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Sample Size

  • Lawrence M. Friedman
  • Curt D. Furberg
  • David L. DeMets
Chapter

Abstract

The size of the study should be considered early in the planning phase. In some instances, no formal sample size is ever calculated. Instead, the number of participants available to the investigators during some period of time determines the size of the study. Many clinical trials that do not carefully consider the sample size requirements turn out to lack the statistical power or ability to detect intervention effects of a magnitude that has clinical importance. In 1978, Freiman and colleagues [1] reviewed the power of 71 published randomized controlled clinical trials, which failed to find significant differences between groups. “Sixty-seven of the trials had a greater than 10% risk of missing a true 25% therapeutic improvement, and with the same risk, 50 of the trials could have missed a 50% improvement.” In other instances, the sample size estimation may assume an unrealistically large intervention effect. Thus, the power for more realistic intervention effects will be low or less than desired. The danger in studies with low statistical power is that interventions that could be beneficial are discarded without adequate testing and may never be considered again. Certainly, many studies do contain appropriate sample size estimates, but many are still too small.

Keywords

Event Rate Sample Size Calculation Sample Size Estimate Annual Event Rate Internal Pilot 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Freiman JA, Chalmers TC, Smith H, Jr, Kuebler RR. The importance of beta, the type II error and sample size in the design and interpretation of the randomized control trial. Survey of 71 “negative” trials. N Engl J Med 1978;299:690–694.CrossRefGoogle Scholar
  2. 2.
    Lachin JM. Introduction to sample size determination and power analysis for clinical trials. Control Clin Trials 1981;2:93–113.CrossRefGoogle Scholar
  3. 3.
    Brown BW, Jr. Statistical controversies in the design of clinical trials – some personal views. Control Clin Trials 1980;1:13–27.CrossRefGoogle Scholar
  4. 4.
    Altman DG. Statistics and ethics in medical research: III. How large a sample? Br Med J 1980;281:1336–1338.CrossRefGoogle Scholar
  5. 5.
    Gore SM. Assessing clinical trials – trial size. Br Med J 1981;282:1687–1689.CrossRefGoogle Scholar
  6. 6.
    Day SJ, Graham DF. Sample size estimation for comparing two or more treatment groups in clinical trials. Stat Med 1991;10:33–43.CrossRefGoogle Scholar
  7. 7.
    Donner A. Approaches to sample size estimation in the design of clinical trials – a review. Stat Med 1984;3:199–214.CrossRefGoogle Scholar
  8. 8.
    Phillips AN, Pocock SJ. Sample size requirements for prospective studies, with examples for coronary heart disease. J Clin Epidemiol 1989;42:639–648.CrossRefGoogle Scholar
  9. 9.
    Steiner DL. Sample size and power in psychiatric research. Can J Psychiatry 1990;35:616–620.Google Scholar
  10. 10.
    Whitehead J. Sample sizes for phase II and phase III clinical trials: An integrated approach. Stat Med 1986;5:459–464.CrossRefGoogle Scholar
  11. 11.
    Schlesselman JJ. Planning a longitudinal study: I. Sample size determination. J Chronic Dis 1973;26:553–560.CrossRefGoogle Scholar
  12. 12.
    Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions (3rd ed.). New York: John Wiley and Sons, 2003.MATHCrossRefGoogle Scholar
  13. 13.
    Snedecor GW, Cochran WG. Statistical Methods (8th ed.). Ames: Iowa State University Press, 1989.MATHGoogle Scholar
  14. 14.
    Brown BW, Hollander M. Statistics – A Biomedical Introduction. New York: John Wiley and Sons, 1977.Google Scholar
  15. 15.
    Remington RD, Schork MA. Statistics with Applications to the Biological and Health Sciences. Englewood Cliffs, NJ: Prentice-Hall, 1970.Google Scholar
  16. 16.
    Dixon WJ, Massey FJ, Jr. Introduction to Statistical Analysis (3rd ed.). New York: McGraw-Hill, 1969.Google Scholar
  17. 17.
