Karnaugh-Map Utility in Medical Studies: The Case of Fetal Malnutrition
Rufaidah Ali Rushdi
Department of Pediatrics, Kasr Al-Ainy School of Medicine, Cairo University Cairo, 11562, Arab Republic of Egypt.
Ali Muhammad Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia.
Received on November 09, 2017
Accepted on January 03, 2018
This paper advocate and demonstrates the utility of the Karnaugh map, as a pictorial manual tool of Boolean algebra, in the exploration of medical problems as exemplified herein by the problem of Fetal Malnutrition (FM). The paper briefly introduces the FM problem, and specifies four metrics or tests used frequently in its study. Clinical data collected about these metrics (as continuous variables or dichotomized versions thereof) are conventionally treated via statistical methods. The Karnaugh map serves as a convenient way for aggregating the set of clinical data available into a pseudo-Boolean function. The map can be used to produce a two-by-two contingency matrix (confusion matrix or frequency matrix) that relates an assessed test or metric to a reference or standard one. Each of these two metrics can be any of the map variables or a function of some or all of these variables. While the map serves in this capacity as a supplement or aid to statistical methods, it is also useful for certain non-statistical methods (specifically Boolean ones). The paper shows how the map entries can be dichotomized via an appropriate threshold for use in Boolean Analysis (BA), which can be conducted despite the lack of a gold standard. The map also implements Qualitative Comparative Analysis (QCA) for the given clinical data. The map variable-handling capability does not pose as a shortcoming for either BA or QCA, since the number of variables involved (not only herein but in other typical medical problems as well) is relatively small. The concepts and methods introduced herein are demonstrated through application to the aforementioned set of clinical data for the FM problem, and can be extended to a wide variety of medical problems.
Keywords- Karnaugh map, Contingency table, Gold standard, Fetal malnutrition, Pseudo-Boolean function, Boolean analysis, Qualitative comparative analysis, Epidemiology, Diagnostic testing.
Rushdi, R. A., & Rushdi, A. M. (2018). Karnaugh-Map Utility in Medical Studies: The Case of Fetal Malnutrition. International Journal of Mathematical, Engineering and Management Sciences, 3(3), 220-244. https://dx.doi.org/10.33889/IJMEMS.2018.3.3-016.
Conflict of Interest
Alonzo, T. A., & Pepe, M. S. (1999). Using a combination of reference tests to assess the accuracy of a new diagnostic test. Statistics in Medicine, 18(22), 2987-3003.
Alturki, A. M., & Rushdi, A. M. A., (2016). Weighted voting systems: a threshold-Boolean perspective, Journal of Engineering Research, 4(1), 125-143.
Anderson, T. W., & Finn, J. D. (1996). Summarizing Multivariate Data: Association between Categorical Variables, Chapter 6 in The New Statistical Analysis of Data. Springer Science & Business Media, pp 177-230
Baumgartner, M. (2009). Uncovering deterministic causal structures: a Boolean approach. Synthese, 170(1), 71-96.
Baumgartner, M., & Thiem, A. (2017). Often trusted but never (properly) tested: evaluating qualitative comparative analysis. Sociological Methods & Research, Online first 3 May, 2017.
Baveja, C. P., & Aggarwal, P. (2017). Statistical analysis of microbiological diagnostic tests. Indian Journal of Medical Microbiology, 35(2), 184-193.
Bhambu, L., & Kumar, D. (2015). A novel approach for classification on breast cancer data set. International Journal of Advanced Research in Computer Science and Software Engineering, 5(7), 1118-1123.
Bradley, E. H., Curry, L. A., & Devers, K. J. (2007). Qualitative data analysis for health services research: developing taxonomy, themes, and theory. Health Services Research, 42(4), 1758-1772.
Broemeling, L. D. (2011). Advanced Bayesian methods for medical test accuracy. CRC Press, Boca Raton, FL, USA.
Brown, F. M. (1990). Boolean reasoning: the logic of Boolean equations, Kluwer Academic Publishers, Boston, USA.
Chughtai, A., Kelly, A. M., & Cronin, P. (2015). How to perform a critical appraisal of diagnostic tests: 7 steps. Pediatric Radiology, 45(6), 793-803.
