Sample Size Procedures Research Articles

Optimization the design of fixed and group sequential three-arm non-inferiority trials with dichotomous endpoints of risk difference and odds ratio

Although the risk difference (RD) is the most common and well explored functional form for testing efficacy with dichotomous endpoint, odds ratio (OR) is also suggested and well applied measure for non-inferiority (NI) trials. Since the construction and interpretation of these function forms are quite different, this study aims to provide detailed discussions and comprehensive comparisons on the design and testing approach for RD and OR scales for the fixed and group sequential three-arm NI trials under various of situations. The sample size determinations and testing approaches for assessing NI of a new treatment in three-arm clinical trials for RD and OR scales were reviewed comprehensively. Simulation studies are conducted for hundreds of scenarios with parameter configurations of the response rates, randomized allocations, NI margins and interim analysis. The operating characteristic (OC) of RD and OR scales based on the MLE and RMLE methods were thoroughly investigated. A trial example was designed and analyzed to demonstrate the methodologies. It is found that sample size determination on OR scale gives smaller sample size and robust procedure compared to RD scale in the majority of situations. When evaluating the behaviors of the attained power, the RMLE methods based on OR scale outperforms the MLE method and tend to have more power to reject the null hypothesis especially under the small sample size situations. Compared to the fixed design, the group sequential design has better OC, which provides a comparable power while needing smaller total average sample sizes for all cases. In addition, we suggest a lower significance level with a higher power for the sample size determination in the superiority test stage in the group sequential design, which can significantly reduce the total sample sizes while the number of subjects in the placebo group does not increase much. It can offer some recommendations for the investigators to choose the optimal endpoints and parameter configurations to design a three-arm NI trial under certain situations.

On the Equivalence Test and Sample Size Procedures for Simple Linear Regressions with Applications to ANCOVA and Moderation Analysis

On the Equivalence Test and Sample Size Procedures for Simple Linear Regressions with Applications to ANCOVA and Moderation Analysis

The Extended Anderson and Hauck Tests and Sample Size Procedures for Equivalence Assessment in Simple Linear Regressions

This study describes extended Anderson and Hauck procedures for equivalence testing of slope coefficients and mean responses in one and two regression lines. The general formulation of asymmetric equivalence ranges permits a wide variety of equivalence questions to be tested for a target magnitude or a negligible value. Specifically, the equivalence tests are useful for assessing negligible trend and similar response in a single regression line, and for evaluating unimportant interaction-moderation effect and comparable simple effect between two linear regression lines. The associated power functions and sample size procedures are also derived and compared under the random and fixed model settings. According to the analytic justification and empirical assessment, the exact approaches have a clear advantage over the approximate formulas for accommodating the full stochastic nature of both the response and predictor variables. Computer algorithms are also provided for conducting the proposed equivalence tests, power calculations, and sample size determinations in simple linear regressions.

Assessment of business resilience abilities: the case of industry and commerce of Santa Maria – RS

Purpose: This research aimed to evaluate the potential for resilient performance in the face of crises and adversities for companies located in the city of Santa Maria – RS (Brazil).Design/methodology/approach: We adopted a quantitative approach. The research was carried out through five sequential steps: (i) construction and validation of the research instrument; (ii) definition of the sample size and sampling procedures; (iii) data collection; (iv) data analysis and interpretation of results; and (v) elaboration of recommendations. A research instrument containing 44 questions was built and distributed among potential respondents, obtaining 87 valid answers, which corresponds to a statistically representative sample for the parameters adopted.Findings: Our sample was divided into two subsets, representing the Industrial and Commerce sectors. Resilience assessment accounted for four dimensions: the ability to respond, to anticipate, to monitor, and to learn. Results for both sectors showed scores ranging from 3.4 to 3.7 (out of 5) in the four dimensions evaluated. This reveals both sectors have been developing practices that positively contribute to the improvement of resilience, although there are opportunities for improvement.Originality/value: This research sheds light on the resilience ability of the cases surveyed, enabling managers to take action to strengthen their organizations and overcome crises and adversities. Additionally, our findings demonstrate how resilience abilities can be assessed, considering the scope of an economic sector and a geographical area. This approach can be adopted by researchers, policymakers, and development agencies to assess the resilience abilities of firms within their sphere of influence.

