highly desirable when the test results contains p $$ p $$ outcome measures. Nonparametric hyperrectangular tolerance regions are attractive in real problems due to their robustness with respect to the underlying distribution of the measurements and ease of intepretation, and methods to construct them have been recently provided by Young and Mathew [Stat Methods Med Res. 2020;29:3569-3585]. However, their validity is supported by a simulation study only. In this paper, nonparametric hyperrectangular tolerance regions are constructed by using Tukey's [Ann Math Stat. 1947;18:529-539; Ann Math Stat. 1948;19:30-39] elegant results of equivalence blocks. The validity of these new tolerance regions is proven mathematically in [Ann Math Stat. 1947;18:529-539; Ann Math Stat. 1948;19:30-39] under the only assumption that the underlying distribution of the measurements is continuous. The methodology is applied to analyze the kidney function problem considered in Young and Mathew [Stat Methods Med Res. 2020;29:3569-3585].

Covariate adjustment in randomized controlled trials: General concepts and practical considerations.

There has been a growing interest in covariate adjustment in the analysis of randomized controlled trials in past years. For instance, the US Food and Drug Administration recently issued guidance that emphasizes the importance of distinguishing between conditional and marginal treatment effects. Although these effects may sometimes coincide in the context of 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 article provides a review of when and how to use 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. In addition, we highlight the potential misalignment of commonly used methods in estimating marginal treatment effects. We hereby advocate for the use of the standardization approach, as it 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.

Testing for similarity of binary efficacy-toxicity responses.

Clinical trials often aim to compare two groups of patients for efficacy and/or toxicity depending on covariates such as dose. Examples include the comparison of populations from different geographic regions or age classes or, alternatively, of different treatment groups. Similarity of these groups can be claimed if the difference in average outcome is below a certain margin over the entire covariate range. In this article, we consider the problem of testing for similarity in the case that efficacy and toxicity are measured as binary outcome variables. We develop a new test for the assessment of similarity of two groups for a single binary endpoint. Our approach is based on estimating the maximal deviation between the curves describing the responses of the two groups, followed by a parametric bootstrap test. Further, using a two-dimensional Gumbel-type model we develop methodology to establish similarity for (correlated) binary efficacy-toxicity outcomes. We investigate the operating characteristics of the proposed methodology by means of a simulation study and present a case study as an illustration.

Optimal test procedures for multiple hypotheses controlling the familywise expected loss.

We consider the problem of testing multiple null hypotheses, where a decision to reject or retain must be made for each one and embedding incorrect decisions into a real-life context may inflict different losses. We argue that traditional methods controlling the Type I error rate may be too restrictive in this situation and that the standard familywise error rate may not be appropriate. Using a decision-theoretic approach, we define suitable loss functions for a given decision rule, where incorrect decisions can be treated unequally by assigning different loss values. Taking expectation with respect to the sampling distribution of the data allows us to control the familywise expected loss instead of the conventional familywise error rate. Different loss functions can be adopted, and we search for decision rules that satisfy certain optimality criteria within a broad class of decision rules for which the expected loss is bounded by a fixed threshold under any parameter configuration. We illustrate the methods with the problem of establishing efficacy of a new medicinal treatment in non-overlapping subgroups of patients.

Rationale for the update algorithm of the graphical approach to sequentially rejective multiple test procedures.

The graphical approach by Bretz et al. is a convenient tool to construct, visualize and perform multiple test procedures that are tailored to structured families of hypotheses while controlling the familywise error rate. A critical step is to update the transition weights following a pre-specified algorithm. In their original publication, however, the authors did not provide a detailed rationale for the update formula. This paper closes the gap and provides three alternative arguments for the update of the transition weights of the graphical approach. It is a legacy of the first author, based on an unpublished technical report from 2014, and after his untimely death reconstructed by the other two authors as a tribute to Willi Maurer's collaboration with Andy Grieve and contributions to biostatistics over many years.

Missing data imputation in clinical trials using recurrent neural network facilitated by clustering and oversampling.

