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A random delayed treatment effect is expected in a confirmatory clinical trial for an immunotherapy due to the individual heterogeneity of physiological conditions. For this reason, the delay time will be assumed to follow a continuous distribution that is difficult to estimate accurately based on the early-phase data, which hinders the specification of the most powerful weighted log-rank test. Therefore, we propose a simulation-based maximum duration design with a robustly powerful Maxcombo test for a group sequential trial for the immunotherapy with the random delayed treatment effect. The design obtains the group sequential boundaries by a simulation procedure and determines the required maximum sample size using a one-dimensional search in which another simulation procedure is used to calculate empirical power. The simulation researches proved the accuracy of the group sequential boundaries and their robustness against the misspecified maximum sample sizes for large samples and revealed their moderate sensitivity against the misspecified survival distributions under the null hypothesis of no difference. The studies investigated whether the type I error rate would inflate under the "inferior" null hypothesis and evaluated the robustness against different distributions of the delay time in terms of the empirical power among the Maxcombo tests and component weighted log-rank tests.

Definitive clinical trials are resource intensive, often requiring a large number of participants over several years. One approach to improving the efficiency of clinical trials is to incorporate historical information into the primary trial analysis. This approach has tremendous potential in the areas of pediatric or rare disease trials, where achieving reasonable power is difficult. In this manuscript, we introduce a novel Bayesian group-sequential trial design based on Multisource Exchangeability Models, which allows for dynamic borrowing of historical information at the interim analyses. Our approach achieves synergy between group sequential and adaptive borrowing methodology to attain improved power and reduced sample size. We explore the frequentist operating characteristics of our design through simulation and compare our method to a traditional group-sequential design. Our method achieves earlier stopping of the primary study while increasing power under the alternative hypothesis but has a potential for type I error inflation under some null scenarios. We discuss the issues of decision boundary determination, power and sample size calculations, and the issue of information accrual. We present our method for a continuous and binary outcome, as well as in a linear regression setting.

Estimation of the dose-response curve for efficacy and subsequent selection of an appropriate dose in phase II trials are important processes in drug development. Various methods have been investigated to estimate dose-response curves. Generally, these methods are used with equal allocation of subjects for simplicity; nevertheless, they may not fully optimize performance metrics because of nonoptimal allocation. Optimal allocation methods, which include adaptive allocation methods, have been proposed to overcome the limitations of equal allocation. However, they rely on asymptotics, and thus sometimes cannot efficiently optimize the performance metric with the sample size in an actual clinical trial. The purpose of this study is to construct an adaptive allocation rule that directly optimizes a performance metric, such as power, accuracy of model selection, accuracy of the estimated target dose, or mean absolute error over the estimated dose-response curve. We demonstrate that deep reinforcement learning with an appropriately defined state and reward can be used to construct such an adaptive allocation rule. The simulation study shows that the proposed method can successfully improve the performance metric to be optimized when compared with the equal allocation, D-optimal, and TD-optimal methods. In particular, when the mean absolute error was set to the metric to be optimized, it is possible to construct a rule that is superior for many metrics.

In planning randomized clinical trials (RCTs) for diseases such as Alzheimer's disease (AD), researchers frequently rely on the use of existing data obtained from only two time points to estimate sample size via the subtraction of baseline from follow-up measurements in each subject. However, the inadequacy of this method has not been reported. The aim of this study is to discuss the limitation of sample size estimation based on the subtraction of available data from only two time points for RCTs. Mathematical equations are derived to demonstrate the condition under which the obtained data pairs with variable time intervals could be used to adequately estimate sample size. The MRI-based hippocampal volume measurements from the Alzheimer's Disease Neuroimaging Initiative (ADNI) and Monte Carlo simulations (MCS) were used to illustrate the existing bias and variability of estimates. MCS results support the theoretically derived condition under which the subtraction approach may work. MCS also show the systematically under- or over-estimated sample sizes by up to 32.27 % bias. Not used properly, such subtraction approach outputs the same sample size regardless of trial durations partly due to the way measurement errors are handled. Estimating sample size by subtracting two measurements should be treated with caution. Such estimates can be biased, the magnitude of which depends on the planned RCT duration. To estimate sample sizes, we recommend using more than two measurements and more comprehensive approaches such as linear mixed effect models.

Globalized drug development studies, such as multiregional clinical trials (MRCTs), have attracted much attention due to their ability to expedite drug development and shorten the time lag of drug release. While observing the overall effect of a new drug, the region-specific effects to support drug registration in constituent regions can also be evaluated. Several challenges arise in conducting MRCTs, such as the heterogeneity in the variability of the primary endpoint across regions. However, most of the existing statistical methods assume a common variability, which may not be valid in practice due to differences across regions (eg, diversities in ethnicity or disparities in medical culture/practice). We present a statistical method for the design and evaluation of MRCTs to consider the heterogeneous variability across regions. We assessed the overall sample size requirement and addressed the region-specific sample size determination to establish the consistency of treatment effects between the specific region and the entire group. We demonstrate the proposed approach with numerical examples.