A Non-Parametric Sequential Procedure for the Generalized Partition Problem

In selection and ranking, the partitioning of treatments by comparing them to a control treatment is an important statistical problem. For over eighty years, this problem has been investigated by a number of researchers via various statistical designs to specify the partitioning criteria and optimal strategies for data collection. Many researchers have proposed designs in order to generalize formulations known at that time. One such generalization adopted the indifference-zone formulation to designate the region between the boundaries for “good” and “bad” treatments as the indifference zone. Since then, this formulation has been adopted by a number of researchers to study various aspects of the partition problem. In this paper, a non-parametric purely sequential procedure is formulated for the partition problem. The “first-order” asymptotic properties of the proposed non-parametric procedure are derived. The performance of the proposed non-parametric procedure for small and moderate sample sizes is studied via Monte Carlo simulations. An example is provided to illustrate the proposed procedure.

Designing individually randomized group treatment trials with repeated outcome measurements using generalized estimating equations.

Individually randomized group treatment (IRGT) trials, in which the clustering of outcome is induced by group-based treatment delivery, are increasingly popular in public health research. IRGT trials frequently incorporate longitudinal measurements, of which the proper sample size calculations should account for correlation structures reflecting both the treatment-induced clustering and repeated outcome measurements. Given the relatively sparse literature on designing longitudinal IRGT trials, we propose sample size procedures for continuous and binary outcomes based on the generalized estimating equations approach, employing the block exchangeable correlation structures with different correlation parameters for the treatment arm and for the control arm, and surveying five marginal mean models with different assumptions of time effect: no-time constant treatment effect, linear-time constant treatment effect, categorical-time constant treatment effect, linear time by treatment interaction, and categorical time by treatment interaction. Closed-form sample size formulas are derived for continuous outcomes, which depends on the eigenvalues of the correlation matrices; detailed numerical sample size procedures are proposed for binary outcomes. Through simulations, we demonstrate that the empirical power agrees well with the predicted power, for as few as eight groups formed in the treatment arm, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator.

Designing multicenter individually randomized group treatment trials.

In an individually randomized group treatment (IRGT) trial, participant outcomes can be positively correlated due to, for example, shared therapists in treatment delivery. Oftentimes, because of limited treatment resources or participants at one location, an IRGT trial can be carried out across multiple centers. This design can be subject to potential correlations in the participant outcomes between arms within the same center. While the design of a single-center IRGT trial has been studied, little is known about the planning of a multicenter IRGT trial. To address this gap, this paper provides analytical sample size formulas for designing multicenter IRGT trials with a continuous endpoint under the linear mixed model framework. We found that accounting for the additional center-level correlation at the design stage can lead to sample size reduction, and the magnitude of reduction depends on the amount of between-therapist correlation. However, if the variance components of therapist-level random effects are considered as input parameters in the design stage, accounting for the additional center-level variance component has no impact on the sample size estimation. We presented our findings through numeric illustrations and performed simulation studies to validate our sample size procedures under different scenarios. Optimal design configurations under the multicenter IRGT trials have also been discussed, and two real-world trial examples are drawn to illustrate the use of ourmethod.

Sample size requirements for testing treatment effect heterogeneity in cluster randomized trials with binary outcomes.

Cluster randomized trials (CRTs) refer to a popular class of experiments in which randomization is carried out at the group level. While methods have been developed for planning CRTs to study the average treatment effect, and more recently, to study the heterogeneous treatment effect, the development for the latter objective has currently been limited to a continuous outcome. Despite the prevalence of binary outcomes in CRTs, determining the necessary sample size and statistical power for detecting differential treatment effects in CRTs with a binary outcome remain unclear. To address this methodological gap, we develop sample size procedures for testing treatment effect heterogeneity in two-level CRTs under a generalized linear mixed model. Closed-form sample size expressions are derived for a binary effect modifier, and in addition, a computationally efficient Monte Carlo approach is developed for a continuous effect modifier. Extensions to multiple effect modifiers are also discussed. We conduct simulations to examine the accuracy of the proposed sample size methods. We present several numerical illustrations to elucidate features of the proposed formulas and to compare our method to the approximate sample size calculation under a linear mixed model. Finally, we use data from the Strategies and Opportunities to Stop Colon Cancer in Priority Populations (STOP CRC) CRT to illustrate the proposed sample size procedure for testing treatment effect heterogeneity.

