Critical positivity ratio
The critical positivity ratio (also known as the Losada ratio or the Losada line) 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 Losada's paper was cited nearly 1,000 times, and Fredrickson wrote a popular book expounding the concept of "the 3-to-1 ratio that will change your life". Fredrickson 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."
However, the critical positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who felt that the paper's mathematical claims underlying the critical positivity ratio were fundamentally flawed. Brown collaborated with physicist Alan Sokal and psychologist Harris Friedman on a re-analysis of the paper's data. They found that Fredrickson and Losada's paper contained "numerous fundamental conceptual and mathematical errors", as did Losada's earlier work on positive psychology, which completely invalidated their claims. 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.
In response, American Psychologist formally retracted the mathematical modeling elements of Fredrickson & Losada's paper, including the specific critical positivity ratio of 2.9013, as invalid. The fundamental nature of the mathematical errors, which 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."
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 cut-off points for the maximum and minimum positivity ratios within which one should observe qualitative changes in an individual's level of flourishing.
Losada's article was critiqued by Andrés Navas in a Note to the French website of the CNRS "Images des Mathématiques".  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 Losada's conclusions in previous papers using modelling from fluid dynamics, and those in his paper co-authored 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 described by Brown et al. in the positivity-ratio theory and its presentation were:
- 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).
- That the differential equations used by Losada to calculate the critical positivity ratio use parameters taken directly from Lorenz's 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 butterfly-like 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 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.
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 Losada's 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.
- Anthony, Andrew (18 January 2014). "The British amateur who debunked the mathematics of happiness". The Guardian.
- Fredrickson BL, Losada MF (2005). "Positive affect and the complex dynamics of human flourishing.". Am Psychol 60 (7): 678–86. doi:10.1037/0003-066X.60.7.678. PMC 3126111. PMID 16221001.
- Wilkinson, Will (16 August 2013). "Barbara Fredrickson’s Bestselling 'Positivity' Is Trashed by a New Study". Daily Beast.
- Fredrickson, Barbara (2009). Positivity: Top-notch research reveals the 3-to-1 ratio that will change your life. Harmony. ISBN 9780307393746.
- Bartlett, Tom (5 August 2013). "The Magic Ratio That Wasn't". Chronicle of Higher Education.
- Brown NJ, Sokal AD, Friedman HL (2013). "The complex dynamics of wishful thinking: the critical positivity ratio.". Am Psychol 68 (9): 801–13. doi:10.1037/a0032850. PMID 23855896.
- Fredrickson BL (2013). "Updated thinking on positivity ratios.". Am Psychol 68 (9): 814–22. doi:10.1037/a0033584. PMID 23855895.
- "Correction to Fredrickson and Losada (2005)". American Psychologist 68 (9): 822. 2013. doi:10.1037/a0034435.
- Cossins, Dan (7 August 2013). "'Positivity Ratio' Debunked". The Scientist.
- Fredrickson, B. L. (2004). The broaden-and-build theory of positive emotions. Philosophical Transactions of the Royal Society of London, Series B: Biological Sciences, 359, 1367-1377.
- Navas, A. (2011). Un cas d'inconscience (?). Images des Mathématiques. )