Index of Independence

  • Ilija Barukčić Internist, Germany
Keywords: study design, study type, measuring technique, publication bias


Objective. Under certain circumstances, the results of multiple investigations – particularly, rigorously-designed trials, can be summarized by systematic reviews and meta-analyses. However, the results of properly conducted meta-analyses can but need not be stronger than single investigations, if (publication) bias is not considered to a necessary extent.

Methods. In assessing the significance of publication bias due to study design simple to handle statistical measures for quantifying publication bias are developed and discussed which can be used as a characteristic of a meta-analysis. In addition, these measures may permit comparisons of publication biases between different meta-analyses.

Results. Various properties and the performance of the new measures of publication bias are studied and illustrated using simulations and clearly described thought experiments. As a result, individual studies can be reviewed with a higher degree of certainty.

Conclusions. Publication bias due to study design is a serious problem in scientific research, which can affect the validity and generalization of conclusions. The index of unfairness and the index of independence are of use to quantify publication bias and to improve the quality of systematic reviews and meta-analyses.


Altman, D. G. (1999). Practical statistics for medical research. Boca Raton, Fla: Chapman & Hall/CRC.
Arbuthnott, J. (1710). An Argument for Divine Providence, Taken from the Constant Regularity Observ’d in the Births of Both Sexes. By Dr. John Arbuthnott, Physitian in Ordinary to Her Majesty, and Fellow of the College of Physitians and the Royal Society. Philosophical Transactions of the Royal Society of London, 27(325-336), 186-190.
Baker, M. (2016). 1,500 scientists lift the lid on reproducibility. Nature, 533(7604), 452-454.
Barukčić I. (2019). Smoking of tobacco is the cause of human lung cancer. Journal of Drug Delivery and Therapeutics, 9(1-s), 148-160.
Barukčić, I. (1989). Die Kausalität (1. Aufl.). Hamburg: Wiss.-Verl.
Barukčić, I. (1997). Die Kausalität (2., völlig überarb. Aufl.). Wilhelmshaven: Scientia.
Barukčić, I. (2005). Causality: New statistical methods. Norderstedt, Germany: Books on Demand GmbH.
Barukčić, I. (2006a). Causality: New statistical methods (2. Aufl.). Norderstedt: Books on Demand.
Barukčić, I. (2006b). New Method for Calculating Causal Relationships. Retrieved from
Barukčić, I. (2009a). Causality I. A theory of energy, time and space (5. ed., 14. rev).
Barukčić, I. (2009b). Causality II. A theory of energy, time and space (5. ed., 13. rev).
Barukčić, I. (2011a). Anti Heisenberg-Refutation Of Heisenberg’s Uncertainty Relation: In: ADVANCES IN QUANTUM THEORY: Proceedings of the International Conference on Advances in Quantum Theory, Växjö, (Sweden), 14-17 June 2010. In American Institute of Physics - Conference Proceedings (Vol. 1327, pp. 322–325).
Barukčić, I. (2011b). The Equivalence of Time and Gravitational Field. Physics Procedia, 22, 56-62.
Barukčić, I. (2012). Anti-Bell - Refutation of Bell’s theorem: In: Quantum Theory: Reconsideration of Foundations-6 (QTRF6), Växjö, (Sweden), 11-14 June 2012. In American Institute of Physics - Conference Proceedings (Vol. 1508, pp. 354-358).
Barukčić, I. (2016a). The Mathematical Formula of the Causal Relationship k. International Journal of Applied Physics and Mathematics, 6(2), 45-65.
Barukčić, I. (2016b). The Physical Meaning of the Wave Function. Journal of Applied Mathematics and Physics, 4(6), 988-1023.
Barukčić, I. (2016c). Unified Field Theory. Journal of Applied Mathematics and Physics, 4(8), 1379-1438.
Barukčić, I. (2017a). Helicobacter pylori—The Cause of Human Gastric Cancer. Journal of Biosciences and Medicines, 5(2), 1-9.