Accounting for expected attrition in the planning of cluster randomized trials for assessing treatment effect heterogeneity

BackgroundDetecting treatment effect heterogeneity is an important objective in cluster randomized trials and implementation research. While sample size procedures for testing the average treatment effect accounting for participant attrition assuming missing completely at random or missing at random have been previously developed, the impact of attrition on the power for detecting heterogeneous treatment effects in cluster randomized trials remains unknown.MethodsWe provide a sample size formula for testing for a heterogeneous treatment effect assuming the outcome is missing completely at random. We also propose an efficient Monte Carlo sample size procedure for assessing heterogeneous treatment effect assuming covariate-dependent outcome missingness (missing at random). We compare our sample size methods with the direct inflation method that divides the estimated sample size by the mean follow-up rate. We also evaluate our methods through simulation studies and illustrate them with a real-world example.ResultsSimulation results show that our proposed sample size methods under both missing completely at random and missing at random provide sufficient power for assessing heterogeneous treatment effect. The proposed sample size methods lead to more accurate sample size estimates than the direct inflation method when the missingness rate is high (e.g., ≥ 30%). Moreover, sample size estimation under both missing completely at random and missing at random is sensitive to the missingness rate, but not sensitive to the intracluster correlation coefficient among the missingness indicators.ConclusionOur new sample size methods can assist in planning cluster randomized trials that plan to assess a heterogeneous treatment effect and participant attrition is expected to occur.

On Fixed-Accuracy Confidence Intervals for the Parameters of Lindley Distribution and Its Extensions

The purpose of the present paper is to deal with sequential estimation of the parameter θ in a Lindley distribution. A fixed-accuracy confidence interval for θ with a preassigned confidence coefficient is developed. It is established that, no fixed sample size procedure can solve the estimation problem and hence a purely sequential methodology is proposed to deal with the situation. The first-order asymptotic efficiency and consistency properties associated with our purely sequential strategy are derived. Similar estimation strategies are also outlined for a few other extensions of the Lindley distribution. Extensive simulation analysis is carried out to validate the theoretical findings. We also provide a real data example, where we estimate the parameter related to the “initial mass function" for a particular cluster of stars.

Assessing standardized contrast effects in ANCOVA: Confidence intervals, precision evaluations, and sample size requirements.

Standardized effect sizes and confidence intervals are useful statistical assessments for comparing results across different studies when measurement units are not directly comparable. This paper aims to describe and compare confidence interval estimation methods for the standardized contrasts of treatment effects in ANCOVA designs. Sample size procedures are also presented to assure that the resulting confidence intervals yield informative estimation with adequate precision. Exact interval estimation approach has theoretical and empirical advantages in coverage probability and interval width over the approximate interval procedures. Numerical investigations of the existing method reveal that the omission of covariate variables has a negative impact on sample size calculations for precise interval estimation, especially when there is disparity in influential covariate variables. The proposed approaches and developed computer programs fully utilize covariate properties in interval estimation and provide accurate sample size determinations under the precision considerations of the expected interval width and the assurance probability of interval width.

Power analyses for stepped wedge designs with multivariate continuous outcomes.

Multivariate outcomes are common in pragmatic cluster randomized trials. While sample size calculation procedures for multivariate outcomes exist under parallel assignment, none have been developed for a stepped wedge design. In this article, we present computationally efficient power and sample size procedures for stepped wedge cluster randomized trials (SW-CRTs) with multivariate outcomes that differentiate the within-period and between-period intracluster correlation coefficients (ICCs). Under a multivariate linear mixed model, we derive the joint distribution of the intervention test statistics which can be used for determining power under different hypotheses and provide an example using the commonly utilized intersection-union test for co-primary outcomes. Simplifications under a common treatment effect and common ICCs across endpoints and an extension to closed-cohort designs are also provided. Finally, under the common ICC across endpoints assumption, we formally prove that the multivariate linear mixed model leads to a more efficient treatment effect estimator compared to the univariate linear mixed model, providing a rigorous justification on the use of the former with multivariate outcomes. We illustrate application of the proposed methods using data from an existing SW-CRT and present extensive simulations to validate the methods.

A nonparametric test for the two-sample problem based on order statistics

We study a nonparametric test procedure based on order statistics for testing the null hypothesis of equality of two continuous distributions. The exact null distribution of the proposed test statistic is obtained using an enumeration method and a novel combinatorial argument. A recurrence relation for the probability generating function and a sequential approach for computing the mean and variance of the distribution are given. Critical values and characteristics of the distribution for selected small sample sizes are presented. For the Lehmann alternative family, the exact power function of the new test is derived, and its power performance is examined. We also study the power performance of the proposed test under the location-shift and scale-shift alternatives using Monte Carlo simulations and observe its superior performance when compared to commonly used nonparametric tests under various scenarios. A generalization of the proposed procedure for unequal sample sizes is discussed. An illustrative example and some concluding remarks are provided.