    Armitage P, Berry G, Matthews JNS. Statistical Methods in Medical Research (4th ed.). Malden, MA: Blackwell Publishing, 2002.CrossRefGoogle Scholar
  18. 18.
    Woolson RF. Statistical Methods for the Analysis of Biomedical Data. New York: John Wiley and Sons, 1987.Google Scholar
  19. 19.
    Fisher L, Van Belle G. Biostatistics – A Methodology for the Health Sciences. New York: John Wiley and Sons, 1993.Google Scholar
  20. 20.
    Canner PL, Klimt CR. Experimental design features. Control Clin Trials 1983;4:313–332.CrossRefGoogle Scholar
  21. 21.
    Dunnett CW. Multiple comparison procedures for comparing several treatments with a control. J Am Stat Assoc 1955;50:1096–1121.MATHCrossRefGoogle Scholar
  22. 22.
    Costigan T. Bonferroni inequalities and intervals. In Armitage P, Colton T (eds.). Encyclopedia of Biostatistics (2nd ed.). New York: John Wiley & Sons, 2007.Google Scholar
  23. 23.
    Brittain E, Schlesselman JJ. Optimal allocation for the comparison of proportions. Biometrics 1982;38:1003–1009.CrossRefGoogle Scholar
  24. 24.
    The Cardiac Arrhythmia Suppression Trial (CAST) Investigators. Preliminary report: Effect of encainide and flecainide on mortality in a randomized trial of arrhythmia suppression after myocardial infarction. N Engl J Med 1989;321:406–412.CrossRefGoogle Scholar
  25. 25.
    Rothman KJ. A show of confidence. N Engl J Med 1978;299:1362–1363.CrossRefGoogle Scholar
  26. 26.
    McHugh RB, Le CT. Confidence estimation and the size of a clinical trial. Control Clin Trials 1984;5:157–163.CrossRefGoogle Scholar
  27. 27.
    Armitage P, McPherson CK, Rowe BC. Repeated significance tests on accumulating data. J R Stat Soc Ser A 1969;132:235–244.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Gail M, Gart JJ. The determination of sample sizes for use with the exact conditional test in 2 × 2 comparative trials. Biometrics 1973;29:441–448.CrossRefGoogle Scholar
  29. 29.
    Gail M. The determination of sample sizes for trials involving several independent 2 × 2 tables. J Chronic Dis 1973;26:669–673.CrossRefGoogle Scholar
  30. 30.
    Haseman JK. Exact sample sizes for use with the Fisher–Irwin Test for 2 × 2 tables. Biometrics 1978;34:106–109.CrossRefGoogle Scholar
  31. 31.
    Feigl P. A graphical aid for determining sample size when comparing two independent proportions. Biometrics 1978;34:111–122.CrossRefGoogle Scholar
  32. 32.
    Casagrande JT, Pike MC. An improved approximate formula for calculating sample sizes for comparing two binomial distributions. Biometrics 1978;34:483–486.MATHCrossRefGoogle Scholar
  33. 33.
    Ury HK, Fleiss JL. On approximate sample sizes for comparing two independent proportions with the use of Yates’ correction. Biometrics 1980;36:347–351.CrossRefGoogle Scholar
  34. 34.
    Fleiss JL, Tytun A, Ury HK. A simple approximation for calculating sample sizes for comparing independent proportions. Biometrics 1980;36:343–346.CrossRefGoogle Scholar
  35. 35.
    Wacholder S, Weinberg CR. Paired versus two-sample design for a clinical trial of treatments with dichotomous outcome: Power considerations. Biometrics 1982;38:801–812.CrossRefGoogle Scholar
  36. 36.
    Day SJ. Optimal placebo response rates for comparing two binomial proportions. Stat Med 1988;7:1187–1194.CrossRefGoogle Scholar
  37. 37.
    Fu YX, Arnold J. A table of exact sample sizes for use with Fisher’s exact test for 2 × 2 tables. Biometrics 1992;48:1103–1112.MATHCrossRefGoogle Scholar
  38. 38.
    Lachenbruch PA. A note on sample size computation for testing interactions. Stat Med 1988;7:467–469.CrossRefGoogle Scholar
  39. 39.