Barukčić, I. (2017b). Theoriae causalitatis principia mathematica. Norderstedt: Books on Demand.
Barukčić, I. (2018a). Epstein Bar Virus—The Cause of Hodgkin’s Lymphoma. Journal of Biosciences and Medicines, 6(1), 75-100.
Barukčić, I. (2018b). Fusobacterium nucleatum—The Cause of Human Colorectal Cancer. Journal of Biosciences and Medicines, 6(3), 31-69.
Barukčić, I. (2018c). Gastric Cancer and Epstein-Barr Virus Infection. Modern Health Science, 1(2), 1-18.
Barukčić, I. (2018d). Helicobacter Pylori is the Cause of Gastric Cancer. Modern Health Science, 1(1), 43-50.
Barukčić, I. (2018e). Human Cytomegalovirus is the Cause of Glioblastoma Multiforme. Modern Health Science, 1(2), 19.
Barukčić, I. (2018f). Human Papillomavirus—The Cause of Human Cervical Cancer. Journal of Biosciences and Medicines, 6(4), 106–125.
Barukčić, I. (2018g). Mycobacterium Avium Subspecies Paratuberculosis: The Cause Of Crohn’s Disease. Modern Health Science, 1(1), 19–34.
Barukčić, I. (2018h). Epstein-Barr virus is the cause of multiple sclerosis. International Journal of Current Medical and Pharmaceutical Research, 4(9(A)), 3674-3682.
Barukčić, K., Barukčić, J. P., & Barukčić, I. (2018). Epstein-Barr virus is the cause of rheumatoid arthritis. Romanian Journal of Rheumatology, 27(4), 148-163.
Bernoulli, J. (1713). Ars conjectandi, Opus posthumus: Accedit Tractatus de seriebus infinitis ; et epistola Gallice scripta De Ludo Pilae Reticularis. Basileae (Basel, Suisse): Impensis Thurnisiorum [Tournes], fratrum.
Bienaymé, I.-J. (1846). Sur les probabilités des erreurs d’après la méthode des moindres carrés. Journal De Mathématiques Pures Et Appliquées, 1(17), 33-78.
Bortkiewicz, L. (Ed.). (1898). Das Gesetz der kleinen Zahlen: [Transl. into English: The law of small numbers]. Leipzig (Germany): B.G. Teubner. Retrieved from
Charan, J.,& Biswas, T. (2013). How to calculate sample size for different study designs in medical research? Indian Journal of Psychological Medicine, 35(121).
Conover, W. J. (1974). Some Reasons for Not Using the Yates Continuity Correction on 2×2 Contingency Tables. Journal of the American Statistical Association, 69(346), 374-376.
Cornfield, J. (1951). A method of estimating comparative rates from clinical data; applications to cancer of the lung, breast, and cervix. Journal of the National Cancer Institute, 11(6), 1269-1275.
DeGroot, M. H., Schervish, M. J., Fang, X., Lu, L., & Li, D. (2005). Probability and Statistics (Third Edition [Rep. & arr. ed.]). Beijing (China): Higher Education Press.
Dorey, F. (2010). In Brief: The P Value: What Is It and What Does It Tell You? Clinical Orthopaedics and Related Research, 468(8), 2297-2298.
Edwards, A. W. F. (1963). The Measure of Association in a 2 × 2 Table. Journal of the Royal Statistical Society. Series A (General), 126(1), 109.
Egger, M., Smith, G. D., Schneider, M., & Minder, C. (1997). Bias in meta-analysis detected by a simple, graphical test. BMJ, 315(7109), 629-634.
Fisher, R. A. (1922). On the Interpretation of χ 2 from Contingency Tables, and the Calculation of P. Journal of the Royal Statistical Society, 85(1), 87.
Fisher, R. A. (1935). The Logic of Inductive Inference. Journal of the Royal Statistical Society, 98(1), 39.