Determining reference ranges and sample sizes in parallel-group studies.

Reference ranges are widely used to locate the major range of the target probability distribution. When future measurements fall outside the reference range, they are classified as atypical and require further investigation. The fundamental principles and statistical properties of reference ranges are closely related to those of tolerance interval procedures. Existing investigations of reference ranges and tolerance intervals mainly devoted to the primitive cases of one- and paired-sample designs. Although reference ranges hold considerable promise for parallel group designs, the corresponding methodological and computational issues for determining reference limits and sample sizes have not been adequately addressed. This paper describes a complete collection of one- and two-sided reference ranges for assessing measurement differences in parallel-group studies that assume variance homogeneity. The problem of sample size determination for precise reference ranges is also examined under the expected half-width and assurance probability considerations. Unlike the current methods, the suggested sample size criteria explicitly accommodate desired interval width in precise interval estimation. Theoretical examinations and empirical assessments are presented to validate the usefulness of the proposed reference range and sample size procedures. To enhance the usages of the recommended techniques in practical applications, computer programs are developed for efficient calculation and exact analysis. A real data example regarding tablet absorption rate and extent is presented to illustrate the suggested assessments between two drug formulations.

Chinese physical fitness standard for campus football players: A pilot study of 765 children aged 9 to 11.

Objectives: In 2022, 55 million Chinese children participate in campus football; however, there is no physical fitness standard, making it a priority task to enhance the current national program. This study aimed to explore a pilot method for the development of a reliable physical fitness standard. Methods: This study examined 765 male football players aged 9 to 11 in 2020 and 2022. The anthropometric and physical fitness assessments were conducted in accordance with the Chinese Football Association’s field manuel. Physical fitness tests include sit and reach test, t test, 30 m run test, and vertical jump test. Physical fitness standard was modeled using the generalized additive models for location, scale, and shape (GAMLSS). Data were fitted with appropriate GAMLSS distributions and smoothing term. P-splines were applied to smooth the model’s parameters using the default local maximum likelihood method and link functions. Following diagnostics of fitted models, age-specific centile estimations were computed for physical fitness tests. In addition, players in each age group were categorized according to their body mass index as normal weight or overweight/obese. Welch’s t-test was utilized to compare the group differences in physical fitness testing. The significance level was chosen at p < 0.05. Results: Sit and reach test, t test, 30 m run test, and vertical jump test data were fitted with original Sinh-Arcsinh, Box-Cox power exponential, Box-Cox power exponential, and Box-Cox Cole and Green, respectively. Physical fitness standard for each age group is presented as tabulated centiles (1p, 3p, 5p, 15p, 25p, 50p, 75p, 85p, 95p, 97p, 99p). Overweight/obese campus football players did significantly worse (p < 0.05) on the t test, 30 m run test, and vertical jump test than their normal-weight peers of the same age. Conclusion: This study developed the first physical fitness standard for 9 to 11-year-old campus football players in China. We made three recommendations to Chinese policymakers on sample size, data management, and field procedure for the creation of a national physical fitness standard.

Design and analysis of cluster randomized trials with time-to-event outcomes under the additive hazards mixed model.

A primary focus of current methods for cluster randomized trials (CRTs) has been for continuous, binary, and count outcomes, with relatively less attention given to right-censored, time-to-event outcomes. In this article, we detail considerations for sample size requirement and statistical inference in CRTs with time-to-event outcomes when the intervention effect parameter is specified through the additive hazards mixed model (AHMM), which includes a frailty term to explicitly account for the dependency between the failure times. First, we discuss improved inference for the treatment effect parameter via bias-corrected sandwich variance estimators and randomization-based test under AHMM, addressing potential small-sample biases in CRTs. Next, we derive a new sample size formula for AHMM analysis of CRTs accommodating both equal and unequal cluster sizes. When the cluster sizes vary, our sample size formula depends on the mean and coefficient of variation of cluster sizes, based on which we articulate the impact of cluster size variation in CRTs with time-to-event outcomes. Furthermore, we obtain the insight that the classical variance inflation factor for CRTs with a non-censored outcome can in fact apply to CRTs with a time-to-event outcome, providing that an appropriate definition of the intraclass correlation coefficient is considered under AHMM. Simulation studies are carried out to illustrate key design and analysis considerations in CRTs with a small to moderate number of clusters. The proposed sample size procedure and analytical methods are further illustrated using the context of the STrategies to Reduce Injuries and Develop Confidence in Elders CRT.

