Optimal Sample Size Allocation Research Articles

Optimal sample size allocation for two-arm superiority and non-inferiority trials with binary endpoints.

The sample size of a clinical trial has to be large enough to ensure sufficient power for achieving the aim the study. On the other side, for ethical and economical reasons it should not be larger than necessary. The sample size allocation is one of the parameters that influences the required total sample size. For two-arm superiority and non-inferiority trials with binary endpoints, we performed extensive computations over a wide range of scenarios to determine the optimal allocation ratio that minimizes the total sample size if all other parameters are fixed. The results demonstrate, that for both superiority and non-inferiority trials the optimal allocation may deviate considerably from the case of equal sample size in both groups. However, the saving in sample size when allocating the total sample size optimally as compared to balanced allocation is typically small.

Optimal allocation of sample size for randomization-based inference from 2 K factorial designs

Abstract Optimizing the allocation of units into treatment groups can help researchers improve the precision of causal estimators and decrease costs when running factorial experiments. However, existing optimal allocation results typically assume a super-population model and that the outcome data come from a known family of distributions. Instead, we focus on randomization-based causal inference for the finite-population setting, which does not require model specifications for the data or sampling assumptions. We propose exact theoretical solutions for optimal allocation in 2 K {2}^{K} factorial experiments under complete randomization with A-, D-, and E-optimality criteria. We then extend this work to factorial designs with block randomization. We also derive results for optimal allocations when using cost-based constraints. To connect our theory to practice, we provide convenient integer-constrained programming solutions using a greedy optimization approach to find integer optimal allocation solutions for both complete and block randomizations. The proposed methods are demonstrated using two real-life factorial experiments conducted by social scientists.

Practical strategies for operationalizing optimal allocationin stratified cluster-based outcome-dependent sampling designs.

Cluster-based outcome-dependent sampling (ODS) has the potential to yield efficiency gains when the outcome of interest is relatively rare, and resource constraints allow only a certain number of clusters to be visited for data collection. Previous research has shown that when the intended analysis is inverse-probability weighted generalized estimating equations, and the number of clusters that can be sampled is fixed, optimal allocation of the (cluster-level) sample size across strata defined by auxiliary variables readily available at the design stage has the potential to increase efficiency in the estimation of the parameter(s) of interest. In such a setting, the optimal allocation formulae depend on quantities that are unknown in practice, currently making such designs difficult to implement. In this paper, we consider a two-wave adaptive sampling approach, in which data is collected from a first wave sample, and subsequently used to compute the optimal second wave stratum-specific sample sizes. We consider two strategies for estimating the necessary components using the first wave data: an inverse-probability weighting (IPW) approach and a multiple imputation (MI) approach. In a comprehensive simulation study, we show that the adaptive sampling approach performs well, and that the MI approach yields designs that are very near-optimal, regardless of the covariate type. The IPW approach, on the other hand, has mixed results. Finally, we illustrate the proposed adaptive sampling procedures with data on maternal characteristics and birth outcomes among women enrolled in the Safer Deliveries program in Zanzibar, Tanzania.

Optimal designs for comparing population curves in hierarchical models

Summary:We construct μp-optimal approximate designs for comparing population curves in hierarchical models with group-specific treatment. Equivalence theorems are given to confirm μp-optimality of designs and the optimal allocation of sample size in each group. An illustrative application to find μ1 and μ∞-optimal designs is presented for the random slope models with two-treatment group assignments.

Lamination Scheme of Curing Degree at Multiple Levels of Temperature With Location-Scale Regression

Solar power has become a key green source of energy. An important factor that affects the reliability and lifetime of solar modules is the quality of encapsulation through the lamination process, which melts the ethylene vinyl acetate (EVA) to make the solar cells combine with the front glass side and the rear side units. The degree of crosslinking or curing degree for EVA sheets, when the EVA sheet gets heated, can affect the efficiency of the performance and power conversion of solar modules. For this reason, motivated by a lamination data, we construct here a statistical model for describing the relationship between the curing degree and the lamination factors (temperature and time). Then, based on some specification limits on the curing degree, the optimal lamination time for solar modules can be determined at different temperatures. Moreover, the optimal sample size allocation in a test for measuring EVA sheets can also be determined. A simulation study is finally carried out to show the closeness of simulation results to the asymptotic results.

Comparison of optimal accelerated life tests with competing risks model under exponential distribution

AbstractAccelerated life testing (ALT) is the process of testing products by subjecting them to strict conditions in order to observe more failure data in a short time period. In this study, we compare the methods of two‐level constant‐stress ALT (CSALT) and simple step‐stress ALT (SSALT) based on competing risks of two or more failure modes with independent exponential lifetime distributions. Optimal sample size allocation during CSALT and the optimal stress change‐time in SSALT are considered based on V‐ and D‐optimality , respectively. Under Type‐I censoring, numerical results show that the optimal SSALT outperforms the optimal CSALT in a wide variety of settings. We theoretically also show that the optimal SSALT is better than the optimal CSALT under a set of conditions. A real data example is analyzed to demonstrate the performance of the optimal plans for both ALTs.

