Applications of asymptotic confidence intervals with continuity corrections for asymmetric comparisons in noninferiority trials

A large‐sample problem of illustrating noninferiority of an experimental treatment over a referent treatment for binary outcomes is considered. The methods of illustrating noninferiority involve constructing the lower two‐sided confidence bound for the difference between binomial proportions corresponding to the experimental and referent treatments and comparing it with the negative value of the noninferiority margin. The three considered methods, Anbar, Falk–Koch, and Reduced Falk–Koch, handle the comparison in an asymmetric way, that is, only the referent proportion out of the two, experimental and referent, is directly involved in the expression for the variance of the difference between two sample proportions. Five continuity corrections (including zero) are considered with respect to each approach. The key properties of the corresponding methods are evaluated via simulations. First, the uncorrected two‐sided confidence intervals can, potentially, have smaller coverage probability than the nominal level even for moderately large sample sizes, for example, 150 per group. Next, the 15 testing methods are discussed in terms of their Type I error rate and power. In the settings with a relatively small referent proportion (about 0.4 or smaller), the Anbar approach with Yates’ continuity correction is recommended for balanced designs and the Falk–Koch method with Yates’ correction is recommended for unbalanced designs. For relatively moderate (about 0.6) and large (about 0.8 or greater) referent proportion, the uncorrected Reduced Falk–Koch method is recommended, although in this case, all methods tend to be over‐conservative. These results are expected to be used in the design stage of a noninferiority study when asymmetric comparisons are envisioned. Copyright © 2013 John Wiley & Sons, Ltd.     

Sample size calculations for noninferiority trials for time-to-event data using the concept of proportional time

Noninferiority trials intend to show that a new treatment is ‘not worse'' than a standard-of-care active control and can be used as an alternative when it is likely to cause fewer side effects compared to the active control. In the case of time-to-event endpoints, existing methods of sample size calculation are done either assuming proportional hazards between the two study arms, or assuming exponentially distributed lifetimes. In scenarios where these assumptions are not true, there are few reliable methods for calculating the sample sizes for a time-to-event noninferiority trial. Additionally, the choice of the non-inferiority margin is obtained either from a meta-analysis of prior studies, or strongly justifiable ‘expert opinion'', or from a ‘well conducted'' definitive large-sample study. Thus, when historical data do not support the traditional assumptions, it would not be appropriate to use these methods to design a noninferiority trial. For such scenarios, an alternate method of sample size calculation based on the assumption of Proportional Time is proposed. This method utilizes the generalized gamma ratio distribution to perform the sample size calculations. A practical example is discussed, followed by insights on choice of the non-inferiority margin, and the indirect testing of superiority of treatment compared to placebo.KEYWORDS: Generalized gamma, noninferiority, non-proportional hazards, proportional time, relative time, sample size     

Blinded sample size re‐estimation in superiority and noninferiority trials: bias versus variance in variance estimation

The internal pilot study design allows for modifying the sample size during an ongoing study based on a blinded estimate of the variance thus maintaining the trial integrity. Various blinded sample size re‐estimation procedures have been proposed in the literature. We compare the blinded sample size re‐estimation procedures based on the one‐sample variance of the pooled data with a blinded procedure using the randomization block information with respect to bias and variance of the variance estimators, and the distribution of the resulting sample sizes, power, and actual type I error rate. For reference, sample size re‐estimation based on the unblinded variance is also included in the comparison. It is shown that using an unbiased variance estimator (such as the one using the randomization block information) for sample size re‐estimation does not guarantee that the desired power is achieved. Moreover, in situations that are common in clinical trials, the variance estimator that employs the randomization block length shows a higher variability than the simple one‐sample estimator and in turn the sample size resulting from the related re‐estimation procedure. This higher variability can lead to a lower power as was demonstrated in the setting of noninferiority trials. In summary, the one‐sample estimator obtained from the pooled data is extremely simple to apply, shows good performance, and is therefore recommended for application. Copyright © 2013 John Wiley & Sons, Ltd.     

Design and analysis of a 3‐arm noninferiority trial with a prespecified margin for the hazard ratio

A 3‐arm trial design that includes an experimental treatment, an active reference treatment, and a placebo is useful for assessing the noninferiority of an experimental treatment. The inclusion of a placebo arm enables the assessment of assay sensitivity and internal validation, in addition to the testing of the noninferiority of the experimental treatment compared with the reference treatment. In 3‐arm noninferiority trials, various statistical test procedures have been considered to evaluate the following 3 hypotheses: (i) superiority of the experimental treatment over the placebo, (ii) superiority of the reference treatment over the placebo, and (iii) noninferiority of the experimental treatment compared with the reference treatment. However, hypothesis (ii) can be insufficient and may not accurately assess the assay sensitivity for the noninferiority of the experimental treatment compared with the reference treatment. Thus, demonstrating that the superiority of the reference treatment over the placebo is greater than the noninferiority margin (the nonsuperiority of the reference treatment compared with the placebo) can be necessary. Here, we propose log‐rank statistical procedures for evaluating data obtained from 3‐arm noninferiority trials to assess assay sensitivity with a prespecified margin Δ. In addition, we derive the approximate sample size and optimal allocation required to minimize the total sample size and that of the placebo treatment sample size, hierarchically.     

