Psychonomic Bulletin & Review

, Volume 14, Issue 5, pp 779–804

A practical solution to the pervasive problems ofp values

Theoretical and Review Articles

Abstract

In the field of psychology, the practice ofp value null-hypothesis testing is as widespread as ever. Despite this popularity, or perhaps because of it, most psychologists are not aware of the statistical peculiarities of thep value procedure. In particular,p values are based on data that were never observed, and these hypothetical data are themselves influenced by subjective intentions. Moreover,p values do not quantify statistical evidence. This article reviews thesep value problems and illustrates each problem with concrete examples. The three problems are familiar to statisticians but may be new to psychologists. A practical solution to thesep value problems is to adopt a model selection perspective and use the Bayesian information criterion (BIC) for statistical inference (Raftery, 1995). The BIC provides an approximation to a Bayesian hypothesis test, does not require the specification of priors, and can be easily calculated from SPSS output.

References

  1. Akaike, H. (1974). A new look at the statistical model identification.IEEE Transactions on Automatic Control,19, 716–723.CrossRefGoogle Scholar
  2. Anscombe, F. J. (1954). Fixed-sample-size analysis of sequential observations.Biometrics,10, 89–100.CrossRefGoogle Scholar
  3. Anscombe, F. J. (1963). Sequential medical trials.Journal of the American Statistical Association,58, 365–383.CrossRefGoogle Scholar
  4. Armitage, P. (1957). Restricted sequential procedures.Biometrika,44, 9–26.Google Scholar
  5. Armitage, P. (1960).Sequential medical trials. Springfield, IL: Thomas.Google Scholar
  6. Armitage, P., McPherson, C. K., &Rowe, B. C. (1969). Repeated significance tests on accumulating data.Journal of the Royal Statistical Society: Series A,132, 235–244.CrossRefGoogle Scholar
  7. Bakan, D. (1966). The test of significance in psychological research.Psychological Bulletin,66, 423–437.PubMedCrossRefGoogle Scholar
  8. Barnard, G. A. (1947). The meaning of a significance level.Biometrika,34, 179–182.Google Scholar
  9. Basu, D. (1964). Recovery of ancillary information.Sankhya: Series A,26, 3–16.Google Scholar
  10. Bayarri, M.-J., &Berger, J. O. (2004). The interplay of Bayesian and frequentist analysis.Statistical Science,19, 58–80.CrossRefGoogle Scholar
  11. Berger, J. O. (1985).Statistical decision theory and Bayesian analysis (2nd ed.). New York: Springer.Google Scholar
  12. Berger, J. O. (2003). Could Fisher, Jeffreys and Neyman have agreed on testing?Statistical Science,18, 1–32.CrossRefGoogle Scholar
  13. Berger, J. O., &Berry, D. A. (1988a). The relevance of stopping rules in statistical inference. In S. S. Gupta & J. O. Berger (Eds.),Statistical decision theory and related topics IV (Vol. 1, pp. 29–72). New York: Springer.Google Scholar
  14. Berger, J. O., &Berry, D. A. (1988b). Statistical analysis and the illusion of objectivity.American Scientist,76, 159–165.Google Scholar
  15. Berger, J. O., Boukai, B., &Wang, Y. (1997). Unified frequentist and Bayesian testing of a precise hypothesis (with discussion).Statistical Science,12, 133–160.CrossRefGoogle Scholar
  16. Berger, J. O., Brown, L., &Wolpert, R. (1994). A unified conditional frequentist and Bayesian test for fixed and sequential hypothesis testing.Annals of Statistics,22, 1787–1807.CrossRefGoogle Scholar
  17. Berger, J. O., &Delampady, M. (1987). Testing precise hypotheses.Statistical Science,2, 317–352.CrossRefGoogle Scholar
  18. Berger, J. O., &Mortera, J. (1999). Default Bayes factors for nonnested hypothesis testing.Journal of the American Statistical Association,94, 542–554.CrossRefGoogle Scholar
  19. Berger, J. O., &Pericchi, L. R. (1996). The intrinsic Bayes factor for model selection and prediction.Journal of the American Statistical Association,91, 109–122.CrossRefGoogle Scholar