Crama, Y., Hammer, P. L., & Ibaraki, T. (1988). Cause-effect relationships and partially defined Boolean functions. Annals of Operations Research, 16(1), 299–325.
De Felice, C., Cortelazzo, A., Leoncini, S., Signorini, C., Hayek, J., & Ciccoli, L. (2016). Statistics, biomedicine and scientific fraud. Journal of the Siena Academy of Sciences, Focus on Biostatistics, 7(1), 15-22.
DeCoster, J., Iselin, A. M. R., & Gallucci, M. (2009). A conceptual and empirical examination of justifications for dichotomization. Psychological Methods, 14(4), 349-366.
Degenne, A., & Lebeaux, M. O. (1996). Boolean analysis of questionnaire data. Social Networks, 18(3), 231-245.
Dendukuri, N., Schiller, I., Joseph, L., & Pai, M. (2012). Bayesian meta-analysis of the accuracy of a test for tuberculous pleuritis in the absence of a gold standard reference. Biometrics, 68(4), 1285-1293.
Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27(8), 861-874.
Feinstein, A. R. (1963). Boolean algebra and clinical taxonomy: analytic synthesis of the general spectrum of a human disease. New England Journal of Medicine, 269(18), 929-938.
Flament, C. (1965). L’analyse Booléenne de questionnaires (Boolean analysis of questionnaires). Mathématiques et Sciences Humaines, 12, 3-10.
Flament, C. (1976). L'analyse Booléenne de questionnaire (the Boolean analysis of a questionnaire), Mouton, Paris, France.
Gigerenzer, G., & Marewski, J. N. (2015). Surrogate science: the idol of a universal method for scientific inference. Journal of Management, 41(2), 421-440.
Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2008). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53-96.
Glantz, S. A. (1980). Biostatistics: how to detect, correct and prevent errors in the medical literature. Circulation, 61(1), 1-7.
Glas, A. S., Lijmer, J. G., Prins, M. H., Bonsel, G. J., & Bossuyt, P. M. (2003). The diagnostic odds ratio: a single indicator of test performance. Journal of Clinical Epidemiology, 56(11), 1129-1135.
Hall, G. H. (1967). The clinical application of Bayes' theorem. The Lancet, 290(7515), 555-557.
Hammer, P. L., & Bonates, T. O. (2006). Logical analysis of data—an overview: from combinatorial optimization to medical applications. Annals of Operations Research, 148(1), 203-225.
Hawkins, R. C. (2005). The evidence based medicine approach to diagnostic testing: practicalities and limitations. Clinical Biochemist Reviews, 26(2), 7-18.
Hoffrage, U., Gigerenzer, G., Krauss, S., & Martignon, L. (2002). Representation facilitates reasoning: what natural frequencies are and what they are not. Cognition, 84(3), 343-352.
Hoffrage, U., Kurzenhäuser, S., & Gigerenzer, G. (2005). Understanding the results of medical tests: why the representation of statistical information matters. In Bibace, R., et al. (Editors), Science and Medicine in Dialogue: Thinking Through particulars and Universals, 83-98.
Jordan, E., Gross, M. E., Javernick-Will, A. N., & Garvin, M. J. (2011). Use and misuse of Qualitative Comparative Analysis. Construction Management and Economics, 29(11), 1159-1173.
Joseph, L., Gyorkos, T. W., & Coupal, L. (1995). Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard. American Journal of Epidemiology, 141(3), 263-272.
Kent, P., & Hancock, M. J. (2016). Interpretation of dichotomous outcomes: sensitivity, specificity, likelihood ratios, and pre-test and post-test probability. Journal of Physiotherapy, 62(4), 231-233.
Kramer, M. S. (1987). Determinants of low birth weight: methodological assessment and meta-analysis. Bulletin of the World Health Organization, 65(5), 663-737.
Kramer, M. S., Olivier, M., McLean, F. H., Dougherty, G. E., Willis, D. M., & Usher, R. H. (1990). Determinants of fetal growth and body proportionality. Pediatrics, 86(1), 18-26.
Lang, T. (2004). Twenty statistical errors even you can find in biomedical research articles. Croatian Medical Journal, 45(4), 361-370.