    McMahon RP, Proschan M, Geller NL, et al. Sample size calculation for clinical trials in which entry criteria and outcomes are counts of events. Stat Med 1994;13:859–870.CrossRefGoogle Scholar
  40. 40.
    Bristol DR. Sample sizes for constructing confidence intervals and testing hypotheses. Stat Med 1989;8:803–811.CrossRefGoogle Scholar
  41. 41.
    Connor RJ. Sample size for testing differences in proportions for the paired-sample design. Biometrics 1987;43:207–211.CrossRefGoogle Scholar
  42. 42.
    Donner A. Statistical methods in ophthalmology: An adjusted chi-square approach. Biometrics 1989;45:605–611.MATHCrossRefGoogle Scholar
  43. 43.
    Gauderman WJ, Barlow WE. Sample size calculations for ophthalmologic studies. Arch Ophthalmol 1992;110:690–692.CrossRefGoogle Scholar
  44. 44.
    Rosner B. Statistical methods in ophthalmology: An adjustment for the intraclass correlation between eyes. Biometrics 1982;38:105–114.CrossRefGoogle Scholar
  45. 45.
    Rosner B, Milton RC. Significance testing for correlated binary outcome data. Biometrics 1988;44:505–512.MATHCrossRefGoogle Scholar
  46. 46.
    Coronary Drug Project Research Group. The Coronary Drug Project. Design, methods, and baseline results. Circulation 1973;47:I1–I50.CrossRefGoogle Scholar
  47. 47.
    Aspirin Myocardial Infarction Study Research Group. A randomized, controlled trial of aspirin in persons recovered from myocardial infarction. JAMA 1980;243:661–669.CrossRefGoogle Scholar
  48. 48.
    Beta-Blocker Heart Attack Trial Research Group. A randomized trial of propranolol in patients with acute myocardial infarction. I. Mortality results. JAMA 1982;247:1707–1714.CrossRefGoogle Scholar
  49. 49.
    The Intermittent Positive Pressure Breathing Trial Group. Intermittent positive pressure breathing therapy of chronic obstructive pulmonary disease – a clinical trial. Ann Intern Med 1983;99:612–620.CrossRefGoogle Scholar
  50. 50.
    CASS Principal Investigators and Their Associates. Coronary Artery Surgery Study (CASS): A randomized trial of coronary artery bypass surgery. Survival data. Circulation 1983;68:939–950.CrossRefGoogle Scholar
  51. 51.
    Hypertension Detection and Follow-up Program Cooperative Group. Five-year findings of the Hypertension Detection and Follow-up Program. I. Reduction in mortality of persons with high blood pressure, including mild hypertension. JAMA 1979;242:2562–2571.CrossRefGoogle Scholar
  52. 52.
    Multiple Risk Factor Intervention Trial Research Group. Multiple risk factor intervention trial. Risk factor changes and mortality results. JAMA 1982;248:1465–1477.CrossRefGoogle Scholar
  53. 53.
    Packer M, Carver JR, Rodeheffer RJ, et al. for the PROMISE Study Research Group. Effect of oral milrinone on mortality in severe chronic heart failure. N Engl J Med 1991;325:1468–1475.CrossRefGoogle Scholar
  54. 54.
    Halperin M, Rogot E, Gurian J, Ederer F. Sample sizes for medical trials with special reference to long-term therapy. J Chronic Dis 1968;21:13–24.CrossRefGoogle Scholar
  55. 55.
    Schork MA, Remington RD. The determination of sample size in treatment–control comparisons for chronic disease studies in which drop-out or non-adherence is a problem. J Chronic Dis 1967;20:233–239.CrossRefGoogle Scholar
  56. 56.
    Wu M, Fisher M, DeMets D. Sample sizes for long-term medical trial with time-dependent dropout and event rates. Control Clin Trials 1980;1:111–124.CrossRefGoogle Scholar
  57. 57.
    Barlow W, Azen S. for the Silicone Study Group. The effect of therapeutic treatment crossovers on the power of clinical trials. Control Clin Trials 1990;11:314–326.CrossRefGoogle Scholar
  58. 58.