Epidemiologic studies conducted to quantify the impact of radiation dose d on an outcome typically model the hazard ratio (HR) for association using a linear term, HR ( d ) = 1 + β d , via a linear excess relative risk (ERR) model, based on biological considerations. To study associations of risk of a second cancer with radiation treatment for a first cancer, several nested case-control designs to estimate β have been proposed that use refined doses received by different locations in the organ of interest. Here we first evaluated the small sample bias in maximum likelihood estimates of β for the linear ERR model using location-specific radiation doses in simulations. As we found substantial upward bias for studies of realistic sample sizes (more than 50% relative bias for studies with 75 cases), we also proposed and investigated several approaches to correct this bias. We studied first and second order jackknife bias corrections and we derived a modified set of score functions under retrospective case-control sampling, from which we directly obtained bias-corrected estimates. In simulations based on doses from a study of stomach cancer among testicular cancer survivors and synthetically generated data, neither the first nor second order jackknife bias correction performed well. Estimates based on the modified score equations corrected the bias much better, albeit not completely, and were numerically much more stable.

Multiple public health and medical research studies have applied matched-pair cluster randomization design to the evaluation of the intervention and/or prevention effects. One of the most common and severe problems faced by researchers when conducting cluster randomized trials (CRTs) is incomplete observations, which are associated with various reasons causing the individuals to discontinue participating in the trials. Although statistical methods to remedy the problems of missing data have already been proposed, there are still methodological gaps in research concerning the determination of sample size in matched-pair CRTs with incomplete binary outcomes. One conventional method for adjusting for missing data in the sample size determination is to divide the sample size under complete data by the expected follow-up rate. However, such crude adjustment ignores the impact of the structure and strength of correlations regarding both outcome data and missing data mechanism. This article provides a closed-form sample size formula for matched-pair CRTs with incomplete binary outcomes, which appropriately accounts for different missing patterns and magnitudes as well as the effects of matching and clustering on the outcome and missing data. The generalized estimating equation (GEE) approach treats incomplete observations as missing data in a marginal logistic regression model, which flexibly accommodates various types of intraclass correlation, missing patterns, and missing proportions. In the presence of missing data, the proposed GEE sample size method provides higher accuracy as compared with the conventional method. The performance of the proposed method is assessed by simulation studies. This article also illustrates how the proposed method can be used to design a real-world matched-pair CRT to examine the effect of a team-based approach on controlling blood pressure (BP).

Adjusting for baseline values and covariates is a recurrent statistical problem in medical science. In particular, variance heteroscedasticity is non-negligible in experimental designs and ignoring it might result in false conclusions. Approximate inference methods are developed to test null hypotheses formulated in terms of adjusted treatment effects and regression parameters in general analysis of covariance designs with arbitrary numbers of factors. Variance homoscedasticity is not assumed. The distributions of the test statistics are approximated using Box-type approximation methods. Extensive simulation studies show that the procedures are particularly suitable when sample sizes are rather small. A real data set illustrates the application of the methods.

In sequential analysis, hypothesis testing is performed repeatedly in a prospective manner as data accrue over time to quickly arrive at an accurate conclusion or decision. In this tutorial paper, detailed explanations are given for both designing and operating sequential testing. We describe the calculation of exact thresholds for stopping or signaling, statistical power, expected time to signal, and expected sample sizes for sequential analysis with Poisson and binary type data. The calculations are run using the package Sequential, constructed in R language. Real data examples are inspired on clinical trials practice, such as the current efforts to develop treatments to face the COVID‐19 pandemic, and the comparison of treatments of osteoporosis. In addition, we mimic the monitoring of adverse events following influenza vaccination and Pediarix vaccination.

To extend marketing exclusivity, a drug manufacture is required by the Food and Drug Administration to conduct an interpretable study of approved drugs that may benefit pediatric patients. In return for conducting an interpretable study, the drug manufacturer is rewarded with an additional 6 months of marketing exclusivity. If the study design uses a fixed sample size and nothing is changed during the study, it is relatively easy to determine whether an interpretable study was conducted. However, if the sample size can be changed based on an unblinded interim analysis, it is difficult to answer the question of whether the study was interpretable. We investigate the optimal design from the manufacturers perspective and find the strategy for sample size adaptation that optimizes the utility including the chance of an interpretable study.

Prediction models often yield inaccurate predictions for new individuals. Large data sets from pooled studies or electronic healthcare records may alleviate this with an increased sample size and variability in sample characteristics. However, existing strategies for prediction model development generally do not account for heterogeneity in predictor-outcome associations between different settings and populations. This limits the generalizability of developed models (even from large, combined, clustered data sets) and necessitates local revisions. We aim to develop methodology for producing prediction models that require less tailoring to different settings and populations. We adopt internal-external cross-validation to assess and reduce heterogeneity in models' predictive performance during the development. We propose a predictor selection algorithm that optimizes the (weighted) average performance while minimizing its variability across the hold-out clusters (or studies). Predictors are added iteratively until the estimated generalizability is optimized. We illustrate this by developing a model for predicting the risk of atrial fibrillation and updating an existing one for diagnosing deep vein thrombosis, using individual participant data from 20 cohorts (N = 10 873) and 11 diagnostic studies (N = 10 014), respectively. Meta-analysis of calibration and discrimination performance in each hold-out cluster shows that trade-offs between average and heterogeneity of performance occurred. Our methodology enables the assessment of heterogeneity of prediction model performance during model development in multiple or clustered data sets, thereby informing researchers on predictor selection to improve the generalizability to different settings and populations, and reduce the need for model tailoring. Our methodology has been implemented in the R package metamisc.

In cancer studies, it is important to understand disease heterogeneity among patients so that precision medicine can particularly target high-risk patients at the right time. Many feature variables such as demographic variables and biomarkers, combined with a patient’s survival outcome, can be used to infer such latent heterogeneity. In this work, we propose a mixture model to model each patient’s latent survival pattern, where the mixing probabilities for latent groups are modelled through a multinomial distribution. The Bayesian information criterion (BIC) is used for selecting the number of latent groups. Furthermore, we incorporate variable selection with the adaptive lasso into inference so that only a few feature variables will be selected to characterize the latent heterogeneity. We show that our adaptive lasso estimator has oracle properties when the number of parameters diverges with the sample size. The finite sample performance is evaluated by the simulation study, and the proposed method is illustrated by two datasets.

