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High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure -- Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.

When designing a clinical trial, one key aspect of the design is the sample size calculation. The sample size calculation tends to rely on a target or expected difference. The expected difference can be based on the observed data from previous studies, which results in bias. It has been reported that large treatment effects observed in trials are often not replicated in subsequent trials. If these values are used to design subsequent studies, the sample sizes may be biased which results in an unethical study. Regression to the mean (RTM) is one explanation for this. If only health technologies which meet a particular continuation criterion (such as p

Phase II clinical trials in oncology are used to initially evaluate the therapeutic efficacy of a new treatment. In the past, the total response was a frequently used endpoint to access the effectiveness of the treatment. When the total response is modest, clinicians may also be interested in disease control (defined as the total response or stable disease) since it may better predict clinical outcomes. Thus, formally testing both the total response and disease control in a phase II trial has an important clinical implication. In this paper, we propose a new method to construct optimal subset designs for one-stage and two-stage phase II clinical trials with two binary endpoints, i.e. the total response and disease control. A new set of hypotheses under the framework of intersection-union tests is provided in which the treatment is considered promising if both the total response and disease control are good. We show that the power function for each test in a large family of tests is nondecreasing in both the total response rate and disease control rate; identify the parameter configurations at which the maximum Type I error rate and minimum power are achieved and derive level- α tests. We also provide optimal one-stage designs with the minimum sample size and optimal two-stage designs with the least expected total sample size.

The cure fraction models are intended to analyze lifetime data from populations where some individuals are immune to the event under study, and allow a joint estimation of the distribution related to the cured and susceptible subjects, as opposed to the usual approach ignoring the cure rate. In situations involving small sample sizes with many censored times, the detection of nonfinite coefficients may arise via maximum likelihood. This phenomenon is commonly known as monotone likelihood (ML), occurring in the Cox and logistic regression models when many categorical and unbalanced covariates are present. An existing solution to prevent the issue is based on the Firth correction, originally developed to reduce the estimation bias. The method ensures finite estimates by penalizing the likelihood function. In the context of mixture cure models, the ML issue is rarely discussed in the literature; therefore, this topic can be seen as the first contribution of our paper. The second major contribution, not well addressed elsewhere, is the study of the ML issue in cure mixture modeling under the flexibility of a semiparametric framework to handle the baseline hazard. We derive the modified score function based on the Firth approach and explore finite sample size properties of the estimators via a Monte Carlo scheme. The simulation results indicate that the performance of coefficients related to the binary covariates are strongly affected to the imbalance degree. A real illustration, in the melanoma dataset, is discussed using a relatively novel data set collected in a Brazilian university hospital.

We consider partially observed binary matched-pair data. We assume that the incomplete subjects are missing at random. Within this missing framework, we propose an EM-algorithm based approach to construct an interval estimator of the proportion difference incorporating all the subjects. In conjunction with our proposed method, we also present two improvements to the interval estimator through some correction factors. The performances of the three competing methods are then evaluated through extensive simulation. Recommendation for the method is given based on the ability to preserve type-I error for various sample sizes. Finally, the methods are illustrated in two real-world data sets. An R-function is developed to implement the three proposed methods.

We introduce a simulation method for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the dynamic value process of a portfolio from a finite sample of its cumulative cash flow. The learned value process is given in closed form thanks to a suitable choice of the kernel. We show asymptotic consistency and derive finite sample error bounds under conditions that are suitable for finance applications. Numerical experiments show good results in large dimensions for a moderate training sample size.

In this study we propose a sequential test for hypothesis testing on the $$C_{pmk}$$ process capability index. Furthermore, we propose a sequential sampling plan for lot acceptance based on $$C_{pmk}$$ . We compare the statistical properties of the sequential procedures with the performance of the corresponding non-sequential methodologies by carrying out an extensive simulation study. The results show that the proposed sequential methods make it possible to reach decisions much more quickly, on average, than the fixed sample size procedures with the same discriminating power.

