Sample size planning for statistical power and accuracy in parameter estimation

Annual Review of Psychology

Volumn 59, Issue , 2008, Pages 537-563

  Maxwell, Scott E ^a   Kelley, Ken ^b   Rausch, Joseph R ^c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b INDIANA UNIVERSITY   (United States)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Confidence intervals; Cumulative science; Effect size

Indexed keywords

CONFIDENCE INTERVAL; EPIDEMIOLOGY; HUMAN; METHODOLOGY; PSYCHOLOGICAL MODEL; PSYCHOLOGY; REVIEW; STATISTICAL MODEL; STATISTICS;

CONFIDENCE INTERVALS; HUMANS; LINEAR MODELS; MODELS, PSYCHOLOGICAL; PSYCHOLOGY; SAMPLING STUDIES;

EID: 41549115181     PISSN: 00664308     EISSN: None     Source Type: Book Series    
DOI: 10.1146/annurev.psych.59.103006.093735     Document Type: Review

References (150)

1. Algina J, Keselman HJ. 2000. Cross-validation sample sizes. Appl. Psychol. Meas. 24(2):173-79
2. Algina J, Moulder BC, Moser BK. 2002. Sample size requirements for accurate estimation of squared semipartial correlation coefficients. Multivariate Behav. Res. 37(1):37-57
3. Algina J, Olejnik S. 2000. Conducting power analyses for ANOVA and ANCOVA in between-subjects designs. Eval. Health Prof 26(3):288-314
4. Algina J, Olejnik S. 2003. Sample size tables for correlation analysis with applications in partial correlation and multiple regression analysis. Multivariate Behav. Res. 38(3):309-23
5. Am. Educ. Res. Assoc. 2006. Standards for Reporting on Empirical Social Science Research in AERA Publications. Washington, DC: Am. Educ. Res. Assoc. http://www.aera.net/ opportunities/?id=1850
6. Am. Psychol. Assoc. 2001. Publication Manual of the American Psychological Association. Washington, DC: Am. Psychol. Assoc. 5th ed.
7. Barthel FMS, Babiker A, Royston P, Parmar MKB. 2006. Evaluation of sample size and power for multi-arm survival trials allowing for nonuniform accrual, nonproportional hazards, loss to follow-up and cross-over. Stat. Med. 25:2521-42
8. Bates DM, Watts DG. 1988. Nonlinear Regression Analysis and Its Applications. New York: Wiley
9. Beal SL. 1989. Sample size determination for confidence intervals on the population mean and on the difference between two population means. Biometrics 45(3):969-77
10. Benjamini Y, Hochberg Y. 1995. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. R. Stat. Soc. B 57(1):289-300
11. Bernardo MVP, Harrington DP. 2001. Sample size calculations for the two-sample problem using the multiplicative intensity model. Stat. Med. 20:557-79
12. Berry DA 2004. Bayesian statistics and the efficiency and ethics of clinical trials. Stat. Sci. 19(1):175-87
13. Berry DA. 2006. Bayesian clinical trials. Nature 5:27-36
14. Bezeau S, Graves R. 2001. Statistical power and effect sizes of clinical neuropsychology research. J. Clin. Exp. Neuropsychol. 23:399-406
15. Biesanz JC, Deeb-Sossa N, Papadakis AA Bollen KA Curran PJ. 2004. The role of coding time in estimating and interpreting growth curve models. Psychol. Methods 9(1):30-52
16. Bollen KA, Curran PJ. 2006. Latent Curve Models: A Structural Equation Perspective. Hoboken, NJ: Wiley
17. Brookes ST, Whitely E, Egger M, Smith GD, Mulheran PA, Peters TJ. 2004. Subgroup analyses in randomized trials: risks of subgroup-specific analyses; power and sample size for the interaction test. J. Clin. Epidemiol. 57:229-36
18. Browne MW, Cudeck R. 1993. Alternative ways of assessing model fit. In Testing Structural Equation Models, ed. K Bollen, S Long, pp. 136-62. Newbury Park, NJ: Sage
19. Campbell MK, Thomson S, Ramsay CR, MacLennan GS, Grimshaw JM. 2004. Sample size calculator for cluster randomized trials. Comput. Biol. Med. 34:113-25
20. Cashen LH, Geiger SW. 2004. Statistical power and the testing of null hypotheses: a review of contemporary management research and recommendations for future studies. Organ. Res. Methods 7(2):151-67
21. Chan A. 2002. Power and sample size determination for noninferiority trials using an exact method. J. Biopharm. Stat. 12(4):457-69
22. Chan A, Altman DG. 2005. Epidemiology and reporting of randomized trials published in PubMed journals. Lancet 365:1159-62
23. Chan A, Hróbjartsson A, Haahr MT, Gotzsche PC, Altman DG. 2004. Empirical evidence for selective reporting of outcomes in randomized trials: comparison of protocols for published articles. JAMA 2291(20):2457-65
24. Coffey CS, Muller KE. 2003. Properties of internal pilots with the univariate approach to repeated measures. Statist. Med. 22:2469-85
25. Cohen J. 1962. The statistical power of abnormal-social psychological research: a review. J. Abnorm. Soc. Psychol. 65(3): 145-5 3