Optimization of Accelerated Destructive Degradation Testing of Cementitious Materials for Their Performances Qualification under Aggressive Environments: The Case of Carbonation

In order to guarantee the performance or to qualify the risk of nonperformance of cementitious materials over time, a significant number of experimental data obtained from tests mimicking various degradation mechanisms are required. The slowness of the materials’ degradation under environmental service conditions is an issue, and thus, acceleration strategies are required to obtain reliable and comprehensive results in a shorter time. The objective of this research work is to provide a generic framework for the design of optimal accelerated destructive degradation tests (ADDTs) for cementitious materials qualification. The definition of the optimal design of experiments depends on the capacity to capture the influence of data variability and uncertainty from any sources; they are extracted either from physical models or from experimental tests. In this research, the evolution of carbonation depth is characterized with the Wiener process formalism and the random effects related to the material heterogeneity are taken into account. Once the process parameters are estimated through the maximum likelihood estimation (MLE), associated with the expectation-maximization (EM) algorithm, we provide a step-by-step and detailed method to investigate the optimal design of ADDTs. The latter is defined as the one for which we can estimate durability indicators such as mean-time-to-failure with the best accuracy based on three criteria (D-V-A optimality) considering constraints of time, total number of samples, or limited costs. The optimal total sample size for the accelerated carbonation test and the optimal sample size allocation proportions for each stress level are determined, and the effects of the stress level on the objective functions and of test time duration constraint are also discussed. A comparison of the relative efficiency of optimal three-level versus optimal two-level ADDT completes this work.

Estimation of optimum sample size allocation: An illustration with body mass index for evaluating the effect of a dietetic supplement

In many medical researches, it is needed to determine the optimal sample size allocation in a heterogeneous population. This paper proposes the algorithm for optimal sample size allocation. We consider the optimal allocation problem as an optimization problem and the solution is obtained by using Bisection, Secant, Regula–Falsi and other numerical methods. The performance of the algorithm for different numerical methods are analyzed and evaluated in terms of computing time, number of iterations and gain in accuracy using stratification. The efficacy of algorithm is evaluated for the response in terms of body mass index (BMI) to the dietetic supplement with diabetes mellitus, HIV/AIDS and cancer post-operatory recovery patients.

Quantifying the trade-off between cost and precision in estimating area of forest loss and degradation using probability sampling in Guyana

Guyana is a high forest cover, low deforestation country. Since 2011–2014 the Guyana Forestry Commission (GFC) has used visual interpretation of 5 m resolution RapidEye imagery to map forest loss and nearby degradation for the entire country. According to the GFC produced national map, 58% of all forest loss events cover less than 1 ha, so forest loss is very rare and occurs at a scale that can be difficult to detect and monitor. Nearly all (~97%) of the forest loss and nearby degradation is due to small-scale mining and its associated infrastructure. For any country wanting to accurately map and monitor forest loss, sample-based area estimation can be considerably less expensive than wall-to-wall mapping. To quantify the tradeoff between precision and cost, we evaluated the standard errors of area estimators for several sample-based strategies using Guyana as a case study. We partitioned Guyana into 374 blocks, with each block 24 km × 24 km to align with the image tiles provided by RapidEye's 3A product. The area of forest loss and area of degradation for each of the 374 blocks were determined from the GFC map to create the population data used to evaluate the precision of different sample-based strategies. We compared the standard errors of estimators of area of forest loss and area of degradation obtained from simple random, stratified random and systematic sampling as applied to these population data. To construct strata for the stratified design, we evaluated two forest loss maps produced from Landsat data, a 30 m global forest loss product and a 30 m national forest loss map produced specifically for Guyana. For both of these maps, several options for defining stratum boundaries and allocating sample size to strata were evaluated. All stratified design options reduced the standard error of the area estimators relative to simple random sampling. The Dalenius-Hodges and Jenks methods for choosing stratum boundaries yielded greater improvements in precision than the Equal Area method. For this Guyana case study, optimal and equal allocation of sample size to strata led to substantially better precision than proportional allocation. Only small improvements in precision were attained by increasing the number of strata from three to five. For simple random sampling, incorporating the global and national forest loss map information in a regression estimator reduced standard errors of estimated area of forest loss and area of degradation relative to the standard error of the Horvitz-Thompson estimator that does not incorporate the forest loss map information. However, the best performing stratified sampling design options had better precision than the regression estimator. RapidEye data cost 800 USD per ‘3A’ tile so the annual imagery cost for mapping Guyana would be 274,000 USD. For an investment of 75,200 USD (the cost of 94 RapidEye tiles, the largest sample size evaluated), the best performing stratified random sample achieved a relative standard error (100% times the standard error divided by area) of 7% for estimating area of forest loss and 6% for estimating area of degradation. This study demonstrates the utility of Landsat-based forest loss maps to provide effective strata to improve precision of area estimators produced from a sample of RapidEye imagery in a country for which forest loss and degradation are very rare features of the landscape.