In this paper an easy to implement method of stochastically weighing short and long-memory linear processes is introduced. The method renders asymptotically exact size confidence intervals for the population mean which are significantly more accurate than their classic counterparts for each fixed sample size n. It is illustrated both theoretically and numerically that the randomization framework of this paper produces randomized (asymptotic) pivotal quantities, for the mean, which admit central limit theorems with smaller magnitudes of error as compared to those of their leading classic counterparts. An Edgeworth expansion result for randomly weighted linear processes whose innovations do not necessarily satisfy the Cramer condition, is established. Numerical illustrations and applications to real world data are also included.

Summary We consider the problem of testing for the presence of linear relationships between large sets of random variables based on a postselection inference approach to canonical correlation analysis. The challenge is to adjust for the selection of subsets of variables having linear combinations with maximal sample correlation. To this end, we construct a stabilized one-step estimator of the Euclidean norm of the canonical correlations maximized over subsets of variables of prespecified cardinality. This estimator is shown to be consistent for its target parameter and asymptotically normal, provided the dimensions of the variables do not grow too quickly with sample size. We also develop a greedy search algorithm to accurately compute the estimator, leading to a computationally tractable omnibus test for the global null hypothesis that there are no linear relationships between any subsets of variables having the prespecified cardinality. We further develop a confidence interval that takes the variable selection into account.

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.

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.

A recently proposed SLOPE estimator (arXiv:1407.3824) has been shown to adaptively achieve the minimax $\ell_2$ estimation rate under high-dimensional sparse linear regression models (arXiv:1503.08393). Such minimax optimality holds in the regime where the sparsity level $k$, sample size $n$, and dimension $p$ satisfy $k/p \rightarrow 0$, $k\log p/n \rightarrow 0$. In this paper, we characterize the estimation error of SLOPE under the complementary regime where both $k$ and $n$ scale linearly with $p$, and provide new insights into the performance of SLOPE estimators. We first derive a concentration inequality for the finite sample mean square error (MSE) of SLOPE. The quantity that MSE concentrates around takes a complicated and implicit form. With delicate analysis of the quantity, we prove that among all SLOPE estimators, LASSO is optimal for estimating $k$-sparse parameter vectors that do not have tied non-zero components in the low noise scenario. On the other hand, in the large noise scenario, the family of SLOPE estimators are sub-optimal compared with bridge regression such as the Ridge estimator., 50 pages, 18 figures

Existing procedures for testing measurement invariance focus mainly on group-level comparisons rather than individual comparisons (i.e., the main conclusion typically concerns the question of comparability of group means). We propose a set of intuitive graphical displays called Person-Level Assessment of Measurement Invariance (PLAMI), which attempt to provide information about the effect of the lack of measurement invariance on the level of person scores. This information complements the group-level information contained in model fit indices and overall effect size indices, and should be especially useful for practitioners using tests for individual diagnostics. PLAMI uses results from the multiple-group factor analysis to estimate and visualize score distortions in terms of expected score difference between members of different groups and the probability of an unacceptably high score difference, conditional for the value of latent trait. The stability of the results can be evaluated by means of resampling. Simulation can be used for an a priori evaluation of the sample size adequacy. We illustrate the proposed plots on two real examples involving testing measurement invariance of the Machiavellianism scale, and we demonstrate the added value of PLAMI compared to the use of fit indices alone.

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.

We provide asymptotic theory for certain functions of the sample autocovariance matrices of a high-dimensional time series with infinite fourth moment. The time series exhibits linear dependence across the coordinates and through time. Assuming that the dimension increases with the sample size, we provide theory for the eigenvectors of the sample autocovariance matrices and find explicit approximations of a simple structure, whose finite sample quality is illustrated for simulated data. We also obtain the limits of the normalized eigenvalues of functions of the sample autocovariance matrices in terms of cluster Poisson point processes. In turn, we derive the distributional limits of the largest eigenvalues and functionals acting on them. In our proofs, we use large deviation techniques for heavy-tailed processes, point process techniques motivated by extreme value theory, and related continuous mapping arguments., 28 pages

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.

