Profile likelihood-based confidence intervals using Monte Carlo integration for population pharmacokinetic parameters

Population pharmacokinetic (PPK) analysis usually employs nonlinear mixed effects models using first-order linearization methods. It is well known that linearization methods do not always perform well in actual situations. To avoid linearization, the Monte Carlo integration method has been proposed. Moreover, we generally utilize asymptotic confidence intervals for PPK parameters based on Fisher information. It is known that likelihood-based confidence intervals are more accurate than those from the usual asymptotic confidence intervals. We propose profile likelihood-based confidence intervals using Monte Carlo integration. We have evaluated the performance of the proposed method through a simulation study, and analyzed the erythropoietin concentration data set by the method.     

Profile Likelihood-Based Confidence Intervals Using Monte Carlo Integration for Population Pharmacokinetic Parameters

Population pharmacokinetic (PPK) analysis usually employs nonlinear mixed effects models using first-order linearization methods. It is well known that linearization methods do not always perform well in actual situations. To avoid linearization, the Monte Carlo integration method has been proposed. Moreover, we generally utilize asymptotic confidence intervals for PPK parameters based on Fisher information. It is known that likelihood-based confidence intervals are more accurate than those from the usual asymptotic confidence intervals. We propose profile likelihood-based confidence intervals using Monte Carlo integration. We have evaluated the performance of the proposed method through a simulation study, and analyzed the erythropoietin concentration data set by the method.     

Tests for inter-subject and total variabilities under crossover designs

In this paper, we consider statistical tests for inter-subject and total variabilities between treatments under crossover designs. Since estimators of variance components for inter-subject variability and total variability in crossover design are not independent, the usual F-test cannot be applied. Alternatively, we propose a test based on the concept of the extension of the modified large sample method to compare inter-subject variability and total variability between treatments under a 2 x 2 m replicated crossover design. An asymptotic power of the proposed test is derived. A sensitivity analysis is performed based on the asymptotic power to determine how the power changes with respect to various parameters such as inter-subject correlation and intra-class correlation. Also the two methods for sample size calculation for testing total variability under 2 x 4 crossover design are discussed. The method based on the Fisher-Cornish inversion shows better performance than the method based on the normal approximation. Several simulation studies were conducted to investigate the finite sample performance of the proposed test. Our simulation results show that the proposed test can control type I error satisfactorily.     

DROPOUTS IN LONGITUDINAL STUDIES: DEFINITIONS AND MODELS

In this paper, we consider statistical tests for inter-subject and total variabilities between treatments under crossover designs. Since estimators of variance components for inter-subject variability and total variability in crossover design are not independent, the usual F-test cannot be applied. Alternatively, we propose a test based on the concept of the extension of the modified large sample method to compare inter-subject variability and total variability between treatments under a 2×2mreplicated crossover design. An asymptotic power of the proposed test is derived. A sensitivity analysis is performed based on the asymptotic power to determine how the power changes with respect to various parameters such as inter-subject correlation and intra-class correlation. Also the two methods for sample size calculation for testing total variability under 2×4 crossover design are discussed. The method based on the Fisher–Cornish inversion shows better performance than the method based on the normal approximation. Several simulation studies were conducted to investigate the finite sample performance of the proposed test. Our simulation results show that the proposed test can control type I error satisfactorily.     

Sample sizes based on exact unconditional tests for phase II clinical trials with historical controls

Investigated in the setting of phase II clinical trials is the two-sample binomial problem of testing H0: pe = pc H1: pe > pc, where pe and pc are the unknown target population response rates for the experimental and control groups, respectively, using the usual Z-statistic with pooled variance estimator. The cornerstones that make this paper unique are as follows. First, the emphasis is on determining the sample size given that the control group information has already been collected (historical control). Second, exact unconditional inference, rather than an asymptotic method, is utilized. Sample size tables, contrasting the exact and asymptotic methods, are provided. Although asymptotic results were usually fairly close to the exact results, some important differences were observed.     

Sample size calculation based on efficient unconditional tests for clinical trials with historical controls

In historical clinical trials, the sample size and the number of success in the control group are often considered as given. The traditional method for sample size calculation is based on an asymptotic approach developed by Makuch and Simon (1980). Exact unconditional approaches may be considered as alternative to control for the type I error rate where the asymptotic approach may fail to do so. We provide the sample size calculation using an efficient exact unconditional testing procedure based on estimation and maximization. The sample size using the exact unconditional approach based on estimation and maximization is generally smaller than those based on the other approaches.     

