Existing Approaches and Development Perspectives for Inferences

Authors

  • K.J. Kachiashvili Georgian Technical University, 77, st. Kostava, Tbilisi, 0175, Georgia

DOI:

https://doi.org/10.6000/1929-6029.2021.10.06

Keywords:

Inference Theory, Hypotheses Testing, p-value test, Frequentist Test, Bayesian Method, Constrained Bayesian Method, Berger’s Test, Wald’s Method

Abstract

Statistical hypotheses testing is one of the basic direction of mathematical statistics the methods of which are widely used in theoretical research and practical applications. These methods are widely used in medical researches too. Scientists of different fields, among them of medical too, that are not experts in statistics, are often faced with the dilemma of which method to use for solving the problem they are interested. The article is devoted to helping the specialists in solving this problem and in finding the optimal resolution. For this purpose, here are very simple and clearly explained the essences of the existed approaches and are shown their positive and negative sides and are given the recommendations about their use depending on existed information and the aim that must be reached as a result of an investigation.

References

Berger JO. Statistical Decision Theory and Bayesian Analysis, NewYork: Springer 1985. https://doi.org/10.1007/978-1-4757-4286-2 DOI: https://doi.org/10.1007/978-1-4757-4286-2

Berger JO. Could Fisher, Jeffreys and Neyman have Agreed on Testing? Statistical Science 2003; 18: 1–32. https://doi.org/10.1214/ss/1056397485 DOI: https://doi.org/10.1214/ss/1056397485

Berger JO, Brown LD, Wolpert RL. A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing. The Annals of Statistics 1994; 22(4): 1787-1807. https://doi.org/10.1214/aos/1176325757 DOI: https://doi.org/10.1214/aos/1176325757

Bernardo JM, Rueda R. Bayesian Hypothesis Testing: A Reference Approach. International Statistical Review 2002; 1-22. https://doi.org/10.2307/1403862 DOI: https://doi.org/10.2307/1403862

Christensen R. Testing Fisher, Neyman, Pearson, and Bayes, The American Statistician 2005; 59(2): 121-126. https://doi.org/10.1198/000313005X20871 DOI: https://doi.org/10.1198/000313005X20871

Hubbard R, Bayarri MJ. Confusion over Measures of Evidence (p’s) Versus Errors (α’s) in Classical Statistical Testing. The American Statistician 2003; 57: 171–177. https://doi.org/10.1198/0003130031856 DOI: https://doi.org/10.1198/0003130031856

Lehmann EL. The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two? American Statistical Association Journal, Theory and Methods 1993; 88(424): 1242-1249. https://doi.org/10.1080/01621459.1993.10476404 DOI: https://doi.org/10.1080/01621459.1993.10476404

Lehmann EL. Testing Statistical Hypotheses, 2nd ed. New York: Springer; 1997. https://doi.org/10.1214/ss/1029963261 DOI: https://doi.org/10.1214/ss/1029963261

Moreno E, Giron FJ. On the Frequentist and Bayesian Approaches to Hypothesis Testing, SORT, 30(1); 2006; p. 3-28.

Wolpert RL. Testing simple hypotheses. Data Analysis and Information Systems. In: Bock HH, Polasek W, editors. 7, Heidelberg: Springer; 1996; p. 289–297. https://doi.org/10.1007/978-3-642-80098-6_24 DOI: https://doi.org/10.1007/978-3-642-80098-6_24

Fisher RA. Statistical Methods for Research Workers, London: Oliver and Boyd; 1925.

Neyman J, Pearson E. On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference. Part I, Biometrica 1928; 20A: 175-240. https://doi.org/10.1093/biomet/20A.1-2.175 DOI: https://doi.org/10.1093/biomet/20A.1-2.175

Neyman J, Pearson E. On the Problem of the Most Efficient Tests of Statistical Hypotheses, Philos. Trans. Roy. Soc., Ser. A 1933; 231: 289-337. DOI: https://doi.org/10.1098/rsta.1933.0009

Jeffreys H. Theory of Probability, 1st ed. Oxford: The Clarendon Press 1939.