A major emphasis in precision medicine is to optimally treat subgroups of patients who may benefit from certain therapeutic agents. And as such, enormous resources and innovative clinical trials designs in oncology are devoted to identifying predictive biomarkers. Predictive biomarkers are ones that will identify patients that are more likely to respond to specific therapies and they are usually discovered through retrospective analysis from large randomized phase II or phase III trials. One important design to consider is the stratified biomarker design, where patients will have their specimens obtained at baseline and the biomarker status will be assessed prior to random assignment. Regardless of their biomarker status, patients will be randomized to either an experimental arm or the standard of care arm. The stratified biomarker design can be used to test for a treatment-biomarker interaction in predicting a time-to event outcome. Many biomarkers, however, are derived from tissues from patients, and their levels may be heterogeneous. As a result, biomarker levels may be measured with error and this would have an adverse impact on the power of a stratified biomarker clinical trial. We present a trial design and an analysis framework for the stratified biomarker design. We show that the naïve test is biased and provide bias-corrected estimators for computing the sample size and the 95% confidence interval when testing for a treatment-biomarker interaction in predicting a time to event outcome. We propose a sample size formula that adjusts for misclassification and apply it in the design of a phase III clinical trial in renal cancer.

The interpretation of randomized clinical trial results is often complicated by intercurrent events. For instance, rescue medication is sometimes given to patients in response to worsening of their disease, either in addition to the randomized treatment or in its place. The use of such medication complicates the interpretation of the intention-to-treat analysis. In view of this, we propose a novel estimand defined as the intention-to-treat effect that would have been observed, had patients on the active arm been switched to rescue medication if and only if they would have been switched when randomized to control. This enables us to disentangle the treatment effect from the effect of rescue medication on a patient's outcome, while tempering the strong extrapolations that are typically needed when inferring what the intention-to-treat effect would have been in the absence of rescue medication. We propose a novel inverse probability weighting method for estimating this effect in settings where the decision to initiate rescue medication is made at one prespecified time point. This estimator relies on specific untestable assumptions, in view of which we propose a sensitivity analysis. We use the method for the analysis of a clinical trial conducted by Janssen Pharmaceuticals, in which patients with type 2 diabetes mellitus can switch to rescue medication for ethical reasons. Monte Carlo simulations confirm that the proposed estimator is unbiased in moderate sample sizes.

In the pharmaceutical industry, in vivo animal experiments are conducted to test the effects of novel preclinical drug compounds. Well-planned animal studies involve a sample size and statistical power analysis to provide a basis for the number of animals allocated into comparator arms of a future study. These calculations require approximate values for the parameters of a statistical model that will be applied to the future data and used to test for differences via statistical hypotheses. If the prestudy parameter estimates are nearly correct, the power analysis guarantees that a difference will be detected from the study data, up to a prespecified probability. Traditional power computations, however, are not calculated with reproducibility in mind. In this work, the issue of reproducibility in drug discovery is tackled from the point of view that study-to-study variability is not included in a typical sample size and power analysis. Three proposed methods that yield a reproducible mean-comparison analysis are derived and compared.

In 2019 we published a pair of articles in Statistics in Medicine that describe how to calculate the minimum sample size for developing a multivariable prediction model with a continuous outcome, or with a binary or time-to-event outcome. As for any sample size calculation, the approach requires the user to specify anticipated values for key parameters. In particular, for a prediction model with a binary outcome, the outcome proportion and a conservative estimate for the overall fit of the developed model as measured by the Cox-Snell R2 (proportion of variance explained) must be specified. This proposal raises the question of how to identify a plausible value for R2 in advance of model development. Our articles suggest researchers should identify R2 from closely related models already published in their field. In this letter, we present details on how to derive R2 using the reported C statistic (AUROC) for such existing prediction models with a binary outcome. The C statistic is commonly reported, and so our approach allows researchers to obtain R2 for subsequent sample size calculations for new models. Stata and R code is provided, and a small simulation study.

In this article, we review and evaluate a number of methods used in the design and analysis of small three-arm parallel cluster randomized trials. We conduct a simulation-based study to evaluate restricted randomization methods including covariate-constrained randomization and a novel method for matched-group cluster randomization. We also evaluate the appropriate modelling of the data and small sample inferential methods for a variety of treatment effects relevant to three-arm trials. Our results indicate that small-sample corrections are required for high (0.05) but not low (0.001) values of the intraclass correlation coefficient and their performance can depend on trial design, number of clusters, and the nature of the hypothesis being tested. The Satterthwaite correction generally performed best at an ICC of 0.05 with a nominal type I error rate for single-period trials, and in trials with repeated measures type I error rates were between 0.04 and 0.06. Restricted randomization methods produce little benefit in trials with repeated measures but in trials with single post-intervention design can provide relatively large gains in power when compared to the most unbalanced possible allocations. Matched-group randomization improves power but is not as effective as covariate-constrained randomization. For model-based analysis, adjusting for fewer covariates than were used in a restricted randomization process under any design can produce non-nominal type I error rates and reductions in power. Where comparisons to two-arm cluster trials are possible, the performance of the methods is qualitatively very similar.