Leeflang, M. M. G. (2014). Systematic reviews and meta analyses of diagnostic test accuracy. Clinical Microbiology and Infection, 20(2), 105-113.
Lewis, F. I., & Torgerson, P. R. (2012). A tutorial in estimating the prevalence of disease in humans and animals in the absence of a gold standard diagnostic. Emerging Themes in Epidemiology, 9, 1-8.
Lin, P. C. K., & Khatri, S. P. (2014). Logic synthesis for genetic diseases: modeling disease behavior using Boolean networks. Springer Science & Business Media, New York, NY, USA.
Lusted, L. B., & Ledley, R. S. (1960). Mathematical models in medical diagnosis. Academic Medicine, 35(3), 214-222.
Marshall, R. J. (1986). Partitioning methods for classification and decision making in medicine. Statistics in Medicine, 5(5), 517-526.
Marshall, R. J. (2001). The use of classification and regression trees in clinical epidemiology. Journal of Clinical Epidemiology, 54(6), 603-609.
Marx, A., Rihoux, B., & Ragin, C. (2014). The origins, development, and application of qualitative comparative analysis: the first 25 years. European Political Science Review, 6(1), 115-142.
O’Neill, D. (2015). Measuring obesity in the absence of a gold standard. Economics & Human Biology, 17, 116-128.
Ogihara, H., Fujita, Y., Hamamoto, Y., Iizuka, N., & Oka, M. (2013, November). Classification based on Boolean algebra and its application to the prediction of recurrence of liver cancer. In IEEE 2013 2nd IAPR Asian Conference on Pattern Recognition (ACPR), pp. 838-841.
Parikh, R., Mathai, A., Parikh, S., Sekhar, G. C., & Thomas, R. (2008). Understanding and using sensitivity, specificity and predictive values. Indian Journal of Ophthalmology, 56(1), 45-50.
Porebski, S., & Straszecka, E. (2018). Extracting easily interpreted diagnostic rules. Information Sciences, 426, 19-37.
Powers, D. M. (2011). Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation. Journal of Machine Learning Technologies. 2(1), 37–63.
Ragin, C. C. (1999). Using qualitative comparative analysis to study causal complexity. Health Services Research, 34(5 Pt 2), 1225-1239.
Ragin, C. C. (1989). The comparative method: moving beyond qualitative and quantitative strategies, Berkeley, Los Angeles and London, University of California Press.
Ragin, C. C., Mayer, S. E., & Drass, K. A. (1984). Assessing discrimination: a Boolean approach. American Sociological Review, 49(2), 221-234.
Reitsma, J. B., Rutjes, A. W., Khan, K. S., Coomarasamy, A., & Bossuyt, P. M. (2009). A review of solutions for diagnostic accuracy studies with an imperfect or missing reference standard. Journal of Clinical Epidemiology, 62(8), 797-806.
Rihoux, B. (2003). Bridging the gap between the qualitative and quantitative worlds? a retrospective and prospective view on qualitative comparative analysis. Field Methods, 15(4), 351-365.
Rihoux, B., & de Meur, G. (2009). Crisp-set qualitative comparative analysis (csQCA), In B. Rihoux, B. and Ragin, C. C. (Editors), Configurational Comparative Methods: Qualitative Comparative Analysis (QCA) and Related Techniques, Thousand Oaks, CA, Sage, pp. 33-69.
Rindskopf, D., & Rindskopf, W. (1986). The value of latent class analysis in medical diagnosis. Statistics in Medicine, 5(1), 21-27.
Royston, P., Altman, D. G., & Sauerbrei, W. (2006). Dichotomizing continuous predictors in multiple regression: a bad idea. Statistics in Medicine, 25(1), 127-141.
Rushdi, A. A. (2010). A mathematical model of DNA replication. International Magazine on Advances in Computer Science and Telecommunications (IMACST), 1(1), 23-30.
Rushdi, A. M., & Rushdi M. A. (2017). Switching-algebraic analysis of system reliability, Chapter 6 in Ram, M. and Davim, P. (Editors). Advances in Reliability and System Engineering. Management and Industrial Engineering Series. Springer International Publishing, Cham, Switzerland, pp.139-161.