Assessing noninferiority: Evaluating efficacy of a new treatment without complete data

The FDA released the final guidance on noninferiority trials in November 2016. In noninferiority trials, validity of the assessment of the efficacy of the test treatment depends on the control treatment's efficacy. Therefore, it is critically important that there be a reliable estimate of the control treatment effect—which is generally obtained from historical trials, and often assumed to hold in the current setting (the assay constancy assumption). Validating the constancy assumption requires clinical data, which are typically lacking. The guidance acknowledges that “lack of constancy can occur for many reasons.” We clarify the objectives of noninferiority trials. We conclude that correction for bias, rather than assay constancy, is critical to conducting valid noninferiority trials. We propose that assay constancy not be assumed and discounting or thresholds be used to address concern about loss of historical efficacy. Examples are provided for illustration.     

A dynamic power prior for borrowing historical data in noninferiority trials with binary endpoint

Traditionally, noninferiority hypotheses have been tested using a frequentist method with a fixed margin. Given that information for the control group is often available from previous studies, it is interesting to consider a Bayesian approach in which information is “borrowed” for the control group to improve efficiency. However, construction of an appropriate informative prior can be challenging. In this paper, we consider a hybrid Bayesian approach for testing noninferiority hypotheses in studies with a binary endpoint. To account for heterogeneity between the historical information and the current trial for the control group, a dynamic P value–based power prior parameter is proposed to adjust the amount of information borrowed from the historical data. This approach extends the simple test‐then‐pool method to allow a continuous discounting power parameter. An adjusted α level is also proposed to better control the type I error. Simulations are conducted to investigate the performance of the proposed method and to make comparisons with other methods including test‐then‐pool and hierarchical modeling. The methods are illustrated with data from vaccine clinical trials.     

Adaptive and repeated cumulative meta‐analyses of safety data during a new drug development process

During a new drug development process, it is desirable to timely detect potential safety signals. For this purpose, repeated meta‐analyses may be performed sequentially on accumulating safety data. Moreover, if the amount of safety data from the originally planned program is not enough to ensure adequate power to test a specific hypothesis (e.g., the noninferiority hypothesis of an event of interest), the total sample size may be increased by adding new studies to the program. Without appropriate adjustment, it is well known that the type I error rate will be inflated because of repeated analyses and sample size adjustment. In this paper, we discuss potential issues associated with adaptive and repeated cumulative meta‐analyses of safety data conducted during a drug development process. We consider both frequentist and Bayesian approaches. A new drug development example is used to demonstrate the application of the methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Testing for noninferiority of binomial distributions referring to a modified equivalence region with piecewise linear boundary

In testing for noninferiority of two binomial distributions, the hypothesis formulation most commonly considered defines equivalence in terms of a constant bound to the difference of the two parameters. In order to avoid some basic logical difficulty entailed in this formulation we use an equivalence region whose boundary has fixed vertical distance from the diagonal for all values of the reference responder rate above some cutoff point and coincides left from this point with the line joining it with the origin. For the corresponding noninferiority hypothesis we derive and compare two different testing procedures. The first one is based on an objective Bayesian decision rule. The other one is obtained through combining the score tests for noninferiority with respect to the difference and the ratio of the two proportions, respectively, by means of the intersection–union principle. Both procedures are extensively studied by means of exact computational methods.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

Optimal sample size for record data and associated cost analysis for exponential distribution

Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] and Doostparast [M. Doostparast, A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. When a random sample Y ₁, …, Y _n is examined sequentially and successive minimum values are recorded, Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] obtained a maximum likelihood estimator of the mean of the population and showed its convergence in probability. We establish here its convergence in mean square error, which is stronger than the convergence in probability. Next, we discuss the optimal sample size for estimating the mean based on a criterion involving a cost function as well as the Fisher information based on records arising from a random sample. Finally, a comparison between complete data and record is carried out and some special cases are discussed in detail.     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.     

Goodness-of-fit tests for progressively Type-II censored data from location–scale distributions

In this article, we propose several goodness-of-fit methods for location–scale families of distributions under progressively Type-II censored data. The new tests are based on order statistics and sample spacings. We assess the performance of the proposed tests for the normal and Gumbel models against several alternatives by means of Monte Carlo simulations. It has been observed that the proposed tests are quite powerful in comparison with an existing goodness-of-fit test proposed for progressively Type-II censored data by Balakrishnan et al. [Goodness-of-fit tests based on spacings for progressively Type-II censored data from a general location–scale distribution, IEEE Trans. Reliab. 53 (2004), pp. 349–356]. Finally, we illustrate the proposed goodness-of-fit tests using two real data from reliability literature.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.     