26. Cohen J. 1988. Statistical Power Analysis for the Behavioral Sciences. Hillsdale, NJ: Erlbaum. 2nd ed.
27. Cohen J. 1994. The earth is round (p < .05). Am. Psychol. 49(12):997-1003
28. Cohn LD, Becker BJ. 2003. How meta-analysis increases statistical power. Psychol. Methods 8(3):243-53
29. Diggle PJ, Heagerty P, Liang KY, Zeger SL. 2002. Analysis of Longitudinal Data. Oxford: Oxford Univ. Press. 2nd ed.
30. Dolan C, Van Der Sluis S, Grasman R. 2005. A note on normal theory power calculation in SEM with data missing completely at random. Struct. Equ. Model. 12(2):245-62
31. Federov V, Jones B. 2005. The design of multicentre trials. Stat. Methods Med. Res. 14:205-48
32. Festinger L, Carlsmith JM. 1959. Cognitive consequences of forced compliance. J. Abnorm. Soc. Psychol. 58(2):203-10
33. Glueck DH, Muller KE. 2003. Adjusting power for a baseline covariate in linear models. Stat. Med. 22:2535-51
34. Goodman SN, Berlin JA. 1994. The use of predicted confidence intervals when planning experiments and the misuse of power when interpreting results. Ann. Intern. Med. 121:200-6
35. Green SB. 1991. How many subjects does it take to do a regression analysis? Multivariate Behav. Res. 26:499-510
36. Greenwald AG. 1975. Consequences of prejudice against the null hypothesis. Psychol. Bull. 82:1-20
37. Hancock GR. 2006. Power analysis in covariance structure modeling. In Structural Equation Modeling: A Second Course, ed. GR Hancock, RO Mueller, pp. 69-115. Greenwich, CT: Inf. Age Publ.
38. Hancock GR, Freeman MJ. 2001. Power and sample size for the root mean square error of approximation test of not close fit in structural equation modeling. Educ. Psychol. Meas. 61(5):741-58
39. Harlow LL, Mulaik SA, Steiger JH. 1997. What If There Were No Significance Tests? Mahwah, NJ: Erlbaum
40. Hedeker D, Gibbons RD, Waternaux C. 1999. Sample size estimation for longitudinal designs with attrition: comparing time-related contrasts between two groups. J. Educ. Behav. Stat. 24(1):70-93
41. Hedges LV, Pigott TD. 2001. The power of statistical tests in meta-analysis. Psychol. Methods 6(3):203-17
42. Hedges LV, Pigott TD. 2004. The power of statistical tests for moderators in meta-analysis. Psychol. Methods 9(4):426-45
43. Hernandez AV, Steyerberg EW, Habbema DF. 2004. Covariate adjustment in randomized controlled trials with dichotomous outcomes increases statistical power and reduces sample size requirements. J. Clin. Epidemiol. 57:454-60
44. Hoenig JM, Heisey DM. 2001. The abuse of power: the pervasive fallacy of power calculations for data analysis. Am. Stat. 55(1):19-24
45. Hogarty KY, Hines CV, Kromrey JD, Ferron JM, Mumford KR. 2005. The quality of factor solutions in explanatory factor analysis: the influence of sample size, communality, and overdetermination. Educ. Psychol. Meas. 65(2):202-26
46. Hsieh FY, Bloch DA Larsen MD. 1998. A simple method of sample size calculation for linear and logistic regression. Stat. Med. 17:1623-34
47. Hsu JC. 1989. Sample size computation for designing multiple comparison experiments. Comput. Stat. Data Anal. 7:79-91
48. Hsu JC. 1996. Multiple Comparisons: Theory and Methods. London: Chapman & Hall
49. Hunter JE, Schmidt FL. 2004. Methods of Meta-Analysis: Correcting Error and Bias in Research Findings. Thousand Oaks, CA: Sage. 2nd ed.
