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Vol. 74. Issue 7.
Pages 641-642 (July 2021)
Vol. 74. Issue 7.
Pages 641-642 (July 2021)
Letter to the Editor
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The use of Bayes factor in clinical cardiology research
El uso del factor Bayes en la investigación clínica de cardiología
Cristian Antony Ramos-Vera
Área de Investigación, Facultad de Ciencias de la Salud, Universidad César Vallejo, Lima, Peru
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Jorge Solano-López, José Luis Zamorano, Ana Pardo Sanz, Ignacio Amat-Santos, Fernando Sarnago, Enrique Gutiérrez Ibañes, Juan Sanchis, Juan Ramón Rey Blas, Joan Antoni Gómez-Hospital, Sandra Santos Martínez, Nicolás Manuel Maneiro-Melón, Roberto Mateos Gaitán, Jessika González D’Gregorio, Luisa Salido, José L. Mestre, Marcelo Sanmartín, Ángel Sánchez-Recalde
Rev Esp Cardiol. 2021;74:642-310.1016/j.rec.2021.03.011
Juan Manuel Monteagudo Ruiz, Jorge Solano-López, José Luis Zamorano, Ángel Sánchez-Recalde
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To the Editor,

I have read with interest the article by Solano-López J et al.,1 an important article that finds a statistically significant association (P < .05) between in-hospital mortality and positive COVID-19 diagnosis in patients with acute myocardial infarction by the measure of association of the odds ratio (OR).

The replication of significance test-based clinical research is recommended to acquire more credible evidence in cardiology. A potential approach is Bayesian inference, which can be used to reanalyze the significant findings reported by Solano-López et al.,1 where the Bayes factor (BF) method is referred to as the likelihood of the data under one hypothesis compared with the other (null vs alternative hypothesis).2,3 In other words, the BF estimates the quantification of the evidence or extent to which the data support the null hypothesis and the alternative hypothesis for comparison beyond the mere dichotomous interpretation of the rejection or acceptance of the null hypothesis.2,3 The statistical repetition of significant findings using the BF strengthens the practical credibility of future articles in the field of cardiology (clinical trials, interventions, and treatments, among others), needed when Bayesian inference produces conclusive (strong) or superior (BF10 > 10) evidence by interpreting the Jeffreys classification4 for BF: anecdotal, moderate, strong, and very strong (figure 1).

Figure 1.

Quantifiable values of the Jeffreys’ Bayes factor.4.


The purpose of this letter is to provide a simple example of Bayesian reanalysis to determine the degree of evidentiary strength of the statistical hypotheses. Therefore, transformation of the OR value (8.23) to correlation effect size (r) using Lenhard and Lenhard's online calculator,5 was first considered, yielding an r=value of 0.502, and the sample size (187) was also considered for BF reproduction.2 This method results in 2 interpretations: BF10 (in favor of the alternative hypothesis of significance) and BF01 (in favor of the null hypothesis), with a credibility interval of 95%.6 The results obtained for the Bayes factor were BF10 = 3.1810 and BF01 = 3.14–11 and a 95% confidence interval of 0.383 to 0.599, which supports the significant findings reported by Solano-López et al.,1 with very strong evidence in favor of the alternative statistical hypothesis (correlation).

Likewise, the maximum Bayes factor parameter (BF10max = 3.56810) was estimated to determine the stability of the results, with the larger value strengthening the Bayesian reanalysis estimate.

The effect-size (ES) transformation and other statistical measures based on a hypothesis contrast methodology (d, f, η2, OR, χ2, Z) using the correlation coefficient (r), used more widely in health sciences, is beneficial for future analyses and Bayesian reanalyses. Additionally, these estimates are easy to perform using Lenhard and Lenhard's calculator.5 The BF is useful in other statistical significance tests7,8 (linear regression, ANOVA, among others) with ES measures that are also transformable. The handbook by Goss-Sampson6 is recommended for Bayesian inference of the analyses most commonly used in cardiology research.

The inclusive use of several transformable ESs supports further research employing various statistical methods for future meta-analyses. Moreover, the use of the BF is beneficial for selecting ESs with the most robust evidence (BF10 > 10) for meta-analytical design, as it strengthens the credibility of clinical meta-analytical conclusions.

In summary, the BF is a useful methodological tool with practical implications for decision-making based on the confirmation of conclusive results, now even more important in the context of COVID-19.


No funding has been received.


No conflicts of interests.

J. Solano-López, J.L. Zamorano, A. Pardo Sanz, et al.
Factores de riesgo de muerte hospitalaria en pacientes con infarto agudo de miocardio durante la pandemia de la COVID-19.
Rev Esp Cardiol., 73 (2020), pp. 985-993
A. Ly, A. Raj, A. Etz, Q.F. Gronau, E.J. Wagenmakers.
Bayesian reanalyses from summary statistics: A guide for academic consumers.
Adv Methods Pract Psychol Sci., 1 (2018), pp. 367-374
M. Marsmamn, E.J. Wagenmakers.
Bayesian benefits with JASP.
Eur J Dev Psychol., 14 (2017), pp. 545-555
H. Jeffreys.
Theory of probability.
Oxford University Press, (1961),
W. Lenhard, A. Lenhard.
Computation of effect sizes.
Dettelbach:, (2016),
M.A. Goss-Sampson.
Bayesian inference in JASP: a guide for students..
University of Amsterdam, JASP team, (2020),
R. Kelter.
Bayesian alternatives to null hypothesis significance testing in biomedical research: a non-technical introduction to Bayesian inference with JASP.
BMC Med Res Methodol., 20 (2020), pp. 142
C.A. Ramos-Vera.
Replicación bayesiana: cuán probable es la hipótesis nula e hipótesis alterna.
Copyright © 2021. Sociedad Española de Cardiología
Revista Española de Cardiología (English Edition)

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