# Sample Design and Sample Size for Single-Stage Surveys

• Richard Valliant
• Jill A. Dever
• Frauke Kreuter
Chapter
Part of the Statistics for Social and Behavioral Sciences book series (SSBS)

## Abstract

Chapter  covers the problem of determining a sample size for single-stage surveys with targets for coefficients of variation, margins of error, and cost constraints. Determining sample sizes based on means, totals, and proportions are emphasized in this chapter. Designs covered are simple random samples selected without replacement, stratified simple random samples, and samples selected with varying probabilities (e.g., probability proportional to size samples, pps). Models are especially useful when analyzing pps sampling as discussed in Sect. 3.2.2. The chapter also covers more specialized topics, including systematic, Poisson, and some other sampling methods, and the estimation of population parameters that are needed in sample size formulas. R code examples are given to illustrate calculations.

## References

1. Barcaroli G. (2014). SamplingStrata: An R package for the optimization of stratified sampling. Journal of Statistical Software 61(4):1–24, URL http://www.jstatsoft.org/v61/i04/
2. Brown L., Cai T., Das Gupta A. (2001). Interval estimation for a binomial proportion. Statistical Science 16:101–133.
3. Bureau of Labor Statistics. (2006). Household Data (A tables, monthly: D tables, quarterly). Employment and Earnings. URL http://www.bls.gov/cps/eetech_methods.pdf
4. Bureau of Labor Statistics. (2018). Economic News Release: CPI Consumer Price Index, June 2009. URL http://www.bls.gov/news.release/cpi.nr0.htm
5. Center for Disease Control and Prevention. (2005). National hospital discharge survey: 2005 annual summary with detailed diagnosis and procedure data. Vital and Health Statistics (165), URL https://www.ncbi.nlm.nih.gov/pubmed/18350768
6. Chromy J. R. (1979). Sequential sample selection methods. In: Proceedings of the Survey Research Methods Section, American Statistical Association, pp 401–406.Google Scholar
7. Cochran W. (1977). Sampling Techniques. John Wiley & Sons, Inc., New York.
8. Council of the European Union. (1998). Council regulation (ec) no. 577/98 on the organization of a labour force survey in the community. Official Journal of the European Communities.Google Scholar
9. Council of the European Union. (2003). Council regulation (ec) no. 1177/2003 concerning community statistics on income and living conditions. Official Journal of the European Communities.Google Scholar
10. Deak M. A., Helba C., Lee K., Rockwell D., Perry S., Simmons R. O., D’Amato-Neff A. L., Ferro G., Lappin B. M. (2002). Tabulations of Responses from the 2000 Survey of Reserve Component Personnel: Defense Manpower Data Center, Volume 2 Military Plans, Military Training, and Military Unit. URL http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA415264&Location=U2&doc=GetTRDoc.pdf
11. Gambino J. G. (2005). pps: Functions for PPS sampling. URL http://CRAN.R-project.org/package=pps
12. Gentle J. (2003). Random Number Generation and Monte Carlo Methods. Springer, New York.
13. Godambe V. P., Joshi V. M. (1965). Admissibility and Bayes estimation in sampling finite populations – I. Annals of Mathematical Statistics 36:1707–1723.
14. Hansen M. H., Hurwitz W. N., Madow W. G. (1953a). Sample Survey Methods and Theory, Volume I. John Wiley & Sons, Inc., New York.Google Scholar
15. Henry K. A., Valliant R. (2009). Comparing sampling and estimation strategies in establishment populations. Survey Research Methods 3:27–44.Google Scholar
16. Henry K. A., Testa V. L., Valliant R. (2008). Variance estimation for an estimator of between-year change in totals from two stratified Bernoulli samples. In: Proceedings of the Survey Research Methods Section, American Statistical Association, pp 1108–1115.Google Scholar
17. Højsgaard S., Halekoh U. (2012). doBy: doBy – Groupwise summary statistics, general linear contrasts, population means (least-squares-means), and other utilities. URL http://CRAN.R-project.org/package=doBy, (contributions from J. Robison-Cox, K. Wright, A. A. Leidi).