There are inconsistencies between the formulas for the variance of standardized mean difference (SMD) in the Cochrane Handbook for Systematic Reviews and the variance reported in other sources. Instead of the variance appropriate for the SMD of a crossover experiment, the Cochrane Handbook uses the variance appropriate for a pre-test post-test experiment. This means that if there is a non-negligible time period effect, the formula reported by the Handbook will underestimate both the effect size and its variance. In addition, the formula for the standard error of SMD reported in the Cochrane Handbook (in section 23.2.7.2) is inconsistent with the variance derived from the variance of the related t-test. Even if the period effect is negligible, the Cochrane Handbook formula is biased toward underestimates. The difference between the estimates from the two formulas will be small if either the correlation between the repeated measures, or the magnitude of the SMD estimate, is small, or if the sample size is large. However, it can be can be quite substantial in other circumstances.

Observational studies often benefit from an abundance of observational units. This can lead to studies that -- while challenged by issues of internal validity -- have inferences derived from sample sizes substantially larger than randomized controlled trials. But is the information provided by an observational unit best used in the analysis phase? We propose the use of `pilot design,' in which observations are expended in the design phase of the study, and the post-treatment information from these observations is used to improve study design. In modern observational studies, which are data rich but control poor, pilot designs can be used to gain information about the structure of post-treatment variation. This information can then be used to improve instrumental variable designs, propensity score matching, doubly-robust estimation, and other observational study designs. We illustrate one version of a pilot design, which aims to reduce within-set heterogeneity and improve performance in sensitivity analyses. This version of a pilot design expends observational units during the design phase to fit a prognostic model, avoiding concerns of overfitting. Additionally, it enables the construction of `Assignment-Control (AC) plots,' which visualize the relationship between propensity and prognostic scores. We first show some examples of these plots, then we demonstrate in a simulation setting how this alternative use of the observations can lead to gains in terms of both treatment effect estimation and sensitivity analyses of unobserved confounding., Comment: 21 pages, 7 figures

Standard network meta-analysis and indirect comparisons combine aggregate data from multiple studies on treatments of interest, assuming that any factors that interact with treatment effects (effect modifiers) are balanced across populations. Population adjustment methods such as multilevel network meta-regression (ML-NMR), matching-adjusted indirect comparison (MAIC), and simulated treatment comparison (STC) relax this assumption using individual patient data from one or more studies, and are becoming increasingly prevalent in health technology appraisals and the applied literature.Motivated by an applied example and two recent reviews of applications, we undertook an extensive simulation study to assess the performance of these methods in a range of scenarios under various failures of assumptions. We investigated the impact of varying sample size, missing effect modifiers, strength of effect modification and validity of the shared effect modifier assumption, validity of extrapolation and varying between-study overlap, and different covariate distributions and correlations.ML-NMR and STC performed similarly, eliminating bias when the requisite assumptions were met. Serious concerns are raised for MAIC, which performed poorly in nearly all simulation scenarios and may even increase bias compared to standard indirect comparisons. All methods incur bias when an effect modifier is missing, highlighting the necessity of careful selection of potential effect modifiers prior to analysis. When all effect modifiers are included, ML-NMR and STC are robust techniques for population adjustment. ML-NMR offers additional advantages over MAIC and STC, including extending to larger treatment networks and producing estimates in any target population, making this an attractive choice in a variety of scenarios.

In a series of articles, Gart and Nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference, relative risk, and odds ratio effect metrics. However, the stratified score methods and their properties are not well understood. We rederive the efficient score tests, which reveals their theoretical relationship with the contrast-based score tests, and provides a basis for adapting the method by using other weighting schemes. The inverse variance weight is optimal for a common treatment effect in large samples. We explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders, and provide guidance on the choice of weights in the analysis of rare events. The score method is recommended for studies with a small number of moderate or large sized strata. A general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority, noninferiority and equivalence clinical trials, or case-control studies. We also describe a nearly exact procedure that underestimates the exact power, but the degree of underestimation can be controlled to a negligible level. The proposed methods are illustrated by numerical examples.

While the traditional clinical trial design lays emphasis on testing the treatment effect between randomly assigned groups, it ignores the role of patient preference for a particular treatment in the trial. Yet, for healthcare providers who seek to optimize the patient-centered treatment strategy, the evaluation of a patient's psychology toward each treatment could be a key consideration. The two-stage randomized trial design allows researchers to test patient's preference and selection effects, in addition to the treatment effect. The current methodology for the two-stage design is limited to continuous and binary outcomes; this article extends the model to include count outcomes. The test statistics for preference, selection, and treatment effects are derived. Closed-form sample size formulae are presented for each effect. Simulations are presented to demonstrate the properties of the unstratified and stratified designs. Finally, we apply methods to the use of antimicrobials at the end of life to demonstrate the applicability of the methods.

Cluster randomized trials (CRTs) refer to experiments with randomization carried out at the cluster or the group level. While numerous statistical methods have been developed for the design and analysis of CRTs, most of the existing methods focused on testing the overall treatment effect across the population characteristics, with few discussions on the differential treatment effect among subpopulations. In addition, the sample size and power requirements for detecting differential treatment effect in CRTs remain unclear, but are helpful for studies planned with such an objective. In this article, we develop a new sample size formula for detecting treatment effect heterogeneity in two-level CRTs for continuous outcomes, continuous or binary covariates measured at cluster or individual level. We also investigate the roles of two intraclass correlation coefficients (ICCs): the adjusted ICC for the outcome of interest and the marginal ICC for the covariate of interest. We further derive a closed-form design effect formula to facilitate the application of the proposed method, and provide extensions to accommodate multiple covariates. Extensive simulations are carried out to validate the proposed formula in finite samples. We find that the empirical power agrees well with the prediction across a range of parameter constellations, when data are analyzed by a linear mixed effects model with a treatment-by-covariate interaction. Finally, we use data from the HF-ACTION study to illustrate the proposed sample size procedure for detecting heterogeneous treatment effects.