  20. Berger, J. O., &Sellke, T. (1987). Testing a point null hypothesis: The irreconcilability of p values and evidence.Journal of the American Statistical Association,82, 112–139.CrossRefGoogle Scholar
  21. Berger, J. O., &Wolpert, R. L. (1988).The likelihood principle (2nd ed.). Hayward, CA: Institute of Mathematical Statistics.Google Scholar
  22. Bernardo, J. M., &Smith, A. F. M. (1994).Bayesian theory. Chichester, U.K.: Wiley.CrossRefGoogle Scholar
  23. Birnbaum, A. (1962). On the foundations of statistical inference (with discussion).Journal of the American Statistical Association,53, 259–326.Google Scholar
  24. Birnbaum, A. (1977). The Neyman—Pearson theory as decision theory, and as inference theory; with a criticism of the Lindley—Savage argument for Bayesian theory.Synthese,36, 19–49.CrossRefGoogle Scholar
  25. Box, G. E. P., &Tiao, G. C. (1973).Bayesian inference in statistical analysis. Reading, MA: Addison-Wesley.Google Scholar
  26. Browne, M. (2000). Cross-validation methods.Journal of Mathematical Psychology,44, 108–132.PubMedCrossRefGoogle Scholar
  27. Burdette, W. J., &Gehan, E. A. (1970).Planning and analysis of clinical studies. Springfield, IL: Thomas.Google Scholar
  28. Burnham, K. P., &Anderson, D. R. (2002).Model selection and multimodel inference: A practical information-theoretic approach (2nd ed.). New York: Springer.Google Scholar
  29. Busemeyer, J. R., &Stout, J. C. (2002). A contribution of cognitive decision models to clinical assessment: Decomposing performance on the Bechara gambling task.Psychological Assessment,14, 253–262.PubMedCrossRefGoogle Scholar
  30. Christensen, R. (2005). Testing Fisher, Neyman, Pearson, and Bayes.American Statistician,59, 121–126.CrossRefGoogle Scholar
  31. Cohen, J. (1994). The earth is round (p <.05).American Psychologist,49, 997–1003.CrossRefGoogle Scholar
  32. Cornfield, J. (1966). Sequential trials, sequential analysis, and the likelihood principle.American Statistician,20, 18–23.CrossRefGoogle Scholar
  33. Cornfield, J. (1969). The Bayesian outlook and its application.Biometrics,25, 617–657.PubMedCrossRefGoogle Scholar
  34. Cortina, J. M., &Dunlap, W. P. (1997). On the logic and purpose of significance testing.Psychological Methods,2, 161–172.CrossRefGoogle Scholar
  35. Cox, D. R. (1958). Some problems connected with statistical inference.Annals of Mathematical Statistics,29, 357–372.CrossRefGoogle Scholar
  36. Cox, D. R. (1971). The choice between alternative ancillary statistics.Journal of the Royal Statistical Society: Series B,33, 251–255.Google Scholar
  37. Cox, R. T. (1946). Probability, frequency and reasonable expectation.American Journal of Physics,14, 1–13.CrossRefGoogle Scholar
  38. Cumming, G. (2007). Replication and p values: p values predict the future vaguely, but confidence intervals do better. Manuscript submitted for publication.Google Scholar
  39. D’Agostini, G. (1999). Teaching statistics in the physics curriculum: Unifying and clarifying role of subjective probability.American Journal of Physics,67, 1260–1268.CrossRefGoogle Scholar
  40. Dawid, A. P. (1984). Statistical theory: The prequential approach.Journal of the Royal Statistical Society: Series A,147, 278–292.CrossRefGoogle Scholar
  41. De Finetti, B. (1974).Theory of probability: A critical introductory treatment (Vols. 1 & 2; A. Machí & A. Smith, Trans.). London: Wiley.Google Scholar
  42. Diamond, G. A., &Forrester, J. S. (1983). Clinical trials and statistical verdicts: Probable grounds for appeal.Annals of Internal Medicine,98, 385–394.PubMedGoogle Scholar
  43. Dickey, J. M. (1973). Scientific reporting and personal probabilities: Student’s hypothesis.Journal of the Royal Statistical Society: Series B,35, 285–305.Google Scholar
  44. Dickey, J. M. (1977). Is the tail area useful as an approximate Bayes factor?Journal of the American Statistical Association,72, 138–142.CrossRefGoogle Scholar