    Lakatos E. Sample size determination in clinical trials with time-dependent rates of losses and noncompliance. Control Clin Trials 1986;7:189–199.CrossRefGoogle Scholar
  59. 59.
    Lavori P. Statistical issues: Sample size and drop out. In Benkert O, Maier W, Rickels K (eds.). Methodology of the Evaluation of Psychotropic Drugs. Berlin: Springer-Verlag, 1990.Google Scholar
  60. 60.
    Newcombe RG. Explanatory and pragmatic estimates of the treatment effect when deviations from allocated treatment occur. Stat Med 1988;7:1179–1186.CrossRefGoogle Scholar
  61. 61.
    Pentico DW. On the determination and use of optimal sample sizes for estimating the difference in means. Am Stat 1981;35:40–42.Google Scholar
  62. 62.
    Schlesselman JJ. Planning a longitudinal study: II. Frequency of measurement and study duration. J Chronic Dis 1973;26:561–570.CrossRefGoogle Scholar
  63. 63.
    Dawson JD, Lagakos SW. Size and power of two-sample tests of repeated measures data. Biometrics 1993;49:1022–1032.MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Kirby AJ, Galai N, Muñoz A. Sample size estimation using repeated measurements on biomarkers as outcomes. Control Clin Trials 1994;15:165–172.CrossRefGoogle Scholar
  65. 65.
    Laird NM, Wang F. Estimating rates of change in randomized clinical trials. Control Clin Trials 1990;11:405–419.CrossRefGoogle Scholar
  66. 66.
    Lipsitz SR, Fitzmaurice GM. Sample size for repeated measures studies with binary responses. Stat Med 1994;13:1233–1239.CrossRefGoogle Scholar
  67. 67.
    Nam J. A simple approximation for calculating sample sizes for detecting linear trend in proportions. Biometrics 1987;43:701–705.CrossRefGoogle Scholar
  68. 68.
    Overall JE, Doyle SR. Estimating sample sizes for repeated measurement designs. Control Clin Trials 1994;15:100–123.CrossRefGoogle Scholar
  69. 69.
    Rochon J. Sample size calculations for two-group repeated-measures experiments. Biometrics 1991;47:1383–1398.MathSciNetCrossRefGoogle Scholar
  70. 70.
    Pasternack BS, Gilbert HS. Planning the duration of long-term survival time studies designed for accrual by cohorts. J Chronic Dis 1971;24:681–700.CrossRefGoogle Scholar
  71. 71.
    Pasternack BS. Sample sizes for clinical trials designed for patient accrual by cohorts. J Chronic Dis 1972;25:673–681.CrossRefGoogle Scholar
  72. 72.
    George SL, Desu MM. Planning the size and duration of a clinical trial studying the time to some critical event. J Chronic Dis 1974;27:15–24.CrossRefGoogle Scholar
  73. 73.
    Schoenfeld DA, Richter JR. Nomograms for calculating the number of patients needed for a clinical trial with survival as an endpoint. Biometrics 1982;38:163–170.CrossRefGoogle Scholar
  74. 74.
    Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics 1983;39:499–503.MATHCrossRefGoogle Scholar
  75. 75.
    Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Stat Med 1982;1:121–129.CrossRefGoogle Scholar
  76. 76.
    Akazawa K, Nakamura T, Moriguchi S. Simulation program for estimating statistical power of Cox’s proportional hazards model assuming no specific distribution for the survival time. Comput Methods Programs Biomed 1991;35:203–212.CrossRefGoogle Scholar
  77. 77.
    Cantor AB. Power estimation for rank tests using censored data: Conditional and unconditional. Control Clin Trials 1991;12:462–473.CrossRefGoogle Scholar
  78. 78.
    Emerich LJ. Required duration and power determinations for historically controlled studies of survival times. Stat Med 1989;8:153–160.CrossRefGoogle Scholar
  79. 79.
    Gail MH. Applicability of sample size calculations based on a comparison of proportions for use with the logrank test. Control Clin Trials 1985;6:112–119.CrossRefGoogle Scholar
  80. 80.