Stein-type estimation using ranked set sampling

In this study, we consider the application of the James–Stein estimator for population means from a class of arbitrary populations based on ranked set sample (RSS). We consider a basis for optimally combining sample information from several data sources. We succinctly develop the asymptotic theory of simultaneous estimation of several means for differing replications based on the well-defined shrinkage principle. We showcase that a shrinkage-type estimator will have, under quadratic loss, a substantial risk reduction relative to the classical estimator based on simple random sample and RSS. Asymptotic distributional quadratic biases and risks of the shrinkage estimators are derived and compared with those of the classical estimator. A simulation study is used to support the asymptotic result. An over-riding theme of this study is that the shrinkage estimation method provides a powerful extension of its traditional counterpart for non-normal populations. Finally, we will use a real data set to illustrate the computation of the proposed estimators.     

Tests for noninferiority trials with binomial endpoints: A guide to modern and quasi‐exact methods for biomedical researchers

Applied statisticians and pharmaceutical researchers are frequently involved in the design and analysis of clinical trials where at least one of the outcomes is binary. Treatments are judged by the probability of a positive binary response. A typical example is the noninferiority trial, where it is tested whether a new experimental treatment is practically not inferior to an active comparator with a prespecified margin δ. Except for the special case of δ = 0, no exact conditional test is available although approximate conditional methods (also called second‐order methods) can be applied. However, in some situations, the approximation can be poor and the logical argument for approximate conditioning is not compelling. The alternative is to consider an unconditional approach. Standard methods like the pooled z‐test are already unconditional although approximate. In this article, we review and illustrate unconditional methods with a heavy emphasis on modern methods that can deliver exact, or near exact, results. For noninferiority trials based on either rate difference or rate ratio, our recommendation is to use the so‐called E‐procedure, based on either the score or likelihood ratio statistic. This test is effectively exact, computationally efficient, and respects monotonicity constraints in practice. We support our assertions with a numerical study, and we illustrate the concepts developed in theory with a clinical example in pulmonary oncology; R code to conduct all these analyses is available from the authors.     

Estimating prediction error for complex samples

With a growing interest in using non-representative samples to train prediction models for numerous outcomes it is necessary to account for the sampling design that gives rise to the data in order to assess the generalized predictive utility of a proposed prediction rule. After learning a prediction rule based on a non-uniform sample, it is of interest to estimate the rule's error rate when applied to unobserved members of the population. Efron (1986) proposed a general class of covariance penalty inflated prediction error estimators that assume the available training data are representative of the target population for which the prediction rule is to be applied. We extend Efron's estimator to the complex sample context by incorporating Horvitz–Thompson sampling weights and show that it is consistent for the true generalization error rate when applied to the underlying superpopulation. The resulting Horvitz–Thompson–Efron estimator is equivalent to dAIC, a recent extension of Akaike's information criteria to survey sampling data, but is more widely applicable. The proposed methodology is assessed with simulations and is applied to models predicting renal function obtained from the large-scale National Health and Nutrition Examination Study survey. The Canadian Journal of Statistics 48: 204–221; 2020 © 2019 Statistical Society of Canada     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Efficient tests for one sample correlated binary data with applications

Four testing procedures are considered for testing the response rate of one sample correlated binary data with a cluster size of one or two, which often occurs in otolaryngologic and ophthalmologic studies. Although an asymptotic approach is often used for statistical inference, it is criticized for unsatisfactory type I error control in small sample settings. An alternative to the asymptotic approach is an unconditional approach. The first unconditional approach is the one based on estimation, also known as parametric bootstrap (Lee and Young in Stat Probab Lett 71(2):143–153, 2005). The other two unconditional approaches considered in this article are an approach based on maximization (Basu in J Am Stat Assoc 72(358):355–366, 1977), and an approach based on estimation and maximization (Lloyd in Biometrics 64(3):716–723, 2008a). These two unconditional approaches guarantee the test size and are generally more reliable than the asymptotic approach. We compare these four approaches in conjunction with a test proposed by Lee and Dubin (Stat Med 13(12):1241–1252, 1994) and a likelihood ratio test derived in this article, in regards to type I error rate and power for sample sizes from small to medium. An example from an otolaryngologic study is provided to illustrate the various testing procedures. The unconditional approach based on estimation and maximization using the test in Lee and Dubin (Stat Med 13(12):1241–1252, 1994) is preferable due to the power advantageous.     

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.