A Formula for Determining Sample Sizes to Study Dose-Response
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0092-86 15/97 Copyright 0 1997 Drug Information Association Inc.
A FORMULA FOR DETERMINING SAMPLE SIZES TO STUDY DOSE-RESPONSE GIL D. FINE,PHD Senior Biostatistician, Inflammatory Diseases Unit, Roche Bioscience, Palo Alto, California, and Instructor, De Anza College, Cupertino, California
Determining the appropriate sample size for a clinical trial requires careful &liberation. Cost and ethical considerations dictate that a minimum number of patients be enrolled to study the safety and eficacy of a new chemical entity, yet the probability of not observing a true treatment effect is overly high if too few patients are enrolled. Recent results regarding adequate sample size for dose-response (DR) studies were based on Monte Carlo simulation. Application of such results is limited to planning DR studies that match the simulation protocol with respect to the desired power, number and spacing of dose levels, level of significance, and actual DR relationship. Complete flexibility in planning DR studies requires a formula relating power to sample size and other design parameters. This paper describes such a formula that is applicable when standard analyses are planned. An example is given to suggest possible alternative hypotheses and to illustrate application of the sample size formula. Key Words: Sample size; Dose-response; Contrast; Power; Polynomial regression
INTRODUCTION TRADITIONAL LINEAR regression is related to the fixed-effect analysis-of-variance (ANOVA) model rather than the randomeffect ANOVA in that values of the independent variable X are measured without error. Polynomial regression, a traditional linear regression in which powers of X appear as independent variables, is also a fixed-effect model. Analysis of dose-response data fits this theoretical framework since: 1. Dose levels of therapeutic agents are predetermined constants (ie, they are not considered observations of a random variable), and
Reprint address: Gil D. Fine, PhD,Mail Stop A2-269, Roche Bioscience, 3401 Hillview Avenue, Palo Alto, CA, 94304-1 397; E-mail:Gil.FineORoche.com.
2. Polynomial equations are adequate to characterize many DR relationships (1).
This analysis is advocated for clinical data (2), and is applicable in many diverse fields. Contrasts of data are used to test a variety of hypotheses including equality of population means, parallelism of regression lines, and adequacy of a polynomial equation to characterize the relationship between quantitative variables. Contrasts also are used to estimate important statistical parameters. In fact, least-squares estimates of regression model parameters are contrasts of observed responses (1). Hence, the common use of orthogonal contrasts (also known as orthogonal polynomials) to test for linear, quadratic, and higher order DR when dose levels are equally spaced on an appropriate scale is simply an application of polynomial regression after a suitable transformation of dose-level
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