Rushdi, A. M., & Rushdi, M. A. (2018). Mathematics and examples of the modern syllogistic method of propositional logic, In Ram, M. (Editor), Mathematics Applied in Information Systems, Bentham Science Publishers, Emirate of Sharjah, United Arab Emirates.
Rushdi, A. M. (1986). Map differentiation of switching functions. Microelectronics and Reliability, 26(5), 891-907.
Rushdi, A. M. A., & Alturki, A. M. (2015). Reliability of coherent threshold systems. Journal of Applied Sciences, 15(3), 431-443.
Rushdi, A. M. A., & Badawi, R. M. S. (2017a). Karnaugh-map utilization in Boolean analysis: The case of war termination. Journal of Qassim University: Engineering and Computer Sciences, 10(1), 53-88.
Rushdi, A. M. A., & Badawi, R. M. S. (2017b). Karnaugh map utilization in coincidence analysis, Journal of King Abdulaziz University: Faculty of Computers and Information Technology, 6(1), in press.
Rushdi, A. M. A. (2018). Utilization of Karnaugh maps in multi-value qualitative comparative analysis, International Journal of Mathematical, Engineering and Management Sciences, 3(1), 28-46.
Rushdi, A. M., & Ba-Rukab, O. M. (2017). Map calculation of the Shapley-Shubik voting powers: an example of the European Economic Community. International Journal of Mathematical, Engineering and Management Sciences, 2(1), 17-29.
Rushdi, R. A. (2017). Fetal Malnutrition: Assessment by the CANS score versus Anthropometry and Impact on Early Neonatal Morbidities, Unpublished Master Thesis, Department of Pediatrics, Kasr Al-Ainy School of Medicine, Cairo University, Cairo, Egypt, Available online at https://www.researchgate.net/profile/Rufaidah_Rushdi/contributions.
Schensul, J. J., Chandran, D., Singh, S. K., Berg, M., Singh, S., & Gupta, K. (2010). The use of qualitative comparative analysis for critical event research in alcohol and HIV in Mumbai, India. AIDS and Behavior, 14(1), 113-125.
Shindo, T., Takahashi, T., Okamoto, T., & Kuraishi, T. (2012). Evaluation of diagnostic results by Bayes' theorem. IEEJ Transactions on Electrical and Electronic Engineering, 7(5), 450-453.
Strasak, A. M., Zaman, Q., Marinell, G., Pfeiffer, K. P., & Ulmer, H. (2007). The use of statistics in medical research: a comparison of The New England Journal of Medicine and Nature Medicine. The American Statistician, 61(1), 47-55.
Theuns, P. (1989). Predicting an optimal threshold in Boolean analysis of questionnaires. In Roskam, E. E. (Editor), Mathematical Psychology in Progress, Springer-Verlag Berlin Heidelberg, 329-343.
Theuns, P. (1994). A dichotomization method for Boolean analysis of quantifiable co-occurrence data. In G. Fischer and D. Laming (Editors), Contributions to Mathematical Psychology, Psychometrics, and Methodology, 2nd Ed., pp. 389-402, New York, USA, Springer.
Theuns, P. (1999). A Boolean approach to hierarchical data analysis: an overview. In 30th Meeting of the European Mathematical Psychology Group, Mannheim, Germany, 1-18.
Thomas, R., Mengersen, K., Parikh, R. S., Walland, M. J., & Muliyil, J. (2011). Enter the reverend: introduction to and application of Bayes' theorem in clinical ophthalmology. Clinical & Experimental Ophthalmology, 39(9), 865-870.
Tsumoto, S. (2009). Contingency matrix theory: statistical dependence in a contingency table. Information Sciences, 179(11), 1615-1627.
Van Loo, H. M., & Romeijn, J. W. (2015). Psychiatric comorbidity: fact or artifact? Theoretical Medicine and Bioethics, 36(1), 41-60.
Winkler, R. L., & Smith, J. E. (2004). On uncertainty in medical testing. Medical Decision Making, 24(6), 654-658.
Zhou, X. H., McClish, D. K., & Obuchowski, N. A. (2009). Statistical methods in diagnostic medicine (Vol. 569). John Wiley & Sons, New York, NY, USA.