Nonparametric Sample Size Estimation for Sensitivity and Specificity with Multiple Observations per Subject

We propose a sample size calculation approach for the estimation of sensitivity and specificity of diagnostic tests with multiple observations per subjects. Many diagnostic tests such as diagnostic imaging or periodontal tests are characterized by the presence of multiple observations for each subject. The number of observations frequently varies among subjects in diagnostic imaging experiments or periodontal studies. Nonparametric statistical methods for the analysis of clustered binary data have been recently developed by various authors. In this paper, we derive a sample size formula for sensitivity and specificity of diagnostic tests using the sign test while accounting for multiple observations per subjects. Application of the sample size formula for the design of a diagnostic test is discussed. Since the sample size formula is based on large sample theory, simulation studies are conducted to evaluate the finite sample performance of the proposed method. We compare the performance of the proposed sample size formula with that of the parametric sample size formula that assigns equal weight to each observation. Simulation studies show that the proposed sample size formula generally yields empirical powers closer to the nominal level than the parametric method. Simulation studies also show that the number of subjects required increases as the variability in the number of observations per subject increases and the intracluster correlation increases.     

Sample Size Calculation for the van Elteren Test Adjusting for Ties

In this article we study sample size calculation methods for the asymptotic van Elteren test. Because the existing methods are only applicable to continuous data without ties, in this article we develop a new method that can be used on ordinal data. The new method has a closed form formula and is very easy to calculate. The new sample size formula performs very well because our simulations show that the corresponding actual powers are close to the nominal powers.     

Power and sample size determination for noninferiority trials using an exact method

Noninferiority studies are frequently conducted to justify the development of new drugs and vaccines that have been shown to offer better safety profiles, easier administration, or lower cost while maintaining similar efficacy as compared to the standard treatment. Recently, exact methods have been developed to address the concern that existing asymptotic methods for analyzing and planning noninferiority may fail because of small sample size or because of skewed or sparse data structure. In this paper, we explore the use of exact methods in determining sample size and power for noninferiority studies that focus on the difference of two proportions. The methodology for sample size and power calculations is developed based on an exact unconditional test of noninferiority. We illustrate this exact method using a clinical trial example in childhood nephroblastoma and briefly discuss the optimal sample-size allocation strategy. This exact unconditional method performs very well in various scenarios and compares favorably to its asymptotic counterpart in terms of sensitivity. Therefore, it is a very desirable tool for planning noninferiority trials, especially in situations where asymptotic methods are likely to fail.     

Sample size calculation for the van Elteren test adjusting for ties

In this article we study sample size calculation methods for the asymptotic van Elteren test. Because the existing methods are only applicable to continuous data without ties, in this article we develop a new method that can be used on ordinal data. The new method has a closed form formula and is very easy to calculate. The new sample size formula performs very well because our simulations show that the corresponding actual powers are close to the nominal powers.     

Two-sided tolerance intervals for balanced and unbalanced random effects models

A procedure for constructing two-sided beta-content, gamma-confidence tolerance intervals is proposed for general random effects models, in both balanced and unbalanced data scenarios. The proposed intervals are based on the concept of effective sample size and modified large sample methods for constructing confidence bounds on functions of variance components. The performance of the proposed intervals is evaluated via simulation techniques. The results indicate that the proposed intervals generally maintain the nominal confidence and content levels. Application of the proposed procedure is illustrated with a one-fold nested design used to evaluate the performance of a quantitative bioanalytical method.     

A NONPARAMETRIC PROCEDURE OF THE SAMPLE SIZE DETERMINATION FOR SURVIVAL RATE TEST

Inthesurvivalstudiesofchronicdiseases,themostfrequentlyusedmethodsofdataanalysisarethenonparametricestimationsandcomparisonsofsurvivalratesatdifferenttimesbuttherehavebeennomatchedmethodsfordeterminingtherequiredsamplesizesofar.Insteadhavebeenoftenusedtheparametricmethodsbasedontheassumptionofexponentialdistributions[1～7]suchthattherelevantsamplesizesmaynotbesatisfiedwiththeprescribedpower.Thispaperreportsanonparametricprocedureofthesamplesizedeterminationforsurvivalratetest.1　SurvivalRateT…     

Exact and Approximate Power and Sample Size Calculations for Analysis of Covariance in Randomized Clinical Trials With or Without Stratification

ABSTRACT

Analysis of covariance (ANCOVA) is commonly used in the analysis of randomized clinical trials to adjust for baseline covariates and improve the precision of the treatment effect estimate. We derive the exact power formulas for testing a homogeneous treatment effect in superiority, noninferiority, and equivalence trials under both unstratified and stratified randomizations, and for testing the overall treatment effect and treatment × stratum interaction in the presence of heterogeneous treatment effects when the covariates excluding the intercept, treatment, and prestratification factors are normally distributed. These formulas also work very well for nonnormal covariates. The sample size methods based on the normal approximation or the asymptotic variance generally underestimate the required size. We adapt the recently developed noniterative and two-step sample size procedures to the above tests. Both methods take into account the nonnormality of the t statistic, and the lower order variance term commonly ignored in the sample size estimation. Numerical examples demonstrate the excellent performance of the proposed methods particularly in small samples. We revisit the topic on the prestratification versus post-stratification by comparing their relative efficiency and power. Supplementary materials for this article are available online.     

