Optimal contrast analysis with heterogeneous variances and budget concerns.

The omnibus test is commonly applied to evaluate the overall disparity between group means in ANOVA. Alternatively, linear contrasts are more informative in detecting specific pattern of mean differences that cannot be obtained via the omnibus test. This article concerns power and sample size calculations for contrast analysis with heterogeneous variances and budget concerns. Optimal allocation procedures for the Welch-Satterthwaite tests of standardized and unstandardized contrasts are presented to minimize the total sample size with the designated ratios, to meet a desirable power level for the least cost, and to attain the maximum power performance under a fixed cost. Currently available methods rely exclusively on simple allocation formula and direct rounding rule. The proposed allocation strategies combine the computing techniques of nonlinear optimization search and iterative screening process. Numerical assessments of a randomized control trial for the overcoming depression on the Internet are conducted to demonstrate and confirm that the approximate procedures do not guarantee optimal solution. The suggested approaches extend and outperform the existing findings in methodological soundness and overall performance. The corresponding computer algorithms are developed to implement the recommended power and sample size calculations for optimal contrast analysis.

Sample size calculations for model validation in linear regression analysis.

BACKGROUND: Linear regression analysis is a widely used statistical technique in practical applications. For planning and appraising validation studies of simple linear regression, an approximate sample size formula has been proposed for the joint test of intercept and slope coefficients. METHODS: The purpose of this article is to reveal the potential drawback of the existing approximation and to provide an alternative and exact solution of power and sample size calculations for model validation in linear regression analysis. RESULTS: A fetal weight example is included to illustrate the underlying discrepancy between the exact and approximate methods. Moreover, extensive numerical assessments were conducted to examine the relative performance of the two distinct procedures. CONCLUSIONS: The results show that the exact approach has a distinct advantage over the current method with greater accuracy and high robustness.

The Bland-Altman range of agreement: Exact interval procedure and sample size determination.

The Bland-Altman analysis has been widely used to assess the agreement between two quantitative methods of measurement. The range of agreement is thus represented by the designated central proportion of the distribution of the paired differences. This article aims to provide concrete guidance on the use of accurate confidence intervals of the range of agreement. Theoretical examination and numerical comparison are conducted to support the exact approach and to demonstrate the disadvantages of the currently suggested methods for confidence interval estimation of the range of agreement. For the purpose of design planning, the associated sample size calculations for precise interval estimation of the range of agreement are delineated under the expected width and assurance probability considerations. To enhance the utility of the recommended confidence interval and sample size procedures in practical applications, both SAS and R computer algorithms are also developed.

A systematic approach to designing statistically powerful heteroscedastic 2 × 2 factorial studies while minimizing financial costs.

BACKGROUND: The 2 × 2 factorial design is widely used for assessing the existence of interaction and the extent of generalizability of two factors where each factor had only two levels. Accordingly, research problems associated with the main effects and interaction effects can be analyzed with the selected linear contrasts. METHODS: To correct for the potential heterogeneity of variance structure, the Welch-Satterthwaite test is commonly used as an alternative to the t test for detecting the substantive significance of a linear combination of mean effects. This study concerns the optimal allocation of group sizes for the Welch-Satterthwaite test in order to minimize the total cost while maintaining adequate power. The existing method suggests that the optimal ratio of sample sizes is proportional to the ratio of the population standard deviations divided by the square root of the ratio of the unit sampling costs. Instead, a systematic approach using optimization technique and screening search is presented to find the optimal solution. RESULTS: Numerical assessments revealed that the current allocation scheme generally does not give the optimal solution. Alternatively, the suggested approaches to power and sample size calculations give accurate and superior results under various treatment and cost configurations. CONCLUSIONS: The proposed approach improves upon the current method in both its methodological soundness and overall performance. Supplementary algorithms are also developed to aid the usefulness and implementation of the recommended technique in planning 2 × 2 factorial designs.

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.

The power of approximate degrees of freedom tests in heteroscedastic factorial designs.

Several viable approaches have been proposed for conducting factorial analysis when the homogeneous variances assumption is violated. Although robust type I error control and excellent power performance are desirable properties of a test procedure for making statistical inferences, the corresponding power calculations must also be considered to extend its applicability in planning research studies. This article presents the power functions of two approximate degrees of freedom tests within the context of two-way factorial designs. Both theoretical examination and empirical investigation are performed to demonstrate the underlying features of the two approaches. Simulation results showed that the suggested procedures provide remarkably good approximations over a wide range of model configurations. In view of the overall accuracy, the power functions are recommended for enhancing the usefulness of the associated test procedures. To facilitate the adoption of the suggested methodology in factorial research designs, computer code is presented for implementing the described power calculations.