Sample size planning for replication studies: The devil is in the design.

Replication is central to scientific progress. Because of widely reported replication failures, replication has received increased attention in psychology, sociology, education, management, and related fields in recent years. Replication studies have generally been assessed dichotomously, designated either a "success" or "failure" based entirely on the outcome of a null hypothesis significance test (i.e., p < .05 or p > .05, respectively). However, alternative definitions of success depend on researchers' goals for the replication. Previous work on alternative definitions for success has focused on the analysis phase of replication. However, the design of the replication is also important, as emphasized with the adage, "an ounce of prevention is better than a pound of cure." One critical component of design often ignored or oversimplified in replication studies is sample size planning, indeed, the details here are crucial. Sample size planning for replication studies should correspond to the method by which success will be evaluated. Researchers have received little guidance, some of which is misguided, on sample size planning for replication goals other than the aforementioned dichotomous null hypothesis significance testing approach. In this article, we describe four different replication goals. Then, we formalize sample size planning methods for each of the four goals. This article aims to provide clarity on the procedures for sample size planning for each goal, with examples and syntax provided to show how each procedure can be used in practice. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Assessing individual equivalence in parallel group and crossover designs: Exact test and sample size procedures.

The consideration of individual equivalence provides an essential alternative to average equivalence in two-group comparative studies. A common procedure for declaring individual equivalence adopts the tolerance intervals of the designated proportions of measurement differences. This statistical practice is a direct generalization of the widely used two one-sided tests (TOST) for average equivalence. Such TOST extensions often do not have adequate control of Type I error and result in excessively conservative tests. To signify and resolve the underlying issues of existing methods, this paper presents exact tests for assessing individual equivalence between two treatments under parallel group and crossover designs. Rigorous evaluations are conducted to clarify the discrepancy of critical values and Type I error probabilities between the equivalence procedures. The findings elucidate the shortcoming of the TOST technique and the advantage of the proposed approach. The associated power and sample size calculations are also justified through simulation studies.

EFFECTS OF MARKETING ON INTRODUCTION OF AUTOMATED TELLER MACHINES (ATMS) ON CUSTOMERS OF BANKING INDUSTRY

This study examined effects of marketing on introduction of automated teller machine (ATM) on customers of banking industry. This study identified lack of network (service) as a problem. The specific objective was to highlight on identification of marketing needs on introduction of automated teller machine on customers of banking industry, among others. Research question was to what extent should identification of marketing needs on introduction of automated teller machine (ATM) on customers banking industry be? among others. Research hypothesis was there is no significant relationship between identification of marketing needs and customers of banking industry, among others. Review of related literature comprised of conceptual review, theoretical framework and empirical review. This study adopted survey research design. Area of the study was Nkwelle-Ezunaka where First bank is situated. Population of the study was customers queuing for automated teller machine (ATM) to render services. Sampling plan comprised of sample designs, sample unit, sample size, sample method and sample procedure. The sample size was one hundred and sixty two (162). This study observed data presentation and analysis. Findings, conclusion and recommendations were discussed.

Accounting for unequal cluster sizes in designing cluster randomized trials to detect treatment effect heterogeneity.

Unequal cluster sizes are common in cluster randomized trials (CRTs). While there are a number of previous investigations studying the impact of unequal cluster sizes on the power for testing the average treatment effect in CRTs, little is known about the impact of unequal cluster sizes on the power for testing the heterogeneous treatment effect (HTE) in CRTs. In this work, we expand the sample size procedures for studying HTE in CRTs to accommodate cluster size variation under the linear mixed model framework. Through analytical derivation and graphical exploration, we show that the sample size for the HTE with an individual-level effect modifier is less affected by unequal cluster sizes than with a cluster-level effect modifier. The impact of cluster size variability jointly depends on the mean and coefficient of variation of cluster sizes, covariate intraclass correlation coefficient (ICC) and the conditional outcome ICC. In addition, we demonstrate that the HTE-motivated analysis of covariance framework can be used for analyzing the average treatment effect, and offer a more efficient sample size procedure for studying the average treatment effect adjusting for the effect modifier. We use simulations to confirm the accuracy of the proposed sample size procedures for both the average treatment effect and HTE in CRTs. Extensions to multivariate effect modifiers are provided and our procedure is illustrated in the context of the Strategies to Reduce Injuries and Develop Confidence in Elders trial.