A nonparametric approach to detecting the difference of survival medians

Some nonparametric methods have been proposed to compare survival medians. Most of them are based on the asymptotic null distribution to estimate the p-value. However, for small to moderate sample sizes, those tests may have inflated Type I error rate, which makes their application limited. In this article, we proposed a new nonparametric test that uses bootstrap to estimate the sample mean and variance of the median. Through comprehensive simulation, we show that the proposed approach can control Type I error rates well. A real data application is used to illustrate the use of the new test.     

Equal‐tailed confidence intervals for comparison of rates

Several methods are available for generating confidence intervals for rate difference, rate ratio, or odds ratio, when comparing two independent binomial proportions or Poisson (exposure‐adjusted) incidence rates. Most methods have some degree of systematic bias in one‐sided coverage, so that a nominal 95% two‐sided interval cannot be assumed to have tail probabilities of 2.5% at each end, and any associated hypothesis test is at risk of inflated type I error rate. Skewness‐corrected asymptotic score methods have been shown to have superior equal‐tailed coverage properties for the binomial case. This paper completes this class of methods by introducing novel skewness corrections for the Poisson case and for odds ratio, with and without stratification. Graphical methods are used to compare the performance of these intervals against selected alternatives. The skewness‐corrected methods perform favourably in all situations—including those with small sample sizes or rare events—and the skewness correction should be considered essential for analysis of rate ratios. The stratified method is found to have excellent coverage properties for a fixed effects analysis. In addition, another new stratified score method is proposed, based on the t‐distribution, which is suitable for use in either a fixed effects or random effects analysis. By using a novel weighting scheme, this approach improves on conventional and modern meta‐analysis methods with weights that rely on crude estimation of stratum variances. In summary, this paper describes methods that are found to be robust for a wide range of applications in the analysis of rates.     

Post-hoc selection of covariates in randomized experiments

Monte Carlo methods are used to compere a number of adaptive strategies for deciding which of several covariates to incorporate into the analysis of a randomized experiment.Sixteen selection strategies in three categories are considered: 1)select covariates correlated with the response, 2)select covariates with means differing across groups, and 3)select covariates with means differing across groups that are also correlated with the response. The criteria examined are the type I error rate of the test for equality of adjusted group means and the variance of the estimated treatment effect. These strategies can result in either inflated or deflated type I errors, depending on the method and the population parameters. The adaptive methods in the first category some times yieldpoint estimates of the treatment effect more precise than estimators derive dusing either all or none of the covariates.     

Robust Small Sample Inference for Generalised Estimating Equations: An Application of the Anova‐type Test

Asymptotically, the Wald‐type test for generalised estimating equations (GEE) models can control the type I error rate at the nominal level. However in small sample studies, it may lead to inflated type I error rates. Even with currently available small sample corrections for the GEE Wald‐type test, the type I error rate inflation is still serious when the tested contrast is multidimensional. This paper extends the ANOVA‐type test for heteroscedastic factorial designs to GEE and shows that the proposed ANOVA‐type test can also control the type I error rate at the nominal level in small sample studies while still maintaining robustness with respect to mis‐specification of the working correlation matrix. Differences of inference between the Wald‐type test and the proposed test are observed in a two‐way repeated measures ANOVA model for a diet‐induced obesity study and a two‐way repeated measures logistic regression for a collagen‐induced arthritis study. Simulation studies confirm that the proposed test has better control of the type I error rate than the Wald‐type test in small sample repeated measures models. Additional simulation studies further show that the proposed test can even achieve larger power than the Wald‐type test in some cases of the large sample repeated measures ANOVA models that were investigated.     

A new kind of stochastic restricted biased estimator for logistic regression model

In the logistic regression model, the variance of the maximum likelihood estimator is inflated and unstable when the multicollinearity exists in the data. There are several methods available in literature to overcome this problem. We propose a new stochastic restricted biased estimator. We study the statistical properties of the proposed estimator and compare its performance with some existing estimators in the sense of scalar mean squared criterion. An example and a simulation study are provided to illustrate the performance of the proposed estimator.KEYWORDS: Logistic regression, maximum likelihood estimator, mean squared error matrix, ridge regression, simulation study, stochastic restricted estimatorMathematics Subject Classifications: Primary 62J05, Secondary 62J07     

Comparison of nonparametric analysis of variance methods: A vote for van der Waerden