50. Ioannidis JPA. 2005. Why most published research findings are false. PLoS Med. 2(8):696-701
51. Inoue LY, Berry DA, Parmigiani G. 2005. Relationship between Bayesian and frequentist sample size determination. Am. Stat. 59(1):79-87
52. Jennison C, Turnbull BW. 2006. Adaptive and nonadaptive group sequential tests. Biometrika 93(1):1-21
53. Jiroutek MR, Muller KE, Kupper LL, Stewart PW. 2003. A new method for choosing sample size for confidence interval-based inferences. Biometrics 59:580-90
54. Jo B. 2002. Statistical power in randomized intervention studies with noncompliance. Psychol. Methods 7(2): 178-93
55. Jones LV, Tukey JW. 2000. A sensible formulation of the significance test. Psychol. Methods 5(4):411-14
56. Julious SA 2004. Sample sizes for clinical trials with normal data. Stat. Med. 23:1921-86
57. Jung S, Ahn C. 2003. Sample size estimation for GEE method for comparing slopes in repeated measurements data. Stat. Med. 22:1305-15
58. Jung S, Ahn C. 2005. Sample size for a two-group comparison of repeated binary measurements using GEE. Stat. Med. 24:2583-96
59. Kelley K. 2007. Sample size planning for the squared multiple correlation coefficient: accuracy in parameter estimation via narrow confidence intervals. Behav. Res. Methods. In press
60. Kelley K, Maxwell SE. 2003. Sample size for multiple regression: obtaining regression coefficients that are accurate, not simply significant. Psychol. Methods 8(3):305-21
61. Kelley K, Maxwell SE. 2008. Sample size planning with applications to multiple regression: power and accuracy for omnibus and targeted effects. In The Sage Handbook of Social Research Methods, ed. P Alasuutari, L Bickman, J Brannen, pp. 166-92. London: Sage
62. Kelley K, Maxwell SE, Rausch JR. 2003. Obtaining power or obtaining precision: delineating methods of sample-size planning. Eval. Health Prof. 26(3):258-87
63. Kelley K, Rausch JR. 2006. Sample size planning for the standardized mean difference: accuracy in parameter estimation via narrow confidence intervals. Psychol. Methods 11(4):363-85
64. Kim KH. 2005. The relation among fit indexes, power, and sample size in structural equation modeling. Struct. Equ. Model. 12(3):368-90
65. Kraemer HC, Gardner C, Brooks JO, Yesavage JA 1998. The advantages of excluding underpowered studies in meta-analysis: inclusionist vs exclusionist viewpoints. Psychol. Methods 3:23-31
66. Kraemer HC, Mints J, Noda A, Tinklenberg J, Yesavage JA. 2006. Caution regarding the use of pilot studies to guide power calculations for study proposals. Arch. Gen. Psychiatry 63:484-89
67. Kraemer HC, Robinson TN. 2005. Are certain multicenter randomized clinical trial structures misleading clinical and policy decisions? Contemp. Clin. Trials 26:518-29
68. Kromrey JD, Rendina-Gobioff G. 2006. On knowing what we do not know. Educ. Psychol. Meas. 66(3):357-73
69. Lagakos SW. 2006. The challenge of subgroup analyses-reporting without distorting. N. Engl. J. Med. 354:1667-70
70. Lau J, Antman EM, Jimenez-Silva J, Kupelnick B, Mosteller F, Chalmers TC. 1992. Cumulative meta-analysis of therapeutic trials for myocardial infarction. N. Engl. J. Med. 327:248-54
71. Lee SJ, Zelen M. 2000. Clinical trials and sample size considerations: another perspective. Rejoinder. Stat. Sci. 15(2): 108-10