A centered $$L^2$$ -norm based test statistic is used for testing if a high-dimensional mean vector equals zero where the data dimension may be much larger than the sample size. Inspired by the fact that under some regularity conditions the asymptotic null distributions of the proposed test are the same as the limiting distributions of a chi-square-mixture, a three-cumulant matched chi-square-approximation is suggested to approximate this null distribution. The asymptotic power of the proposed test under a local alternative is established and the effect of data non-normality is discussed. A simulation study under various settings demonstrates that in terms of size control, the proposed test performs significantly better than some existing competitors. Several real data examples are presented to illustrate the wide applicability of the proposed test to a variety of high-dimensional data analysis problems, including the one-sample problem, paired two-sample problem, and MANOVA for correlated samples or independent samples.

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.

As a reaction to the pandemic of the severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2), a multitude of clinical trials for the treatment of SARS‐CoV‐2 or the resulting corona disease 2019 (COVID‐19) are globally at various stages from planning to completion. Although some attempts were made to standardize study designs, this was hindered by the ferocity of the pandemic and the need to set up clinical trials quickly. We take the view that a successful treatment of COVID‐19 patients (i) increases the probability of a recovery or improvement within a certain time interval, say 28 days; (ii) aims to expedite favorable events within this time frame; and (iii) does not increase mortality over this time period. On this background, we discuss the choice of endpoint and its analysis. Furthermore, we consider consequences of this choice for other design aspects including sample size and power and provide some guidance on the application of adaptive designs in this particular context.

One of the key aspects of designing a clinical trial is sample size estimation. While the statistical power improves with the increased sample size, one has to be considerate about the ethical, sci...

We consider the General Linear Model (GLM) which includes multivariate analysis of variance (MANOVA) and multiple linear regression as special cases. In practice, there are several widely used criteria for GLM: Wilks’ lambda, Bartlett–Nanda–Pillai test, Lawley–Hotelling test, and Roy maximum root test. Limiting distributions for the first three mentioned tests are known under different asymptotic settings. In the present paper, we obtain computable error bounds for the normal approximation of the Lawley–Hotelling statistic when the dimension grows proportionally to the sample size. This result enables us to get more precise calculations of the p-values in applications of multivariate analysis. In practice, more and more often analysts encounter situations where the number of factors is large and comparable with the sample size. Examples include medicine, biology (i.e., DNA microarray studies), and finance.

Whereas the theory of confirmatory adaptive designs is well understood for uncensored data, implementation of adaptive designs in the context of survival trials remains challenging. Commonly used adaptive survival tests are based on the independent increments structure of the log-rank statistic. This implies some relevant limitations: On the one hand, essentially only the interim log-rank statistic may be used for design modifications (such as data-dependent sample size recalculation). Furthermore, the treatment arm allocation ratio in these classical methods is assumed to be constant throughout the trial period. Here, we propose an extension of the independent increments approach to adaptive survival tests that addresses some of these limitations. We present a confirmatory adaptive two-sample log-rank test that allows rejection regions and sample size recalculation rules to be based not only on the interim log-rank statistic, but also on point-wise survival rate estimates, simultaneously. In addition, the possibility is opened to adapt the treatment arm allocation ratio after each interim analysis in a data-dependent way. The ability to include point-wise survival rate estimators in the rejection region of a test for comparing survival curves might be attractive, e.g., for seamless phase II/III designs. Data-dependent adaptation of the allocation ratio could be helpful in multi-arm trials in order to successively steer recruitment into the study arms with the greatest chances of success. The methodology is motivated by the LOGGIC Europe Trial from pediatric oncology. Distributional properties are derived using martingale techniques in the large sample limit. Small sample properties are studied by simulation.

We consider a random censoring model for survival analysis, allowing the possibility that only a proportion of individuals in the population are susceptible to death or failure, and the remainder are immune or cured. Susceptibles suffer the event under study eventually, but the time at which this occurs may not be observed due to censoring. Immune individuals have infinite lifetimes which are always censored in the sample. Assuming that the distribution of the susceptibles’ lifetimes as well as the censoring distribution have infinite right endpoints and are in the domain of attraction of the Gumbel distribution, we obtain asymptotic distributions, as sample size tends to infinity, of statistics relevant to testing for the possible existence of immunes in the population.