    Gross AJ, Hunt HH, Cantor AB, Clark BC. Sample size determination in clinical trials with an emphasis on exponentially distributed responses. Biometrics 1987;43:875–883.MATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    Halperin M, Johnson NJ. Design and sensitivity evaluation of follow-up studies for risk factor assessment. Biometrics 1981;37:805–810.MATHCrossRefGoogle Scholar
  82. 82.
    Hsieh FY. Sample size tables for logistic regression. Stat Med 1989;8:795–802.CrossRefGoogle Scholar
  83. 83.
    Lachin JM, Foulkes MA. Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics 1986;42:507–519. (Correction: 42:1009, 1986).MATHCrossRefGoogle Scholar
  84. 84.
    Lakatos E. Sample sizes based on the log-rank statistic in complex clinical trials. Biometrics 1988;44:229–241.MATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Lui K-J. Sample size determination under an exponential model in the presence of a confounder and type I censoring. Control Clin Trials 1992;13:446–458.CrossRefGoogle Scholar
  86. 86.
    Morgan TM. Nonparametric estimation of duration of accrual and total study length for clinical trials. Biometrics 1987;43:903–912.MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    Palta M, Amini SB. Consideration of covariates and stratification in sample size determination for survival time studies. J Chronic Dis 1985;38:801–809.CrossRefGoogle Scholar
  88. 88.
    Rubenstein LV, Gail MH, Santner TJ. Planning the duration of a comparative clinical trial with loss to follow-up and a period of continued observation. J Chronic Dis 1981;34:469–479.CrossRefGoogle Scholar
  89. 89.
    Taulbee JD, Symons MJ. Sample size and duration for cohort studies of survival time with covariables. Biometrics 1983;39:351–360.CrossRefGoogle Scholar
  90. 90.
    Wu MC. Sample size for comparison of changes in the presence of right censoring caused by death, withdrawal, and staggered entry. Control Clin Trials 1988;9:32–46.CrossRefGoogle Scholar
  91. 91.
    Zhen B, Murphy JR. Sample size determination for an exponential survival model with an unrestricted covariate. Stat Med 1994;13:391–397.CrossRefGoogle Scholar
  92. 92.
    Hjaimarson A, Herlitz J, Malek I, et al. Effect on mortality of metoprolol in acute myocardial infarction: A double-blind randomized trial. Lancet 1981;ii:823–827.CrossRefGoogle Scholar
  93. 93.
    The Norwegian Multicenter Study Group. Timolol-induced reduction in mortality and reinfarction in patients surviving acute myocardial infarction. N Engl J Med 1981;304:801–807.CrossRefGoogle Scholar
  94. 94.
    Nocturnal Oxygen Therapy Trial Group. Continuous or nocturnal oxygen therapy in hypoxemic chronic obstructive lung disease: A clinical trial. Ann Intern Med 1980;93:391–398.CrossRefGoogle Scholar
  95. 95.
    Ingle JN, Ahmann DL, Green SJ, et al. Randomized clinical trial of diethylstilbestrol versus tamoxifen in postmenopausal women with advanced breast cancer. N Engl J Med 1981;304:16–21.CrossRefGoogle Scholar
  96. 96.
    Spriet A, Beiler D. When can “non significantly different” treatments be considered as “equivalent”? (Letter to the editors). Br J Clin Pharmacol Ther 1979;7:623–624.CrossRefGoogle Scholar
  97. 97.
    Makuch R, Simon R. Sample size requirements for evaluating a conservative therapy. Cancer Treat Rep 1978;62:1037–1040.Google Scholar
  98. 98.
    Blackwelder WC. “Proving the null hypothesis” in clinical trials. Control Clin Trials 1982;3:345–353.CrossRefGoogle Scholar
  99. 99.
    Blackwelder WC, Chang MA. Sample size graphs for “proving the null hypothesis”. Control Clin Trials 1984;5:97–105.CrossRefGoogle Scholar
  100. 100.
    Anderson S, Hauck WW. A new procedure for testing equivalence in comparative bioavailability and other clinical trials. Commun Stat Theory Methods 1983;12:2663–2692.MATHCrossRefGoogle Scholar
  101. 101.
