Logicist statistics. I: Models and modeling. (English) Zbl 1099.62501
Summary: Arguments are presented to support increased emphasis on logical aspects of formal methods of analysis, depending on probability in the sense of R. A. Fisher. Formulating probabilistic models that convey uncertain knowledge of objective phenomena and using such models for inductive reasoning are central activities of individuals that introduce limited but necessary subjectivity into science. Statistical models are classified into overlapping types called here empirical, stochastic and predictive, all drawing on a common mathematical theory of probability, and all facilitating statements with logical and epistemic content. Contexts in which these ideas are intended to apply are discussed via three major examples.
MSC:
| 62A01 | Foundations and philosophical topics in statistics |
Keywords:
Logicism and proceduralism; specificity of analysis; formal subjective probability; complementarity; subjective and objective; formal and informal; empirical, stochastic and predictive models; U.S. national census; screening for chronic disease; global climate changeReferences:
| [1] | Aitkin, M. (1997). The calibration of P-values, posterior Bay es factors and the AIC from the posterior distribution of likelihood. Statist. Comput. 7 253-272. [Includes discussion by M. Stone of Aitkin (1997) and Dempster (1997), and replies by Dempster and Aitkin.] |
| [2] | Belin, T. R. and Rolph, J. E. (1994). Can we reach consensus on census adjustment? Statist. Sci. 9 486-508. |
| [3] | Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. Wiley, New York. · Zbl 0796.62002 |
| [4] | Bloomfield, P. (1992). Trends in global temperature. Climate Change 21 1-16. |
| [5] | Bloomfield, P. and Ny chka, D. (1992). Climate spectra and detecting climate change. Climate Change 21 275-287. |
| [6] | Boole, G. (1854). An Investigation into the Laws of Thought. Walton and Maberly, London. [Reprinted (1951) Dover, New York.] |
| [7] | Box, G. E. P. (1976). Science and statistics. J. Amer. Statist. Assoc. 71 791-799. JSTOR: · Zbl 0335.62002 · doi:10.2307/2286841 |
| [8] | Box, G. E. P. (1980). Sampling and Bay es’ inference in scientific modelling and robustness. J. Roy. Statist. Soc. Ser. A 143 383-430. JSTOR: · Zbl 0471.62036 · doi:10.2307/2982063 |
| [9] | Breiman, L. (1994). The 1991 census adjustment: undercount or bad data? Statist. Sci. 9 458-475. |
| [10] | Bry ant, B. E. (1993). Guest commentary. Census-taking for a litigious data driven society. Chance 6 44-49. |
| [11] | Chalmers, T. C. (1993). Screening for breast cancer: What should national policy be? Journal of the National Cancer Institute 85 1619-1621. |
| [12] | Cochran, W. G. (1967). Footnote by William G. Cochran. Science 156 1462-1463. |
| [13] | Cox, D. R. (1990). Role of models in statistical analysis. Statist. Sci. 5 179-174. · Zbl 0955.62518 · doi:10.1214/ss/1177012165 |
| [14] | Cox, D. R. (1995). The relation between theory and application in statistics (with discussion). TEST 4 207-261. · Zbl 0844.62001 · doi:10.1007/BF02562627 |
| [15] | Cox, D. R. and Wermuth, N. (1996). Multivariate Dependencies. Chapman and Hall, London. · Zbl 0880.62124 |
| [16] | Crowley, T. J. and North, G. R. (1991). Paleoclimatology. Oxford Univ. Press. |
| [17] | Day, N. E. and Duffy, S. W. (1996). Trial design based on surrogate end points-application to comparison of different breast screening frequencies. J. Roy. Statist. Soc. Ser. A 159 49-60. |
| [18] | De Angeles, D., Gilks, W. R. and Day, N. E. (1998). Bayesian projection of the acquired immune deficiency sy ndrome epidemic. J. Roy. Statist. Soc. Ser. C 47. · Zbl 0935.62121 · doi:10.1111/1467-9876.00123 |
| [19] | DeMenocal, P. B., Ruddiman, W. F. and Pokras, E. M. (1993). Influences of highand low-latitude processes on African terrestrial climate: pleistocene eolian records from equatorial, Atlantic Ocean Drilling Program Site 663. Paleoceanography 8 209-242. |