Cluster randomized designs are frequently employed in pragmatic clinical trials which test interventions in the full spectrum of everyday clinical settings in order to maximize applicability and generalizability. In this study, we propose to directly incorporate pragmatic features into power analysis for cluster randomized trials with count outcomes. The pragmatic features considered include arbitrary randomization ratio, overdispersion, random variability in cluster size, and unequal lengths of follow-up over which the count outcome is measured. The proposed method is developed based on generalized estimating equation (GEE) and it is advantageous in that the sample size formula retains a closed form, facilitating its implementation in pragmatic trials. We theoretically explore the impact of various pragmatic features on sample size requirements. An efficient Jackknife algorithm is presented to address the problem of underestimated variance by the GEE sandwich estimator when the number of clusters is small. We assess the performance of the proposed sample size method through extensive simulation and an application example to a real clinical trial is presented.

Three-level cluster randomized trials (CRTs) are increasingly used in implementation science, where 2fold-nested-correlated data arise. For example, interventions are randomly assigned to practices, and providers within the same practice who provide care to participants are trained with the assigned intervention. Teerenstra et al proposed a nested exchangeable correlation structure that accounts for two levels of clustering within the generalized estimating equations (GEE) approach. In this article, we utilize GEE models to test the treatment effect in a two-group comparison for continuous, binary, or count data in three-level CRTs. Given the nested exchangeable correlation structure, we derive the asymptotic variances of the estimator of the treatment effect for different types of outcomes. When the number of clusters is small, researchers have proposed bias-corrected sandwich estimators to improve performance in two-level CRTs. We extend the variances of two bias-corrected sandwich estimators to three-level CRTs. The equal provider and practice sizes were assumed to calculate number of practices for simplicity. However, they are not guaranteed in practice. Relative efficiency (RE) is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal provider and practice sizes. The expressions of REs are obtained from both asymptotic variance estimation and bias-corrected sandwich estimators. Their performances are evaluated for different scenarios of provider and practice size distributions through simulation studies. Finally, a percentage increase in the number of practices is proposed due to efficiency loss from unequal provider and/or practice sizes.

Longitudinal biomarker data are often collected in studies, providing important information regarding the probability of an outcome of interest occurring at a future time. With many new and evolving technologies for biomarker discovery, the number of biomarker measurements available for analysis of disease progression has increased dramatically. A large amount of data provides a more complete picture of a patient's disease progression, potentially allowing us to make more accurate and reliable predictions, but the magnitude of available data introduces challenges to most statistical analysts. Existing approaches suffer immensely from the curse of dimensionality. In this article, we propose methods for making dynamic risk predictions using repeatedly measured biomarkers of a large dimension, including cases when the number of biomarkers is close to the sample size. The proposed methods are computationally simple, yet sufficiently flexible to capture complex relationships between longitudinal biomarkers and potentially censored events times. The proposed approaches are evaluated by extensive simulation studies and are further illustrated by an application to a data set from the Nephrotic Syndrome Study Network.

Individual randomized trials (IRTs) and cluster randomized trials (CRTs) with binary outcomes arise in a variety of settings and are often analyzed by logistic regression (fitted using generalized estimating equations for CRTs). The effect of stratification on the required sample size is less well understood for trials with binary outcomes than for continuous outcomes. We propose easy-to-use methods for sample size estimation for stratified IRTs and CRTs and demonstrate the use of these methods for a tuberculosis prevention CRT currently being planned. For both IRTs and CRTs, we also identify the ratio of the sample size for a stratified trial versus a comparably-powered unstratified trial, allowing investigators to evaluate how stratification will affect the required sample size when planning a trial. For CRTs, these can be used when the investigator has estimates of the within-stratum intra-cluster correlation coefficients (ICCs) or by assuming a common within-stratum ICC. Using these methods, we describe scenarios where stratification may have a practically important impact on the required sample size. We find that in the two-stratum case, for both IRTs and for CRTs with very small cluster sizes, there are unlikely to be plausible scenarios in which an important sample size reduction is achieved when the overall probability of a subject experiencing the event of interest is low. When the probability of events is not small, or when cluster sizes are large, however, there are scenarios where practically important reductions in sample size result from stratification.

Conventional seamless phase 2/3 design with fixed sample size determination (SSD) has gained its popularity in oncology drug development due to attractive features such as significantly shortening the development timeline, minimizing sample size, as well as early decision making. However, this design is not immune to inaccurate treatment effect assumption when only limited efficacy data are available at study design stage. We propose an innovative seamless phase 2/3 study design with flexible SSD for oncology trials, in which the trial is designed under a distribution of treatment effect instead of one single assumption due to huge uncertainty of treatment effect at design stage and the sample size for end of phase 3 analysis is not predetermined at design stage, but rather dynamically determined based on observed treatment effect at phase 2 portion. Some practical sample size determination rules for end of phase 3 analysis will be discussed. The proposed design can lead to reduced sample size or/and improved power compared with conventional seamless phase 2/3 design with fixed SSD. This innovative study design can be especially useful for programs with aggressive development strategy to expedite the process in delivering efficacious treatment to patients.