In clinical practice, the composition of missing data may be complex, for example, a mixture of missing at random (MAR) and missing not at random (MNAR) assumptions. Many methods under the assumption of MAR are available. Under the assumption of MNAR, likelihood-based methods require specification of the joint distribution of the data, and the missingness mechanism has been introduced as sensitivity analysis. These classic models heavily rely on the underlying assumption, and, in many realistic scenarios, they can produce unreliable estimates. In this paper, we develop a machine learning based missing data prediction framework with the aim of handling more realistic missing data scenarios. We use an imbalanced learning technique (i.e., oversampling of minority class) to handle the MNAR data. To implement oversampling in longitudinal continuous variable, we first perform clustering via k$k$ -mean trajectories. And use the recurrent neural network (RNN) to model the longitudinal data. Further, we apply bootstrap aggregating to improve the accuracy of prediction and also to consider the uncertainty of a single prediction. We evaluate the proposed method using simulated data. The prediction result is evaluated at the individual patient level and the overall population level. We demonstrate the powerful predictive capability of RNN for longitudinal data and its flexibility for nonlinear modeling. Overall, the proposed method provides an accurate individual prediction for both MAR and MNAR data and reduce the bias of missing data in treatment effect estimation when compared to standard methods and classic models. Finally, we implement the proposed method in a real dataset from an antidepressant clinical trial. In summary, this paper offers an opportunity to encourage the integration of machine learning strategies for handling of missing data in the analysis of randomized clinical trials.

Simultaneous confidence tubes for comparing several multivariate linear regression models.

Much of the research on multiple comparison and simultaneous inference in the past 60 years or so has been for the comparisons of several population means. Spurrier seems to have been the first to investigate multiple comparisons of several simple linear regression lines using simultaneous confidence bands. In this paper, we extend the work of Liu et al. for finite comparisons of several univariate linear regression models using simultaneous confidence bands to finite comparisons of several multivariate linear regression models 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 methods are illustrated with examples.

[Multiple Endpoints in Oncology Clinical Trials].

Like most complex(or multifactorial)diseases, cancer results not from a single factor, but rather from the interaction of multiple genes and environmental factors. Thus patients can experience different signs and symptoms that reflect more than one consequence of suffering the disease. When evaluating the effects of new treatments in cancer clinical trials, the multidimensional assessment using multiple outcomes to measure improvements in the patients' signs and symptoms associated with treatments would be preferred. Most cancer clinical trials use more than one clinical outcome as multiple primary, or primary and(key)secondary endpoints, such as overall survival, endpoints based on tumor assessments(e.g., disease-free survival, event-free survival, objective response rate, time to progression, progression-free survival), and endpoints involving symptom assessment. Utilizing multiple endpoints may provide the opportunity for characterizing the intervention's multidimensional effects, but also creates challenges, specifically controlling the Type Ⅰ and/or Type Ⅱ errors in hypotheses testing and trial designs associated with multiple endpoints. In this article, we review issues in design, monitoring, analysis and reporting of clinical trials with multiple endpoints, with illustrating examples in oncology disease settings. We outline several methods for controlling the Type Ⅰ error associated multiple tests, which have been commonly used in clinical trials. We also briefly discuss issues in interim analyses and group sequential designs for clinical trials with multiple endpoints.

[What Is an Estimand?-The Most Important Clinical Question in Oncology Trials].

Patients can experience different disease journeys and clinical trials that investigate the benefit of oncology treatments need to account for this diversity. When defining the treatment effect of interest in a trial, researchers thus have to account for events occurring after treatment initiation, such as the start of a new therapy, before observing the variable of interest. We review the estimand framework recently introduced by the International Council for Harmonisation(ICH, 2019)to structure discussions on the relationship between patient journeys and the treatment effect of interest in oncology trials. This framework is expected to improve coherence between trial objectives, design, analysis, reporting and interpretation, as illustrated in this article by examples in oncology disease settings.

Connecting Instrumental Variable methods for causal inference to the Estimand Framework.

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 article 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.

Comparison of normal distribution-based and nonparametric decision limits on the GH-2000 score for detecting growth hormone misuse (doping) in sport.

This paper is motivated by the GH-2000 biomarker test, though the discussion is applicable to other diagnostic tests. The GH-2000 biomarker test has been developed as a powerful technique to detect growth hormone misuse by athletes, based on the GH-2000 score. Decision limits on the GH-2000 score have been developed and incorporated into the guidelines of the World Anti-Doping Agency (WADA). These decision limits are constructed, however, under the assumption that the GH-2000 score follows a normal distribution. As it is difficult to affirm the normality of a distribution based on a finite sample, nonparametric decision limits, readily available in the statistical literature, are viable alternatives. In this paper, we compare the normal distribution-based and nonparametric decision limits. We show that the decision limit based on the normal distribution may deviate significantly from the nominal confidence level 1-α or nominal FPR γ when the distribution of the GH-2000 score departs only slightly from the normal distribution. While a nonparametric decision limit does not assume any specific distribution of the GH-2000 score and always guarantees the nominal confidence level and FPR, it requires a much larger sample size than the normal distribution-based decision limit. Due to the stringent FPR of the GH-2000 biomarker test used by WADA, the sample sizes currently available are much too small, and it will take many years of testing to have the minimum sample size required, in order to use the nonparametric decision limits. Large sample theory about the normal distribution-based and nonparametric decision limits is also developed in this paper to help understanding their behaviours when the sample size is large.