Power and sample size calculations for testing linear combinations of group means under variance heterogeneity with applications to meta and moderation analyses

The general formulation of a linear combination of population means permits a wide range of research questions to be tested within the context of ANOVA. However, it has been stressed in many research areas that the homogeneous variances assumption is frequently violated. To accommodate the heterogeneity of variance structure, the Welch–Satterthwaite procedure is commonly used as an alternative to the t test for detecting the substantive significance of a linear combination of mean effects. This article presents two approaches to power and sample size calculations for the Welch–Satterthwaite test. The usefulness and diversity of the suggested techniques are illustrated with two of the potential applications in meta and moderation analyses. The numerical assessments showed that the proposed approaches outperform the existing methods on the accuracy of power calculations and sample size determinations for meta and moderation studies. Computer algorithms are also developed to implement the recommended procedures in actual research designs (AU)

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Sample size determinations for Welch's test in one-way heteroscedastic ANOVA.

For one-way fixed effects ANOVA, it is well known that the conventional F test of the equality of means is not robust to unequal variances, and numerous methods have been proposed for dealing with heteroscedasticity. On the basis of extensive empirical evidence of Type I error control and power performance, Welch's procedure is frequently recommended as the major alternative to the ANOVA F test under variance heterogeneity. To enhance its practical usefulness, this paper considers an important aspect of Welch's method in determining the sample size necessary to achieve a given power. Simulation studies are conducted to compare two approximate power functions of Welch's test for their accuracy in sample size calculations over a wide variety of model configurations with heteroscedastic structures. The numerical investigations show that Levy's (1978a) approach is clearly more accurate than the formula of Luh and Guo (2011) for the range of model specifications considered here. Accordingly, computer programs are provided to implement the technique recommended by Levy for power calculation and sample size determination within the context of the one-way heteroscedastic ANOVA model.

Optimal sample sizes for precise interval estimation of Welch's procedure under various allocation and cost considerations.

Welch's (Biometrika 29: 350-362, 1938) procedure has emerged as a robust alternative to the Student's t test for comparing the means of two normal populations with unknown and possibly unequal variances. To facilitate the advocated statistical practice of confidence intervals and further improve the potential applicability of Welch's procedure, in the present article, we consider exact approaches to optimize sample size determinations for precise interval estimation of the difference between two means under various allocation and cost considerations. The desired precision of a confidence interval is assessed with respect to the control of expected half-width, and to the assurance probability of interval half-width within a designated value. Furthermore, the design schemes in terms of participant allocation and cost constraints include (a) giving the ratio of group sizes, (b) specifying one sample size, (c) attaining maximum precision performance for a fixed cost, and (d) meeting a specified precision level for the least cost. The proposed methods provide useful alternatives to the conventional sample size procedures. Also, the developed programs expand the degree of generality for the existing statistical software packages and can be accessed at brm.psychonomic-journals.org/content/ supplemental.

Optimal sample sizes for Welch's test under various allocation and cost considerations.

The issue of the sample size necessary to ensure adequate statistical power has been the focus of considerableattention in scientific research. Conventional presentations of sample size determination do not consider budgetary and participant allocation scheme constraints, although there is some discussion in the literature. The introduction of additional allocation and cost concerns complicates study design, although the resulting procedure permits a practical treatment of sample size planning. This article presents exact techniques for optimizing sample size determinations in the context of Welch (Biometrika, 29, 350-362, 1938) test of the difference between two means under various design and cost considerations. The allocation schemes include cases in which (1) the ratio of group sizes is given and (2) one sample size is specified. The cost implications suggest optimally assigning subjects (1) to attain maximum power performance for a fixed cost and (2) to meet adesignated power level for the least cost. The proposed methods provide useful alternatives to the conventional procedures and can be readily implemented with the developed R and SAS programs that are available as supplemental materials from brm.psychonomic-journals.org/content/supplemental.

Nonparametric procedures for simultaneous identification of the minimum effective dose in each of several groups.

This paper considers the extension of the dose finding problem in a one-way layout setting to the case of several groups, in which the main goal is to identify the minimum effective dose (MED) of each group simultaneously. We propose two nonparametric procedures using the respective pairwise- and Helmert-type two-sample Mann-Whitney statistics, which are applied in a step-down testing scheme for identifying the MED simultaneously. The computation for the associated p-value of the identified MED vector is discussed. One numerical example is given to illustrate the proposed procedures.