Sample size calculations for randomized controlled trials.

Most informed consent documents for randomized controlled trials implicitly or explicitly promise the prospective participant that the trial has a reasonable chance of answering a medically important question. The medical literature, however, is replete with descriptions of trials that provided equivocal answers to the questions they addressed. Papers describing the results of such studies may clearly imply that the trial required a much larger sample size to adequately address the questions it posed. Hidden in file drawers, undoubtedly, are data from other trials whose results never saw the light of day—some, perhaps, victims of inadequate sample size. Although many inadequate-sized studies are performed in a single institution with patients who happen to be available, some are multicenter trials designed with overly optimistic assumptions about the effectiveness of therapy, too high an estimate of the event rate in the control group, or unrealistic assumptions about follow-up and compliance. In this review, I discuss statistical considerations in the choice of sample size and statistical power for randomized controlled trials. Underlying the discussion is the view that investigators should hesitate before embarking on a trial that is unlikely to detect a biologically reasonable effect of therapy. Such studies waste both time and resources. The number of participants in a randomized controlled trial can vary over several orders of magnitude. Rather than choose an arbitrary sample size, an investigator should allow both the variability of response to therapy and the assumed degree of effectiveness of therapy to drive the number of people to be studied in order to answer a scientific question. The more variable the response, the larger the sample size necessary to assess whether an observed effect of therapy represents a true effect of treatment or simply reflects random variation. On the other hand, the more effective or harmful the therapy, the smaller the trial required to detect that benefit or harm. As is often pointed out, only a few observations sufficed to demonstrate the dramatic benefit of penicillin; however, few therapies provide such unequivocal evidence of cure, so study of a typical medical intervention requires a large sample size. Lack of resources often constrains sample size. When they are limited by a restricted budget or a small patient pool, investigators should calculate the power of the trial to detect various outcomes of interest given the feasible sample size. A trial with very low statistical power may not be worth pursuing. Typical first trials of a new drug include only a handful of people. Trials that study the response of a continuous variable to an effective therapy—for example, blood pressure change in response to administration of an antihypertensive agent—may include several tens of people. Controlled trials of diseases with high event rates—for example, trials of therapeutic agents for cancer—may study several hundred patients. Trials of prevention of complications of disease in slowly progressing diseases such as diabetes mellitus may enroll a few thousand people. Trials comparing agents of similar effectiveness—for instance, different thrombolytic interventions after a heart attack—may include tens of thousands of patients. The poliomyelitis vaccine trial included approximately a half-million participants (1). This review begins with some general ideas about approaches to calculation of sample size for controlled trials. It then presents a generic formula for sample size that can be specialized to continuous, binary, and time-to-failure variables. The discussion assumes a randomized trial comparing two groups but indicates approaches to more than two groups. An example from a hypothetical controlled trial that tests the effect of a therapy on levels of high density lipoprotein (HDL) cholesterol is used to illustrate each case. Having introduced a basic formula for sample size, the review discusses each element of the formula in relation to its applicability to controlled trials and then points to special complexities faced by many controlled trials—how the use of multiple primary endpoints, multiple treatment arms, and sequential monitoring affects the type I error rate and hence how these considerations should influence the choice of sample size; how staggered entry and lag time to effect of therapy affect statistical power in studies with binary or time-to-failure endpoints; how noncompliance with prescribed therapy attenuates the difference between treated groups and control groups; and how to adjust sample size during the course of the trial to maintain desired power. The review discusses the consequences to sample size calculation of projected rates of loss to follow-up and competing risks. It suggests strategies for determining reasonable values to assume for the different parameters in the formulas. Finally, the review addresses three special types of studies: equivalence trials, multiarm trials, and factorial designs. Calculation of sample size is fraught with imprecision, Received for publication November 1, 2001, and accepted for publication April 16, 2002. Abbreviation: HDL, high density lipoprotein. From Statistics Collaborative, Inc., 1710 Rhode Island Avenue NW, Suite 200, Washington, DC 20036 (e-mail: janet@ statcollab.com). (Reprint requests to Dr. Janet Wittes at this address). Epidemiologic Reviews Vol. 24, No. 1 Copyright © 2002 by the Johns Hopkins Bloomberg School of Public Health Printed in U.S.A. All rights reserved