Conservative Sample Size Estimation in Nonparametrics

Due to the uncertainty of the results of phase II trials, underpowered phase III trials are often planned. In recent literature the conservative approach for sample size estimation was proposed. Some authors, in the parametric framework, make use of the lower bound of the effect size for conservatively estimating the true power, and so the sample sizes. Here, we present a general bootstrap method for conservatively estimating, on the basis of phase II data, the sample size needed for a phase III trial. The method we propose is based on the use of nonparametric lower bounds for the true power of the test. A wide study is shown for comparing the performances of the new method in estimating the power of the Wilcoxon rank-sum test with those given by standard techniques based on the asymptotic normality of the test statistic. Results indicate that when the phase II sample size is around the ideal sample size for the phase III, the bootstrap provides better results than the other techniques. Since the method is general, it could be used for planning clinical trials for testing superiority, for testing noninferiority, and for more complicated situations, e.g., for testing multiple endpoints.     

Conservative sample size estimation in nonparametrics

Due to the uncertainty of the results of phase II trials, underpowered phase III trials are often planned. In recent literature the conservative approach for sample size estimation was proposed. Some authors, in the parametric framework, make use of the lower bound of the effect size for conservatively estimating the true power, and so the sample sizes. Here, we present a general bootstrap method for conservatively estimating, on the basis of phase II data, the sample size needed for a phase III trial. The method we propose is based on the use of nonparametric lower bounds for the true power of the test. A wide study is shown for comparing the performances of the new method in estimating the power of the Wilcoxon rank-sum test with those given by standard techniques based on the asymptotic normality of the test statistic. Results indicate that when the phase II sample size is around the ideal sample size for the phase III, the bootstrap provides better results than the other techniques. Since the method is general, it could be used for planning clinical trials for testing superiority, for testing noninferiority, and for more complicated situations, e.g., for testing multiple endpoints.     

Statistical Monitoring of Clinical Trials: Fundamentals for Investigators,L. A. Moyé

We consider collecting the measurements of a gold standard and two methods with error from each site of subjects. We propose an asymptotic test statistic to compare the concordance rates of two methods with the gold standard and a closed-form sample size formula. Through simulations, we show that the test statistic accurately controls the type I error in small sample sizes, and the sample size formula accurately maintains the power in various settings. The proposed test statistic and the sample size formula can be easily modified for other indices for agreement such as sensitivity and specificity. A real eye study is taken as an example.     

Sample size for comparing correlated concordance rates

We consider collecting the measurements of a gold standard and two methods with error from each site of subjects. We propose an asymptotic test statistic to compare the concordance rates of two methods with the gold standard and a closed-form sample size formula. Through simulations, we show that the test statistic accurately controls the type I error in small sample sizes, and the sample size formula accurately maintains the power in various settings. The proposed test statistic and the sample size formula can be easily modified for other indices for agreement such as sensitivity and specificity. A real eye study is taken as an example.     

Sample Size Calculations for Combination Drugs of Two Monotherapies With One Approved Dose Level

In this article, we consider sample size calculations for combination drugs of two monotherapies that each has only one approved dose level. We modify the method of Laska and Meiner by employing the asymptotic joint distribution of test statistics to derive the power function and using unequal allocation to minimize the total sample sizes. Two cases are investigated. In the first case, each monotherapy has the same indication. A heuristic method, the method of Laska and Meiner, and the proposed method are compared in terms of the total sample sizes. We show that the proposed method produces the smallest total sample sizes. In the second case, each monotherapy has a different indication. While the method of Laska and Meiner cannot be applied in this case, we show that the proposed method can be employed and that it produces smaller total sample sizes than a heuristic method.     

Sample size calculations for comparing two groups of count data

A sample size formula for comparing two groups of count data is derived using the method of moments by matching the first and second moments of the distribution of the count data, and it does not need any further distributional assumption. Compared to sample size formulas derived using a likelihood-based approach or using simulations, the proposed sample size formula applies to count data following any distribution in addition to the negative binomial distribution. The proposed sample size formula can be used even when the study is analyzed with a likelihood-based approach. Because asymptotically, the method of moments is no more efficient than likelihood-based approaches, the proposed sample size formula can be viewed as an upper bound of the required sample size by likelihood-based approaches to start the study. Applications of the sample size formula are illustrated using an asthma study design.     

Estimation of spatial autoregressive models with randomly missing data in the dependent variable

Summary We suggest and compare different methods for estimating spatial autoregressive models with randomly missing data in the dependent variable. Aside from the traditional expectation‐maximization (EM) algorithm, a nonlinear least squares method is suggested and a generalized method of moments estimation is developed for the model. A two‐stage least squares estimation with imputation is proposed as well. We analytically compare these estimation methods and find that generalized nonlinear least squares, best generalized two‐stage least squares with imputation and best method of moments estimators have identical asymptotic variances. These methods are less efficient than maximum likelihood estimation implemented with the EM algorithm. When unknown heteroscedasticity exists, however, EM estimation produces inconsistent estimates. Under this situation, these methods outperform EM. We provide finite sample evidence through Monte Carlo experiments.