In standard clinical trial designs, the required sample size is fixed in the planning stage based on initial parameter assumptions. It is intuitive that the correct choice of the sample size is of major importance for an ethical justification of the trial. The required parameter assumptions should be based on previously published results from the literature. In clinical practice, however, historical data often do not exist or show highly variable results. Adaptive group sequential designs allow a sample size recalculation after a planned unblinded interim analysis in order to adjust the sample size during the ongoing trial. So far, there exist no unique standards to assess the performance of sample size recalculation rules. Single performance criteria commonly reported are given by the power and the average sample size; the variability of the recalculated sample size and the conditional power distribution are usually ignored. Therefore, the need for an adequate performance score combining these relevant performance criteria is evident. To judge the performance of an adaptive design, there exist two possible perspectives, which might also be combined: Either the global performance of the design can be addressed, which averages over all possible interim results, or the conditional performance is addressed, which focuses on the remaining performance conditional on a specific interim result. In this work, we give a compact overview of sample size recalculation rules and performance measures. Moreover, we propose a new conditional performance score and apply it to various standard recalculation rules by means of Monte-Carlo simulations.

Measurement error and misclassification of variables frequently occur in epidemiology and involve variables important to public health. Their presence can impact strongly on results of statistical analyses involving such variables. However, investigators commonly fail to pay attention to biases resulting from such mismeasurement. We provide, in two parts, an overview of the types of error that occur, their impacts on analytic results, and statistical methods to mitigate the biases that they cause. In this first part, we review different types of measurement error and misclassification, emphasizing the classical, linear and Berkson models, and on the concepts of non-differential and differential error. We describe the impacts of these types of error in covariates and in outcome variables on various analyses, including estimation and testing in regression models and estimating distributions. We outline types of ancillary studies required to provide information about such errors and discuss the implications of covariate measurement error for study design. Methods for ascertaining sample size requirements are outlined, both for ancillary sub-studies designed to provide information about measurement error and for main studies where the exposure of interest is measured with error. We describe two of the simpler methods, regression calibration and simulation extrapolation (SIMEX), that adjust for bias in regression coefficients caused by measurement error in continuous covariates, and illustrate their use through examples drawn from the Observing Protein and Energy (OPEN) dietary validation study. Finally, we review software available for implementing these methods. The second part of the paper deals with more advanced topics.

In randomized clinical trials, it is standard to include baseline variables in the primary analysis as covariates, as it is recommended by international guidelines. For the study design to be consistent with the analysis, these variables should also be taken into account when calculating the sample size to appropriately power the trial. Because assumptions made in the sample size calculation are always subject to some degree of uncertainty, a blinded sample size reestimation (BSSR) is recommended to adjust the sample size when necessary. In this article, we introduce a BSSR approach for count data outcomes with baseline covariates. Count outcomes are common in clinical trials and examples include the number of exacerbations in asthma and chronic obstructive pulmonary disease, relapses, and scan lesions in multiple sclerosis and seizures in epilepsy. The introduced methods are based on Wald and likelihood ratio test statistics. The approaches are illustrated by a clinical trial in epilepsy. The BSSR procedures proposed are compared in a Monte Carlo simulation study and shown to yield power values close to the target while not inflating the type I error rate.

When calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials, it is critical that the planned sampling structure be accurately specified. One common assumption is that participants will provide measurements in each trial period, that is, a closed cohort, and another is that each participant provides only one measurement during the course of the trial. However some studies have an "open cohort" sampling structure, where participants may provide measurements in variable numbers of periods. To date, sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts. Feldman and McKinlay (1994) provided some guidance, stating that the participant-level autocorrelation could be varied to account for the degree of overlap in different periods of the study, but did not indicate precisely how to do so. We present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of "openness" on sample size and power. We consider designs where the number of participants in each cluster will be maintained throughout the trial, but individual participants may provide differing numbers of measurements. Our results are a unification of closed cohort and repeated cross-sectional sample results of Hooper et al (2016), and indicate precisely how participant autocorrelation of Feldman and McKinlay should be varied to account for an open cohort sampling structure. We discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within-cluster correlations and autoregressive participant-level errors.

To control the SARS-CoV-2 pandemic and future pathogen outbreaks requires an understanding of which non-pharmaceutical interventions are effective at reducing transmission. Observational studies, however, are subject to biases, even when there is no true effect. Cluster randomized trials provide a means to conduct valid hypothesis tests of the effect of interventions on community transmission. While they may only require a short duration, they often require large sample sizes to achieve adequate power. However, the sample sizes required for such tests in an outbreak setting are largely undeveloped and the question of whether these designs are practical remains unanswered. We develop approximate sample size formulae and simulation-based sample size methods for cluster randomized trials in infectious disease outbreaks. We highlight key relationships between characteristics of transmission and the enrolled communities and the required sample sizes, describe settings where cluster randomized trials powered to detect a meaningful true effect size may be feasible, and provide recommendations for investigators in planning such trials. The approximate formulae and simulation banks may be used by investigators to quickly assess the feasibility of a trial, and then more detailed methods may be used to more precisely size the trial. For example, we show that community-scale trials requiring 220 clusters with 100 tested individuals per cluster are powered to identify interventions that reduce transmission by 40% in one generation interval, using parameters identified for SARS-CoV-2 transmission. For more modest treatment effects, or settings with extreme overdispersion of transmission, however, much larger sample sizes are required.