Optimal sample size planning for the Wilcoxon-Mann-Whitney test.

There are many different proposed procedures for sample size planning for the Wilcoxon‐Mann‐Whitney test at given type‐I and type‐II error rates α and β, respectively. Most methods assume very specific models or types of data to simplify calculations (eg, ordered categorical or metric data, location shift alternatives, etc). We present a unified approach that covers metric data with and without ties, count data, ordered categorical data, and even dichotomous data. For that, we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following “synthetic data” approach. We evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities. Then, well‐known relations for the ranks of the data are used for the calculations.In addition to computing the necessary sample size N for a fixed allocation proportion t = n 1/N, where n 1 is the sample size in the first group and N = n 1 + n 2 is the total sample size, we provide an interval for the optimal allocation rate t, which minimizes the total sample size N. It turns out that, for certain distributions, a balanced design is optimal. We give a characterization of such distributions. Furthermore, we show that the optimal choice of t depends on the ratio of the two variances, which determine the variance of the Wilcoxon‐Mann‐Whitney statistic under the alternative. This is different from an optimal sample size allocation in case of the normal distribution model.

Optimal sample size allocation and go/no-go decision rules for phase II/III programs where several phase III trials are performed.

The conduct of phase II and III programs is costly, time-consuming and, due to high failure rates in late development stages, risky. There is a strong connection between phase II and III trials as the go/no-go decision and the sample size chosen for phase III are based on the results observed in phase II. An integrated planning of phase II and III is therefore reasonable. The success of phase II/III programs crucially depends on the allocation of the resources to phase II and III in terms of sample size and the rule applied to decide whether to stop or to proceed with phase III. Recently, a utility-based approach was proposed, where optimal planning of phase II/III programs is achieved by taking fixed and variable costs of the drug development program and potential gains after a successful launch into account. However, this method is restricted to programs with a single phase III trial, while regulatory authorities usually require statistical significance in two or more phase III trials. We present a generalization of this procedure to programs where two or more phase III trials are performed. Optimal phase II sample sizes and go/no-go decision rules are provided for time-to-event outcomes and cases, where at least one, two, or three phase III trials need to be successful. Different drug development program strategies (e.g. one large vs. two phase III trials) are compared within these different cases. Application to practical examples typically met in oncology trials illustrates the proposed method.

Optimal design for constant-stress accelerated degradation test based on gamma process

ABSTRACTThis paper addresses the optimal design problems for constant-stress accelerated degradation test (CSADT) based on gamma processes with fixed effect and random effect. For three optimization criteria, we prove that optimal CSADT plans with multiple stress levels degenerate to two-stress-level test plans only using the minimum and maximum stress levels under model assumptions. Under each optimization criterion, the optimal sample size allocation proportions for the minimum and maximum stress levels are determined theoretically. The effect of the stress level on the objective functions is also discussed. A numerical example and a simulation study are provided to illustrate the obtained results.

Optimal Allocation in Domains Mean Estimation Using Double Sampling with Non-Linear Cost Function in the Presence of Non-Response

Studies have been carried out on domain mean estimation using non-linear cost function. However little has been done on domain stratum estimation using non-linear cost function using ratio estimation in the presence of non-response. This study develops a method of optimal stratum sample size allocation in domain mean estimation using double sampling with non-linear cost function in the presence of non- response. To obtain an optimum sample size, Lagrangian multiplier technique is employed by minimizing precision at a specified cost. In the estimation of the domain mean, auxiliary variable information in which the study and auxiliary variables both suffers from non-response in the second phase sampling is used. The expressions of the biases and mean square errors of proposed estimator has also been obtained.

Multiple sensitive estimation and optimal sample size allocation in the item sum technique.

For surveys of sensitive issues in life sciences, statistical procedures can be used to reduce nonresponse and social desirability response bias. Both of these phenomena provoke nonsampling errors that are difficult to deal with and can seriously flaw the validity of the analyses. The item sum technique (IST) is a very recent indirect questioning method derived from the item count technique that seeks to procure more reliable responses on quantitative items than direct questioning while preserving respondents' anonymity. This article addresses two important questions concerning the IST: (i) its implementation when two or more sensitive variables are investigated and efficient estimates of their unknown population means are required; (ii) the determination of the optimal sample size to achieve minimum variance estimates. These aspects are of great relevance for survey practitioners engaged in sensitive research and, to the best of our knowledge, were not studied so far. In this article, theoretical results for multiple estimation and optimal allocation are obtained under a generic sampling design and then particularized to simple random sampling and stratified sampling designs. Theoretical considerations are integrated with a number of simulation studies based on data from two real surveys and conducted to ascertain the efficiency gain derived from optimal allocation in different situations. One of the surveys concerns cannabis consumption among university students. Our findings highlight some methodological advances that can be obtained in life sciences IST surveys when optimal allocation is achieved.