The classical two-sample t-test is not robust to skewed populations, and the large sample approximation has low accuracy for finite sample sizes. This paper presents two new types of tests, the TCF...

Recent minimum sample size formula (Riley et al.) for developing clinical prediction models help ensure that development datasets are of sufficient size to minimise overfitting. While these criteria are known to avoid excessive overfitting on average, the extent of variability in overfitting at recommended sample sizes is unknown. We investigated this through a simulation study and empirical example to develop logistic regression clinical prediction models using unpenalised maximum likelihood estimation, and various post-estimation shrinkage or penalisation methods. While the mean calibration slope was close to the ideal value of one for all methods, penalisation further reduced the level of overfitting, on average, compared to unpenalised methods. This came at the cost of higher variability in predictive performance for penalisation methods in external data. We recommend that penalisation methods are used in data that meet, or surpass, minimum sample size requirements to further mitigate overfitting, and that the variability in predictive performance and any tuning parameters should always be examined as part of the model development process, since this provides additional information over average (optimism-adjusted) performance alone. Lower variability would give reassurance that the developed clinical prediction model will perform well in new individuals from the same population as was used for model development.

A family of self-normalized CUSUM tests for structural change under long memory is proposed. The test statistics apply non-parametric kernel-based long-run variance estimators and have well-defined limiting distributions that only depend on the long-memory parameter. A Monte Carlo simulation shows that these tests provide finite sample size control while outperforming competing procedures in terms of power.

In many orphan diseases and pediatric indications, the randomized controlled trials may be infeasible because of their size, duration, and cost. Leveraging information on the control through a prior can potentially reduce sample size. However, unless an objective prior is used to impose complete ignorance for the parameter being estimated, it results in biased estimates and inflated type-I error. Hence, it is essential to assess both the confirmatory and supplementary knowledge available during the construction of the prior to avoid "cherry-picking" advantageous information. For this purpose, propensity score methods are employed to minimize selection bias by weighting supplemental control subjects according to their similarity in terms of pretreatment characteristics to the subjects in the current trial. The latter can be operationalized through a proposed measure of overlap in propensity-score distributions. In this paper, we consider single experimental arm in the current trial and the control arm is completely borrowed from the supplemental data. The simulation experiments show that the proposed method reduces prior and data conflict and improves the precision of the of the average treatment effect.

Two-stage single arm designs are widely used in phase II clinical trials with binary endpoints. The trial may be stopped early due to insufficient positive responses in the first stage. There may be some enrolled subjects who have yet to respond by the end of the first stage, and their data are ignored if the first stage results in rejection of the trial. It is possible that the result after the first stage is rejection by a slim margin, while the results of pipeline subjects are quite positive. In this case, combining the data from the two sources may provide a valuable opportunity to rescue a promising treatment that was mistakenly rejected. We propose a novel double-check design to take advantage of the pipeline subjects' data to establish a rescue criterion based on two-stage design. When the rescue criterion is met, the decision to reject the trial at the end of the first stage can be reversed, allowing the trial to continue. A derivation based on a binomial distribution shows that the double-check strategy can strictly preserve the type I error rate. Further examination shows that the strategy can provide a slight increase in overall power and a substantial increase in conditional power when the proportion of positive response at the end of the first stage is at the margin. The extra rescue opportunity's cost is pretty low, only a slight increasing in the expected sample size.

