Efficient testing of the biomarker positive and negative subgroups in a biomarker-stratified trial.

In many randomized placebo-controlled trials with a biomarker defined subgroup, it is believed that this subgroup has the same or higher treatment effect compared with its complement. These subgroups are often referred to as the biomarker positive and negative subgroups. Most biomarker-stratified pivotal trials are aimed at demonstrating a significant treatment effect either in the biomarker positive subgroup or in the overall population. A major shortcoming of this approach is that the treatment can be declared effective in the overall population even though it has no effect in the biomarker negative subgroup. We use the isotonic assumption about the treatment effects in the two subgroups to construct an efficient way to test for a treatment effect in both the biomarker positive and negative subgroups. A substantial reduction in the required sample size for such a trial compared with existing methods makes evaluating the treatment effect in both the biomarker positive and negative subgroups feasible in pivotal trials especially when the prevalence of the biomarker positive subgroup is less than 0.5.

Shape restricted additive hazards models: Monotone, unimodal, and U-shape hazard functions.

We consider estimation of the semiparametric additive hazards model with an unspecified baseline hazard function where the effect of a continuous covariate has a specific shape but otherwise unspecified. Such estimation is particularly useful for a unimodal hazard function, where the hazard is monotone increasing and monotone decreasing with an unknown mode. A popular approach of the proportional hazards model is limited in such setting due to the complicated structure of the partial likelihood. Our model defines a quadratic loss function, and its simple structure allows a global Hessian matrix that does not involve parameters. Thus, once the global Hessian matrix is computed, a standard quadratic programming method can be applicable by profiling all possible locations of the mode. However, the quadratic programming method may be inefficient to handle a large global Hessian matrix in the profiling algorithm due to a large dimensionality, where the dimension of the global Hessian matrix and number of hypothetical modes are the same order as the sample size. We propose the quadratic pool adjacent violators algorithm to reduce computational costs. The proposed algorithm is extended to the model with a time-dependent covariate with monotone or U-shape hazard function. In simulation studies, our proposed method improves computational speed compared to the quadratic programming method, with bias and mean square error reductions. We analyze data from a recent cardiovascular study.

Bootstrap tests for simultaneous monotone ordering of effects in a two-way ANOVA.

In a two-way additive analysis of variance (ANOVA) model, we consider the problem of testing for homogeneity of both row and column effects against their simultaneous ordering. The error variances are assumed to be heterogeneous with unbalanced samples in each cell. Two simultaneous test procedures are developed-the first one using the likelihood ratio test (LRT) statistics of two independent hypotheses and another based on the consecutive pairwise differences of estimators of effects. The parametric bootstrap (PB) approach is used to find critical points of both the tests and the asymptotic accuracy of the bootstrap is established. An extensive simulation study shows that the proposed tests achieve the nominal size and have very good power performance. The robustness of the tests is also analyzed under deviation from normality. An "R" package is developed and shared on "GitHub" for ease of implementation of users. The proposed tests are illustrated using a real data set on the mortality due to alcoholic liver disease and it is shown that age and gender have a significant impact on the increasing incidence of mortality.

Fully order restricted multi-arm multi-stage clinical trial design.

We consider a multi-arm trial with two or more active treatments plus a control where it is reasonable to assume an order for the treatment effects of the active arms compared to control. For example, the arms could be a high dose and low dose of a new drug and a placebo. The objective of the trial is to compare each active arm to control while maintaining strong control of the type 1 error rate. We show that when the study is powered to identify all promising treatments, a design that uses the order of the treatment effects to calculate the test statistic and to set the order of testing requires a smaller sample size than a design where each active arm is tested against the control arm independently. Under the considered settings, the sample size for a single-stage trial and a two-stage trial was reduced by at least 20%.

Joint inference about the AUC and Youden index for paired biomarkers.

