# Calculating Sample Size in Anthropometry

• Carine A. Bellera
• Bethany J. Foster
• James A. Hanley
Chapter

## Abstract

Sample size estimation is a fundamental step when designing clinical trials and epidemiological studies for which the primary objective is the estimation or the comparison of parameters. One may be interested in the prevalence of overweight children in a given population; however, the true prevalence will remain unknown and cannot be observed unless the whole population is studied. Statistical inference is the use of statistics and random sampling to make inferences concerning the true parameters of a population. By choosing a representative sample, inference based on the observed prevalence leads to an estimation of the true parameter. But how many subjects should be sampled to obtain an accurate estimate of the prevalence? Similarly, how many subjects should we sample to show that this parameter is different from some fixed value? We first review basic statistical concepts including random variables, population and sample statistics, as well as probability distributions such as the binomial and normal distributions. Principles of point and interval estimation, as well as hypothesis testing, are presented. We consider several commonly used statistics: single proportions, differences between two proportions, single means, differences between two means, and reference limits. For each parameter, point estimators are presented as well as methods for constructing confidence intervals. We then review general methods for calculating sample sizes. We first consider precision-based estimation procedures, where the sample size is estimated as a function of the desired degree of precision. Next, although there is greater emphasis on precision-driven estimation procedures, we also briefly describe power-based estimation methods. This approach requires defining a priori the difference one wishes to detect, the desired significance level, and the desired power of the test. Sample size estimation procedures are presented for each parameter, and examples are systematically provided.

### Keywords

Cholesterol Obesity Cytel

### Abbreviations and Notations

N

Population size

n

Sample size

ME

Margin of error

e

Precision

m

Population mean

m

Sample mean

s 2

Population variance

s2

Sample variance

p

Population proportion

p

Sample proportion

H0

Null hypothesis

HA

Alternative hypothesis

a

Type I error rate

b

Type II error rate

zp

100p% standard normal deviate

BMI

Body mass index

DBP

Diastolic blood pressure

### References

1. Altman DG, Machin D, Bryant T, Gardner S. Statistics with confidence: Confidence Intervals and statistical Guidelines. 2nd ed. BMJ Books; 2000.Google Scholar
2. Armitage P, Berry G, Matthews JNS. Statistical methods in medical research. 4 ed. Blackwell Science; 2002.
3. Bellera CA, Hanley JA. A method is presented to plan the required sample size when estimating regression-based reference limits. J Clin Epidemiol. 2007;60:610–5.
4. Bonett DG. Sample size requirements for estimating intraclass correlations with desired precision. Stat Med. 2002;21:1331–5.
5. Cole TJ. The International Growth Standard for Preadolescent and Adolescent Children: Statistical considerations. Food Nutr Bull. 2006;27:S237–3.
6. Dixon DO, Simon R. Sample size considerations for studies comparing survival curves using historical controls. J Clin Epidemiol 1988;41:1209–13.
7. Donner A. Sample size requirements for the comparison of two or more coefficients of inter-observer agreement. Stat Med. 1998;17:1157–68.
8. Donner A, Birkett N, Buck C. Randomization by cluster. Sample size requirements and analysis. Am J Epidemiol. 1981;114:906–14.
9. Elveback LR, Guillier CL, Keating FR. Health, normality, and the ghost of Gauss. JAMA. 1970;211:69–75.
10. Fleming TR. Current issues in non-inferiority trials. Stat Med. 2008;27:317–32.
11. Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Stat Med. 1982;1:121–9.
12. Friedman L, Furberg C, DeMets DL. Fundamentals of clinical trials. 3rd ed. New York: Springer-Verlag; 1998.Google Scholar
13. Gardner MJ, Altman DG. Confidence intervals rather than P values: estimation rather than hypothesis testing. BMJ. 1986;292:746–50.
14. Gardner MJ, Altman DG. Estimating with confidence. BMJ. 1988;296:1210–1.
15. Goldstein H. Sampling for growth studies. In: Falkner F, Tanner JM, eds. Human growth: a comprehensive treatise, 2nd ed. New-York: Plenum Press. 1986. p. 59–78.Google Scholar
16. Gong J, Pinheiro JC, DeMets DL. Estimating significance level and power comparisons for testing multiple endpoints in clinical trials. Control Clin Trials. 2000;21:313–29.
17. Harris EK, Boyd JC. Statistical bases of reference values in laboratory medicine, Vol 146 of Statistics: textbooks and Monographs. New York: Marcel Dekker; 1995.Google Scholar
18. Hsieh FY. Sample size formulae for intervention studies with the cluster as unit of randomization. Stat Med. 1988;7:1195–201.
19. Lachin JM. Introduction to sample size determination and power analysis for clinical trials. Control Clin Trials. 1981;2:93–113.
20. Last J. A Dictionnary of Epidemiology. 4th ed. Oxford University Press; 2001.Google Scholar
21. Machin D, Campbell M, Fayers P, Pinol A. Sample size tables for clinical studies. 2nd ed. London: Blackwell Science; 1997.Google Scholar
22. Piantadosi S. Clinical Trials: A Methodologic Perspective. 2nd ed. Hoboken, New Jersey: John Wiley and Sons; 2005.
23. Sackett DL. Why randomized controlled trials fail but needn’t: 2. Failure to employ physiological statistics, or the only formula a clinician-trialist is ever likely to need (or understand!). Canadian Medical Association Journal 2001;165:1226–36.
24. Schoenfeld D. Sample-Size Formula for the Proportional-Hazards Regression Model. Biometrics. 1983;39:499–503.
25. World Health Organization. Department of nutrition for health and development. Length/height-for-age, weight-for-age, weight-for-length, weight-for-height and body mass index-for age: Methods and development. WHO Press; 2006.Google Scholar
26. Wright EM, Royston P. Calculating reference intervals for laboratory measurements. Stat Methods Med Res. 1999;8:93–112.Google Scholar