    Dunnett CW, Gent M. Significance testing to establish equivalence between treatments, with special reference to data in the form of 2 × 2 tables. Biometrics 1977;33:593–602.CrossRefGoogle Scholar
  102. 102.
    Westlake WJ. Statistical aspects of comparative bioavailability trials. Biometrics 1979;35:273–280.CrossRefGoogle Scholar
  103. 103.
    Kirkwood TBL, Westlake WJ. Response to “Bioequivalence testing-a need to rethink”. Biometrics 1981;37:589–594.CrossRefGoogle Scholar
  104. 104.
    Donner A, Birkett N, Buck C. Randomization by cluster. Sample size requirements and analysis. Am J Epidemiol 1981;114:906–914.Google Scholar
  105. 105.
    Hsieh FY. Sample size formulae for intervention studies with the cluster as unit of randomization. Stat Med 1988;7:1195–1201.CrossRefGoogle Scholar
  106. 106.
    Lee EW, Dubin, N. Estimation and sample size considerations for clustered binary responses. Stat Med 1994;13:1241–1252.CrossRefGoogle Scholar
  107. 107.
    Cornfield J. Randomization by group: A formal analysis. Am J Epidemiol 1978;108:100–102.Google Scholar
  108. 108.
    Church TR, Ederer F, Mandel JS, et al. Estimating the duration of ongoing prevention trials. Am J Epidemiol 1993;137:797–810.Google Scholar
  109. 109.
    Ederer F, Church TR, Mandel JS. Sample sizes for prevention trials have been too small. Am J Epidemiol 1993;137:787–796.Google Scholar
  110. 110.
    Neaton JD, Bartsch GE. Impact of measurement error and temporal variability on the estimation of event probabilities for risk factor intervention trials. Stat Med 1992;11:1719–1729.CrossRefGoogle Scholar
  111. 111.
    Patterson BH. The impact of screening and eliminating preexisting cases on sample size requirements for cancer prevention trials. Control Clin Trials 1987;8:87–95.CrossRefGoogle Scholar
  112. 112.
    Shih WJ. Sample size reestimation in clinical trials. In Peace KE (ed.). Biopharmaceutical Sequential Statistical Applications. New York: Marcel Dekker, 1992, pp. 285–301.Google Scholar
  113. 113.
    Wittes J, Brittain E. The role of internal pilot studies in increasing the efficiency of clinical trials. Stat Med 1990;9:65–72.CrossRefGoogle Scholar
  114. 114.
    Steering Committee of the Physicians’ Health Study Research Group. Final report on the aspirin component of the ongoing Physicians’ Health Study. N Engl J Med 1989;321:129–135.CrossRefGoogle Scholar
  115. 115.
    The Diabetes Control and Complications Trial Research Group. The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus. N Engl J Med 1993;329:977–986.CrossRefGoogle Scholar
  116. 116.
    Urokinase Pulmonary Embolism Trial Study Group. Urokinase–streptokinase embolism trial. Phase 2 results. JAMA 1974;229:1606–1613.CrossRefGoogle Scholar
  117. 117.
    Roberts R, Croft C, Gold HK, et al. Effect of propranolol on myocardial-infarct size in a randomized blinded multicenter trial. N Engl J Med 1984;311:218–225.CrossRefGoogle Scholar
  118. 118.
    Miller RG. Simultaneous Statistical Inference. New York: McGrawHill, 1966.MATHGoogle Scholar
  119. 119.
    The Coronary Drug Project Research Group. Clofibrate and niacin in coronary heart disease. JAMA 1975;231:360–381.CrossRefGoogle Scholar

Copyright information

© Springer New York 2010

Authors and Affiliations

  • Lawrence M. Friedman
    • 1
  • Curt D. Furberg
    • 2
  • David L. DeMets
    • 3
  1. 1.BethesdaUSA
  2. 2.School of MedicineWake Forest UniversityWinston-SalemUSA
  3. 3.Department of Biostatistics & Medical InformaticsUniversity of WisconsinMadisonUSA

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