| [20] | Dempster, A. P. (1964). On the difficulties inherent in Fisher’s fiducial argument. J. Amer. Statist. Assoc. 59 56-66. JSTOR: · Zbl 0149.14905 · doi:10.2307/2282858 |
| [21] | Dempster, A. P. (1971). Model searching and estimation in the logic of inference. In Foundations of Statistical Inference (V. P. Godambe and D. A. Sprott, eds.) 56-78. Holt, Rinehart and Winston, Toronto. |
| [22] | Dempster, A. P. (1990). Causality and statistics. J. Statist. Plann. Inference 25 261-278. |
| [23] | Dempster, A. P. (1997). The direct use of likelihood for significance testing. Statist. Comput. 7 247-252. [Reprinted from Proceedings of the Conference on Foundational Issues in Statistical Inference (O. Barndorff-Nielsen et al., eds.) Aarhus, Denmark, 1974.] Dempster, A. P. (1998a). Comment on ”R. A. Fisher in the 21st century,” by Bradley Efron. Statist. Sci. 13 120-121. Dempster, A. P. (1998b). Logicist statistics II. Inference. In preparation. · Zbl 0367.62004 |
| [24] | Dempster, A. P. and Liu, C. (1995). Trend and drift in climatological time series. In Conference Proceedings, 6th International Meeting on Statistical Climatology 21-24. University College, Galway, Ireland. |
| [25] | Draper, D. (1995). Assessment and propagation of model uncertainty (with discussion). J. Roy. Statist. Soc. Ser. B 57 45-97. JSTOR: · Zbl 0812.62001 |
| [26] | Edgeworth, F. Y. (1884). The philosophy of chance. Mind 9 223- 235. |
| [27] | Efron, B. (1998). R. A. Fisher in the 21st century (with discussion). Statist. Sci. 13 95-114. · Zbl 1074.01536 · doi:10.1214/ss/1028905930 |
| [28] | Ericksen, E. P., Fienberg, S. E. and Kadane, J. B. (1994). Comment on ”The 1991 Census Adjustment, Undercount or Bad Data” by Leo Breiman, ”Heterogeneity and Census Adjustment for the Intercensal Base” by D. Freedman and K. Wachter, and ”Can We Reach Consensus on Census Adjustment?” by Thomas E. Belin and John E. Rolph. Statist. Sci. 9 511-515. |
| [29] | Fienberg, S. E. (1993). The New York City Census adjustment trial: witness for the plaintiffs. Jurimetrics Journal 34 65- 83. |
| [30] | Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boy d, Edinburgh. (Many later editions exist, up to the 14th in 1973.) · JFM 51.0414.08 |
| [31] | Fisher, R. A. (1955). Statistical methods and scientific induction. J. Roy. Statist. Soc. Ser. B 17 69-78. JSTOR: · Zbl 0066.38008 |
| [32] | Fisher, R. A. (1956). Statistical Methods and Scientific Inference. Oliver and Boy d, Edinburgh. (Slightly revised versions appeared in 1958 and 1960.) · Zbl 0070.36903 |
| [33] | Fisher, R. A. (1958). The nature of probability. Centennial Review 2 261-274. |
| [34] | Fisher, R. A. (1959). Mathematical probability in the natural sciences. Technometrics 1 21-29. · Zbl 0094.31701 · doi:10.1007/BF02613720 |
| [35] | Flehinger, B. J. and Melamed, M. R. (1994). Current status of screening for lung cancer. Current Perspectives in Thoracic Oncology 4 1-15. |
| [36] | Freedman, D. and Wachter, K. (1994). Heterogeneity and census adjustment for the intercensal base. Statist. Sci. 9 476- 485. |
| [37] | Glantz, M. H., Katz, R. W. and Nicholls, N. (1991). Teleconnections Linking Worldwide Climate Anomalies. Cambridge Univ. Press. |
| [38] | Good, I. J. (1950). Probability and the Weighing of Evidence. Griffin, London. Good, I. J. (1961-62). A causal calculus. British J. Philos. Sci. 11 305-318, 12 43-51, 13 88. [Reprinted (1983) in Good Thinking 197-217. Univ. Minnesota Press.] · Zbl 0036.08402 |
| [39] | Hasselmann, K. (1979). On the signal-to-noise problem in atmospheric response studies. In Meteorology of Tropical Oceans (D. B. Shaw, ed.) 251-259. Roy al Meteorological Society, London. |
| [40] | Hasselmann, K. (1993). Optimal fingerprints for the detection of climate change. J. Climate 6 1957-1971. |