Optimal sample size allocation for accelerated life test under progressive type-II censoring with competing risks

ABSTRACTIn this article, we study the optimization problem of sample size allocation when the competing risks data are from a progressive type-II censoring in a constant-stress accelerated life test with multiple levels. The failure times of the individual causes are assumed to be statistically independent and exponentially distributed with different parameters. We obtain the estimates of the unknown parameters through a maximum likelihood method, and also derive the Fisher information matrix. We propose three optimization criteria and two search scenarios to obtain the sample size allocation at each stress level. Some numerical results are studied to illustrate the usage of the proposed methods.

Assessing the consistency of the treatment effect under the discrete random effects model in multiregional clinical trials

In recent years, developing pharmaceutical products via multiregional clinical trials (MRCTs) has become standard. Traditionally, an MRCT would assume that a treatment effect is uniform across regions. However, heterogeneity among regions may have impact upon the evaluation of a medicine's effect. In this study, we consider a random effects model using discrete distribution (DREM) to account for heterogeneous treatment effects across regions for the design and evaluation of MRCTs. We derive an power function for a treatment that is beneficial under DREM and illustrate determination of the overall sample size in an MRCT. We use the concept of consistency based on Method 2 of the Japanese Ministry of Health, Labour, and Welfare's guidance to evaluate the probability for treatment benefit and consistency under DREM. We further derive an optimal sample size allocation over regions to maximize the power for consistency. Moreover, we provide three algorithms for deriving sample size at the desired level of power for benefit and consistency. In practice, regional treatment effects are unknown. Thus, we provide some guidelines on the design of MRCTs with consistency when the regional treatment effect are assumed to fall into a specified interval. Numerical examples are given to illustrate applications of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

Optimal sample size allocation for multi-level stress testing with exponential regression under type-I censoring

ABSTRACTIn this article, we discuss the optimal allocation problem in a multi-level accelerated life-testing experiment under Type-I censoring when an exponential regression model is used for the analysis. We derive the expected Fisher information matrix and use it to obtain the asymptotic variance–covariance matrix of the maximum likelihood estimators (MLEs). We then consider the optimal allocation under the D-optimality criterion, and present an algorithm for determining the optimal allocation. A numerical example is presented for the purpose of illustration. The optimal allocation depends on the model parameters and so a sensitivity analysis of the optimal allocation to misspecification of the model parameters is carried out as well.

Optimal generalized case-cohort analysis with Cox’s proportional hazards model

Generalized case-cohort design has been proved to be a cost-effective way to enhance the efficiency of large epidemiological cohort. In this article, we propose an inference procedure for estimating the unknown parameters in Cox’s proportional hazards model in generalized case-cohort design and establish an optimal sample size allocation to achieve the maximum power at a given budget. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm’s Tumor Study Group.

Ratio Estimators Using Coefficient of Variation and Coefficient of Correlation

This paper introduces ratio estimators of the population mean using the coefficient of variation of study variable and auxiliary variables together with the coefficient of correlation between the study and auxiliary variables under simple random sampling and stratified random sampling. These ratio estimators are almost unbiased. The mean square errors of the estimators and their estimators are given. Sample size estimation in both sampling designs are presented. An optimal sample size allocation in stratified random sampling is also suggested. Based on theoretical study, it can be shown that these ratio estimators have smaller MSE than the unbiased estimators. Moreover, the empirical study indicates that these ratio estimators have smallest MSE compared to the existing ones.

Optimal sample size allocation for Welch's test in one-way heteroscedastic ANOVA.

The determination of an adequate sample size is a vital aspect in the planning stage of research studies. A prudent strategy should incorporate all of the critical factors and cost considerations into sample size calculations. This study concerns the allocation schemes of group sizes for Welch's test in a one-way heteroscedastic ANOVA. Optimal allocation approaches are presented for minimizing the total cost while maintaining adequate power and for maximizing power performance for a fixed cost. The commonly recommended ratio of sample sizes is proportional to the ratio of the population standard deviations or the ratio of the population standard deviations divided by the square root of the ratio of the unit sampling costs. Detailed numerical investigations have shown that these usual allocation methods generally do not give the optimal solution. The suggested procedures are illustrated using an example of the cost-efficiency evaluation in multidisciplinary pain centers.