Equivalence of regression curves sharing common parameters.

In clinical trials, the comparison of two different populations is a common problem. Nonlinear (parametric) regression models are commonly used to describe the relationship between covariates, such as concentration or 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 significance level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters, a considerably more powerful test can be constructed compared with the test that does not use this assumption. Finally, we illustrate the potential applications of the new methodology by a clinical trial example.

Graphical approaches for the control of generalized error rates.

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 generalized error rates, particularly when the number of hypotheses tested is no longer small. We consider the generalized 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, which is the expectation of the FDP. In this article, we show how to control these generalized 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.

Symmetric graphs for equally weighted tests, with application to the Hochberg procedure.

The graphical approach to multiple testing provides a convenient tool for designing, visualizing, and performing multiplicity adjustments in confirmatory clinical trials while controlling the familywise error rate. It assigns a set of weights to each intersection null hypothesis within the closed test framework. These weights form the basis for intersection tests using weighted individual p-values, such as the weighted Bonferroni test. In this paper, we extend the graphical approach to intersection tests that assume equal weights for the elementary null hypotheses associated with any intersection hypothesis, including the Hochberg procedure as well as omnibus tests such as Fisher's combination, O'Brien's, and F tests. More specifically, we introduce symmetric graphs that generate sets of equal weights so that the aforementioned tests can be applied with the graphical approach. In addition, we visualize the Hochberg and the truncated Hochberg procedures in serial and parallel gatekeeping settings using symmetric component graphs. We illustrate the method with two clinical trial examples.

Assessing the similarity of dose response and target doses in two non-overlapping subgroups.

We consider 2 problems of increasing importance in clinical dose finding studies. First, we assess the similarity of 2 non-linear regression models for 2 non-overlapping subgroups of patients over a restricted covariate space. To this end, we derive a confidence interval for the maximum difference between the 2 given models. If this confidence interval excludes the pre-specified equivalence margin, similarity of dose response can be claimed. Second, we address the problem of demonstrating the similarity of 2 target doses for 2 non-overlapping subgroups, using again an approach based on a confidence interval. We illustrate the proposed methods with a real case study and investigate their operating characteristics (coverage probabilities, Type I error rates, power) via simulation.

Flexible alpha allocation strategies for confirmatory adaptive enrichment clinical trials with a prespecified subgroup.

Adaptive enrichment designs have recently received considerable attention as they have the potential to make drug development process for personalized medicine more efficient. Several statistical approaches have been proposed so far in the literature and the operating characteristics of these approaches are extensively investigated using simulation studies. In this paper, we improve on existing adaptive enrichment designs by assigning unequal weights to the significance levels associated with the hypotheses of the overall population and a prespecified subgroup. More specifically, we focus on the standard combination test, a modified combination test, the marginal combination test, and the partial conditional error rate approach and explore the operating characteristics of these approaches by a simulation study. We show that these approaches can lead to power gains, compared to existing approaches, if the weights are chosen carefully.

Estimands in clinical trials - broadening the perspective.

Defining the scientific questions of interest in a clinical trial is crucial to align its planning, design, conduct, analysis, and interpretation. However, practical experience shows that oftentimes specific choices in the statistical analysis blur the scientific question either in part or even completely, resulting in misalignment between trial objectives, conduct, analysis, and confusion in interpretation. The need for more clarity was highlighted by the Steering Committee of the International Council for Harmonization (ICH) in 2014, which endorsed a Concept Paper with the goal of developing a new regulatory guidance, suggested to be an addendum to ICH guideline E9. Triggered by these developments, we elaborate in this paper what the relevant questions in drug development are and how they fit with the current practice of intention-to-treat analyses. To this end, we consider the perspectives of patients, physicians, regulators, and payers. We argue that despite the different backgrounds and motivations of the various stakeholders, they all have similar interests in what the clinical trial estimands should be. Broadly, these can be classified into estimands addressing (a) lack of adherence to treatment due to different reasons and (b) efficacy and safety profiles when patients, in fact, are able to adhere to the treatment for its intended duration. We conclude that disentangling adherence to treatment and the efficacy and safety of treatment in patients that adhere leads to a transparent and clinical meaningful assessment of treatment risks and benefits. We touch upon statistical considerations and offer a discussion of additional implications. Copyright © 2016 John Wiley & Sons, Ltd.