Fisher, Ronald A. (1925). Statistical Methods for Research Workers. Edinburgh: Oliver and Boyd. Retrieved from
Gomes, G. (2009). Are Necessary and Sufficient Conditions Converse Relations. Australasian Journal of Philosophy, 87, 375-387.
Gonin, H. T. (1936). XIV. The use of factorial moments in the treatment of the hypergeometric distribution and in tests for regression. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(139), 215-226.
Grimes, D. A., & Schulz, K. F. (2002). Bias and causal associations in observational research. Lancet (London, England), 359(9302), 248-252.
Grizzle, J. E. (1967). Continuity Correction in the χ 2 -Test for 2 × 2 Tables. The American Statistician, 21(4), 28.
Hanley, J. A. (1983). If Nothing Goes Wrong, Is Everything All Right? JAMA, 249(13), 1743.
Hogg, R. V., & Craig, A. T. (2004). Introduction to Mathematical Statistics. Beijing: Higher Education Press.
Hopewell, S., Loudon, K., Clarke, M. J., Oxman, A. D., & Dickersin, K. (2009a). Publication bias in clinical trials due to statistical significance or direction of trial results. Cochrane Database of Systematic Reviews, (1).
Hume, D. (1739). David Hume: A Treatise of Human Nature (Second Edition). Oxford University Press.
Hunter, J. P., Saratzis, A., Sutton, A. J., Boucher, R. H., Sayers, R. D., & Bown, M. J. (2014). In meta-analyses of proportion studies, funnel plots were found to be an inaccurate method of assessing publication bias. Journal of Clinical Epidemiology, 67(8), 897-903.
Huygens, C. (1629-1695), & van Schooten, F. (1615-1660). (1657). De ratiociniis in ludo alae: In: Exercitationum mathematicarum liber primus [- quintus]. Lugdunum Batavorum (Leiden, The Netherlands): ex officina Johannis Elsevirii.
Isserlis, L. (1918). On the Value of a Mean as Calculated from a Sample. Journal of the Royal Statistical Society, 81(1), 75.
Joober, R., Schmitz, N., Annable, L., & Boksa, P. (2012). Publication bias: What are the challenges and can they be overcome? Journal of Psychiatry & Neuroscience, 37(3), 149-152.
Jovanovic, B. D., & Levy, P. S. (1997). A Look at the Rule of Three. The American Statistician, 51(2), 137-139.
Kaplan, R. M., Chambers, D. A., & Glasgow, R. E. (2014). Big Data and Large Sample Size: A Cautionary Note on the Potential for Bias. Clinical and Translational Science, 7(4), 342-346.
Kleijnen, J., & Knipschild, P. (1992). Review articles and publication bias. Arzneimittelforschung, 45(2), 587-591.
Kolmogoroff, A. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin, Heidelberg: Springer Berlin Heidelberg.
LaPlace, Pierre Simon de. (1812). Théorie analytique des probabilités. Paris (France): Courcier.
Lau, J., Ioannidis, J. P. A., Terrin, N., Schmid, C. H., & Olkin, I. (2006). The case of the misleading funnel plot. BMJ, 333(7568), 597-600.
Light, R. J., & Pillemer, D. B. (1984). Summing up. The science of reviewing research. Cambridge, Mass., USA: Harvard University Press.
Louis, T. A. (1981). Confidence Intervals for a Binomial Parameter after Observing No Successes. The American Statistician, 35(3), 154-154.
Lyapunov, A. M. (1901). Nouvelle forme du théorème sur la limite de probabilité. Mémoires de l’Académie Impériale Des Sciences de St.-Pétersbourg. Série VIIIe. Classe Physico-Mathématique, 12, 1-24.
Moher, D., Dulberg, C. S., & Wells, G. A. (1994). Statistical Power, Sample Size, and Their Reporting in Randomized Controlled Trials. JAMA: The Journal of the American Medical Association, 272(2), 122.
Moivre, A. (1733). Approximatio ad summam terminorum binomii (a+b)n in seriem expansi. London: Privately (Publisher not identified). Retrieved from
Moivre, A. de [1667-1754]. (1718). The Doctrine of Chances or a Method of Calculating the Probability of Events in Play. London: printed by W. Pearson for the author.
