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Published October 11, 2020 | Version 3
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Causal relationship k

Authors/Creators

  • 1. Internist

Description

Objectives: Analyzing experimental data in basic and clinical pharmacology and other research areas for cause effect relationships requires appropriate statistical methods too.

Methods: The axiom lex identitatis or + 1 = +1 was the foundation of a hypothetico-deductive method used to describe the relationship between a cause and an effect by the tools of probability theory. Let p(Ut) denote the probability of a quantum mechanical observable, a random variable Ut (a cause) at a single Bernoulli trial t. Let E(Ut) = Ut*p(Ut) denote the expectation value of Ut (a cause) at a single Bernoulli trial t. Let E(Ut2) = Ut*Ut*p(Ut) denote the expectation value of Ut2 (a cause squared) at a single Bernoulli trial t. Let p(Wt) denote the probability of an effect Wt (a quantum mechanical observable, a random variable). The variance of the cause is defined as s(Ut)2 =E(Ut2) – E(Ut)2= (Ut*Ut)*p(Ut)*(1- p(Ut)). From this follows that Ut = s(Ut) / (p(Ut)*(1- p(Ut)))1/2. The variance of an effect is defined as s(Wt)2 = E(Wt2) – E(Wt)2 = (Wt*Wt)´p(Wt)*(1- p(Wt)). From this follows that Wt = s(Wt) / (p(Wt)*(1- p(Wt)))1/2. The co-variance of a cause and an effect at a single Bernoulli trial t is defined as s(Ut,Wt) =E(Ut,Wt) – (E(Ut)*(E(Wt))= (Ut*Wt)*(p(Ut,Wt) - (p(Ut)*p(Wt))). This leads to the relationship that Ut*Wt = s(Ut,Wt) / (p(Ut,Wt) - (p(Ut)*p(Wt))).  

Results: In general, it is +1 = +1. Multiplying by (Ut*Wt), it is (Ut*Wt) = (Ut*Wt). Substituting the definitions above into equation, it is (Ut*Wt) = s(Ut) / (p(Ut)*(1-p(Ut)))1/2 * Wt or (Ut*Wt)=(s(Ut)/(p(Ut)*(1-p(Ut)))1/2)*(s(Wt)/(p(Wt)´(1- p(Wt)))1/2) and equally s(Ut,Wt) / (p(Ut,Wt) - (p(Ut)*p(Wt))) =(s(Ut) / (p(Ut)*(1- p(Ut)))1/2)*(s(Wt) / (p(Wt)*(1- p(Wt)))1/2). The causal (1) relationship k(Ut,Wt) is determined as k(Ut,Wt) = s(Ut,Wt)/(s(Ut)*s(Wt)) = (p(Ut,Wt) - (p(Ut)*p(Wt))) / (p(Ut)*(1- p(Ut))´p(Wt)*(1- p(Wt)) )1/2. In contrast to Bravais (2) - Pearson's coefficient of correlation (3) and Pearson’s phi (4), the causal relationship k (1) is defined at every single Bernoulli trial t.

Conclusions: Biotechnological, medical engineering and other data can be analyzed for causal relationships by the method as described, derived and proved before.

References:

(1) Barukčić I. The Mathematical Formula of the Causal Relationship k. IJAPM. 2016;6(2):45-65. https://publons.com/researcher/3501739/ilija-barukcic/

(2) Bravais A. Analyse mathématique sur les probabilités d es erreurs de situation d’un point. Mémoires Présentées Par Divers Savants À L’Académie Royale Des Sciences De L’Institut De France. 1846;9:255–332. https://orcid.org/0000-0002-6988-2780

(3) Pearson K. VII. Mathematical contributions to the theory of evolution.—III. Regression, heredity, and panmixia. Philosophical Transactions of the Royal Society of London Series A, Containing Papers of a Mathematical or Physical Character. 1896;187: 253–318. https://www.scopus.com/authid/detail.uri?authorId=37099674500

(4) Pearson K. Mathematical Contributions to the Theory of Evolution. XIII. On the Theory of Contingency and Its Relation to Association and Normal Correlation. Dulau and Co.; 1904. https://www.researchgate.net/profile/Ilija_Barukcic2

Notes

Paper is still not published by a web of science indexed journal.

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