It is common to compare biomarkers' diagnostic or prognostic performance using some summary ROC measures such as the area under the ROC curve (AUC) or the Youden index. We propose to compare two paired biomarkers using both the AUC and the Youden index since the two indices describe different aspects of the ROC curve. This comparison can be made by estimating the joint confidence region (an elliptical area) of the differences of the paired AUCs and the Youden indices. Furthermore, for deciding if one marker is better than the other in terms of both the AUC and the Youden index (J), we can test H0:AUCa≤AUCb or Ja≤Jb against Ha:AUCa>AUCb and Ja>Jb using the paired differences. The construction of such a joint hypothesis is an example of the multivariate order-restricted hypotheses. For such a hypothesis, we propose and compare three testing procedures: (1) the intersection-union test ( IUT ); (2) the conditional test; and (3) the joint test. The performance of the proposed inference methods was evaluated and compared through simulations. The simulation results demonstrate that the proposed joint confidence region maintains the desired confidence level, and all three tests maintain the type I error under the null. Furthermore, among the three proposed testing methods, the conditional test is the preferred approach with markedly larger power consistently than the other two competing methods.

More powerful goodness-of-fit tests for uniform stochastic ordering.

The ordinal dominance curve (ODC) is a useful graphical tool to compare two population distributions. These distributions are said to satisfy uniform stochastic ordering (USO) if the ODC for them is star-shaped. A goodness-of-fit test for USO was recently proposed when both distributions are unknown. This test involves calculating the L p distance between an empirical estimator of the ODC and its least star-shaped majorant. The least favorable configuration of the two distributions was established so that proper critical values could be determined; i.e., to control the probability of type I error for all star-shaped ODCs. However, the use of these critical values can lead to a conservative test and minimal power to detect certain non-star-shaped alternatives. Two new methods for determining data-dependent critical values are proposed. Simulation is used to show both methods can provide substantial increases in power while still controlling the size of the distance-based test. The methods are also applied to a data set involving premature infants. An R package that implements all tests is provided.

NONPARAMETRIC TESTING FOR MULTIPLE SURVIVAL FUNCTIONS WITH NON-INFERIORITY MARGINS.

New nonparametric tests for the ordering of multiple survival functions are developed with the possibility of right censorship taken into account. The motivation comes from non-inferiority trials with multiple treatments. The proposed tests are based on nonparametric likelihood ratio statistics, which are known to provide more powerful tests than Wald-type procedures, but in this setting have only been studied for pairs of survival functions or in the absence of censoring. We introduce a novel type of pool adjacent violator algorithm that leads to a complete solution of the problem. The limit distributions can be expressed as weighted sums of squares involving projections of certain Gaussian processes onto the given ordered alternative. A simulation study shows that the new procedures have superior power to a competing combined-pairwise Cox model approach. We illustrate the proposed methods using data from a three-arm non-inferiority trial.

Modeling semicontinuous longitudinal data with order constraints.

Semicontinuous longitudinal data are characterized by within-subjects repeated measurements that either indicate absence of abnormality or reflect different amount of abnormality. Joint models for semicontinuous longitudinal data have been increasingly receiving attention in the literature. Such models permit flexible characterization of covariates-outcome associations. Order-restricted statistical inference has been well established in the literature but has not yet been applied to joint models for semicontinuous longitudinal data. We incorporate general order-restricted inference into the general joint models for semicontinuous longitudinal data previously proposed. We develop computational methods to address general order restrictions. Through simulations and a real-data example, we demonstrate the advantages of order-restricted inference in terms of increased power in hypothesis testing and increased precision in parameter estimation.

Revisiting isotonic phase I design in the era of model-assisted dose-finding.

Background/aims In the conduct of phase I trials, the limited use of innovative model-based designs in practice has led to an introduction of a class of "model-assisted" designs with the aim of effectively balancing the trade-off between design simplicity and performance. Prior to the recent surge of these designs, methods that allocated patients to doses based on isotonic toxicity probability estimates were proposed. Like model-assisted methods, isotonic designs allow investigators to avoid difficulties associated with pre-trial parametric specifications of model-based designs. The aim of this work is to take a fresh look at an isotonic design in light of the current landscape of model-assisted methods. Methods The isotonic phase I method of Conaway, Dunbar, and Peddada was proposed in 2004 and has been regarded primarily as a design for dose-finding in drug combinations. It has largely been overlooked in the single-agent setting. Given its strong simulation performance in application to more complex dose-finding problems, such as drug combinations and patient heterogeneity, as well as the recent development of user-friendly software to accompany the method, we take a fresh look at this design and compare it to a current model-assisted method. We generated operating characteristics of the Conaway-Dunbar-Peddada method using a new web application developed for simulating and implementing the design and compared it to the recently proposed Keyboard design that is based on toxicity probability intervals. Results The Conaway-Dunbar-Peddada method has better performance in terms of accuracy of dose recommendation and safety in patient allocation in 17 of 20 scenarios considered. The Conaway-Dunbar-Peddada method also allocated fewer patients to doses above the maximum tolerated dose than the Keyboard method in many of scenarios studied. Overall, the performance of the Conaway-Dunbar-Peddada method is strong when compared to the Keyboard method, making it a viable simple alternative to the model-assisted methods developed in recent years. Conclusion The Conaway-Dunbar-Peddada method does not rely on the specification and fitting of a parametric model for the entire dose-toxicity curve to estimate toxicity probabilities as other model-based designs do. It relies on a similar set of pre-trial specifications to toxicity probability interval-based methods, yet unlike model-assisted methods, it is able to borrow information across all dose levels, increasing its efficiency. We hope this concise study of the Conaway-Dunbar-Peddada method, and the availability of user-friendly software, will augment its use in practice.