Berger JO, Wolpert RL. The Likelihood Principle. Institute of Mathematical Statistics Monograph Series (IMS), Hayward: CA; 1984.

Berger JO, Wolpert RL. The Likelihood Principle, 2nd ed. (with discussion). IMS, Hayward: CA; 1988. DOI: https://doi.org/10.1007/978-1-4613-8768-8_15

Berger JO, Boukai B, Wang Y. Unified Frequentist and Bayesian Testing of a Precise Hypothesis. Statistical Science 1997; 12(3): 133-160. https://doi.org/10.1214/ss/1030037904 DOI: https://doi.org/10.1214/ss/1030037904

Bernardo JM. A Bayesian analysis of classical hypothesis testing. Universidad de Valencia 1980; 605-617. https://doi.org/10.1007/BF02888370 DOI: https://doi.org/10.1007/BF02888370

Delampady M, Berger JO. Lower bounds on Bayes factors for the multinomial distribution, with application to chi-squared tests of fit. Ann. Statist. 1990; 18: 1295-1316. https://doi.org/10.1214/aos/1176347750 DOI: https://doi.org/10.1214/aos/1176347750

Kiefer J. Conditional confidence statement and confidence estimations (with discussion). J. Amer. Statist. Assoc. 1977; 72(360): 789-808. https://doi.org/10.1080/01621459.1977.10479956 DOI: https://doi.org/10.1080/01621459.1977.10479956

Bansal NK, Sheng R. Beyesian Decision Theoretic Approach to Hypothesis Problems with Skewed Alternatives. Journal of Statistical Planning and Inference 2010; 140, 2894-2903. https://doi.org/10.1016/j.jspi.2010.03.013 DOI: https://doi.org/10.1016/j.jspi.2010.03.013

Bansal NK, Miescke KJ. A Bayesian decision theoretic approach to directional multiple hypotheses problems, Journal of Multivariate Analysis 2013; 120, 205–215. https://doi.org/10.1016/j.jmva.2013.05.012 DOI: https://doi.org/10.1016/j.jmva.2013.05.012

Bansal NK, Hamedani GG, Maadooliat M. Testing Multiple Hypotheses with Skewed Alternatives. Biometrics 2016; 72(2): 494-502. https://doi.org/10.1111/biom.12430 DOI: https://doi.org/10.1111/biom.12430

Kachiashvili KJ. Comparison of Some Methods of Testing Statistical Hypotheses. Part II. Sequential Methods. International Journal of Statistics in Medical Research 2014; 3: 189-197. https://doi.org/10.6000/1929-6029.2014.03.02.11

Kachiashvili KJ. Constrained Bayesian Methods of Hypotheses Testing: A New Philosophy of Hypotheses Testing in Parallel and Sequential Experiments. New York: Nova Science Publishers 2018.

Rao CR. Linear Statistical Inference and Its Application, 2nd ed. New York: Wiley 2006.

Kachiashvili KJ. Generalization of Bayesian Rule of Many Simple Hypotheses Testing. Int. J. of Information Technology and Decision Making 2003; 2(1): 41-70. https://doi.org/10.1142/S0219622003000525 DOI: https://doi.org/10.1142/S0219622003000525

Kachiashvili KJ. Bayesian algorithms of many hypothesis testing, Tbilisi: Ganatleba 1989.

Sage AP, Melse JL. Estimation Theory with Application to Communication and Control. New York: McGraw-Hill 1972.

Duda RO, Hart EH, Stork DG. Pattern Classification. 2nd Edition. 2006.

Dass SC, Berger JO. Unified Conditional Frequentist and Bayesian Testing of Composite Hypotheses. Scandinavian Journal of Statistics 2003; 30(1): 193-210. https://doi.org/10.1111/1467-9469.00326 DOI: https://doi.org/10.1111/1467-9469.00326

Kachiashvili KJ. Investigation and Computation of Unconditional and Conditional Bayesian Problems of Hypothesis Testing. ARPN Journal of Systems and Software 2011; 1(2): 47-59.