72. Lei PW, Dunbar SB. 2004. Effects of score discreteness and estimating alternative model parameters on power estimation methods in structural equation modeling. Struct. Equ. Model. 11(1):20-44
73. Leon AC. 2004. Sample size requirements for comparisons of two groups on repeated observations of a binary outcome. Eval. Health Prof. 27(1):34-44
74. Liang KY, Zeger SL. 1986. Longitudinal data analysis using generalized linear models. Biometrika 73:13-22
75. Lipsey MW. 1990. Design Sensitivity: Statistical Power for Experimental Research. Thousand Oaks, CA Sage
76. Liu J, Hsueh H, Hsieh E, Chen JJ. 2002. Tests for equivalence or noninferiority for paired binary data. Stat. Med. 21:231-45
77. Lubke G, Neale MC. 2006. Distinguishing between latent classes and continuous factors: resolution by maximum likelihood? Multivariate Behav. Res. 41:499-532
78. Lyles RH, Lin H, Williamson JM. 2007. A practical approach to computing power for generalized linear models with nominal, count, or ordinal responses. Stat. Med. 26:1632-48
79. MacCallum RC, Browne MW, Cai L. 2006. Testing differences between nested covariance structure models: power analysis and null hypotheses. Psychol. Methods 11(1):19-35
80. MacCallum RC, Browne MW, Sugawara HM. 1996. Power analysis and determination of sample size for covariance structure modeling. Psychol. Methods 1(2):130-49
81. MacCallum RC, Hong S. 1997. Power analysis in covariance structure modeling using GFI and AGFI. Multivariate Behav. Res. 32(2):193-210
82. MacCallum RC, Widaman KF, Preacher KJ, Hong S. 2001. Sample size in factor analysis: the role of model error. Multivariate Behav. Res. 36(4):611-37
83. MacCallum RC, Widaman KF, Zhang S, Hong S. 1999, Sample size in factor analysis. Psychol. Methods 4(1):84-99
84. Maggard MA, O'Connell JB, Liu JH, Etzioni DA, Ko CK. 2003. Sample size calculations in surgery: Are they done correctly? Surgery 134(2):275-79
85. Maki E. 2006. Power and sample size considerations in clinical trials with competing risk endpoints. Pharm. Stat. 5:159-71
86. Maxwell SE. 2000. Sample size and multiple regression analysis. Psychol. Methods 5(4):434-58
87. Maxwell SE. 2004. The persistence of underpowered studies in psychological research: causes, consequences, and remedies. Psychol. Methods 9(2): 147-63
88. McArdle JJ. 1988. Dynamic but structural equation modeling of repeated measures data. In The Handbook of Multivariate Experimental Psychology, Vol. 2, ed. JR Nesselroade, RB Cattell, pp. 561-614. New York: Plenum
89. McArdle JJ, Epstein D. 1987. Latent growth curves within developmental structural equation models. Child. Dev. 58:110-33
90. McCartney K, Rosenthal R. 2000. Effect size, practical importance, and social policy for children. Chid. Dev. 71(1):173-80
91. McClelland GH. 2000. Increasing statistical power without increasing sample size. Am. Psychol. 55(8):963-64
92. McLachlan GJ, Peel D. 2000. Finite Mixture Models. New York: Wiley
93. Meehl PE. 1978. Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald, and the slow progress of soft psychology. J. Consult. Clin. Psychol. 46(4):806-34
94. Mehta CR, Patel NR. 2006. Adaptive, group sequential and decision theoretic approaches to sample size determination. Stat. Med. 25:3250-69
