Nonparametric Versus Parametric Reasoning Based on 2×2 Contingency Tables
The Pomeranian University, Institute of Mathematics
76-200 Słupsk, Poland
E-mail: piotr.sulewski@apsl.edu.pl
Received:
Received: 19 January 2018; revised: 27 March 2018; accepted: 08 May 2018; published online: 30 June 2018
DOI: 10.12921/cmst.2018.0000009
Abstract:
This paper proposes scenarios of generating contingency tables (CTs) with the probability flow parameter (PFP). It also defines measures of untruthfulness of H0 that involve PFP for all proposed scenarios. This paper is an attempt to
replace a nonparametric statistical inference method by the parametric one. The paper applies the maximum likelihood method to estimate PFP and presents instructions to generate CTs by means of the bar method. The Monte Carlo method is used to carry out computer simulations.
Key words:
contingency tables, likelihood function, parametric test, probability flow parameter, statistical inference
References:
[1] A. Agresti, Categorical Data Analysis, Wiley, New Jersey
2002.
[2] G. Van Belle, L.D. Fisher, P.J. Heagerty, T. Lumley, Categori-
cal Data: Contingency Tables, John Wiley & Sons 2004.
[3] Y. Bishop, S. Fienberg, P. Holland, Discrete Multivariate Anal-
ysis – Theory and Practice, Cambridge, MA: MIT Press 1975.
[4] T.D. Wickens, Multiway Contingency Tables Analysis for The
Social Sciences, Psychology Press 2014.
[5] T. Dickhaus, K. Straßburger, D. Schunk, C. Morcillo-Suarez,
T. Illig, A. Navarro, How to Analyze Many Contingency Ta-
bles Simultaneously in Genetic Association Studies, Stat. Appl.
Genet. Mol. Biol. 11(4), 12 (2012).
[6] R. El Galta, T. Stijnen, J.J. Houwing-Duistermaat. Testing
for Genetic Association: A Powerful Score Test, Statistics in
Medicine 27(22), 4596–4609 (2008).
[7] B. Cung, Crime and Demographics: An Analysis of LAPD
Crime Data, UCLA Electronic Theses and Dissertations 2013.
[8] R. Iossifova, F. Marmolejo-Ramos, When The Body Is Time:
Spatial and Temporal Deixis in Children with Visual Impair-
ments and Sighted Children, Research in developmental dis-
abilities 34(7), 2173–2184, (2013).
[9] P.D. Allison, J.K. Liker, Analyzing Sequential Categorical Data
on Dyadic Interaction: A comment on Gottman, Psychological
Bulletin, 91(2) 393–403 (1982).
[10] H.H. Bock, Two-Way Clustering for Contingency Tables: Max-
imizing A Dependence Measure, Between data science and
applied data analysis, 143–154 (2003).
[11] S. Kaski, J. Nikkila, J. Sinkkonen, L. Lahti, J.E. Knuuttila,
C. Roos, Associative Clustering for Exploring Dependencies
Between Functional Genomics Data Sets, IEEE/ACM Transac-
tions on Computational Biology and Bioinformatics (TCBB),
2(3), 203–216 (2005).
[12] M. Steinle, K. Aberer, S. Girdzijauskas, C. Lovis, Mapping
Moving Landscapes by Mining Mountains of Logs: Novel Tech-
niques for Dependency Model Generation, [in:] Proceedings
of the 32nd international conference on Very large data bases
VLDB Endowment, pp. 1093–1102 (2006).
[13] P.J. Haas, F. Hueske, V. Markl, Detecting Attribute Depen-
dencies from Query Feedback, [in:] Proceedings of the 33rd
international conference on Very large data bases VLDB En-
dowment, pp. 830–841 (2007)
[14] I.F. Ilyas, V. Markl, P. Haas, P. Brown, A. Aboulnaga, CORDS:
Automatic Discovery of Correlations and Soft Functional De-
pendencies, [in:] Proceedings of the 2004 ACM SIGMOD
international conference on Management of data, ACM, pp.
647–658 (2004).
[15] T. Oates, P.R. Cohen, Searching for Structure in Multiple
Streams of Data, [in:] ICML 96, pp. 346 (1996).