Sparse high-dimensional massive sample size (sHDMSS) time-to-event data present multiple challenges to quantitative researchers as most current sparse survival regression methods and software will grind to a halt and become practically inoperable. This paper develops a scalable l0 -based sparse Cox regression tool for right-censored time-to-event data that easily takes advantage of existing high performance implementation of l2 -penalized regression method for sHDMSS time-to-event data. Specifically, we extend the l0 -based broken adaptive ridge (BAR) methodology to the Cox model, which involves repeatedly performing reweighted l2 -penalized regression. We rigorously show that the resulting estimator for the Cox model is selection consistent, oracle for parameter estimation, and has a grouping property for highly correlated covariates. Furthermore, we implement our BAR method in an R package for sHDMSS time-to-event data by leveraging existing efficient algorithms for massive l2 -penalized Cox regression. We evaluate the BAR Cox regression method by extensive simulations and illustrate its application on an sHDMSS time-to-event data from the National Trauma Data Bank with hundreds of thousands of observations and tens of thousands sparsely represented covariates.

Immunotherapies are increasingly used for treating patients with advanced-stage cancers. However, cancer immunotherapy trials often present delayed treatment effects and long-term survivors which result non-proportional hazard models and challenge the immunotherapy trial designs. In this article, we proposed a general random delayed cure rate model for designing cancer immunotherapy trials. A sample size formula is derived for a weighted log-rank test. The accuracy of sample size estimation is assessed and compared with the existing methods via simulation studies. The sensitivities for misspecifying the random delay time are also studied through simulations.

Cancer immunotherapy trials have two special features: a delayed treatment effect and a cure rate. Both features violate the proportional hazards model assumption and ignoring either one of the two features in an immunotherapy trial design will result in substantial loss of statistical power. To properly design immunotherapy trials, we proposed a piecewise proportional hazards cure rate model to incorporate both delayed treatment effect and cure rate into the trial design consideration. A sample size formula is derived for a weighted log-rank test under a fixed alternative hypothesis. The accuracy of sample size calculation using the new formula is assessed and compared with the existing methods via simulation studies. A real immunotherapy trial is used to illustrate the study design along with practical consideration of balance between sample size and follow-up time.

Statistical inference based on correlated count measurements are frequently performed in biomedical studies. Most of existing sample size calculation methods for count outcomes are developed under the Poisson model. Deviation from the Poisson assumption (equality of mean and variance) has been widely documented in practice, which indicates urgent needs of sample size methods with more realistic assumptions to ensure valid experimental design. In this study, we investigate sample size calculation for clinical trials with correlated count measurements based on the negative binomial distribution. This approach is flexible to accommodate overdispersion and unequal measurement intervals, as well as arbitrary randomization ratios, missing data patterns, and correlation structures. Importantly, the derived sample size formulas have closed forms both for the comparison of slopes and for the comparison of time-averaged responses, which greatly reduces the burden of implementation in practice. We conducted extensive simulation to demonstrate that the proposed method maintains the nominal levels of power and type I error over a wide range of design configurations. We illustrate the application of this approach using a real epileptic trial.

Many sample size criteria exist. These include power calculations and methods based on confidence interval widths from a frequentist viewpoint, and Bayesian methods based on credible interval widths or decision theory. Bayesian methods account for the inherent uncertainty of inputs to sample size calculations through the use of prior information rather than the point estimates typically used by frequentist methods. However, the choice of prior density can be problematic because there will almost always be different appreciations of the past evidence. Such differences can be accommodated a priori by robust methods for Bayesian design, for example, using mixtures or ϵ-contaminated priors. This would then ensure that the prior class includes divergent opinions. However, one may prefer to report several posterior densities arising from a "community of priors," which cover the range of plausible prior densities, rather than forming a single class of priors. To date, however, there are no corresponding sample size methods that specifically account for a community of prior densities in the sense of ensuring a large-enough sample size for the data to sufficiently overwhelm the priors to ensure consensus across widely divergent prior views. In this paper, we develop methods that account for the variability in prior opinions by providing the sample size required to induce posterior agreement to a prespecified degree. Prototypic examples to one- and two-sample binomial outcomes are included. We compare sample sizes from criteria that consider a family of priors to those that would result from previous interval-based Bayesian criteria.

Two-way enriched design (TED) is a novel approach addressing placebo response in clinical trials. It is a two-stage, randomized, placebo-controlled trial design with enrichment in placebo non-responders and treatment responders at the second stage. All data from the first stage and data from placebo non-responders and treatment responders in the second stage are used for the final analysis of the treatment effect. The existing methods for the analysis of TED data include score tests with one, two, and three degrees of freedom. All these methods are only applicable to binary outcomes. However, there is an interest in continuous outcomes in clinical trials in psychiatry. In this manuscript, we apply some novel methods, including a repeated measures model, a weighted repeated measures model with weights from propensity score, and weights from K-means clustering, to analyze TED data for both binary outcomes and continuous outcomes. The simulation study indicates that the repeated measures model performs consistently well in preserving the type I error and achieving the minimum mean standard error as well as a higher power. The performance of the weighted repeated measures model with weights from K-means clustering improves with increasing sample size. Investigators can choose from these analytic approaches under different scenarios.

Stopping rules for clinical trials are primarily intended to control Type I error rates if interim analyses are planned, but less is known about the impact that potential stopping has on estimating treatment benefit. In this paper, we derive analytic expressions for (1) the over-estimation of benefit in studies that stop early, (2) the under-estimation of benefit in completed studies, and (3) the overall bias in studies with a stopping rule. We also examine the probability of stopping early and the situation in meta-analyses. Numerical evaluations show that the greatest concern is with over-estimation of benefit in stopped studies, especially if the probability of stopping early is small. The overall bias is usually less than 10% of the true benefit, and under-estimation in completed studies is also typically small. The probability of stopping depends on the true treatment effect and sample size. The magnitude of these effects depends on the particular rule adopted, but we show that the maximum overall bias is the same for all stopping rules. We also show that an essentially unbiased meta-analysis estimate of benefit can be recovered, even if some component studies have stopping rules. We illustrate these methods using data from three clinical trials. The results confirm our earlier empirical work on clinical trials. Investigators may consult our numerical results for guidance on potential mis-estimation and bias in the treatment effect if a stopping rule is adopted. Particular concern is warranted in studies that actually stop early, where interim results may be quite misleading.