18. Internal Revenue Service. (2004). Internal Revenue Bulletin: 2004-20, Meals and Entertainment Expenses. URL http://www.irs.gov/irb/2007-23_IRB/ar10.html
19. Internal Revenue Service. (2005). Audit Techniques Guide: Credit for Increasing Research Activities (i.e. Research Tax Credit). URL https://www.irs.gov/pub/irs-utl/rc2005atg2irsgovrepublished1_2008.pdf
20. Internal Revenue Service. (2007). Cost Segregation Audit Techniques Guide. URL http://www.irs.gov/Businesses/Cost-Segregation-Audit-Techniques-Guide-Table-of-Contents
21. Isaki C. T., Fuller W. A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association 77(377):89–96.
22. Jenkins S. (2005). SAMPLEPPS: Stata module to draw a random sample with probabilities proportional to size. Statistical Software Components, Boston College Department of Economics, URL https://ideas.repec.org/c/boc/bocode/s454101.html
23. Jovanovic B. D., Levy P. S. (1997). A look at the rule of three. The American Statistician 51:137–139.Google Scholar
24. Kalton G. (1993). Sampling rare and elusive populations. Tech. Rep. INT-92-P80-16E, Department for Economic and Social Information and Policy Analysis, United Nations.Google Scholar
25. Kish L. (1987a). Statistical Design for Research. John Wiley & Sons, Inc., New York.Google Scholar
26. Korn E. L., Graubard B. I. (1998). Confidence intervals for proportions with small expected number of positive counts estimated from survey data. Survey Methodology 24:193–201.Google Scholar
27. Kott P. S. (1988). Model-based finite population correction for the Horvitz-Thompson estimator. Biometrika 75:797–799.
28. Kott P. S., Liu Y. (2009). One-sided coverage intervals for a proportion estimated from a stratified simple random sample. International Statistical Review 77:251–265.
29. Lohr S. L. (1999). Sampling: Design and Analysis. Duxbury Press, Pacific Grove, CA.
30. Lumley T. (2017). survey: analysis of complex survey samples R package v. 3.32. URL http://CRAN.R-project.org/package=survey
31. Manitz J. (2012). samplingbook: Survey Sampling Procedures. URL http://CRAN.R-project.org/package=samplingbook, (contributions by M. Hempelmann, G. Kauermann, H. Kuechenhoff, S. Shao, C. Oberhauser, N. Westerheide, M. Wiesenfarth).
32. Matsumoto M., Nishimura T. (1998). Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation 8:3–30.
33. Newcombe R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine 17(8):857–872.
34. RTI International. (2012). SUDAAN Language Manual, Release 11.0. Research Triangle Park, NC.Google Scholar
35. Särndal C., Swensson B., Wretman J. (1992). Model Assisted Survey Sampling. Springer, New York.
36. Thayer W. C., Diamond G. L. (2002). Blood Lead Concentrations of U.S. Adult Females: Summary Statistics from Phases 1 and 2 of the National Health and Nutrition Evaluation Survey (NHANES III). URL http://www.epa.gov/superfund/lead/products/nhanes.pdf
37. Therneau T. (2012). survival: Survival analysis, including penalised likelihood. URL http://CRAN.R-project.org/package=survival
38. Tillé Y., Matei A. (2012). sampling: Survey Sampling. URL http://CRAN.R-project.org/package=sampling
39. US Center for Disease Control. (2007). Health Insurance Coverage: Early Release of Estimates from the National Health Interview Survey, 2006. URL https://www.cdc.gov/nchs/data/nhis/earlyrelease/insur200706.pdf
40. US Center for Disease Control. (2010). Healthy People 2010 Criteria for Data Suppression. URL www.cdc.gov/nchs/data/statnt/statnt24.pdf
41. Valliant R., Dever J. A. (2018). Survey Weights: A Step-by-Step Guide to Calculation. Stata Press, College Station, TX.Google Scholar
42. Valliant R., Dorfman A. H., Royall R. M. (2000). Finite Population Sampling and Inference: A Prediction Approach. John Wiley & Sons, Inc., New York.
43. Wilson E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association 22:209–212.

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Richard Valliant
• 1
• 2
• Jill A. Dever
• 3
• Frauke Kreuter
• 2
• 4
1. 1.University of MichiganAnn ArborUSA
2. 2.University of MarylandCollege ParkUSA
3. 3.RTI InternationalWashington, DCUSA
4. 4.University of MannheimMannheimGermany