ON OPTIMIZING MULTI‐LEVEL DESIGNS: POWER UNDER BUDGET CONSTRAINTS

Summary This paper derives a procedure for efficiently allocating the number of units in multi-level designs given prespecified power levels. The derivation of the procedure is based on a constrained optimization problem that maximizes a general form of a ratio of expected mean squares subject to a budget constraint. The procedure makes use of variance component estimates to optimize designs during the budget formulating stages. The method provides more general closed form solutions than other currently available formulae. As such, the proposed procedure allows for the determination of the optimal numbers of units for studies that involve more complex designs. A method is also described for optimizing designs when variance component estimates are not available. Case studies are provided to demonstrate the method.

[1]  R. Kirk.  Practical Significance: A Concept Whose Time Has Come , 1996 .

[2]  Martijn P. F. Berger,et al.  Optimal experimental designs for multilevel logistic models , 2001 .

[3]  W. Deming.  On a Probability Mechanism to Attain an Economic Balance Between the Resultant Error of Response and the Bias of Nonresponse , 1953 .

[4]  Samuel H. Brooks.  The Estimation of an Optimum Subsampling Number , 1955 .

[5]  F. Hsieh,et al.  Sample size formulae for intervention studies with the cluster as unit of randomization. , 1988, Statistics in medicine.

[6]  D. Rubin,et al.  Contrasts and Effect Sizes in Behavioral Research , 1999 .

[7]  C. Jm.  The use of components of variance in preparing schedules for sampling of baled wool. , 1951 .

[8]  S. Raudenbush.  Statistical analysis and optimal design for cluster randomized trials , 1997 .

[9]  G. Glass.  Primary, Secondary, and Meta-Analysis of Research1 , 1976 .

[10]  OPTIMIZING EXPERIMENTAL DESIGNS RELATIVE TO VARIANCE COMPONENTS AND PRICE INPUTS , 2001 .

[11]  K. Muller,et al.  Power Calculations for General Linear Multivariate Models Including Repeated Measures Applications. , 1992, Journal of the American Statistical Association.

[12]  Martijn P. F. Berger,et al.  Design Issues for Experiments in Multilevel Populations , 2000 .

[13]  A. C. Chiang.  Fundamental methods of mathematical economics , 1974 .

[14]  M. J. Allen.  Introduction to Measurement Theory , 1979 .

[15]  P. Lachenbruch.  Statistical Power Analysis for the Behavioral Sciences (2nd ed.) , 1989 .

[16]  George A. Marcoulides,et al.  Maximizing Power in Generalizability Studies Under Budget Constraints , 1993 .

[17]  J. Overall,et al.  Design of experiments to maximize power relative to cost. , 1965, Psychological bulletin.

[18]  A Donner,et al.  A methodological review of non-therapeutic intervention trials employing cluster randomization, 1979-1989. , 1990, International journal of epidemiology.

[19]  Roel Bosker,et al.  Standard Errors and Sample Sizes for Two-Level Research , 1993 .

[20]  B. J. Winer.  Statistical Principles in Experimental Design , 1992 .

[21]  D. C. Howell.  Statistical Methods for Psychology , 1987 .

[22]  L. Hedges.  Distribution Theory for Glass's Estimator of Effect size and Related Estimators , 1981 .