ABSTRACT

For two-way layouts in a between-subjects analysis of variance design, the parametric F-test is compared with seven nonparametric methods: rank transform (RT), inverse normal transform (INT), aligned rank transform (ART), a combination of ART and INT, Puri & Sen's L statistic, Van der Waerden, and Akritas and Brunners ANOVA-type statistics (ATS). The type I error rates and the power are computed for 16 normal and nonnormal distributions, with and without homogeneity of variances, for balanced and unbalanced designs as well as for several models including the null and the full model. The aim of this study is to identify a method that is applicable without too much testing for all the attributes of the plot. The Van der Waerden test shows the overall best performance though there are some situations in which it is disappointing. The Puri & Sen's and the ATS tests show generally very low power. These two and the other methods cannot keep the type I error rate under control in too many situations. Especially in the case of lognormal distributions, the use of any of the rank-based procedures can be dangerous for cell sizes above 10. As already shown by many other authors, nonnormal distributions do not violate the parametric F-test, but unequal variances do, and heterogeneity of variances leads to an inflated error rate more or less also for the nonparametric methods. Finally, it should be noted that some procedures show rising error rates with increasing cell sizes, the ART, especially for discrete variables, and the RT, Puri & Sen, and the ATS in the cases of heteroscedasticity.     

The impact of missing data and how it is handled on the rate of false-positive results in drug development

In drug development, a common choice for the primary analysis is to assess mean changes via analysis of (co)variance with missing data imputed by carrying the last or baseline observations forward (LOCF, BOCF). These approaches assume that data are missing completely at random (MCAR). Multiple imputation (MI) and likelihood-based repeated measures (MMRM) are less restrictive as they assume data are missing at random (MAR). Nevertheless, LOCF and BOCF remain popular, perhaps because it is thought that the bias in these methods lead to protection against falsely concluding that a drug is more effective than the control. We conducted a simulation study that compared the rate of false positive results or regulatory risk error (RRE) from BOCF, LOCF, MI, and MMRM in 32 scenarios that were generated from a 2(5) full factorial arrangement with data missing due to a missing not at random (MNAR) mechanism. Both BOCF and LOCF inflated RRE were compared to MI and MMRM. In 12 of the 32 scenarios, BOCF yielded inflated RRE compared with eight scenarios for LOCF, three scenarios for MI and four scenarios for MMRM. In no situation did BOCF or LOCF provide adequate control of RRE when MI and MMRM did not. Both MI and MMRM are better choices than either BOCF or LOCF for the primary analysis.     

A generalized pivotal quantity approach to portfolio selection

The major problem of mean–variance portfolio optimization is parameter uncertainty. Many methods have been proposed to tackle this problem, including shrinkage methods, resampling techniques, and imposing constraints on the portfolio weights, etc. This paper suggests a new estimation method for mean–variance portfolio weights based on the concept of generalized pivotal quantity (GPQ) in the case when asset returns are multivariate normally distributed and serially independent. Both point and interval estimations of the portfolio weights are considered. Comparing with Markowitz's mean–variance model, resampling and shrinkage methods, we find that the proposed GPQ method typically yields the smallest mean-squared error for the point estimate of the portfolio weights and obtains a satisfactory coverage rate for their simultaneous confidence intervals. Finally, we apply the proposed methodology to address a portfolio rebalancing problem.     

A novel equivalence probability weighted power prior for using historical control data in an adaptive clinical trial design: A comparison to standard methods

A standard two-arm randomised controlled trial usually compares an intervention to a control treatment with equal numbers of patients randomised to each treatment arm and only data from within the current trial are used to assess the treatment effect. Historical data are used when designing new trials and have recently been considered for use in the analysis when the required number of patients under a standard trial design cannot be achieved. Incorporating historical control data could lead to more efficient trials, reducing the number of controls required in the current study when the historical and current control data agree. However, when the data are inconsistent, there is potential for biased treatment effect estimates, inflated type I error and reduced power. We introduce two novel approaches for binary data which discount historical data based on the agreement with the current trial controls, an equivalence approach and an approach based on tail area probabilities. An adaptive design is used where the allocation ratio is adapted at the interim analysis, randomising fewer patients to control when there is agreement. The historical data are down-weighted in the analysis using the power prior approach with a fixed power. We compare operating characteristics of the proposed design to historical data methods in the literature: the modified power prior; commensurate prior; and robust mixture prior. The equivalence probability weight approach is intuitive and the operating characteristics can be calculated exactly. Furthermore, the equivalence bounds can be chosen to control the maximum possible inflation in type I error.     

Adjusted homogeneity tests of odds ratios when data are clustered

Three modified tests for homogeneity of the odds ratio for a series of 2 × 2 tables are studied when the data are clustered. In the case of clustered data, the standard tests for homogeneity of odds ratios ignore the variance inflation caused by positive correlation among responses of subjects within the same cluster, and therefore have inflated Type I error. The modified tests adjust for the variance inflation in the three existing standard tests: Breslow–Day, Tarone and the conditional score test. The degree of clustering effect is measured by the intracluster correlation coefficient, ρ. A variance correction factor derived from ρ is then applied to the variance estimator in the standard tests of homogeneity of the odds ratio. The proposed tests are an application of the variance adjustment method commonly used in correlated data analysis and are shown to maintain the nominal significance level in a simulation study. Copyright © 2004 John Wiley & Sons, Ltd.     