  45. Dixon, P. (2003). The p value fallacy and how to avoid it.Canadian Journal of Experimental Psychology,57, 189–202.PubMedGoogle Scholar
  46. Djurić, P. M. (1998). Asymptotic MAP criteria for model selection.IEEE Transactions on Signal Processing,46, 2726–2735.CrossRefGoogle Scholar
  47. Edwards, A. W. F. (1992).Likelihood. Baltimore: Johns Hopkins University Press.Google Scholar
  48. Edwards, W., Lindman, H., &Savage, L. J. (1963). Bayesian statistical inference for psychological research.Psychological Review,70, 193–242.CrossRefGoogle Scholar
  49. Efron, B. (2005). Bayesians, frequentists, and scientists.Journal of the American Statistical Association,100, 1–5.CrossRefGoogle Scholar
  50. Efron, B., &Tibshirani, R. (1997). Improvements on cross-validation: The.6321 bootstrap method.Journal of the American Statistical Association,92, 548–560.CrossRefGoogle Scholar
  51. Feller, W. (1940). Statistical aspects of ESP.Journal of Parapsychology,4, 271–298.Google Scholar
  52. Feller, W. (1970).An introduction to probability theory and its applications: Vol. 1 (2nd ed.). New York: Wiley.Google Scholar
  53. Fine, T. L. (1973).Theories of probability: An examination of foundations. New York: Academic Press.Google Scholar
  54. Firth, D., &Kuha, J. (1999). Comments on “A critique of the Bayesian information criterion for model selection.”Sociological Methods & Research,27, 398–402.CrossRefGoogle Scholar
  55. Fisher, R. A. (1934).Statistical methods for research workers (5th ed.). London: Oliver & Boyd.Google Scholar
  56. Fisher, R. A. (1935a).The design of experiments. Edinburgh: Oliver & Boyd.Google Scholar
  57. Fisher, R. A. (1935b). The logic of inductive inference (with discussion).Journal of the Royal Statistical Society,98, 39–82.CrossRefGoogle Scholar
  58. Fisher, R. A. (1958).Statistical methods for research workers (13th ed.). New York: Hafner.Google Scholar
  59. Freireich, E. J., Gehan, E., Frei, E., III,Schroeder, L. R., Wolman, I. J., Anbari, R., et al. (1963). The effect of 6-mercaptopurine on the duration of steroid-induced remissions in acute leukemia: A model for evaluation of other potentially useful therapy.Blood,21, 699–716.Google Scholar
  60. Frick, R. W. (1996). The appropriate use of null hypothesis testing.Psychological Methods,1, 379–390.CrossRefGoogle Scholar
  61. Friedman, L. M., Furberg, C. D., &DeMets, D. L. (1998).Fundamentals of clinical trials (3rd ed.). New York: Springer.Google Scholar
  62. Galavotti, M. C. (2005).A philosophical introduction to probability. Stanford: CSLI Publications.Google Scholar
  63. Geisser, S. (1975). The predictive sample reuse method with applications.Journal of the American Statistical Association,70, 320–328.CrossRefGoogle Scholar
  64. Gelman, A., &Rubin, D. B. (1999). Evaluating and using statistical methods in the social sciences.Sociological Methods & Research,27, 403–410.CrossRefGoogle Scholar
  65. Gigerenzer, G. (1993). The superego, the ego, and the id in statistical reasoning. In G. Keren & C. Lewis (Eds.),A handbook for data analysis in the behavioral sciences: Methodological issues (pp. 311–339). Hillsdale, NJ: Erlbaum.Google Scholar
  66. Gigerenzer, G. (1998). We need statistical thinking, not statistical rituals.Behavioral & Brain Sciences,21, 199–200.CrossRefGoogle Scholar
  67. Gilks, W. R., Richardson, S., &Spiegelhalter, D. J. (Eds.) (1996).Markov chain Monte Carlo in practice. Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
  68. Gill, J. (2002).Bayesian methods: A social and behavioral sciences approach. Boca Raton, FL: CRC Press.Google Scholar
  69. Glover, S., &Dixon, P. (2004). Likelihood ratios: A simple and flexible statistic for empirical psychologists.Psychonomic Bulletin & Review,11, 791–806.Google Scholar
  70. Good, I. J. (1983).Good thinking: The foundations of probability and its applications. Minneapolis: University of Minnesota Press.Google Scholar