A sequential multiple assignment randomized trial (SMART) facilitates the comparison of multiple adaptive treatment strategies (ATSs) simultaneously. Previous studies have established a framework to test the homogeneity of multiple ATSs by a global Wald test through inverse probability weighting. SMARTs are generally lengthier than classical clinical trials due to the sequential nature of treatment randomization in multiple stages. Thus, it would be beneficial to add interim analyses allowing for an early stop if overwhelming efficacy is observed. We introduce group sequential methods to SMARTs to facilitate interim monitoring based on the multivariate chi-square distribution. Simulation studies demonstrate that the proposed interim monitoring in SMART (IM-SMART) maintains the desired type I error and power with reduced expected sample size compared to the classical SMART. Finally, we illustrate our method by reanalyzing a SMART assessing the effects of cognitive behavioral and physical therapies in patients with knee osteoarthritis and comorbid subsyndromal depressive symptoms.

The win odds is a distribution-free method of comparing locations of distributions of two independent random variables. Introduced as a method for analyzing hierarchical composite endpoints, it is well suited to be used in the analysis of ordinal scale endpoints in COVID-19 clinical trials. For a single outcome, we provide power and sample size calculation formulas for the win odds test. We also provide an implementation of the win odds analysis method for a single ordinal outcome in a commonly used statistical software to make the win odds analysis fully reproducible.

Restricted mean survival time (RMST) is one measure now used to summarize time-to-event type data, but it has been pointed out that the distribution of differences in RMST deviates markedly from a normal distribution for controlled clinical trials with small sample sizes. Therefore, we conducted a numerical simulation of the RMST in which the one-sample survival time follows a Weibull distribution, comparing eight different confidence intervals combining two types of variance with four types of variable transformations, including no transformation. The evaluation items were the coverage probability and the above and below error probabilities for the true value. The variance types were based on Greenwood's formula and its Kaplan-Meier correction. The arcsine square root transformation, logit transformation, and complementary log-log transformation were used as the variable transformations. When the sample size was small and the event rate was low, the confidence interval of the untransformed RMST tended to have a small coverage probability and to be overestimated. Variance by Kaplan-Meier correction improved the coverage. The problems of coverage and overestimation were also improved by variable transformations, and in particular, applying the logit transformation and the complementary log-log transformation both resulted in substantial improvements. Our study suggested that it is preferable to construct the confidence intervals of RMST using the logit transformation for variances based on Greenwood's formula in small sample size trials. The SAS code to replicate the analyses is available at https://github.com/HiroyaHashimoto/SAS-Programs.

The case-control design is known for its efficiency at a fixed sample size when the disease under study is rare. Such a design is usually carried out for the inference of disease odds ratio in the ...

Sepehrifar and Yarahmadian (Stat Pap 58:1115–1124, 2017) had developed a non-parametric test for testing exponentiality against decreasing renewal dichotomous Markov noise shock model (DRDMNS) alternatives which when subjected to scrutiny under delivers. Hence, we propose a non-parametric test for testing exponentiality against a class of distributions belonging to DRDMNS models. The asymptotic properties of the test statistic are discussed. An exact null distribution is derived and critical values with different sample sizes are obtained. The proposed test is applied to the censored data also. The results of the Monte Carlo simulations are used to further manifest the quality of the proposed test. Finally, the proposed test is illustrated using two real data sets.

Ultrahigh-dimensional gene features are often collected in modern cancer studies in which the number of gene features p is extremely larger than sample size n. While gene expression patterns have been shown to be related to patients' survival in microarray-based gene expression studies, one has to deal with the challenges of ultrahigh-dimensional genetic predictors for survival predicting and genetic understanding of the disease in precision medicine. The problem becomes more complicated when two types of survival endpoints, distant metastasis-free survival and overall survival, are of interest in the study and outcome data can be subject to semi-competing risks due to the fact that distant metastasis-free survival is possibly censored by overall survival but not vice versa. Our focus in this paper is to extract important features, which have great impacts on both distant metastasis-free survival and overall survival jointly, from massive gene expression data in the semi-competing risks setting. We propose a model-free screening method based on the ranking of the correlation between gene features and the joint survival function of two endpoints. The method accounts for the relationship between two endpoints in a simply defined utility measure that is easy to understand and calculate. We show its favorable theoretical properties such as the sure screening and ranking consistency, and evaluate its finite sample performance through extensive simulation studies. Finally, an application to classifying breast cancer data clearly demonstrates the utility of the proposed method in practice. Ministry of Education (MOE) Submitted/Accepted version Xiang’s research was supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant RG98/20 and Peng’s research was supported by National Natural Science Foundation of China (NSFC Grant No. 92046005).

