Ratio for a good life exposed as ‘nonsense’

Highly touted measure of emotional health criticized as math disaster

FLUID HAPPINESS  Edward Lorenz was studying fluid dynamics 50 years ago when he developed a set of equations that, when graphed, trace out an oscillating pattern that is extremely sensitive to small perturbations. The equations, which became a foundation of chaos theory, were improperly applied in 2005 to research on happiness.


A 52-year-old, part-time graduate student with no previous training in psychology and little math education beyond high school has knocked a celebrated measure of the emotional mix needed to live well off its mathematical pedestal.

Nicholas Brown, who is completing a master’s degree in applied positive psychology at the University of East London in England, teamed up with two colleagues to demolish the math at the heart of a widely cited October 2005 American Psychologist paper that claimed to identify the precise ratio of positive to negative emotions that enables life success. The researchers’ takedown of what’s known as the critical positivity ratio appears July 15 in American Psychologist.

“It’s slightly worrying to discover that a leading journal could publish an article with so many obvious errors in it,” Brown says.

His report joins a movement in psychology to clean up research practices (SN: 6/1/13, p. 26).

One of Brown’s coauthors is physicist Alan Sokal of New York University. Sokal gained notoriety in 1996 by publishing an intentionally nonsensical paper in a leading journal of cultural studies.

Psychologist Barbara Fredrickson of the University of North Carolina at Chapel Hill and psychologist Marcial Losada, head of Losada Line Consulting in Brasilia, Brazil, coauthored the 2005 paper. In a response published alongside the new critique, psychologist Fredrickson acknowledges that their paper employed “now questionable mathematics.” But she devotes most of her response to shoring up the argument of the 2005 paper by describing evidence that people do best when positive feelings exceed negative feelings by a factor of about 3 to 1 — roughly equivalent to the contested critical positivity ratio.

Responding to a request for comment, Fredrickson told Science News that she’s “not speculating any further on these issues.” Losada, whose firm uses the ratio when advising companies on improving employee productivity, declined to write a response to Brown’s paper.

Brown first read Fredrickson and Losada’s paper in November 2011 as a class assignment. The study used mathematical equations known as Lorenz equations to calculate how positive and negative human emotions change over time. In 1963, the mathematician Edward Lorenz published the equations  to model how fluids change over time. Fredrickson and Losada used the equations with emotion data from volunteers tracked for 28 days. The researchers determined that creativity, helpfulness to others and other elements of “flourishing” characterized people who displayed a ratio of positive to negative emotions above 2.9013:1 and below 11.6346:1.

People whose balance of emotions fell outside that range “languished” in an unproductive state, the two psychologists concluded. Their report emphasized the lower ratio as the critical threshold to cross in order to flourish in life.

Upon reading a 1963 paper on Lorenz equations “with some difficulty,” Brown realized that the equation Fredrickson and Losada used to calculate the critical positivity ratio had no connection to their emotion data: Regardless of the volunteers’ data points, the equation would simply generate the same, meaningless number.

Brown then asked Sokal and psychologist Harris Friedman of the University of Florida in Gainesville to analyze the 2005 paper more completely.

“What’s shocking is not just that this piece of pseudomathematical nonsense received 322 scholarly citations and 164,000 web mentions, but that no one criticized it publicly for eight years, not even supposed experts in the field,” Sokal says.

Brown and his colleagues’ sacking of the critical positivity ratio is on the mark, comments mathematician Colin Sparrow of the University of Warwick in England, who studies Lorenz equations.

The equations belong to the field of nonlinear dynamics, which describes how small changes in a few variables that evolve independently — mainly in physics and chemistry — can lead to complex consequences.

In the 2005 report, Fredrickson and Losada failed to show how individuals’ self-reported feelings could be mathematically described as quantities that vary smoothly over time, as the Lorenz equations require, Sokal says. Two earlier papers by Losada that examined emotional changes in groups suffered from the same problem, he adds.

Losada and Fredrickson also plugged into key parts of their calculation values that had been adopted by Lorenz for his fluid analysis, Sokal says. There’s no reason to assume those values also apply to emotional changes, rendering the critical positivity ratio “entirely fanciful,” he concludes.

No retraction of the 2005 paper is planned, says psychologist Norman Anderson, editor-in-chief of American Psychologist and CEO of the American Psychological Association in Washington, D.C.

Editor’s Note: This story was corrected on August 14, 2013 to remove the assertion that the journal that published Alan Sokal’s 1996 paper was peer reviewed.


People who feel good because they try to achieve noble, meaningful goals display healthy, gene-regulated levels of key immune substances. In contrast, those whose happiness stems from personal indulgences show unhealthy levels of the same immune agents.

That’s the conclusion that psychologist Barbara Fredrickson of the University of North Carolina at Chapel Hill and her colleagues reached July 29 in the Proceedings of the National Academy of Sciences. Unlike her 2005 paper on the critical positivity ratio, it took only a week for the new paper to draw fire.

Fredrickson’s group used vague measures of self- and meaning-related happiness that tap into the same mental disposition, argued psychologist James Coyne of the University of Pennsylvania in an August 5 post on his blog, Mind the Brain. Any associations of these well-being measures with immune activity “are likely to be artificial and not replicated in future studies,” Coyne wrote.

Two study groups: Fredrickson and her colleagues classified 80 happy people into two groups. People were termed “hedonic” if their good feelings generally stemmed from their own pleasant experiences, or “eudaimonic” if their happiness derived from “striving toward meaning and a noble purpose beyond self-gratification.”

The test: Then the researchers measured something called “conserved transcriptional response to adversity,” or CTRA, in both groups. This number reflects the activity of a set of 53 genes that are associated with stress.

The conclusion: People in the eudaimonic group had lower CTRA values than people in the hedonic group, as the graph above illustrates. That pattern suggests that eudaimonic happiness may be much more effective at fostering a long and healthy life, Fredrickson and her colleagues conclude.

Bruce Bower has written about the behavioral sciences for Science News since 1984. He writes about psychology, anthropology, archaeology and mental health issues.

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