Critical positivity ratio
The critical positivity ratio is a largely discredited concept in positive psychology positing an exact ratio of positive to negative emotions which distinguishes "flourishing" people from "languishing" people. The ratio was proposed by psychologists Barbara Fredrickson and Marcial Losada, who believed that they had identified an experimental measure of affect whose model-derived positive-to-negative ratio of 2.9013 defined a critical separation between flourishing and languishing individuals, as reported in their 2005 paper in American Psychologist. This concept of a critical positivity ratio was widely embraced by academic psychologists and the lay public; Fredrickson and Losada's paper had been cited more than 320 times by January 2014, and Fredrickson wrote a popular book expounding the concept of "the 3-to-1 ratio that will change your life". In it she wrote, "just as zero degrees Celsius is a special number in thermodynamics, the 3-to-1 positivity ratio may well be a magic number in human psychology."
The first consequential re-evaluation of the mathematical modeling behind the critical positivity ratio was published in 2008 by a group of Finnish researchers from the Systems Analysis Laboratory at Aalto University. The authors noted that "only very limited explanations are given about the modeling process and the meaning and interpretation of its parameters... 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... the predictive validity of the model also becomes problematic." Losada's 1999 modeling article was also critiqued by Andrés Navas in a French language publication, a note in the CNRS publication, Images des Mathématiques. Neither of these articles received broad attention at the times of their publication.
Later, but of critical importance, 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 underlying the critical positivity ratio were suspect. Brown contacted and ultimately collaborated with physics and maths professor Alan Sokal and psychology professor Harris Friedman on a re-analysis of the paper's data. They argued that Losada's earlier work on positive psychology and Fredrickson and Losada's 2005 critical positivity ratio paper contained "numerous fundamental conceptual and mathematical errors", errors of a magnitude that completely invalidated their claims.
Fredrickson wrote a response in which she conceded that the mathematical aspects of the critical positivity ratio were "questionable" and that she had "neither the expertise nor the insight" to defend them, but she maintained that the empirical evidence for the existence of a critical positivity ratio was solid. Brown, Sokal, and Friedman, the rebuttal authors, published their response to Fredrickson's "Update" the next year, maintaining that there was no evidence for a critical positivity ratio. Losada declined to respond to the criticism. Hämäläinen and colleagues responded later, passing over the Brown-Sokal-Friedman rebuttal claim 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.
A formal retraction for the mathematical modeling elements of the Losada and Fredrickson paper was issued by the journal, American Psychologist, concluding that both the specific critical positivity ratio of 2.9013 and its upper limit were invalid. The fact that the problems with the paper went unnoticed for years despite the widespread adulatory publicity surrounding the critical positivity ratio concept contributed to a perception of social psychology as a field lacking scientific soundness and rigorous critical thinking. Sokal later stated, "The main claim made by Fredrickson and Losada is so implausible on its face that some red flags ought to have been raised", as would only happen broadly in graduate student Brown's initiating the collaboration that resulted in the Brown-Sokal-Friedman rebuttal.
Background
Building on research by Barbara Fredrickson suggesting that individuals with a higher ratio of positive to negative emotions tend to have more successful life outcomes, and on studies by Marcial Losada applying differential equations from fluid dynamics to human emotions, Fredrickson and Losada proposed as informative a ratio of positive to negative affect derived from nonlinear dynamics modelling, which appeared in 2005 in a paper in American Psychologist. The derived combination of expressions and default parameters led them to conclude that a critical ratio of positive to negative affect of exactly 2.9013 separated flourishing from languishing individuals, and to argue that the ideal positivity/negativity ratio lies between 2.9013 and an upper limit ratio of 11.6346. Hence, they claimed that their model predicted cut-off points for the minimum and maximum positivity ratios within which one should observe qualitative changes in an individual's level of flourishing, specifically, that those within this range of ratios would "flourish", and those outside would "languish". As of January 2014, the 2005 Fredrickson and Losada's paper had been cited more than 320 times in the psychology literature.Criticism
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 "nly very limited explanations are given about the modeling process and the meaning and interpretation of its parameters... 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... 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., Andrés Navas kindly drew our attention to his article in which a very similar critique of Losada was made.
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 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 ;
- 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, 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.
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".