Mosteller, F. (1968). Association and Estimation in Contingency Tables. Journal of the American Statistical Association, 63(321), 1.
Murad, M. H., Chu, H., Lin, L., & Wang, Z. (2018). The effect of publication bias magnitude and direction on the certainty in evidence. BMJ Evidence-Based Medicine, 23(3), 84-86.
Noordzij, M., Dekker, F. W., Zoccali, C., & Jager, K. J. (2010). Measures of Disease Frequency: Prevalence and Incidence. Nephron Clinical Practice, 115, c17-c20. https://doi.org10.1159/000286345
Pagano, M., & Gauvreau, K. (2018). Principles of Biostatistics (2nd ed.). Milton: CRC Press. Retrieved from
Panagiotakos, D. B. (2008). The Value of p-Value in Biomedical Research. The Open Cardiovascular Medicine Journal, 2, 97-99.
Pearson, K. (1900). X. On the criterion that a given system of deviations 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. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302), 157-175.
Pearson, K., & Heron, D. (1913). On Theories of Association. Biometrika, 9(1-2), 159-315.
Pearson, Karl. (1899). XV. On certain properties of the hypergeometrical series, and on the fitting of such series to observation polygons in the theory of chance. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47(285), 236-246.
Poisson, S. D. (1837). Recherches sur la Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés. Paris, France: Bachelier. Retrieved from
Pólya, G. (1920). Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem: [In English: On the central limit theorem of probability calculation and the problem of moments]. Mathematische Zeitschrift, 8(3-4), 171-181.
Quine, M. P., & Robinson, J. (1985). Efficiencies of Chi-Square and Likelihood Ratio Goodness-of-Fit Tests. The Annals of Statistics, 13(2), 727-742.
Rahme, E., & Joseph, L. (1998). Exact sample size determination for binomial experiments. Journal of Statistical Planning and Inference, 66, 83-93.
Retrieved from
Rumke, C. L. (1975). Implications of the Statement: No Side Effects Were Observed. The New England Journal of Medicine, 292(7), 372-373.
Sachs, L. (1992). Angewandte Statistik. Berlin, Heidelberg: Springer Berlin Heidelberg.
Scheid, H. (1992). Wahrscheinlichkeitsrechnung (Vol. 6). Mannheim: BI-Wiss.-Verl.
Schervish, M. J. (1996). P Values: What They are and What They are Not. The American Statistician, 50(3), 203-206.
Szumilas, M. (2010). Explaining Odds Ratios. J Can Acad Child Adolesc Psychiatry, 19, 227-229.
Tchébychef, P. L. (1867). Des valeurs moyennes. Journal de Mathématiques Pures et Appliquées, 2(12), 177-184.
von Leibniz, G. W. F. (1765). Oeuvres philosophiques latines & francoises de feu Mr. de Leibniz. Amsterdam (NL): Chez Jean Schreuder.
Warrens, M. J. (2008). On Association Coefficients for 2x2 Tables and Properties That Do Not Depend on the Marginal Distributions. Psychometrika, 73(4), 777-789.
Wertheimer, R. (1968). Conditions. Journal of Philosophy, 65, 355-364.
Yates, F. (1934). Contingency Tables Involving Small Numbers and the χ 2 Test. The Journal of the Royal Statistical Society (Supplement), 1(2), 217-235.
Yule, G. U. (1900). On the Association of Attributes in Statistics: With Illustrations from the Material of the Childhood Society, &c. Philosophical Transactions of the Royal Society A Mathematical, Physical and Engineering Sciences, 194(252-261), 257-319.
Zwetsloot, P. P., Van Der Naald, M., Sena, E. S., Howells, D. W., IntHout, J., De Groot, J. A., … Wever, K. E. (2017). Standardized mean differences cause funnel plot distortion in publication bias assessments. ELife, 6.