ⓘ 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 Marcial Losada and psychologist Barbara Fredrickson, who identified a ratio of positive to negative affect of exactly 2.9013 as separating flourishing from languishing individuals in a 2005 paper in American Psychologist. The concept of a critical positivity ratio was widely embraced by both academic psychologists and the lay public; Fredrickson and Losadas paper was cited nearly 1.000 times, and Fredrickson wrote a popular book expounding the concept of "the 3to1 ratio that will change your life". Fredrickson wrote: "Just as zero degrees Celsius is a special number in thermodynamics, the 3to1 positivity ratio may well be a magic number in human psychology."
The first critical evaluation of the mathematical modeling behind the critical positivity ratio was published by a group of Finnish researchers Luoma, Hamalainen and Saarinen in 2008. The authors noted that "Only very limited explanations are given about the modeling process and the meaning and interpretation of its parameters. Thus, the reasoning behind the model equations remains unclear to the reader." Moreover, "the model also produces strange and previously unreported behavior under certain conditions. Thus, the predictive validity of the model also becomes problematic." Later in 2013, the critical positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who felt that the papers mathematical claims underlying the critical positivity ratio were fundamentally flawed. Brown collaborated with physicist Alan Sokal and psychologist Harris Friedman on a reanalysis of the papers data. They argued that Fredrickson and Losadas paper contained "numerous fundamental conceptual and mathematical errors", as did Losadas earlier work on positive psychology, which completely invalidated their claims. In their response, Hamalainen and colleagues argued that there were no fundamental errors in the mathematics itself, but the problems relate to the interpretation and justification of the use of the model. Losada declined to respond to the criticism, indicating that he was too busy running his consulting business. 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 was solid. Brown and colleagues, whose response was published the next year, maintain that there is no evidence for the critical positivity ratio.
In response, American Psychologist formally retracted the mathematical modeling elements of Fredrickson & Losadas paper, including the specific critical positivity ratio of 2.9013, as invalid. The problems with the paper went unnoticed for years despite the widespread publicity surrounding the critical positivity ratio, contributed to a perception that social psychology as a field lacked 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."
1. Concept
Building on research by Barbara Fredrickson indicating 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 used nonlinear dynamics modelling based on Lorenz systems to argue that the ideal positivity/negativity ratio lies between 2.9013 and 11.6346. They argued that those with ratios within this range will "flourish", whereas those with values outside it will "languish". They claimed that their model predicted cutoff points for the maximum and minimum positivity ratios within which one should observe qualitative changes in an individuals level of flourishing.
2. Criticism
Losadas article was critiqued by Andres Navas in a Note to the French website of the CNRS "Images des Mathematiques". The whole theory of the critical positivity ratio was strongly critiqued by Nicholas Brown, Alan Sokal, and Harris Friedman, in a 2013 article published in American Psychologist. Brown et al. argue that Losadas conclusions in previous papers using modelling from fluid dynamics, and those in his paper coauthored with Fredrickson, are not only based on poorly reported experiments – they argue that it is difficult to draw any conclusions from some previous studies by Losada because critical details are omitted, and "interpretations of results are made with little or no justification" p. 5 – but are based on elementary errors in the use of differential equations.
Among the severe flaws claimed by Brown et al. in the positivityratio theory and its presentation were:
 That even if one takes the idea of the precise positivity/negativity ratio numbers seriously, there should exist not just a single ratio band in which "flourishing" should occur, but several "windows" of desirable and undesirable positivity/negativity ratios above a certain value. That is, it is likely that Fredrickson and Losada did not fully grasp the implications of applying nonlinear dynamics to their data.
 That the butterflylike Figure 1 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" p. 11.
 That the differential equations used by Losada to calculate the critical positivity ratio use parameters taken directly from Lorenzs simplified, illustrative, and most importantly, arbitrary models for fluid dynamics. Losada gives no rationale for his choice of parameters. Using different arbitrary parameters gives different positivity ratios, and thus the precise values for the ratio given by Fredrickson and Losada 2.9013 to 11.6346 are meaningless.
 That the data used by Losada in several analyses do not meet the basic criteria for the use of differential equations such as the use of continuous variables that evolve smoothly and deterministically over time.
Brown et al. 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. p. 8
They "urge future researchers to exercise caution in the use of advanced mathematical tools, such as nonlinear dynamics" p. 1.
Fredrickson responded to the critique by agreeing that Losadas mathematical modelling is "questionable" and does 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. Sokal was openly critical about this partial retraction, and in 2014, American Psychologist published their response to Fredriksons retraction, where they emphatically argue that there is no evidence for the critical positivity ratio whatsoever. Responding to comments on their original critique, they conclude the entire affair by lamenting that instead of replacing the "unbridled romanticism" of humanist psychology with a rigorous evidencebased psychology, as Seligman and Csikszentmihalyi promised in their founding manifesto of positive psychology, the widespread acceptance of the critical positivity ratio shows, in their view, that positive psychology has betrayed this promise: "That the sin is now romantic scientism rather than pure romanticism is not, in our view, a great advance."

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