Author(s): Zhang, Yuqian; Bradic, Jelena | Abstract: We provide a high-dimensional semi-supervised inference framework focused on the mean and variance of the response. Our data are comprised of an extensive set of observations regarding the covariate vectors and a much smaller set of labeled observations where we observe both the response as well as the covariates. We allow the size of the covariates to be much larger than the sample size and impose weak conditions on a statistical form of the data. We provide new estimators of the mean and variance of the response that extend some of the recent results presented in low-dimensional models. In particular, at times we will not necessitate consistent estimation of the functional form of the data. Together with estimation of the population mean and variance, we provide their asymptotic distribution and confidence intervals where we showcase gains in efficiency compared to the sample mean and variance. Our procedure, with minor modifications, is then presented to make important contributions regarding inference about average treatment effects. We also investigate the robustness of estimation and coverage and showcase widespread applicability and generality of the proposed method.

Markov networks are frequently used in sciences to represent conditional independence relationships underlying observed variables arising from a complex system. It is often of interest to understand how an underlying network differs between two conditions. In this paper, we develop methods for comparing a pair of high-dimensional Markov networks where we allow the number of observed variables to increase with the sample sizes. By taking the density ratio approach, we are able to learn the network difference directly and avoid estimating the individual graphs. Our methods are thus applicable even when the individual networks are dense as long as their difference is sparse. We prove finite-sample Gaussian approximation error bounds for the estimator we construct under significantly weaker assumptions than are typically required for model selection consistency. Furthermore, we propose bootstrap procedures for estimating quantiles of a max-type statistics based on our estimator, and show how they can be used to test the equality of two Markov networks or construct simultaneous confidence intervals. The performance of our methods is demonstrated through extensive simulations. The scientific usefulness is illustrated with an analysis of a new fMRI data set.

For the general linear hypothesis testing problem for high-dimensional data, several interesting tests have been proposed in the literature. Most of them have imposed strong assumptions on the underlying covariance matrix so that their test statistics under the null hypothesis are asymptotically normally distributed. In practice, however, these strong assumptions may not be satisfied or hardly be checked so that these tests are often applied blindly in real data analysis. Their empirical sizes may then be much larger or smaller than the nominal size. For these tests, this is a size control problem which cannot be overcome via purely increasing the sample size to infinity. To overcome this difficulty, in this paper, a new normal-reference test using the centralized $$L^2$$ -norm based test statistic with three cumulant matched chi-square approximation is proposed and studied. Some theoretical discussion and two simulation studies demonstrate that in terms of size control, the new normal-reference test performs very well regardless of if the high-dimensional data are nearly uncorrelated, moderately correlated, or highly correlated and it outperforms two existing competitors substantially. Two real high-dimensional data examples motivate and illustrate the new normal-reference test.

In RCTs with an interest in a long-term efficacy endpoint, the follow-up time necessary to observe the endpoint may be substantial. In order to reduce the expected duration of such trials, early-outcome data may be collected to enrich an interim analysis aimed at stopping the trial early for efficacy. We propose to extend such a design with an additional interim analysis using solely early-outcome data in order to expedite the evaluation of treatment's efficacy. We evaluate the potential gain in operating characteristics (power, expected trial duration, and expected sample size) when introducing such an early interim analysis, in function of the properties of the early outcome as a surrogate for the long-term endpoint. In the context of a longitudinal age-related macular degeneration (ARMD) ophthalmology trial, results show potentially substantial gains in both the expected trial duration and the expected sample size. A prerequisite, though, is that the treatment effect on the early outcome has to be strongly correlated with the treatment effect on the long-term endpoint, that is, that the early outcome is a validated surrogate for the long-term endpoint.