| [41] | Hasselmann, K. (1997). Multi-pattern fingerprint method for detection and attribution of climate change. Climate Dy namics 13 601-611. Hegerl, G. C., von Storch, H., Hasselmann, K., Santer, B. D., |
| [42] | Cubasch, U. and Jones, P. D. (1996). Detecting greenhousegas-induced climate change with an optimal fingerprint. J. Climate 9 2281-2306. Hegerl, G. C., Hasselmann, K., Cubasch, V., Mitchell, J. F. B., |
| [43] | Roeckner, E., Voss, R. and Waskewitz, J. (1997). Multifingerprint detection and attribution analysis of greenhouse gas, greenhouse gas-plus-aerosol and solar forced climate change. Climate Dy namics 13 613-634. |
| [44] | Hegerl, G. C. and North, G. R. (1997). Statistically optimal approaches to detecting anthropogenic climate change. J. Climate 10 1125-1133. |
| [45] | Houghton, J. T., Jenkins, G. J. and Ephraums, J. J., eds. (1991). Climate Change. Cambridge Univ. Press. (The Intergovernmental Panel on Climate Change Scientific Assessment.) Houghton, J. T., Callander, B. A. and Varney, S. K., eds. |
| [46] | . Climate Change 1992. Cambridge Univ. Press. (Supplementary Report to the Intergovernmental Panel on Climate Change Scientific Assessment. Prepared for IPCC by Working Group I.) Houghton, J. T., Meira Filho, L. G., Callander, B. A., Harris, |
| [47] | N., Kattenberg, A. and Maskell, K., eds. (1996). Climate Change 1995. The Science of Climate Change. Cambridge Univ. Press. (Contribution of Working Group I to the Second Assessment Report of the Intergovernmental Panel on Climate Change.) |
| [48] | Jones, P. D. (1994). Hemispheric surface air temperature variations: a reanalysis and update to 1993. J. Climate 7 1794- 1802. |
| [49] | Jones, P. D., Osborn, T. J. and Briffa, K. R. (1997). Estimating sampling errors in large-scale temperature averages. J. Climate 10 2548-2568. |
| [50] | Kass, R. E. and Raftery, A. E. (1995). Bay es factors. J. Amer. Statist. Assoc. 90 773-795. · Zbl 0846.62028 · doi:10.2307/2291091 |
| [51] | Kerr, R. A. (1997). Greenhouse forecasting still cloudy. Science 276 1040-1042. (16 May 1997.) |
| [52] | Laplace, P. S. (1814). Essai Philosophique sur les Probabilities. Courcier, Paris. [A 1902 translation by F. W. Truscott and F. L. Emory was reprinted (1951) by Dover, New York.] · JFM 48.0008.05 |
| [53] | Lindzen, R. S. (1994). Climate Dy namics and global change. Ann. Rev. Fluid Mech. 26 353-378. · Zbl 0808.76016 |
| [54] | Lindzen, R. S. (1995). The importance and nature of the water vapor budget in nature and models. In Climate Sensitivity to Radiative Perturbations: physical Mechanisms and Their Validation (H. Le Treut, ed.) 51-66. Springer, Berlin. |
| [55] | Lindzen, R. S. (1997). Can increasing carbon dioxide cause climate change? Proc. Nat. Acad. Sci. U.S.A. 94 8335-8342. |
| [56] | Madden, R. A. and Julian, P. R. (1972). Description of globalscale circulation cells in the tropics with a 40-50 day period. J. Atmospheric Sci. 29 1109-1123. |
| [57] | Morrison, A. S. (1992). Screening in Chronic Disease, 2nd ed. Oxford Univ. Press. |
| [58] | Moskowitz, M. (1986). Breast cancer: age-specific growth rates and screening strategies. Radiology 161 37-41. |
| [59] | Nelder, J. (1986). Statistics, science, and technology. J. Roy. Statist. Soc. Ser. A 149 109-121. |
| [60] | Ney man, J. (1935). Statistical problems in agricultural experimentation. J. Roy. Statist. Soc. Suppl. 2 107-180. · JFM 63.1103.02 |
| [61] | Ney man, J. (1955). The problem of inductive inference. Comm. Pure Appl. Math. 8 13-46. · Zbl 0064.38701 · doi:10.1002/cpa.3160080103 |
| [62] | Ney man, J. (1957). Inductive behavior as a basic concept of the philosophy of science. Rev. Internat. Statist. Inst. 25 22-35. · Zbl 0087.24404 |
| [63] | Ney man, J. (1960). Indeterminism in science and new demands on statisticians. J. Amer. Statist. Assoc. 55 625-639. JSTOR: · Zbl 0104.12102 · doi:10.2307/2281586 |
| [64] | Ney man, J. (1961). Silver jubilee of my dispute with Fisher. J. Oper. Res. Soc. Japan 3 145-154. |
