The Magic Ratio That Wasn’t

The 2009 book Positivity: Top-Notch Research Reveals the 3 to 1 Ratio That Will Change Your Life, by Barbara Fredrickson, was praised by the heavyweights of psychology. Daniel Gilbert said it provided a “scientifically sound prescription for joy.” Daniel Goleman extolled its “surefire methods for transforming our lives.” Martin E.P. Seligman, often called the father of positive psychology, raved that “this book, like Barb, is the ‘real thing.’”

But the top-notchness of the research that underpins the book has been called into serious question. Even Fredrickson, a professor of psychology at the University of North Carolina at Chapel Hill, has now backed away from the ratio in the book’s subtitle, saying she didn’t really understand the mathematics behind it and had relied instead on the fact that it had been peer-reviewed.

The book grew out of a 2005 paper by Fredrickson and Marcial Losada, a Chilean psychologist and consultant, the findings of which suggest that “a set of general mathematical principles may describe the relations between positive affect and human flourishing.” The idea is that you count up the number of positive statements (“Good idea, Bob”) and negative statements (“That stinks, Phil”) made in certain situations, and arrive at a ratio. That, in itself, isn’t terribly novel, but they took it a step further, using a mathematical model derived from nonlinear dynamics, and asserted that the ratio must be above 2.9013 for human beings to flourish.

The paper has been cited 350 times, according to Thomson Reuters’s Web of Science database, and has been mentioned in prominent books, including Seligman’s Flourish, published last year. And it’s not hard to see why: The ratio appears to lend some quantitative rigor to a field sometimes criticized for its mushiness. Fredrickson and Losada had apparently stumbled on the magic number for human happiness.

Then along came Nick Brown. Brown, a graduate student in applied positive psychology at the University of East London, read the paper as part of a course. It seemed like bunk to him. Brown is not a math genius; as an undergraduate he skipped math courses because they were too hard. Still, it immediately appeared to him as if the paper was making outlandish and completely unsupported claims. “I realized that Fredrickson and Losada’s approach was always going to come up with the same number, like a stopped clock, because in that equation there is no connection between the data and the math,” he recalls. So he set about attempting to see whether the numbers added up. He found that they did not.

Next he sent an e-mail to Alan Sokal—he of the infamous 1990s hoax in Social Text—a professor of physics at New York University and a man who enjoys poking the softer sciences in their softest spots, to see if he was intrigued. Of course he was. Brown, Sokal, and Harris Friedman, a professor of psychology at the University of Florida, put together a paper, published in July in American Psychologist, dissecting the 2005 paper by Fredrickson and Losada. They concluded that it had been “based on a series of erroneous and, for the most part, completely illusory ‘applications’ of mathematics.” They titled their takedown “The Complex Dynamics of Wishful Thinking.” The paper doesn’t just assert that errors were made, though it does assert that. It argues that the authors concocted a transparently ridiculous fiction.

One passage they single out for ridicule comes from a 1999 paper by Losada, the findings of which provide the basis for the 2005 paper he wrote with Fredrickson. In that passage, Losada explains how human interaction is a lot like fluid viscosity. People, he writes, “could be characterized as being stuck in a viscous atmosphere highly resistant to flow.” The authors of “Wishful Thinking” offer their opinion of this metaphorical approach in a few sentences worth quoting in full:

One could describe a team’s interactions as “sparky” and confidently predict that their emotions would be subject to the same laws that govern the dielectric breakdown of air under the influence of an electric field. Alternatively, the interactions of a team of researchers whose journal articles are characterized by “smoke and mirrors” could be modeled using the physics of airborne particulate combustion residues, combined in some way with classical optics.

Could be, in other words, but shouldn’t be.

Fredrickson, in a response to their paper also published in American Psychologist, doesn’t argue the mathematics.

She says she has come to see “sufficient reason to question the particular mathematical framework Losada and I adopted.” She writes that she has “neither the expertise nor the insight” to defend that framework. Still, she says, much of the paper remains valid, even if the ratio must be abandoned or, as she puts it, the researchers must “unthread the now-questionable element of mathematical modeling from this braid.”

The blog Neuroskeptic, reviewing Fredrickson’s response, advised retracting the 2005 paper, calling it “complete rubbish.”

Fredrickson points to the failure of the peer-reviewed journals that published Losada in the first place, writing in an e-mail that “what this process shows yet again is that peer review is not fail-safe. It is the best we have, though.” Likewise, Seligman, who cites the ratio multiple times in Flourish and in his presentations on achieving happiness, says that he relied on those peer reviewers and that his understanding of the ratio was “very superficial.”

Sokal, however, argues that the mathematics in the ratio is not that complicated and that basing a theory or a book on an idea just because it’s published in a journal is unwise. “I would say that it’s rather naïve for a researcher in psychology to assume that anything published in an applied-mathematics journal is automatically completely correct—much less that it can be applied in psychology as a ‘black box’ without even bothering to understand what is inside it,” he writes in an e-mail.

The researcher who came up with the ratio to begin with, Marcial Losada, has so far been silent on the controversy. Even Fredrickson has had trouble getting in touch with her co-author and says now that she is “troubled by Losada’s lack of attention to this matter.”

He did respond to e-mail questions from The Chronicle. He writes that he started to read the Brown paper but “lost interest” when he discovered what he says were gross miscalculations. He also points out that his paper has been cited favorably many times and that this is the first negative appraisal. However, Losada, who is chief executive of Losada Line Consulting and has a Ph.D. in psychology from the University of Michigan, doesn’t address the serious questions raised by the paper, which is perhaps understandable considering that he never finished reading it.

American Psychologist asked him to write a response, but he declined. “I didn’t have the time to prepare a proper response,” he writes in an e-mail. “I am not an academic, I am a consultant to business and I am fully booked for the rest of 2013. The demand for training in my model has taken all my available time. My priority is not to publish, but to attend my clients properly.”

Papers turn out to be flawed all the time. But this was a widely cited paper that has remained a powerful talking point in the how-humans-flourish literature for years. And the timing of the Brown paper is not good for social psychology, which is struggling with the problem of results that can’t be replicated, with high-profile researchers—like Diederik Stapel—who turn out to be con artists. Having two big names in the field, Fredrickson and Seligman, admit that they didn’t even understand the ratio they featured in presentations and popular books doesn’t exactly inspire confidence.

Fredrickson says she has no regrets and sees what happened as a consequence of “pushing the boundaries of what the science of emotion can offer.”

Both Sokal and Brown say they are surprised that no one, before now, had taken a more skeptical look at such a revolutionary ratio. “The main claim made by Fredrickson and Losada is so implausible on its face that some red flags ought to have been raised,” Sokal writes in an e-mail. “At this point I can’t resist drawing the analogy with the reaction of the editors of Social Text to a certain strange manuscript that appeared on their desks in the fall of 1995.”