  71. Good, I. J. (1985). Weight of evidence: A brief survey. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds.),Bayesian statistics 2: Proceedings of the Second Valencia International Meeting, September 6/10, 1983 (pp. 249–269). Amsterdam: North-Holland.Google Scholar
  72. Goodman, S. N. (1993).p values, hypothesis tests, and likelihood: Implications for epidemiology of a neglected historical debate.American Journal of Epidemiology,137, 485–496.PubMedGoogle Scholar
  73. Grünwald, P. [D.] (2000). Model selection based on minimum description length.Journal of Mathematical Psychology,44, 133–152.PubMedCrossRefGoogle Scholar
  74. Grünwald, P. D., Myung, I. J., &Pitt, M. A. (Eds.) (2005).Advances in minimum description length: Theory and applications. Cambridge, MA: MIT Press.Google Scholar
  75. Hacking, I. (1965).Logic of statistical inference. Cambridge: Cambridge University Press.Google Scholar
  76. Hagen, R. L. (1997). In praise of the null hypothesis statistical test.American Psychologist,52, 15–24.CrossRefGoogle Scholar
  77. Haldane, J. B. S. (1945). On a method of estimating frequencies.Biometrika,33, 222–225.PubMedCrossRefGoogle Scholar
  78. Hannan, E. J. (1980). The estimation of the order of an ARMA process.Annals of Statistics,8, 1071–1081.CrossRefGoogle Scholar
  79. Helland, I. S. (1995). Simple counterexamples against the conditionality principle.American Statistician,49, 351–356.CrossRefGoogle Scholar
  80. Hill, B. M. (1985). Some subjective Bayesian considerations in the selection of models.Econometric Reviews,4, 191–246.CrossRefGoogle Scholar
  81. Howson, C., &Urbach, P. (2005).Scientific reasoning: The Bayesian approach (3rd. ed.). Chicago: Open Court.Google Scholar
  82. Hubbard, R., &Bayarri, M.-J. (2003). Confusion over measures of evidence (p’s) versus errors (a’s) in classical statistical testing.American Statistician,57, 171–182.CrossRefGoogle Scholar
  83. Jaynes, E. T. (1968). Prior probabilities.IEEE Transactions on Systems Science & Cybernetics,4, 227–241.CrossRefGoogle Scholar
  84. Jaynes, E. T. (2003).Probability theory: The logic of science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  85. Jeffreys, H. (1961).Theory of probability. Oxford: Oxford University Press.Google Scholar
  86. Jennison, C., &Turnbull, B. W. (1990). Statistical approaches to interim monitoring of medical trials: A review and commentary.Statistical Science,5, 299–317.CrossRefGoogle Scholar
  87. Kadane, J. B., Schervish, M. J., &Seidenfeld, T. (1996). Reasoning to a foregone conclusion.Journal of the American Statistical Association,91, 1228–1235.CrossRefGoogle Scholar
  88. Karabatsos, G. (2006). Bayesian nonparametric model selection and model testing.Journal of Mathematical Psychology,50, 123–148.CrossRefGoogle Scholar
  89. Kass, R. E. (1993). Bayes factors in practice.Statistician,42, 551–560.CrossRefGoogle Scholar
  90. Kass, R. E., &Raftery, A. E. (1995). Bayes factors.Journal of the American Statistical Association,90, 377–395.Google Scholar
  91. Kass, R. E., &Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion.Journal of the American Statistical Association,90, 928–934.CrossRefGoogle Scholar
  92. Kass, R. E., &Wasserman, L. (1996). The selection of prior distributions by formal rules.Journal of the American Statistical Association,91, 1343–1370.CrossRefGoogle Scholar
  93. Killeen, P. R. (2005a). An alternative to null-hypothesis significance tests.Psychological Science,16, 345–353.PubMedCrossRefGoogle Scholar
  94. Killeen, P. R. (2005b). Replicability, confidence, and priors.Psychological Science,16, 1009–1012.PubMedCrossRefGoogle Scholar
  95. Killeen, P. R. (2006). Beyond statistical inference: A decision theory for science.Psychonomic Bulletin & Review,13, 549–562.Google Scholar
  96. Klugkist, I., Laudy, O., &Hoijtink, H. (2005). Inequality constrained analysis of variance: A Bayesian approach.Psychological Methods,10, 477–493.PubMedCrossRefGoogle Scholar