| [65] | Ney man, J. (1967). R. A. Fisher (1890-1962): an appreciation. Science 156 1456-1462. |
| [66] | Ney man, J. (1977). Frequentist probability and frequentist statistics. Sy nth ese 36 97-131. · Zbl 0372.60002 · doi:10.1007/BF00485695 |
| [67] | North, G. R. and Stevens, M. J. (1998). Detecting climate signals in the surface temperature record. J. Climate. 11 563- 577. |
| [68] | O’Hagan, A. (1995). Fractional Bay es factors for model comparisons (with discussion). J. Roy. Statist. Soc. Ser. B 57 99-138. JSTOR: · Zbl 0813.62026 |
| [69] | Pais, A. (1991). Niels Bohr’s Times, in physics, Philosophy, and Polity. Clarendon, Oxford. |
| [70] | Parker, D. E., Folland, C. K. and Jackson, M. (1995). Marine surface temperature: observed variations and data requirements. Climate Change 31 559-600. |
| [71] | Pearl, J. (1997). Structural and probabilistic causality. The Psy chology of Learning and Motivation 34 393-435. |
| [72] | Pearson, E. S. (1962). Thoughts on statistical inference. Ann. Math. Statist. 33 394-403. · Zbl 0109.13002 · doi:10.1214/aoms/1177704566 |
| [73] | Pearson, K. (1920). The fundamental problem of practical statistics. Biometrika 13 1-20, 300-301. |
| [74] | Porter, T. M. (1986). The Rise of Statistical Thinking, 1820- 1900. Princeton Univ. Press. |
| [75] | Rolph, J. (1993). The census adjustment trial: reflections of a witness for the plaintiffs. Jurimetrics Journal 34 85-97. Santer, B. D., Wigley, T. P., Barnett, T. P. and Any amba, E. |
| [76] | . Detection of climate change and attribution of causes. In Climate Change 1995. The Science of Climate Change (J. T. Houghton, et al., eds.) 407-444. Cambridge Univ. Press. |
| [77] | Savage, L. J. (1962). The Foundations of Statistical Inference. Methuen, London. |
| [78] | Schneider, S. H. (1994). Detecting climatic change signals: are there any ”fingerprints”? Science 263 341-347. |
| [79] | Smart, C. R., Hendrick, R. E. and Rutledge, J. H. (1995). Benefit of mammography screening in women ages 40-49 years. Cancer 75 1619-1626. |
| [80] | Smith, A. F. M. (1995). Discussion of ”Fractional Bay es factors for model comparison,” by A. O’Hagan. J. Roy. Statist. Soc. Ser. B 57 120-122. |
| [81] | Stigler, S. (1986). The History of Statistics. Harvard Univ. Press. · Zbl 0656.62005 |
| [82] | Stigler, S. (1989). The role of probability models in statistical inference in 19th century Europe. Bull. Internat. Statist. Inst. 53(Book 3), 157-162. |
| [83] | Strauss, G. M. (1997). Measuring effectiveness of lung cancer screening. Chest 112 216S-228S. |
| [84] | Strauss, G. M., Gleason, R. E. and Sugarbaker, D. J. (1997). Screening for lung cancer. Chest 111 754-768. |
| [85] | Strauss, G. M. (1998). Randomized population trials: implications for cancer early detection. Unpublished manuscript. |
| [86] | Suppes, P. (1970). A Probabilistic Theory of Causality. NorthHolland, Amsterdam. Tett, S. F. B., Mitchell, J. F. B., Parker, D. E. and Allen, M. |
| [87] | . Human influence on the atmospheric vertical temperature structure: detection and observations. Science 274 1169-1173. |
| [88] | Tukey, J. W. (1962). The future of data analysis. Ann. Math. Statist. 33 1-67. · Zbl 0107.36401 · doi:10.1214/aoms/1177704711 |
| [89] | Tukey, J. W. (1977). Exploratory Data Analy sis. Addison-Wesley, Reading, MA. · Zbl 0409.62003 |
| [90] | U.S. Bureau of the Census (1996). The Plan for Census 2000. Economics and Statistics Administration, U.S. Dept. Commerce, Washington, DC. (Revised and reissued February 28, 1996.) |
| [91] | Wald, A. (1950). Statistical Decision Functions. Wiley, New York. · Zbl 0040.36402 |
| [92] | Whitaker, A. (1996). Einstein, Bohr and the Quantum Dilemma. Cambridge Univ. Press. · Zbl 0847.01011 |
| [93] | Zelen, M. (1986). Case-control studies and Bayesian inference. Statistics and Medicine 5 261-269. |
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