[16] M. Schrepp, A Method for the Analysis of Hierarchical De-
pendencies Between Items of a Questionnaire, Methods of
Psychological Research Online 19, 43–79 (2003).
[17] J. Blitzstein, P. Diaconis, A Sequential Importance Sampling
Algorithm for Generating Random Graphs with Prescribed
Degrees, Internet mathematics 6(4), 489–522 (2011).
[18] N. Cressie, T.R. Read, Multinomial Goodness-Of-Fit Tests,
Journal of the Royal Statistical Society. Series B (Methodolog-
ical), 440–464 (1984).
[19] K. Pearson, On the Criterion that a given System of Devia-
tions from the Probable in the Case of a Correlated System of
Variables is such that it can be Reasonably Supposed to have
Arisen from Random Sampling, Philosophy Magazine Series
50, 157–172 (1900).
[20] M.F. Freeman, J.W. Tukey, Transformations Related to the An-
gular and the Square Root, Annals of Mathematical Statistics
21, 607–611 (1950).
[21] R.R. Sokal, F.J. Rohlf, Biometry: the Principles and Practice
of Statistics in Biological Research, Freeman, New York 2012.
[22] S. Kullback, Information Theory and Statistics, Wiley, New
York 1959.
[23] J. Neyman, Contribution to the Theory of the Chi-Square Test,
Proc. (First) Berkeley Symp. on Math. Statist. and Prob. (Univ.
of Calif. Press), 239–273 (1949).
[24] P. Sulewski, Modyfikacja testu niezależności (Modification of
the Independence Test), Statistical News 10, 1–19 (2013) (in
Polish).
[25] P. Diaconis, B. Sturmfels, Algebraic Algorithms for Sampling
from Conditional Distributions, The Annals of Statistics 26(1),
363–397 (1998).
[26] M. Cryan, M. Dyer, A Polynomial-Time Algorithm to Approxi-
mately Count Contingency Tables When the Number of Rows
Is Constant, Journal of Computer and System Sciences 67(2),
291–310 (2003).
[27] M. Cryan, M. Dyer, L.A. Goldberg, M. Jerrum, R. Martin,
Rapidly Mixing Markov Chains for Sampling Contingency
Tables with A Constant Number of Rows, SIAM Journal on
Computing 36(1), 247–278 (2006).
[28] Y. Chen, P. Diaconis, S.P. Holmes, J.S. Liu. Sequential Monte
Carlo Methods for Statistical Analysis of Tables, Journal of the
American Statistical Association 100(469), 109–120 (2005).
[29] G.S. Fishman, Counting Contingency Tables Via Multistage
Markov Chain Monte Carlo, Journal of Computational and
Graphical Statistics 21(3), 713–738 (2012).
[30] Y. Chen, I.H. Dinwoodie, S. Sullivant Sequential Importance
Sampling for Multiway Tables, The Annals of Statistics, 523–
545 (2006).
[31] R. Yoshida, J. Xi, S. Wei, F. Zhou, D. Haws, Semigroups and
Sequential Importance Sampling for Multiway Tables, arXiv
preprint arXiv:1111.6518 (2011).
[32] S. DeSalvo, J.Y. Zhao, Random Sampling of Contingency Ta-
bles Via Probabilistic Divide-And-Conquer. arXiv preprint
arXiv:1507.00070 (2015).
[33] P. Sulewski, Two-By-Two Contingency Table as a Goodness-
Of-Fit Test, CMST 15(2), 203–211 (2009).
[34] P. Sulewski, R. Motyka, Power Analysis of Independence Test-
ing for Contingency Tables, Scientific Journal of Polish Naval
Academy 56, 37–46 (2015).
[35] P. Sulewski, Moc testów niezależności w tablicy dwudzielczej
większej niż 2×2 (Power Analysis of Independence Testing
for Two-way Contingency Tables bigger than 2×2), Przegląd
statystyczny 63(2), 191–209 (2016) (in Polish).
[36] P. Sulewski, Moc testów niezależności w tablicy trójdzielczej
2x2x2 (Power Analysis of Independence Testing for Three-way
Contingency Table 2x2x2), Przegląd statystyczny 63(4), 431–
447 (2016) (in Polish).
