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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Critical positivity ratio | 2/3 | https://en.wikipedia.org/wiki/Critical_positivity_ratio | reference | science, encyclopedia | 2026-05-05T09:18:23.760203+00:00 | kb-cron |
=== Initially ignored questioning === The first critical evaluation of the mathematical modeling behind the critical positivity ratio was published by a group of Finnish researchers—Luoma, Hämäläinen, and Saarinen of the Systems Analysis Laboratory at Aalto University—in 2008. The authors noted that "[o]nly very limited explanations are given about the modeling process and the meaning and interpretation of its parameters... [so that] the reasoning behind the model equations remains unclear to the reader"; moreover, they noted that "the model also produces strange and previously unreported behavior under certain conditions... [so that] the predictive validity of the model also becomes problematic." Not widely impactful at the time, Losada's earlier modeling article was also critiqued by Andrés Navas in a French language publication, a note in the CNRS publication, "Images des Mathématiques", which also failed to attract a wide readership. In their followup to Fredrickson's immediate response to the rebuttal, Brown, Sokal, and Friedman note as a footnote to their submission:After the publication of Brown et al. (2013), Andrés Navas kindly drew our attention to his article (Navas, 2011) in which a very similar (though briefer) critique of Losada (1999) was made. [This footnote was unfortunately omitted from the published version of this article, due to space limitations.]
=== The Brown-Sokal-Friedman rebuttal === The Fredrickson and Losada work on modeling the positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who questioned whether such work could reliably make such broad claims, and perceived that the paper's mathematical claims were suspect. Brown contacted and ultimately collaborated with physics and maths professor Alan Sokal and psychology professor Friedman on a re-analysis of the paper's data. The result was a strong critique of the critical positivity ratio in its entirety by Brown, Sokal, and Friedman, that appeared in a 2013 article in American Psychologist, here referred to as the Brown-Sokal-Friedman rebuttal. These authors argued that Losada's conclusions in previous papers using modelling from fluid dynamics, and those in his paper co-authored with Fredrickson, were not only based on poorly reported experiments, but also that it was difficult to draw conclusions from Losada's previous cited studies because critical details were omitted, "interpretations of results [were] made with little or no justification", and that elementary errors were made in the application of differential equations. Among the severe flaws claimed by Brown et al. in the positivity-ratio theory and its presentation were that:
data used by Losada in several analyses do not meet basic criteria for the use of differential equations (such as the use of continuous variables that evolve smoothly and deterministically over time); differential equations used by Losada to calculate the critical positivity ratio use parameters taken directly from Lorenz's simplified, illustrative, and arbitrary models for fluid dynamics, with Losada giving no rationale for his choice of parameters; use of different arbitrary parameters would give different positivity ratios, and thus the precise values for the lower and upper critical ratios based on the arbitrary parameters, Fredrickson and Losada's 2.9013 to 11.6346 ratios, are meaningless; the butterfly-like first figure provided by Fredrickson and Losada is not a model of the data taken from their human participants, but "the results of computer simulations of the Lorenz equations, nothing more"; and based on the maths, even if precise positivity/negativity ratios could be derived, several "windows" of desirable and undesirable positivity/negativity ratios above a certain value should exist, rather than a simple range of ratios in which "flourishing" should occur. With regard to these, and especially the last, the Brown-Sokal-Friedman rebuttal argues that it is likely that Fredrickson and Losada did not fully grasp the implications of applying nonlinear dynamics to their data. Brown, Sokal, and Friedman state that one can:only marvel at the astonishing coincidence that human emotions should turn out to be governed by exactly the same set of equations that were derived in a celebrated article several decades ago as a deliberately simplified model of convection in fluids, and whose solutions happen to have visually appealing properties. An alternative explanation – and, frankly, the one that appears most plausible to us – is that the entire process of "derivation" of the Lorenz equations has been contrived to demonstrate an imagined fit between some rather limited empirical data and the scientifically impressive world of nonlinear dynamics. They "urge future researchers to exercise caution in the use of advanced mathematical tools, such as nonlinear dynamics".
== Responses to the Brown-Sokal-Friedman rebuttal ==
=== Fredrickson and Am. Psychol. retraction === Fredrickson responded to the critique by agreeing that Losada's mathematical modelling was "questionable" and did not show that there are precise values of the ratio, but also arguing that the evidence for the benefits of a high positivity/negativity ratio is solid. Fredrickson noted that Losada declined to respond to the criticism. The American Psychologist proceeded to formally retract as invalid the mathematical modeling elements of Fredrickson and Losada's paper, including the specific critical positivity ratios of 2.9013 and its upper limit.
=== Other respondents ===
In a follow-up to the 2013 papers—the Brown-Sokal-Friedman rebuttal, and the Fredrickson response—American Psychologist published further scholarly responses, mostly supportive, but some critical of at least some aspects of the rebuttal. The series of responses culminated in a further response to these from Brown, Sokal, and Friedman. C.A. Nickerson, an independent scholar formerly at the University of Colorado, Boulder, concurred with the Brown-Sokal-Friedman rebuttal conclusion of the lack of empirical evidence for a critical positivity ratio, and noted the necessity of distinguishing between within-person-across-time versus within-time-across-persons theories. Emeritus professor Raimo Hämäläinen and colleagues responded, passing over the Brown-Sokal-Friedman rebuttal claims of failed criteria for use of differential equations in modeling, instead arguing that there were no fundamental errors in the mathematics itself, only problems related to the model's justification and interpretation.