Order-restricted inference for means with missing values.

Missing values appear very often in many applications, but the problem of missing values has not received much attention in testing order-restricted alternatives. Under the missing at random (MAR) assumption, we impute the missing values nonparametrically using kernel regression. For data with imputation, the classical likelihood ratio test designed for testing the order-restricted means is no longer applicable since the likelihood does not exist. This article proposes a novel method for constructing test statistics for assessing means with an increasing order or a decreasing order based on jackknife empirical likelihood (JEL) ratio. It is shown that the JEL ratio statistic evaluated under the null hypothesis converges to a chi-bar-square distribution, whose weights depend on missing probabilities and nonparametric imputation. Simulation study shows that the proposed test performs well under various missing scenarios and is robust for normally and nonnormally distributed data. The proposed method is applied to an Alzheimer's disease neuroimaging initiative data set for finding a biomarker for the diagnosis of the Alzheimer's disease.

NONPARAMETRIC GOODNESS-OF-FIT TESTS FOR UNIFORM STOCHASTIC ORDERING.

We propose Lp distance-based goodness-of-fit (GOF) tests for uniform stochastic ordering with two continuous distributions F and G, both of which are unknown. Our tests are motivated by the fact that when F and G are uniformly stochastically ordered, the ordinal dominance curve R = FG-1 is star-shaped. We derive asymptotic distributions and prove that our testing procedure has a unique least favorable configuration of F and G for p ∈ [1,∞]. We use simulation to assess finite-sample performance and demonstrate that a modified, one-sample version of our procedure (e.g., with G known) is more powerful than the one-sample GOF test suggested by Arcones and Samaniego (2000, Annals of Statistics). We also discuss sample size determination. We illustrate our methods using data from a pharmacology study evaluating the effects of administering caffeine to prematurely born infants.

Multiple confidence intervals for selected parameters adjusted for the false coverage rate in monotone dose-response microarray experiments.

Benjamini and Yekutieli () introduced the concept of the false coverage-statement rate (FCR) to account for selection when the confidence intervals (CIs) are constructed only for the selected parameters. Dose-response analysis in dose-response microarray experiments is conducted only for genes having monotone dose-response relationship, which is a selection problem. In this paper, we consider multiple CIs for the mean gene expression difference between the highest dose and control in monotone dose-response microarray experiments for selected parameters adjusted for the FCR. A simulation study is conducted to study the performance of the method proposed. The method is applied to a real dose-response microarray experiment with 16, 998 genes for illustration.

CLME: An R Package for Linear Mixed Effects Models under Inequality Constraints.

In many applications researchers are typically interested in testing for inequality constraints in the context of linear fixed effects and mixed effects models. Although there exists a large body of literature for performing statistical inference under inequality constraints, user friendly statistical software for implementing such methods is lacking, especially in the context of linear fixed and mixed effects models. In this article we introduce CLME, a package in the R language that can be used for testing a broad collection of inequality constraints. It uses residual bootstrap based methodology which is reasonably robust to non-normality as well as heteroscedasticity. The package is illustrated using two data sets. The package also contains a graphical interface built using the shiny package.

Tests for stochastic ordering under biased sampling.