Asymptotic properties of robust regression estimators are well known. However, it is not always clear what is the best strategy for confidence intervals and hypothesis testing when the sample size is not very large, since the distribution of residuals coming from robust estimates has unknown properties for small samples. In the present work we propose an analysis of various strategies for estimating the variance-covariance matrix of the S estimators at the variation of n and p, considering different �� functions. An adaptive correction strategy is proposed. In addition to the simulation study, an example on a benchmark dataset is shown.

Sample size calculations for cluster-randomised trials require inclusion of an inflation factor taking into account the intra-cluster correlation coefficient. Often, estimates of the intra-cluster correlation coefficient are taken from pilot trials, which are known to have uncertainty about their estimation. Given that the value of the intra-cluster correlation coefficient has a considerable influence on the calculated sample size for a main trial, the uncertainty in the estimate can have a large impact on the ultimate sample size and consequently, the power of a main trial. As such, it is important to account for the uncertainty in the estimate of the intra-cluster correlation coefficient. While a commonly adopted approach is to utilise the upper confidence limit in the sample size calculation, this is a largely inefficient method which can result in overpowered main trials. In this paper, we present a method of estimating the sample size for a main cluster-randomised trial with a continuous outcome, using numerical methods to account for the uncertainty in the intra-cluster correlation coefficient estimate. Despite limitations with this initial study, the findings and recommendations in this paper can help to improve sample size estimations for cluster randomised controlled trials by accounting for uncertainty in the estimate of the intra-cluster correlation coefficient. We recommend this approach be applied to all trials where there is uncertainty in the intra-cluster correlation coefficient estimate, in conjunction with additional sources of information to guide the estimation of the intra-cluster correlation coefficient.

Interrupted time series (ITS) design is commonly used to evaluate the impact of interventions in healthcare settings. Segmented regression (SR) is the most commonly used statistical method and has been shown to be useful in practical applications involving ITS designs. Nevertheless, SR is prone to aggregation bias, which leads to imprecision and loss of power to detect clinically meaningful differences. The objective of this article is to present a weighted SR method, where variability across patients within the healthcare facility and across time points is incorporated through weights. We present the methodological framework, provide optimal weights associated with data at each time point and discuss relevant statistical inference. We conduct extensive simulations to evaluate performance of our method and provide comparative analysis with the traditional SR using established performance criteria such as bias, mean square error and statistical power. Illustrations using real data is also provided. In most simulation scenarios considered, the weighted SR method produced estimators that are uniformly more precise and relatively less biased compared to the traditional SR. The weighted approach also associated with higher statistical power in the scenarios considered. The performance difference is much larger for data with high variability across patients within healthcare facilities. The weighted method proposed here allows us to account for the heterogeneity in the patient population, leading to increased accuracy and power across all scenarios. We recommend researchers to carefully design their studies and determine their sample size by incorporating heterogeneity in the patient population.

In medical researches such as case-control studies with contaminated controls, frequently encountered is a particular two-sample testing problem in which one sample has a mixture structure. It is a very common case that the exposure in a case-control study may have a positive (or negative) effect on the response variable if the effect exists. This is often ignored by existing tests, which would lead to potentially power loss. Meanwhile, it is of much practical importance to determine a minimal sample size to reach a target power. Based on empirical likelihood and density ratio model, we develop a new EM-test by incorporating the inequality information in the alternative. We show that the proposed EM-test has a mixture of zero and χ 1 2 limiting distribution under the null hypothesis. Its local power analysis and sample size calculations are also investigated. A simulation study and two real data analyses are provided to illustrate the proposed EM-test and sample size formula.

This work studies the properties of the maximum likelihood estimator (MLE) of a multidimensional parameter in a nonlinear model with additive Gaussian errors. The observations are collected in a two-stage experimental design and are dependent because the second stage design is determined by the observations at the first stage. The MLE maximizes the total likelihood. Unlike most theory in the literature, the approximation made to the distribution of the MLE only involves taking the second stage sample size to infinity, as the resulting approximate model retains the dependency between stages, and therefore, more closely reflects the actual two-stage experiment. It is proved that the MLE is consistent and that its asymptotic distribution is a specific Gaussian mixture, via stable convergence. Finally, the efficiency of the adaptive procedure relative to the fixed procedure is illustrated by a simulation study under three parameter dose–response Emax and Exponential models.