Sequential analysis hypothesis testing is now an important tool for postmarket drug and vaccine safety surveillance. When the number of adverse events accruing in time is assumed to follow a Poisson distribution, and if the baseline Poisson rate is assessed only with uncertainty, the conditional maximized sequential probability ratio test, CMaxSPRT, is a formal solution. CMaxSPRT is based on comparing monitored data with historical matched data, and it was primarily developed under a flat signaling threshold. This paper demonstrates that CMaxSPRT can be performed under nonflat thresholds too. We pose the discussion in the light of the alpha spending approach. In addition, we offer a rule of thumb for establishing the best shape of the signaling threshold in the sense of minimizing expected time to signal and expected sample size. An example involving surveillance for adverse events after influenza vaccination is used to illustrate the method.

A requirement for calculating sample sizes for cluster randomized trials (CRTs) conducted over multiple periods of time is the specification of a form for the correlation between outcomes of subjects within the same cluster, encoded via the within-cluster correlation structure. Previously proposed within-cluster correlation structures have made strong assumptions; for example, the usual assumption is that correlations between the outcomes of all pairs of subjects are identical ("uniform correlation"). More recently, structures that allow for a decay in correlation between pairs of outcomes measured in different periods have been suggested. However, these structures are overly simple in settings with continuous recruitment and measurement. We propose a more realistic "continuous-time correlation decay" structure whereby correlations between subjects' outcomes decay as the time between these subjects' measurement times increases. We investigate the use of this structure on trial planning in the context of a primary care diabetes trial, where there is evidence of decaying correlation between pairs of patients' outcomes over time. In particular, for a range of different trial designs, we derive the variance of the treatment effect estimator under continuous-time correlation decay and compare this to the variance obtained under uniform correlation. For stepped wedge and cluster randomized crossover designs, incorrectly assuming uniform correlation will underestimate the required sample size under most trial configurations likely to occur in practice. Planning of CRTs requires consideration of the most appropriate within-cluster correlation structure to obtain a suitable sample size.

In some diseases, such as multiple sclerosis, lesion counts obtained from magnetic resonance imaging (MRI) are used as markers of disease progression. This leads to longitudinal, and typically overdispersed, count data outcomes in clinical trials. Models for such data invariably include a number of nuisance parameters, which can be difficult to specify at the planning stage, leading to considerable uncertainty in sample size specification. Consequently, blinded sample size re-estimation procedures are used, allowing for an adjustment of the sample size within an ongoing trial by estimating relevant nuisance parameters at an interim point, without compromising trial integrity. To date, the methods available for re-estimation have required an assumption that the mean count is time-constant within patients. We propose a new modeling approach that maintains the advantages of established procedures but allows for general underlying and treatment-specific time trends in the mean response. A simulation study is conducted to assess the effectiveness of blinded sample size re-estimation methods over fixed designs. Sample sizes attained through blinded sample size re-estimation procedures are shown to maintain the desired study power without inflating the Type I error rate and the procedure is demonstrated on MRI data from a recent study in multiple sclerosis.

Pooling data from multiple studies improves estimation of exposure-disease associations through increased sample size. However, biomarker exposure measurements can vary substantially across laboratories and often require calibration to a reference assay prior to pooling. We develop two statistical methods for aggregating biomarker data from multiple studies: the full calibration method and the internalized method. The full calibration method calibrates all biomarker measurements regardless of the availability of reference laboratory measurements while the internalized method calibrates only non-reference laboratory measurements. We compare the performance of these two aggregation methods to two-stage methods. Furthermore, we compare the aggregated and two-stage methods when estimating the calibration curve from controls only or from a random sample of individuals from the study cohort. Our findings include the following: (1) Under random sampling for calibration, exposure effect estimates from the internalized method have a smaller mean squared error than those from the full calibration method. (2) Under the controls-only calibration design, the full calibration method yields effect estimates with the least bias. (3) The two-stage approaches produce average effect estimates that are similar to the full calibration method under a controls only calibration design and the internalized method under a random sample calibration design. We illustrate the methods in an application evaluating the relationship between circulating vitamin D levels and stroke risk in a pooling project of cohort studies.

In multilevel populations, there are two types of population means of an outcome variable ie, the average of all individual outcomes ignoring cluster membership and the average of cluster-specific means. To estimate the first mean, individuals can be sampled directly with simple random sampling or with two-stage sampling (TSS), that is, sampling clusters first, and then individuals within the sampled clusters. When cluster size varies in the population, three TSS schemes can be considered, ie, sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster; sampling clusters with equal probability and then sampling the same percentage of individuals per cluster; and sampling clusters with equal probability and then sampling the same number of individuals per cluster. Unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative. Furthermore, the three TSS schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size. The relative efficiency of the sampling schemes is shown to vary across different cluster size distributions. However, sampling clusters with probability proportional to size is the most efficient TSS scheme for many cluster size distributions. Model-based and design-based inference are compared and are shown to give similar results. The results are applied to the distribution of high school size in Italy and the distribution of patient list size for general practices in England.