95. Meredith W, TisakJ. 1984. "Tuckerizing" curves. Presented at Annu. Meet. Psychometric Soc., Santa Barbara, Calif.
96. Meredith W, Tisak J. 1990. Latent curve analysis. Psychometrika 55:107-22
97. Miller RG. 1981. Simultaneous Statistical Inference. New York: Springer-Verlag. 2nd ed.
98. M'Lan CE, Joseph L, Wolfson DB. 2006. Bayesian sample size determination for case-control studies. J. Am. Stat. Assoc. 101(474):760-72
99. MoerbeekM. 2006. Power and money in cluster randomized trials: When is it worth measuring a covariate? Stat. Med. 25:2607-17
100. Mooijaart A. 2003. Estimating the statistical power in small samples by empirical distributions. In New Development in Psychometrics, ed. H Yanai, A Okada, K Shigemasu, Y Kano, JJ Meulman, pp. 149-56. Tokyo: Springer-Verlag
101. Muller KE, LaVange LM, Ramey SL, Ramey CT. 1992. Power calculations for general linear multivariate models including repeated measures applications. J. Am. Stat. Assoc. 87:1209-26
102. Muller KE, Pasour VB. 1997. Bias in linear model power and sample size due to estimating variance. Commun. Stat: Theory Methods 26:839-51
103. Muñoz MA, Acuña JD. 1999. Sample size requirements of a mixture analysis method with applications in systematic biology. J. Theor. Biol. 196:263-165
104. Murray DM, van Horn ML, Hawkins JD, Arthur MW. 2006. Analysis strategies for a community trial to reduce adolescent ATOD use: a comparison of random coefficient and ANOVA/ANCOVA models. Contemp. Clin. Trials 27:188-206
105. Muthén BO, Curran PJ. 1997. General longitudinal modeling of individual differences in experimental designs: a latent variable framework for analysis and power estimation. Psychol. Methods 24(4):371-402
106. Muthén LK, Muthén BO. 2002. How to use a Monte Carlo study to decide on sample size and determine power. Struct. Equ. Model. 9(4):599-620
107. Nasser F, Wisenbaker J. 2001. Modeling the observation-to-variable ratio necessary for determining the number of factors by the standard error scree procedure using logistic regression. Educ. Psychol. Meas. 61(3):3 87-403
108. Newson R. 2004. Generalized power calculations for generalized linear models and more. Stata J. 4(4):379-401
109. Nickerson RS. 2000. Null hypothesis significance testing: a review of an old and continuing controversy. Psychol. Methods 5(2):241-301
110. O'Brien RG, Castelloe JM. 2007. Sample size analysis for traditional hypothesis testing: concepts and issues. In Pharmaceutical Statistics Using SAS: A Practical Guide, ed. A Dmitrienko, C Chuang-Stein, R D'Agostino, pp. 237-71. Cary, NC: SAS
111. O'Brien RG, Muller KE. 1993. Unified power analysis for t-tests through multivariate hypotheses. In Applied Analysis of Variance in Behavioral Science, ed. LK Edwards, pp. 297-344. New York: Marcel Dekker
112. O'Hagan A Steves JW, Campbell MJ. 2005. Assurance in clinical trial design. Pharm. Stat. 4:187-201
113. Pan Z, Kupper LL. 1999. Sample size determination for multiple comparison studies treating confidence interval width as random. Stat. Med. 18:1475-88
114. Prentice DA Miller DT. 1992. When small effects are impressive. Psychol. Bull. 112(1):160-64
115. Proschan MA. 2005. Two-stage sample size re-estimation based on a nuisance parameter: a review. J. Biopharm. Stat. 15(4):559-74
116. Raudenbush SW. 1997. Statistical analysis and optimal design for cluster randomized trials. Psychol. Methods 2:173-85
117. Raudenbush SW, Bryk AS. 2002. Hierarchical Linear Models: Applications and Data Analysis Methods. Thousand Oaks, CA: Sage. 2nd ed.
118. Raudenbush SW, Xiao-Feng L. 2 001. Effects of study duration, frequency of observation, and sample size on power in studies of group differences in polynomial change. Psychol. Methods 6(4):387-401
119. Rochon J. 1998. Application of GEE procedures for sample size calculations in repeated measures experiments. Stat. Med. 17:1643-58
120. Satorra A Saris WE. 1985. Power of the likelihood ratio test in covariance structure analysis. Psychometrika 50(1):83-90
121. Schmidt FL. 1996. Statistical significance testing and cumulative knowledge in psychology: implications for training of researchers. Psychol. Methods 1:115-29
122. Schulgen G, Olschewski M, Krane V, Wanner C, Ruff G, Schumacher M. 2005. Sample size for clinical trials with time-to-event endpoints and competing risks. Contemp. Clin. Trials 26:386-96
123. Senn SJ. 2002. Power is indeed irrelevant in interpreting completed studies. BMJ 325:1304
124. Serlin RC, Lapsley DK. 1985. Rationality in psychological research: the good-enough principle. Am. Psychol. 40(1):73-83