  97. Lee, M. D. (2002). Generating additive clustering models with limited stochastic complexity.Journal of Classification,19, 69–85.CrossRefGoogle Scholar
  98. Lee, M. D., &Pope, K. J. (2006). Model selection for the rate problem: A comparison of significance testing, Bayesian, and minimum description length statistical inference.Journal of Mathematical Psychology,50, 193–202.CrossRefGoogle Scholar
  99. Lee, M. D., &Wagenmakers, E.-J. (2005). Bayesian statistical inference in psychology: Comment on Trafimow (2003).Psychological Review,112, 662–668.PubMedCrossRefGoogle Scholar
  100. Lee, P. M. (1989).Bayesian statistics: An introduction. New York: Oxford University Press.Google Scholar
  101. Lindley, D. V. (1957). A statistical paradox.Biometrika,44, 187–192.Google Scholar
  102. Lindley, D. V. (1972).Bayesian statistics: A review. Philadelphia: Society for Industrial & Applied Mathematics.Google Scholar
  103. Lindley, D. V. (1977). The distinction between inference and decision.Synthese,36, 51–58.CrossRefGoogle Scholar
  104. Lindley, D. V. (1982). Scoring rules and the inevitability of probability.International Statistical Review,50, 1–26.CrossRefGoogle Scholar
  105. Lindley, D. V. (1993). The analysis of experimental data: The appreciation of tea and wine.Teaching Statistics,15, 22–25.CrossRefGoogle Scholar
  106. Lindley, D. V. (2004). That wretched prior.Significance,1, 85–87.CrossRefGoogle Scholar
  107. Lindley, D. V., &Phillips, L. D. (1976). Inference for a Bernoulli process (a Bayesian view).American Statistician,30, 112–119.CrossRefGoogle Scholar
  108. Lindley, D. V., &Scott, W. F. (1984).New Cambridge elementary statistical tables. Cambridge: Cambridge University Press.Google Scholar
  109. Loftus, G. R. (1996). Psychology will be a much better science when we change the way we analyze data.Current Directions in Psychological Science,5, 161–171.CrossRefGoogle Scholar
  110. Loftus, G. R. (2002). Analysis, interpretation, and visual presentation of experimental data. In H. Pashler (Ed. in Chief) & J. Wixted (Vol. Ed.),Stevens’ Handbook of experimental psychology: Vol. 4. Methodology in experimental psychology (3rd ed., pp. 339–390). New York: Wiley.Google Scholar
  111. Ludbrook, J. (2003). Interim analyses of data as they accumulate in laboratory experimentation.BMC Medical Research Methodology,3, 15.PubMedCrossRefGoogle Scholar
  112. McCarroll, D., Crays, N., &Dunlap, W. P. (1992). Sequential ANOVAs and Type I error rates.Educational & Psychological Measurement,52, 387–393.CrossRefGoogle Scholar
  113. Myung, I. J. (2000). The importance of complexity in model selection.Journal of Mathematical Psychology,44, 190–204.PubMedCrossRefGoogle Scholar
  114. Myung, I. J., Forster, M. R., & Browne, M. W. (Eds.) (2000). Model selection [Special issue].Journal of Mathematical Psychology,44(1).Google Scholar
  115. Myung, I. J., Navarro, D. J., &Pitt, M. A. (2006). Model selection by normalized maximum likelihood.Journal of Mathematical Psychology,50, 167–179.CrossRefGoogle Scholar
  116. Myung, I. J., &Pitt, M. A. (1997). Applying Occam’s razor in modeling cognition: A Bayesian approach.Psychonomic Bulletin & Review,4, 79–95.Google Scholar
  117. Nelson, N., Rosenthal, R., &Rosnow, R. L. (1986). Interpretation of significance levels and effect sizes by psychological researchers.American Psychologist,41, 1299–1301.CrossRefGoogle Scholar
  118. Neyman, J. (1977). Frequentist probability and frequentist statistics.Synthese,36, 97–131.CrossRefGoogle Scholar
  119. Neyman, J., &Pearson, E. S. (1933). On the problem of the most efficient tests of statistical hypotheses.Philosophical Transactions of the Royal Society: Series A,231, 289–337.CrossRefGoogle Scholar
  120. Nickerson, R. S. (2000). Null hypothesis statistical testing: A review of an old and continuing controversy.Psychological Methods,5, 241–301.PubMedCrossRefGoogle Scholar