Variance estimation for sparse ultra-high dimensional varying coefficient models

This paper considers the problem of variance estimation for sparse ultra-high dimensional varying coefficient models. We first use B-spline to approximate the coefficient functions, and discuss the asymptotic behavior of a naive two-stage estimator of error variance. We also reveal that this naive estimator may significantly underestimate the error variance due to the spurious correlations, which are even higher for nonparametric models than linear models. This prompts us to propose an accurate estimator of the error variance by effectively integrating the sure independence screening and the refitted cross-validation techniques. The consistency and the asymptotic normality of the resulting estimator are established under some regularity conditions. The simulation studies are carried out to assess the finite sample performance of the proposed methods.     

Short confidence intervals for variance components

Four approximate methods are proposed to construct confidence intervals for the estimation of variance components in unbalanced mixed models. The first three methods are modifications of the Wald, arithmetic and harmonic mean procedures, see Harville and Fenech (1985), while the fourth is an adaptive approach, combining the arithmetic and harmonic mean procedures. The performances of the proposed methods were assessed by a Monte Carlo simulation study. It was found that the intervals based on Wald's method maintained the nominal confidence levels across all designs and values of the parameters under study. On the other hand, the arithmetic (harmonic) mean method performed well for small (large) values of the variance component, relative to the error variance component. The adaptive procedure performed rather well except for extremely unbalanced designs. Further, compared with equal tails intervals, the intervals which use special tables, e.g., Table 678 of Tate and Klett (1959), provided adequate coverage while having much shorter lengths and are thus recommended for use in practice.     

Statistical considerations in bioequivalence of two area under the concentration–time curves obtained from serial sampling data

In this paper, we study the bioequivalence (BE) inference problem motivated by pharmacokinetic data that were collected using the serial sampling technique. In serial sampling designs, subjects are independently assigned to one of the two drugs; each subject can be sampled only once, and data are collected at K distinct timepoints from multiple subjects. We consider design and hypothesis testing for the parameter of interest: the area under the concentration–time curve (AUC). Decision rules in demonstrating BE were established using an equivalence test for either the ratio or logarithmic difference of two AUCs. The proposed t-test can deal with cases where two AUCs have unequal variances. To control for the type I error rate, the involved degrees-of-freedom were adjusted using Satterthwaite's approximation. A power formula was derived to allow the determination of necessary sample sizes. Simulation results show that, when the two AUCs have unequal variances, the type I error rate is better controlled by the proposed method compared with a method that only handles equal variances. We also propose an unequal subject allocation method that improves the power relative to that of the equal and symmetric allocation. The methods are illustrated using practical examples.     

Optimal determination of the parameters of some biased estimators using genetic algorithm

In this paper, some new algorithms for estimating the biasing parameters of the ridge, Liu and two-parameter estimators are introduced with the help of genetic algorithm (GA). The proposed algorithms are based on minimizing some statistical measures such as mean square error (MSE), mean absolute error (MAE) and mean absolute prediction error (MAPE). At the same time, the new algorithms allow one to keep the condition number and variance inflation factors to be less than or equal to ten by means of the GA. A numerical example is presented to show the utility of the new algorithms. In addition, an extensive Monte Carlo experiment is conducted. The numerical findings prove that the proposed algorithms enable to eliminate the problem of multicollinearity and minimize the MSE, MAE and MAPE.     

A Bayesian sequential design with binary outcome

Several researchers have proposed solutions to control type I error rate in sequential designs. The use of Bayesian sequential design becomes more common; however, these designs are subject to inflation of the type I error rate. We propose a Bayesian sequential design for binary outcome using an alpha‐spending function to control the overall type I error rate. Algorithms are presented for calculating critical values and power for the proposed designs. We also propose a new stopping rule for futility. Sensitivity analysis is implemented for assessing the effects of varying the parameters of the prior distribution and maximum total sample size on critical values. Alpha‐spending functions are compared using power and actual sample size through simulations. Further simulations show that, when total sample size is fixed, the proposed design has greater power than the traditional Bayesian sequential design, which sets equal stopping bounds at all interim analyses. We also find that the proposed design with the new stopping for futility rule results in greater power and can stop earlier with a smaller actual sample size, compared with the traditional stopping rule for futility when all other conditions are held constant. Finally, we apply the proposed method to a real data set and compare the results with traditional designs.     