[37] P. Sulewski, Power Analysis of Independence Testing for
the Three-way Contingency Tables of Small Sizes, Jour-
nal of Applied Statistics, http://dx.doi.org/10.1080/02664763.
2018.1424122 (2018).
This paper proposes scenarios of generating contingency tables (CTs) with the probability flow parameter (PFP). It also defines measures of untruthfulness of H0 that involve PFP for all proposed scenarios. This paper is an attempt to
replace a nonparametric statistical inference method by the parametric one. The paper applies the maximum likelihood method to estimate PFP and presents instructions to generate CTs by means of the bar method. The Monte Carlo method is used to carry out computer simulations.
Key words:
contingency tables, likelihood function, parametric test, probability flow parameter, statistical inference
References:
[1] A. Agresti, Categorical Data Analysis, Wiley, New Jersey
2002.
[2] G. Van Belle, L.D. Fisher, P.J. Heagerty, T. Lumley, Categori-
cal Data: Contingency Tables, John Wiley & Sons 2004.
[3] Y. Bishop, S. Fienberg, P. Holland, Discrete Multivariate Anal-
ysis – Theory and Practice, Cambridge, MA: MIT Press 1975.
[4] T.D. Wickens, Multiway Contingency Tables Analysis for The
Social Sciences, Psychology Press 2014.
[5] T. Dickhaus, K. Straßburger, D. Schunk, C. Morcillo-Suarez,
T. Illig, A. Navarro, How to Analyze Many Contingency Ta-
bles Simultaneously in Genetic Association Studies, Stat. Appl.
Genet. Mol. Biol. 11(4), 12 (2012).
[6] R. El Galta, T. Stijnen, J.J. Houwing-Duistermaat. Testing
for Genetic Association: A Powerful Score Test, Statistics in
Medicine 27(22), 4596–4609 (2008).
[7] B. Cung, Crime and Demographics: An Analysis of LAPD
Crime Data, UCLA Electronic Theses and Dissertations 2013.
[8] R. Iossifova, F. Marmolejo-Ramos, When The Body Is Time:
Spatial and Temporal Deixis in Children with Visual Impair-
ments and Sighted Children, Research in developmental dis-
abilities 34(7), 2173–2184, (2013).
[9] P.D. Allison, J.K. Liker, Analyzing Sequential Categorical Data
on Dyadic Interaction: A comment on Gottman, Psychological
Bulletin, 91(2) 393–403 (1982).
[10] H.H. Bock, Two-Way Clustering for Contingency Tables: Max-
imizing A Dependence Measure, Between data science and
applied data analysis, 143–154 (2003).
[11] S. Kaski, J. Nikkila, J. Sinkkonen, L. Lahti, J.E. Knuuttila,
C. Roos, Associative Clustering for Exploring Dependencies
Between Functional Genomics Data Sets, IEEE/ACM Transac-
tions on Computational Biology and Bioinformatics (TCBB),
2(3), 203–216 (2005).
[12] M. Steinle, K. Aberer, S. Girdzijauskas, C. Lovis, Mapping
Moving Landscapes by Mining Mountains of Logs: Novel Tech-
niques for Dependency Model Generation, [in:] Proceedings
of the 32nd international conference on Very large data bases
VLDB Endowment, pp. 1093–1102 (2006).
[13] P.J. Haas, F. Hueske, V. Markl, Detecting Attribute Depen-
dencies from Query Feedback, [in:] Proceedings of the 33rd
international conference on Very large data bases VLDB En-
dowment, pp. 830–841 (2007)
[14] I.F. Ilyas, V. Markl, P. Haas, P. Brown, A. Aboulnaga, CORDS:
Automatic Discovery of Correlations and Soft Functional De-
pendencies, [in:] Proceedings of the 2004 ACM SIGMOD
international conference on Management of data, ACM, pp.
647–658 (2004).
[15] T. Oates, P.R. Cohen, Searching for Structure in Multiple
Streams of Data, [in:] ICML 96, pp. 346 (1996).
[16] M. Schrepp, A Method for the Analysis of Hierarchical De-
pendencies Between Items of a Questionnaire, Methods of
Psychological Research Online 19, 43–79 (2003).