Kachiashvili KJ. Comparison of Some Methods of Testing Statistical Hypotheses. Part I. Parallel Methods. International Journal of Statistics in Medical Research 2014; 3, 174-189. https://doi.org/10.6000/1929-6029.2014.03.02.11 DOI: https://doi.org/10.6000/1929-6029.2014.03.02.11

Kachiashvili KJ. Constrained Bayesian Method for Testing Multiple Hypotheses in Sequential Experiments. Sequential Analysis: Design Methods and Applications 2015; 34(2), 171-186. https://doi.org/10.1080/07474946.2015.1030973 DOI: https://doi.org/10.1080/07474946.2015.1030973

Kachiashvili KJ. Constrained Bayesian Method of Composite Hypotheses Testing: Singularities and Capabilities. International Journal of Statistics in Medical Research 2016; 5(3): 135-167. https://doi.org/10.6000/1929-6029.2016.05.03.1 DOI: https://doi.org/10.6000/1929-6029.2016.05.03.1

Kachiashvili KJ. On One Aspect of Constrained Bayesian Method for Testing Directional Hypotheses. Biomed J Sci &Tech Res 2018; 2(5). https://doi.org/10.26717/BJSTR.2018.02.000821 DOI: https://doi.org/10.26717/BJSTR.2018.02.000821

Kachiashvili GK, Kachiashvili KJ, Mueed A. Specific Features of Regions of Acceptance of Hypotheses in Conditional Bayesian Problems of Statistical Hypotheses Testing. Sankhya: The Indian Journal of Statistics 2012: 74(1): 112-125. https://doi.org/10.1007/s13171-012-0014-8 DOI: https://doi.org/10.1007/s13171-012-0014-8

Kachiashvili KJ, Hashmi MA, Mueed A. Sensitivity Analysis of Classical and Conditional Bayesian Problems of Many Hypotheses Testing. Communications in Statistics—Theory and Methods 2012; 41(4): 591–605. https://doi.org/10.1080/03610926.2010.510255 DOI: https://doi.org/10.1080/03610926.2010.510255

Kachiashvili KJ, Hashmi MA, Mueed A. The Statistical Risk Analysis as the Basis of the Sustainable Development. Int. J. of Innovation and Technol. Management (World Scientific Publishing Company) 2012: 9(3). https://doi.org/10.1142/S0219877012500241 DOI: https://doi.org/10.1142/S0219877012500241

Kachiashvili KJ, Mueed A. Conditional Bayesian Task of Testing Many Hypotheses. Statistics 2013; 47(2): 274-293. https://doi.org/10.1080/02331888.2011.602681 DOI: https://doi.org/10.1080/02331888.2011.602681

Kachiashvili KJ, Bansal NK, Prangishvili IA. Constrained Bayesian Method for Testing the Directional Hypotheses. Journal of Mathematics and System Science 2018; 8: 96-118. https://doi.org/10.17265/2159-5291/2018.04.002

Kachiashvili KJ, Hashmi MA. About Using Sequential Analysis Approach for Testing Many Hypotheses. Bulletin of the Georgian Academy of Sciences 2010; 4(2): 20-25.

Wald A. Sequential analysis. New-York: Wiley 1947.

Wald A. Foundations of a General Theory of Sequential Decision Functions. Econometrica 1947, 15: 279-313. https://doi.org/10.2307/1905331 DOI: https://doi.org/10.2307/1905331

Siegmund D. Sequential Analysis. Springer Series in Statistics. New York: Springer-Verlag 1985. https://doi.org/10.1007/978-1-4757-1862-1 DOI: https://doi.org/10.1007/978-1-4757-1862-1

Ghosh BK. Sequential Tests of Statistical Hypotheses. Massachusetts: Addison-Wesley 1970.