Power and stability comparisons of multiple testing procedures with false discovery rate control

High-throughput data analyses are widely used for examining differential gene expression, identifying single nucleotide polymorphisms, and detecting methylation loci. False discovery rate (FDR) has been considered a proper type I error rate to control for discovery-based high-throughput data analysis. Various multiple testing procedures have been proposed to control the FDR. The power and stability properties of some commonly used multiple testing procedures have not been extensively investigated yet, however. Simulation studies were conducted to compare power and stability properties of five widely used multiple testing procedures at different proportions of true discoveries for various sample sizes for both independent and dependent test statistics. Storey's two linear step-up procedures showed the best performance among all tested procedures considering FDR control, power, and variance of true discoveries. Leukaemia and ovarian cancer microarray studies were used to illustrate the power and stability characteristics of these five multiple testing procedures with FDR control.     

Evaluation of false discovery rate and power via sample size in microarray studies

Microarray studies are now common for human, agricultural plant and animal studies. False discovery rate (FDR) is widely used in the analysis of large-scale microarray data to account for problems associated with multiple testing. A well-designed microarray study should have adequate statistical power to detect the differentially expressed (DE) genes, while keeping the FDR acceptably low. In this paper, we used a mixture model of expression responses involving DE genes and non-DE genes to analyse theoretical FDR and power for simple scenarios where it is assumed that each gene has equal error variance and the gene effects are independent. A simulation study was used to evaluate the empirical FDR and power for more complex scenarios with unequal error variance and gene dependence. Based on this approach, we present a general guide for sample size requirement at the experimental design stage for prospective microarray studies. This paper presented an approach to explicitly connect the sample size with FDR and power. While the methods have been developed in the context of one-sample microarray studies, they are readily applicable to two-sample, and could be adapted to multiple-sample studies.     

Two-parameter ridge estimator in the binary logistic regression

The binary logistic regression is a commonly used statistical method when the outcome variable is dichotomous or binary. The explanatory variables are correlated in some situations of the logit model. This problem is called multicollinearity. It is known that the variance of the maximum likelihood estimator (MLE) is inflated in the presence of multicollinearity. Therefore, in this study, we define a new two-parameter ridge estimator for the logistic regression model to decrease the variance and overcome multicollinearity problem. We compare the new estimator to the other well-known estimators by studying their mean squared error (MSE) properties. Moreover, a Monte Carlo simulation is designed to evaluate the performances of the estimators. Finally, a real data application is illustrated to show the applicability of the new method. According to the results of the simulation and real application, the new estimator outperforms the other estimators for all of the situations considered.     

A Rank-Based Test for Comparison of Multidimensional Outcomes

For comparison of multiple outcomes commonly encountered in biomedical research, Huang et al. (2005) improved O'Brien's (1984) rank-sum tests through the replacement of the ad hoc variance by the asymptotic variance of the test statistics. The improved tests control the Type I error rate at the desired level and gain power when the differences between the two comparison groups in each outcome variable fall into the same direction. However, they may lose power when the differences are in different directions (e.g., some are positive and some are negative). These tests and the popular Bonferroni correction failed to show important significant difference when applied to compare heart rates from a clinical trial to evaluate the effect of a procedure to remove the cardioprotective solution HTK. We propose an alternative test statistic, taking the maximum of the individual rank-sum statistics, which controls the type I error and maintains satisfactory power regardless of the directions of the differences. Simulation studies show the proposed test to be of higher power than other tests in certain alternative parameter space of interest. Furthermore, when used to analyze the heart rates data the proposed test yields more satisfactory results.     

Confidence intervals for the difference between independent binomial proportions: comparison using a graphical approach and moving averages

This paper uses graphical methods to illustrate and compare the coverage properties of a number of methods for calculating confidence intervals for the difference between two independent binomial proportions. We investigate both small‐sample and large‐sample properties of both two‐sided and one‐sided coverage, with an emphasis on asymptotic methods. In terms of aligning the smoothed coverage probability surface with the nominal confidence level, we find that the score‐based methods on the whole have the best two‐sided coverage, although they have slight deficiencies for confidence levels of 90% or lower. For an easily taught, hand‐calculated method, the Brown‐Li ‘Jeffreys’ method appears to perform reasonably well, and in most situations, it has better one‐sided coverage than the widely recommended alternatives. In general, we find that the one‐sided properties of many of the available methods are surprisingly poor. In fact, almost none of the existing asymptotic methods achieve equal coverage on both sides of the interval, even with large sample sizes, and consequently if used as a non‐inferiority test, the type I error rate (which is equal to the one‐sided non‐coverage probability) can be inflated. The only exception is the Gart‐Nam ‘skewness‐corrected’ method, which we express using modified notation in order to include a bias correction for improved small‐sample performance, and an optional continuity correction for those seeking more conservative coverage. Using a weighted average of two complementary methods, we also define a new hybrid method that almost matches the performance of the Gart‐Nam interval. Copyright © 2014 John Wiley & Sons, Ltd.