[17] J. Blitzstein, P. Diaconis, A Sequential Importance Sampling
Algorithm for Generating Random Graphs with Prescribed
Degrees, Internet mathematics 6(4), 489–522 (2011).
[18] N. Cressie, T.R. Read, Multinomial Goodness-Of-Fit Tests,
Journal of the Royal Statistical Society. Series B (Methodolog-
ical), 440–464 (1984).
[19] K. Pearson, On the Criterion that a given System of Devia-
tions from the Probable in the Case of a Correlated System of
Variables is such that it can be Reasonably Supposed to have
Arisen from Random Sampling, Philosophy Magazine Series
50, 157–172 (1900).
[20] M.F. Freeman, J.W. Tukey, Transformations Related to the An-
gular and the Square Root, Annals of Mathematical Statistics
21, 607–611 (1950).
[21] R.R. Sokal, F.J. Rohlf, Biometry: the Principles and Practice
of Statistics in Biological Research, Freeman, New York 2012.
[22] S. Kullback, Information Theory and Statistics, Wiley, New
York 1959.
[23] J. Neyman, Contribution to the Theory of the Chi-Square Test,
Proc. (First) Berkeley Symp. on Math. Statist. and Prob. (Univ.
of Calif. Press), 239–273 (1949).
[24] P. Sulewski, Modyfikacja testu niezależności (Modification of
the Independence Test), Statistical News 10, 1–19 (2013) (in
Polish).
[25] P. Diaconis, B. Sturmfels, Algebraic Algorithms for Sampling
from Conditional Distributions, The Annals of Statistics 26(1),
363–397 (1998).
[26] M. Cryan, M. Dyer, A Polynomial-Time Algorithm to Approxi-
mately Count Contingency Tables When the Number of Rows
Is Constant, Journal of Computer and System Sciences 67(2),
291–310 (2003).
[27] M. Cryan, M. Dyer, L.A. Goldberg, M. Jerrum, R. Martin,
Rapidly Mixing Markov Chains for Sampling Contingency
Tables with A Constant Number of Rows, SIAM Journal on
Computing 36(1), 247–278 (2006).
[28] Y. Chen, P. Diaconis, S.P. Holmes, J.S. Liu. Sequential Monte
Carlo Methods for Statistical Analysis of Tables, Journal of the
American Statistical Association 100(469), 109–120 (2005).
[29] G.S. Fishman, Counting Contingency Tables Via Multistage
Markov Chain Monte Carlo, Journal of Computational and
Graphical Statistics 21(3), 713–738 (2012).
[30] Y. Chen, I.H. Dinwoodie, S. Sullivant Sequential Importance
Sampling for Multiway Tables, The Annals of Statistics, 523–
545 (2006).
[31] R. Yoshida, J. Xi, S. Wei, F. Zhou, D. Haws, Semigroups and
Sequential Importance Sampling for Multiway Tables, arXiv
preprint arXiv:1111.6518 (2011).
[32] S. DeSalvo, J.Y. Zhao, Random Sampling of Contingency Ta-
bles Via Probabilistic Divide-And-Conquer. arXiv preprint
arXiv:1507.00070 (2015).
[33] P. Sulewski, Two-By-Two Contingency Table as a Goodness-
Of-Fit Test, CMST 15(2), 203–211 (2009).
[34] P. Sulewski, R. Motyka, Power Analysis of Independence Test-
ing for Contingency Tables, Scientific Journal of Polish Naval
Academy 56, 37–46 (2015).
[35] P. Sulewski, Moc testów niezależności w tablicy dwudzielczej
większej niż 2×2 (Power Analysis of Independence Testing
for Two-way Contingency Tables bigger than 2×2), Przegląd
statystyczny 63(2), 191–209 (2016) (in Polish).
[36] P. Sulewski, Moc testów niezależności w tablicy trójdzielczej
2x2x2 (Power Analysis of Independence Testing for Three-way
Contingency Table 2x2x2), Przegląd statystyczny 63(4), 431–
447 (2016) (in Polish).
[37] P. Sulewski, Power Analysis of Independence Testing for
the Three-way Contingency Tables of Small Sizes, Jour-
nal of Applied Statistics, http://dx.doi.org/10.1080/02664763.
2018.1424122 (2018).