  121. O’Hagan, A. (1997). Fractional Bayes factors for model comparison.Journal of the Royal Statistical Society: Series B,57, 99–138.Google Scholar
  122. O’Hagan, A. (2004). Dicing with the unknown.Significance,1, 132–133.CrossRefGoogle Scholar
  123. O’Hagan, A., &Forster, J. (2004).Kendall’s advanced theory of statistics: Vol. 2B. Bayesian inference (2nd ed.). London: Arnold.Google Scholar
  124. Pauler, D. K. (1998). The Schwarz criterion and related methods for normal linear models.Biometrika,85, 13–27.CrossRefGoogle Scholar
  125. Peto, R., Pike, M. C., Armitage, P., Breslow, N. E., Cox, D. R., Howard, S. V., et al. (1976). Design and analysis of randomized clinical trials requiring prolonged observation of each patient: I. Introduction and design.British Journal of Cancer,34, 585–612.PubMedGoogle Scholar
  126. Pitt, M. A., Myung, I. J., &Zhang, S. (2002). Toward a method of selecting among computational models of cognition.Psychological Review,109, 472–491.PubMedCrossRefGoogle Scholar
  127. Pocock, S. J. (1983).Clinical trials: A practical approach. New York: Wiley.Google Scholar
  128. Pratt, J. W. (1961). [Review of Lehmann, E. L., Testing statistical hypotheses].Journal of the American Statistical Association,56, 163–167.CrossRefGoogle Scholar
  129. Pratt, J. W. (1962). On the foundations of statistical inference: Discussion.Journal of the American Statistical Association,57, 314–315.Google Scholar
  130. Raftery, A. E. (1993). Bayesian model selection in structural equation models. In K. A. Bollen & J. S. Long (Eds.),Testing structural equation models (pp. 163–180). Newbury Park, CA: Sage.Google Scholar
  131. Raftery, A. E. (1995). Bayesian model selection in social research. In P. V. Marsden (Ed.),Sociological methodology 1995 (pp. 111–196). Cambridge, MA: Blackwell.Google Scholar
  132. Raftery, A. E. (1996). Hypothesis testing and model selection. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.),Markov chain Monte Carlo in practice (pp. 163–187). Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
  133. Raftery, A. E. (1999). Bayes factors and BIC.Sociological Methods & Research,27, 411–427.CrossRefGoogle Scholar
  134. Rissanen, J. (2001). Strong optimality of the normalized ML models as universal codes and information in data.IEEE Transactions on Information Theory,47, 1712–1717.CrossRefGoogle Scholar
  135. Robert, C. P., &Casella, G. (1999).Monte Carlo statistical methods. New York: Springer.Google Scholar
  136. Rosenthal, R., &Gaito, J. (1963). The interpretation of levels of significance by psychological researchers.Journal of Psychology,55, 33–38.CrossRefGoogle Scholar
  137. Rouder, J. N., &Lu, J. (2005). An introduction to Bayesian hierarchical models with an application in the theory of signal detection.Psychonomic Bulletin & Review,12, 573–604.Google Scholar
  138. Rouder, J. N., Lu, J., Speckman, P., Sun, D., &Jiang, Y. (2005). A hierarchical model for estimating response time distributions.Psychonomic Bulletin & Review,12, 195–223.Google Scholar
  139. Royall, R. M. (1997).Statistical evidence: A likelihood paradigm. London: Chapman & Hall.Google Scholar
  140. Savage, L. J. (1954).The foundations of statistics. New York: Wiley.Google Scholar
  141. Schervish, M. J. (1996).P values: What they are and what they are not.American Statistician,50, 203–206.CrossRefGoogle Scholar
  142. Schmidt, F. L. (1996). Statistical significance testing and cumulative knowledge in psychology: Implications for training of researchers.Psychological Methods,1, 115–129.CrossRefGoogle Scholar
  143. Schwarz, G. (1978). Estimating the dimension of a model.Annals of Statistics,6, 461–464.CrossRefGoogle Scholar
  144. Sellke, T., Bayarri, M.-J., &Berger, J. O. (2001). Calibration of p values for testing precise null hypotheses.American Statistician,55, 62–71.CrossRefGoogle Scholar
  145. Shafer, G. (1982). Lindley’s paradox.Journal of the American Statistical Association,77, 325–351.CrossRefGoogle Scholar