In two-sample comparison problems it is often of interest to examine whether one distribution function majorizes the other, i.e., for the presence of stochastic ordering. This paper develops a nonparametric test for stochastic ordering from size-biased data, allowing the pattern of the size bias to differ between the two samples. The test is formulated in terms of a maximally-selected local empirical likelihood statistic. A Gaussian multiplier bootstrap is devised to calibrate the test. Simulation results show that the proposed test outperforms an analogous Wald-type test, and that it provides substantially greater power over ignoring the size bias. The approach is illustrated using data on blood alcohol concentration of drivers involved in car accidents, where the size bias is due to drunker drivers being more likely to be involved in accidents. Further, younger drivers tend to be more affected by alcohol, so in making comparisons with older drivers the analysis is adjusted for differences in the patterns of size bias.

Estimation for two Gompertz populations under a balanced joint progressive Type-II censoring scheme.

Comparative lifetime experiments are remarkable when the study is to ascertain the relative merits of two competing products regarding the duration of their service life. This paper considers the comparative lifetime experiments of two Gompertz populations under a balanced joint progressive Type-II censoring scheme. The lifetime distributions of the units are assumed to follow the Gompertz distribution with a common shape but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived. The existence of the maximum likelihood estimates is proved. Expectation-maximization and stochastic expectation-maximization algorithms are provided to calculate the estimates. The bootstrap-p, bootstrap-t, and approximate confidence intervals are established. To obtain the Bayesian estimates, it is assumed that the prior of scale parameters is a Beta-Gamma distribution and the prior of the common shape parameter is an independent Gamma distribution. Under squared error loss and LINEX loss functions, the Metropolis-Hastings algorithm is provided to compute the Bayes estimates and the credible intervals. Further, the statistical inferences with order restriction are studied when it is known a priori that the expectation of the lifespan of one population is shorter than that of the other population. A wide range of simulation experiments is conducted to evaluate the performance of the proposed methods. Finally, the lifetimes of white organic light-emitting diodes and the breaking strengths of jute fiber of gauge lengths are analyzed to illustrate the practical application of the proposed model and methods.

Empirical likelihood-based tests for stochastic ordering.

This paper develops an empirical likelihood approach to testing for the presence of stochastic ordering among univariate distributions based on independent random samples from each distribution. The proposed test statistic is formed by integrating a localized empirical likelihood statistic with respect to the empirical distribution of the pooled sample. The asymptotic null distribution of this test statistic is found to have a simple distribution-free representation in terms of standard Brownian bridge processes. The approach is used to compare the lengths of rule of Roman Emperors over various historical periods, including the "decline and fall" phase of the empire. In a simulation study, the power of the proposed test is found to improve substantially upon that of a competing test due to El Barmi and Mukerjee.

Model comparison in ANOVA.

Analysis of variance (ANOVA), the workhorse analysis of experimental designs, consists of F-tests of main effects and interactions. Yet, testing, including traditional ANOVA, has been recently critiqued on a number of theoretical and practical grounds. In light of these critiques, model comparison and model selection serve as an attractive alternative. Model comparison differs from testing in that one can support a null or nested model vis-a-vis a more general alternative by penalizing more flexible models. We argue this ability to support more simple models allows for more nuanced theoretical conclusions than provided by traditional ANOVA F-tests. We provide a model comparison strategy and show how ANOVA models may be reparameterized to better address substantive questions in data analysis.

Empirical likelihood based tests for stochastic ordering under right censorship.

This paper develops an empirical likelihood approach to testing for stochastic ordering between two univariate distributions under right censorship. The proposed test is based on a maximally selected local empirical likelihood statistic. The asymptotic null distribution is expressed in terms of a Brownian bridge. The new procedure is shown via a simulation study to have superior power to the log-rank and weighted Kaplan-Meier tests under crossing hazard alternatives. The approach is illustrated using data from a randomized clinical trial involving the treatment of severe alcoholic hepatitis.

Bayesian Evaluation of inequality-constrained Hypotheses in SEM Models using Mplus.

Researchers in the behavioural and social sciences often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model resulting in an informative hypothesis. The question they would like an answer to is "Is the Hypothesis Correct" or "Is the hypothesis incorrect?". We demonstrate a Bayesian approach to compare an inequality-constrained hypothesis with its complement in an SEM framework. The method is introduced and its utility is illustrated by means of an example. Furthermore, the influence of the specification of the prior distribution is examined. Finally, it is shown how the approach proposed can be implemented using Mplus.