Separation or monotone-likelihood can be observed in fitting process of a multinomial logistic model using maximum likelihood estimation (MLE) when sample size is small and/or one of the outcome categories is rare and/or there is one or more influential covariates, resulting in infinite or biased estimate of at least one regression coefficient of the model. This study investigated empirically to identify the optimal data condition to define both ‘separation’ and ‘near-to-separation’ (partial separation) and explored their consequences in MLE and provided a solution by applying a penalized likelihood approach, which has been proposed in the literature, by adding a Jeffreys prior-based penalty term to the original likelihood function to remove the first-order bias in the MLEs of the multinomial logit model via equivalent Poisson regression. Furthermore, the penalized estimating equation (PMLE) is extended to a weighted estimating equation allowing for survey-weight for analyzing data from a complex survey. The simulation study suggests that the PMLE outperforms the MLE by providing smaller amount of bias and mean squared of error and better coverage. The methods are applied to analyze data on choice of health facility for treatment of childhood diseases.

This paper considers the portfolio problem for high dimensional data when the dimension and size are both large. We analyze the traditional Markowitz mean-variance (MV) portfolio by large dimension matrix theory, and find the spectral distribution of the sample covariance is the main factor to make the expected return of the traditional MV portfolio overestimate the theoretical MV portfolio. A correction is suggested to the spectral construction of the sample covariance to be the sample spectrally corrected covariance, and to improve the traditional MV portfolio to be spectrally corrected. In the expressions of the expected return and risk on the MV portfolio, the population covariance matrix is always a quadratic form, which will direct MV portfolio estimation. We provide the limiting behavior of the quadratic form with the sample spectrally-corrected covariance matrix, and explain the superior performance to the sample covariance as the dimension increases to infinity proportionally with the sample size. Moreover, this paper deduces the limiting behavior of the expected return and risk on the spectrally-corrected MV portfolio, and illustrates the superior properties of the spectrally-corrected MV portfolio. In simulations, we compare the spectrally-corrected estimates with the traditional and bootstrap-corrected estimates, and show the performance of the spectrally-corrected estimates are the best in portfolio returns and portfolio risk. We also compare the performance of the new proposed estimation with deferent optimal portfolio estimates for real data from S&P 500. The empirical findings are consistent with the theory developed in the paper.

Summary As a generalization of the classical linear factor model, generalized latent factor models are useful for analysing multivariate data of different types, including binary choices and counts. This paper proposes an information criterion to determine the number of factors in generalized latent factor models. The consistency of the proposed information criterion is established under a high-dimensional setting, where both the sample size and the number of manifest variables grow to infinity, and data may have many missing values. An error bound is established for the parameter estimates, which plays an important role in establishing the consistency of the proposed information criterion. This error bound improves several existing results and may be of independent theoretical interest. We evaluate the proposed method by a simulation study and an application to Eysenck’s personality questionnaire.

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.

The timings of visits in observational longitudinal data may depend on the study outcome, and this can result in bias if ignored. Assessing the extent of visit irregularity is important because it can help determine whether visits can be treated as repeated measures or as irregular data. We propose plotting the mean proportions of individuals with 0 visits per bin against the mean proportions of individuals with >1 visit per bin as bin width is varied and using the area under the curve (AUC) to assess the extent of irregularity. The AUC is a single score which can be used to quantify the extent of irregularity and assess how closely visits resemble repeated measures. Simulation results confirm that the AUC increases with increasing irregularity while being invariant to sample size and the number of scheduled measurement occasions. A demonstration of the AUC was performed on the TARGet Kids! study which enrolls healthy children aged 0–5 years with the aim of investigating the relationship between early life exposures and later health problems. The quality of statistical analyses can be improved by using the AUC as a guide to select the appropriate analytic outcome approach and minimize the potential for biased results.