Tartakovsky A, Nikiforov I, Basseville M. Sequential Analysis. Hypothesis Testing and Chalengepoint Detection. New York: Taylor & Francis 2015. https://doi.org/10.1201/b17279 DOI: https://doi.org/10.1201/b17279

Arrow KJ, Blackwell D, Girshick MA.. Bayes and minimax solutions of sequential decision problems. Econometrica 1949; 17(2): 213-244. https://doi.org/10.2307/1905525 DOI: https://doi.org/10.2307/1905525

Ghosh BK, Sen PK. Eds. Handbook of Sequential Analysis. New York: Dekker 1991.

Kachiashvili KJ, Melikdzhanian DI. Identification of River Water Excessive Pollution Sources. International Journal of Information Technology & Decision Making 2006; 5(2): 397-417. https://doi.org/10.1142/S0219622006001988 DOI: https://doi.org/10.1142/S0219622006001988

Kachiashvili KJ, Gordeziani DG, Lazarov RG, Melikdzhanian DI. Modeling and simulation of pollutants transport in rivers. International Journal of Applied Mathematical Modelling (AMM) 2007; 31: 1371-1396. https://doi.org/10.1016/j.apm.2006.02.015 DOI: https://doi.org/10.1016/j.apm.2006.02.015

Kachiashvili KJ, Hashmi MA, Mueed A. The statistical risk analysis as the basis of the sustainable development. Proceedings of the 4th IEEE International Conference on Management of Innovation & Technology (ICMIT2008); 2008: Bangkok, Thailand, p. 1210-1215.

Kachiashvili KJ, Hashmi MA, Mueed A. Bayesian Methods of Statistical Hypothesis Testing for Solving Different Problems of Human Activity. Applied Mathematics and Informatics (AMIM) 2009; 14(2): 3-17.

Kachiashvili KJ, Prangishvili AI. Verification in biometric systems: problems and modern methods of their solution, Journal of Applied Statistics 2018; 45(1): 43-62. https://doi.org/10.1080/02664763.2016.1267122 DOI: https://doi.org/10.1080/02664763.2016.1267122

Kachiashvili KJ, Prangishvili IA, Kachiashvili JK. Constrained Bayesian Methods for Testing Directional Hypotheses Restricted False Discovery Rates. Biostat Biometrics Open Acc J 2019: 9(3).

Kachiashvili KJ. An Example of Application of CBM to Intersection-Union Hypotheses Testing. Biomed J Sci & Tech Res 2019; 19(3). https://doi.org/10.26717/BJSTR.2019.19.003304 DOI: https://doi.org/10.26717/BJSTR.2019.19.003304

Kachiashvili KJ. Modern State of Statistical Hypotheses Testing and Perspectives of its Development. Biostat Biometrics Open Acc J. 2019; 9(2).

Kachiashvili KJ, Kachiashvili JK, Prangishvili IA. CBM for Testing Multiple Hypotheses with Directional Alternatives in Sequential Experiments. Sequential Analysis 2020; 39(1): 115-131. https://doi.org/10.1080/07474946.2020.1727166 DOI: https://doi.org/10.1080/07474946.2020.1727166

Kachiashvili KJ, Bansal NK, Prangishvili IA. Constrained Bayesian Method for Testing the Directional Hypotheses. Journal of Mathematics and System Science 2018; 8: 96-118. https://doi.org/10.17265/2159-5291/2018.04.002 DOI: https://doi.org/10.17265/2159-5291/2018.04.002

Bishop ChM. Pattern Recognition and Machine Learning. Springer Verlag 2006.

Efron B. Large-Scale Simultaneous Hypothesis Testing. Journal of the American Statistical Association 2004; 99(465): 96-104. https://doi.org/10.1198/016214504000000089 DOI: https://doi.org/10.1198/016214504000000089

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Published

2021-05-28

How to Cite

Kachiashvili, K. . (2021). Existing Approaches and Development Perspectives for Inferences. International Journal of Statistics in Medical Research, 10, 63–71. https://doi.org/10.6000/1929-6029.2021.10.06

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General Articles