125. Serlin RC, Lapsley DK. 1993. Rational appraisal of psychological research and the goodenough principle. In A Handbook for Data Analysis in the Behavioral Sciences: Methodological Issues, ed. G Keren, C Lewis, pp. 199-228. Hillsdale, NJ: Erlbaum
126. Shadish WR, Cook TD, Campbell DT. 2002. Experimental and Quasi-Experimental Designs for Generalized Causal Inference. Boston, MA: Houghton Mifflin
127. Shieh G. 2005. Power and sample size calculations for multivariate linear models with randomized explanatory variables. Psychometrika 70(2):347-58
128. Singer JD, Willett JB. 2003. Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. New York: Oxford Univ. Press
129. Steiger JH. 1990. Structural model evaluation and modification: an interval estimation approach. Multivariate Behav. Res. 25:173-80
130. Steiger JH, Lind JM. 1980. Statistically Based Tests for the Number of Common Factors. Presented at Annu. Meet. Psychometric Soc., Iowa City, IA
131. Sternberg RJ, Williams WW. 1997. Does the Graduate Record Examination predict meaningful success in the graduate training of psychology? A case study. Am. Psychol. 52(6):630-41
132. Strickland PA, Lu S. 2003. Estimates, power and sample size calculations for two-sample ordinal outcomes under before-after study designs. Stat. Med. 22:1807-18
133. Sutton AJ, Cooper NJ, Jones DR, Lambert PC, Thompson JR, Abrams KR. 2007. Evidence-based sample size calculations based upon updated meta-analysis. Stat. Med. 26:2479-500
134. Tang M, Tang N, Chan IS, Chan BP. 2002. Sample size determination for establishing equivalence/noninferiority via ratio of two proportions in matched-pair design. Biometrics 58:957-63
135. Taylor AB, West SG, Aiken LS. 2006. Loss of power in logistic, ordinal logistic, and probit regression when an outcome variable is coarsely categorized. Educ. Psychol. Meas. 66(2):228-39
136. Tsonaka R, Rizopoulos D, Lesaffre E. 2006. Power and sample size calculations for discrete bounded outcome scores. Stat. Med. 25:4241-52
137. Tryon WW. 2001. Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: an integrated alternative method of conducting null hypothesis statistical tests. Psychol. Methods 6(4):371-86
138. Vaeth M, Skovlund E. 2004. A simple approach to power and sample size calculations in logistic regression and Cox regression models. Stat. Med. 23:1781-92
139. Velicer WF, Fava JL. 1998. Effects of variable and subject sampling on factor pattern recovery. Psychol. Methods 3(2):231-51
140. Vergouwe Y, Steyerberg EW, Eijkemans MJC, Habbema JDF. 2005. Substantial effective sample sizes were required for external validation studies of predictive logistic regression models. J. Clin. Epidemiol. 58:475-83
141. Wang F, Gelfand AE. 2 002. A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models. Stat. Sci. 17:193-208
142. West SG, Biesanz JC, Pitts SC. 2000. Causal inference and generalization in field settings: experimental and quasi-experimental designs. In Handbook of Research Methods in Sodaland Personality Psychology, ed. HT Reis, CM Judd, pp. 40-84. New York: Cambridge Univ. Press
143. Wilkinson L, TaskForce Statistical Inference. 1999. Statistical methods in psychology journals: guidelines and explanations. Am. Psychol. 54(8):594-604
144. Williams VSL, Jones LV, Tukey JW. 1999. Controlling error in multiple comparisons, with examples from state-to-state differences in educational achievement. J. Educ. Behav. Stat. 24:42-69
145. Wilson DB, Lipsey MW. 2001. The role of method in treatment effectiveness research: evidence from meta-analysis. Psychol. Methods 6:413-29
146. Winkens B, Schouten HJA, van Breukelen GJP, Berger MPF. 2006. Optimal number of repeated measures and group sizes in clinical trials with linearly divergent treatment effects. Contemp. Clin. Trials 27:57-69
147. Yan X, Su X. 2006. Sample size determination for clinical trials in patients with nonlinear disease progression. J. Biopharm. Stat. 16:91-105
148. Yuan K, Hayashi K. 2003. Bootstrap approach to inference and power analysis based on three test statistics for covariance structure models. Br. J. Math. Stat. Psychol. 56(1):93-110
149. Yuan K, Maxwell SE. 2005. On the post hoc power in testing mean differences. J. Educ. Behav. Stat. 30(2): 141-67
150. Zheng J, Frey HC. 2004. Quantification of variability and uncertainty using mixture distributions: evaluation of sample size, mixing weights, and separation between components. Risk Anal. 24(1):553-71