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On the Determination of Sample Size

Umbach, David M.

doi: 10.1097/01.EDE.0000044327.46102.64

From the Biostatistics Branch, National Institute of Environmental Health Sciences, Research Triangle Park, NC.

Address correspondence to: Biostatistics Branch, National Institute of Environmental Health Sciences, Mail Drop A3–03, P.O. Box 12233, Research Triangle Park, NC 27709–2233;

Epidemiologists face issues of sample size every time they design a study. Sample size determination has two distinct aspects. One is the technical aspect of how to calculate the sample size required to meet the desired Type I error rate and the power for any specified state of nature. (By “state of nature,” I mean the exact point in the relevant multidimensional parameter space for the statistical testing problem, the point that corresponds to the hypothesized effect that one seeks to detect. If the specified state of nature is “close” to any state that satisfies the null hypothesis, a relatively large sample size will be required to achieve the desired power, whereas if the specified state of nature is “far” from the null hypothesis, a relatively small sample size will do.)

The second and more philosophic aspect of sample size determination is the question of how to specify the state of nature that is most relevant to the study. The paper by Yang et al. 1 is noteworthy in that it provides an avenue for thinking about this more philosophic aspect. Their core idea is that, in designing a study to detect a certain phenomenon (in this case, “gene-environment interaction”), the size of the effect under investigation can be usefully examined from multiple points of view. The authors consider two: one based on the relative odds ratio for interaction (R i), and the other based on a population attributable fraction attributable to interaction (PAF i). Both points of view describe the same phenomenon, the same state of nature. Mathematically, to fix one is to determine the other. Even so, for a given state of nature, the relation between the value of R i and the corresponding value of PAF i is not intuitively obvious. Consequently, it is useful to calculate both. A comparison of their values provides perspective on the size of interaction that one views as reasonable to detect. The authors’ core idea can be useful for design issues in addition to interaction and is worthy of epidemiologists’ attention.

How do we choose a state of nature for sample size calculation? On one hand, we would like the choice to be “realistic.” On the other hand, if we actually knew the true state in advance, any study would be superfluous. Thus, we desire a plausible state that, although not necessarily true, embodies a scientifically interesting or relevant possibility. Certain practical pressures also arise, such as making the study look worthy to grant reviewers or defending a study as worthwhile within a circumscribed budget. Choosing states of nature to determine sample size is a multifaceted process in which competing interests and intuitions confront the investigator.

The authors’ main contribution is in helping us to make an informed decision about what constitutes a reasonable state of nature. Suppose an investigator selects values that include a reasonable relative risk for gene-environment interaction to specify a state of nature and then independently chooses a reasonable population attributable fraction to describe the same interaction. As Yang and colleagues 1 show, these two separately chosen measures of the interaction effect may seem equally reasonable a priori but turn out to correspond to quite distinct states of nature. The investigator’s task would then be to reexamine the beliefs, intuitions, values, etc., that led to those conflicting a priori specifications. The investigator must reconcile or re-calibrate his or her thinking to arrive at a single state of nature that represents both a reasonable R i and a reasonable PAF i. Alternatively, an investigator with an eye to public health impact would more likely specify a state of nature in terms of population attributable fraction. That investigator might be well advised to reexpress the state in terms of relative risk. If the corresponding R i value were unrealistically large, that investigator might reevaluate the scientific relevance of the assumptions that generated the PAF i. Similarly, a grant reviewer faced with a proposal justified in terms of R i might translate to PAF i, especially if public health relevance were a criterion of merit. Then the magnitude of PAF i could be used as an aid to assessing the likely public health relevance of the proposed study.

In designing a study to detect a certain phenomenon. . .the size of the effect can be usefully examined from multiple points of views.

The point here is that the ability to describe any chosen state of nature in terms of either R i or PAF i gives investigators a useful way to calibrate their intuitive valuation of one measure of effect based on relative odds ratios and another measure of effect related to public health impact. In this way, researchers may find a more rational basis for determining what will be an adequate sample size for their study.

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About the Author

DAVID UMBACH is a statistician whose research focuses on the development of statistical tools for detecting and characterizing gene-environment interactions in epidemiologic research. He is also interested in the statistical analysis of data arising from newer technologies, such as gene expression microarrays or SELDI-TOF mass spectrometry. A substantial portion of his time is devoted to collaborations with environmental and molecular epidemiologists.

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1. Yang Q, Khoury MJ, Friedman JM, Flanders WD. On the use of population attributable fraction to determine sample size for case-control studies of gene-environment interaction. Epidemiology 2003; 14: 161–167.
© 2003 Lippincott Williams & Wilkins, Inc.