  146. Siegmund, D. (1985).Sequential analysis: Tests and confidence intervals. New York: Springer.Google Scholar
  147. Smith, A. F. M., &Spiegelhalter, D. J. (1980). Bayes factors and choice criteria for linear models.Journal of the Royal Statistical Society: Series B,42, 213–220.Google Scholar
  148. Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions (with discussion).Journal of the Royal Statistical Society: Series B,36, 111–147.Google Scholar
  149. Strube, M. J. (2006). SNOOP: A program for demonstrating the consequences of premature and repeated null hypothesis testing.Behavior Research Methods,38, 24–27.PubMedGoogle Scholar
  150. Stuart, A., Ord, J. K., &Arnold, S. (1999).Kendall’s advanced theory of statistics: Vol. 2A. Classical inference and the linear model (6th ed.). London: Arnold.Google Scholar
  151. Trafimow, D. (2003). Hypothesis testing and theory evaluation at the boundaries: Surprising insights from Bayes’s theorem.Psychological Review,110, 526–535.PubMedCrossRefGoogle Scholar
  152. Vickers, D., Lee, M. D., Dry, M., &Hughes, P. (2003). The roles of the convex hull and the number of potential intersections in performance on visually presented traveling salesperson problems.Memory & Cognition,31, 1094–1104.Google Scholar
  153. Wagenmakers, E.-J. (2003). How many parameters does it take to fit an elephant? [Book review].Journal of Mathematical Psychology,47, 580–586.CrossRefGoogle Scholar
  154. Wagenmakers, E.-J., &Farrell, S. (2004). AIC model selection using Akaike weights.Psychonomic Bulletin & Review,11, 192–196.Google Scholar
  155. Wagenmakers, E.-J., &Grünwald, P. (2006). A Bayesian perspective on hypothesis testing: A comment on Killeen (2005).Psychological Science,17, 641–642.PubMedCrossRefGoogle Scholar
  156. Wagenmakers, E.-J., Grünwald, P., &Steyvers, M. (2006). Accumulative prediction error and the selection of time series models.Journal of Mathematical Psychology,50, 149–166.CrossRefGoogle Scholar
  157. Wagenmakers, E.-J., Ratcliff, R., Gomez, P., &Iverson, G. J. (2004). Assessing model mimicry using the parametric bootstrap.Journal of Mathematical Psychology,48, 28–50.PubMedCrossRefGoogle Scholar
  158. Wagenmakers, E.-J., & Waldorp, L. (Eds.) (2006). Model selection: Theoretical developments and applications [Special issue].Journal of Mathematical Psychology,50(2).Google Scholar
  159. Wainer, H. (1999). One cheer for null hypothesis significance testing.Psychological Methods,4, 212–213.CrossRefGoogle Scholar
  160. Wallace, C. S., &Dowe, D. L. (1999). Refinements of MDL and MML coding.Computer Journal,42, 330–337.CrossRefGoogle Scholar
  161. Ware, J. H. (1989). Investigating therapies of potentially great benefit: ECMO.Statistical Science,4, 298–340.CrossRefGoogle Scholar
  162. Wasserman, L. (2000). Bayesian model selection and model averaging.Journal of Mathematical Psychology,44, 92–107.PubMedCrossRefGoogle Scholar
  163. Wasserman, L. (2004).All of statistics: A concise course in statistical inference. New York: Springer.Google Scholar
  164. Weakliem, D. L. (1999). A critique of the Bayesian information criterion for model selection.Sociological Methods & Research,27, 359–397.CrossRefGoogle Scholar
  165. Wilkinson, L., &the Task Force on Statistical Inference (1999). Statistical methods in psychology journals: Guidelines and explanations.American Psychologist,54, 594–604.CrossRefGoogle Scholar
  166. Winship, C. (1999). Editor’s introduction to the special issue on the Bayesian information criterion.Sociological Methods & Research,27, 355–358.CrossRefGoogle Scholar
  167. Xie, Y. (1999). The tension between generality and accuracy.Sociological Methods & Research,27, 428–435.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2007

Authors and Affiliations

  1. 1.Department of Psychology, Methodology UnitUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations