Comment Contents
Nelson on Peace Psychology special issue ....................................................626 On Brown et al. (2013) and Fredrickson (2013) Nickerson .......................................... 626 Guastello ........................................... 628 Brown et al.......................................... 629 Musau ................................................ 632 Hamalainen et al................................. 633 Lefebvre & Schwartz ......................634 Brown et al. reply ............................. 636 http://dx.doi.org/10.1037/a0037415
Peace P sych o lo g y Should Include th e S tu d y o f P eaceful In d iv id u a ls Linden L. Nelson California Polytechnic State University The selection of topics for the special issue on peace psychology (October 2013) prob ably gave readers the impression that peace psychology should be defined as the study of conflict and peace at intergroup, societal, and global levels. The titles for each of the first five articles included the words war or intergroup conflict, and the titles for each of the other three articles included the words terrorism, group violence, or global well-being. It seems odd that the psycho logical discipline focused on peace would be more concerned with societal and global issues, normally addressed by sociology and political science, than with intraper sonal and interpersonal peacefulness. Although it is true that the literature commonly identified as “peace psychol ogy” has focused mostly on intergroup conflict, there are reasons for defining peace psychology more broadly to include intrapersonal and interpersonal peaceful ness. One reason for studying peacefulness at all levels is that peacefulness at one level affects peacefulness at other levels. The final article in the special issue (Cohrs, Christie, White, & Das, 2013) suggested that positive emotions may increase peace fulness at the intrapersonal, interpersonal, and intergroup levels. To the extent that 626
positive emotions are peaceful inner states, it could be said that intrapersonal peace increases peacefulness at the interpersonal and intergroup levels. In addition to positive emotions, there are other components of inner peace such as self-acceptance, congruence between as pects of self, and nonviolence toward self. Many studies using both diverse and com prehensive measures of inner peace have found a positive correlation between inner peace and the interpersonal peacefulness of individuals (for a list and description of such studies, see Nelson, 2014). Also, mea sures of happiness and life satisfaction cor related positively with holding values of universalism (e.g., equality, social justice, a world at peace, protecting the environ ment) in a study with large representative samples of people from 26 countries (Schwartz, 2010). Furthermore, there is considerable evidence showing that interpersonally peaceful individuals tend to hold relatively peaceful attitudes about in tergroup and international relations (Nel son, 2014). The peace psychology literature sup ports my contention that the study of both intrapersonal and interpersonal peaceful ness should be included in the definition of peace psychology even though the litera ture shows more evidence of peace psy chologists’ interest in interpersonal peace fulness than in intrapersonal peacefulness. For example, 41% of the 297 essays in the Encyclopedia o f Peace Psychology (Chris tie, 2012) address interpersonal conflict or peacefulness to at least some extent, and 12% of the entries include discussion about intrapersonal peacefulness or distress. Among the 21 books in the Peace Psychol ogy Book Series published by Springer, 14 address interpersonal peacefulness to some extent, and nine of the books in the series include discussion concerning intraper sonal peacefulness. Creating a peaceful world would seem to entail the development of peaceful individuals, and peace psychology can most effectively contribute toward that ob jective by studying and promoting peace fulness at all levels, including intraper sonal, interpersonal, and intergroup.
REFERENCES
Christie, D. J. (Ed.). (2012). Encyclopedia of peace psychology. Malden, MA: WileyBlackwell. Cohrs, J. C., Christie, D. J., White, M. P., & Das, C. (2013). Contributions of positive psychol ogy to peace: Toward global well-being and resilience. American Psychologist, 68, 590600. doi: 10.1037/a0032089 Nelson, L. L. (2014). Peacefulness as a person ality trait. In G. K. Sims, L. L. Nelson, & M. R. Puopolo (Eds.), Personal peacefulness: Psychological perspectives (pp. 7-43). New York, NY: Springer, doi: 10.1007/978-1-46149366-2_2 Schwartz, S. H. (2010). Basic values: How they motivate and inhibit prosocial behavior. In M. Mikulincer & P. R. Shaver (Eds.), Prosocial motives, emotions, and behaviors: The better angels of our nature (pp. 221-241). Washing ton, DC: American Psychological Associa tion. Correspondence concerning this comment should be addressed to Linden L. Nelson, 290 Ramona Drive, San Luis Obispo, CA 93405. E-mail: [email protected] http://dx.doi.org/10.1037/a0036961
N o Em pirical Evidence fo r Critical P o sitivity Ratios Carol A. Nickerson Champaign, Illinois Fredrickson and Losada (2005) theorized that a ratio of positive affect to negative affect (positivity ratio) of 2.9013 acts as a critical minimum or “tipping point” for well-being; a person must meet or surpass this ratio to flourish. Fredrickson and Losada (2005, pp. 683-684) tested this the ory with two samples of college students, reporting that flourishing and nonflourish ing students had mean positivity ratios of 3.2 and 2.3, and 3.4 and 2.1, respectively, in the two samples; they considered their results empirical evidence for their theory because the two means for each sample flanked the theorized critical minimum positivity ratio. Fredrickson and Losada also theorized, but did not test, that flour ishing disintegrates at a critical maximum positivity ratio of 11.6346.
Septem ber 2014 • A m erican P sychologist © 2014 American Psychological Association 0003-066X/14/$ 12.00 Vol. 69, No. 6, 6 26-637
Recently, Brown, Sokal, and Fried man (December 2013) convincingly demonstrated that the mathematical work underlying Fredrickson and Losada’s (2005) critical minimum positivity ratio was both flawed and misapplied, con cluding that their “claim to have demon strated the existence of a critical mini mum positivity ratio of 2.9013 is entirely unfounded” (p. 801). Fredrickson (De cember 2013) replied that, although the mathematical work might be flawed, sub stantial empirical evidence exists for crit ical minimum and maximum positivity ratios and, more generally, for an (un specified) nonlinear relation between the positivity ratio and well-being. In fact, neither Fredrickson and Losada’s study nor the other studies cited by Fredrickson (2013, pp. 818-819) provided empirical evidence for either tipping points or a more general nonlinear relation between the positivity ratio and well-being, for one or the other or both of two reasons. First, even if there existed a critical minimum positivity ratio of 2.9013, the fact that the two mean positivity ratios obtained in each of the two samples in Fredrickson and Losada’s (2005) study flanked it would not provide empirical evidence for the theorized tipping point between nonflourishing and flourishing. The design of this study made it impos sible to test for the existence of the tip ping point because the well-being scores were dichotomized to create two groups of respondents, those who were flourish ing and those who were not. Dichotomiz ing well-being completely obscured the (possible) existence of a tipping point or any other nonlinearity; two points deter mine a line. Therefore, the results of Fredrickson and Losada’s study and those of the studies they cited that also used two well-being groups (p. 683) were compatible with either a linear or a non linear relation between the positivity ra tio and well-being. Second, there was a mismatch be tween Fredrickson and Losada’s (2005) theory and the data they used to test it. The theory clearly described a psycho logical process that occurs within person across time. Testing a within-person across-time theory requires observations on two (or more) variables for one person over many time points; the analysis can be repeated for any number of persons. Fredrickson and Losada’s data and anal yses, however, tested a within-time across-persons theory. Testing a withintime across-persons theory involves ob servations on two (or more) variables over many persons for one time point; the
analysis can be repeated for any number of time points. Despite the common belief that the result of a within-time across-persons analysis is some sort of an “on average” equivalent of the result of a within-per son across-time analysis, there is in fact no necessary relation between the results of the two different types of analyses even if performed on the same “persons by time points” data set. Because a with in-time across-persons analysis can, and probably will, give a result different from that of a within-person across-time anal ysis, it should not be used to test a with in-person across-time theory unless it can be shown (a) that persons are essentially replicates of one another or “exchange able,” with observations that differ being due to random variation only, and (b) that the structure of the across-persons varia tion between the variables under consid eration is the same as the structure of the within-person variation, in which case the two analyses should give similar re sults. This situation is very unlikely in psychological studies involving process, development, or change. Unfortunately, all of the studies cited by Fredrickson and Losada (2005, p. 683) and Fredrickson (2013, pp. 818-819) used within-time across-persons data and analyses, and so, like Fredrickson and Losada’s study, did not provide empirical evidence for Fredrickson and Losada’s within-person across-time theory, contrary to Fredrick son’s claim. An appropriate test of one or the other or both of Fredrickson and Losada’s (2005) theorized critical positivity ratios requires that each person’s positivity ratio and well being be assessed at many time points. The analysis should begin with a visual inspection for each person of the relation between the positivity ratio and well-being over these time points; this might include creating a three-dimensional (time by positivity ratio by well-being) scatterplot. Given the strong pre dictions of the theory, visual inspection might be sufficient to determine whether critical minimum and maximum positivity ratios ex ist, and if so, whether they equal 2.9013 and 11.6346, respectively, or some other values (which might not be the same for each per son). If the data are so noisy that it is difficult to identify critical positivity ratios via visual inspection, or if a more formal analysis is needed or desired, the statistical methods de veloped for identifying “change points” in regression and time-series models should be useful. The results can then be summarized across all persons. O f course, the appropriate withinperson across-time research strategy is
September 2014 • American Psychologist
more costly, labor-intensive, and timeconsuming than that used by Fredrickson and Losada (2005). Positive affect and negative affect would need to be assessed daily for many months to enable the com putation of multiple monthly positivity ratios; well-being would need to be as sessed once at the beginning of each month. These assessments are not as ar duous as they first seem, however, be cause recent advances in technology (e.g., mobile phones, electronically acti vated voice recorders, Internet websites) make recording many observations over a long time period much easier than has been the case in the past. A more serious problem is that some respondents— per haps most— might never achieve tippingpoint values of the positivity ratios (if they exist, and whatever they might be), or never move from one state of well being to the other, or both, so that a very large sample of respondents would be needed for a long period of time to be able to collect enough data for an ade quate test of the theory, necessitating consideration of ways to maintain re spondent motivation and limit respondent attrition. Such practical problems do not justify collecting within-time across-per sons data to test a within-person acrosstime theory, however. Fredrickson (2013) herself, in reply to Brown et al. (2013), remarked that “most valuable to the maturation of this work will be longitudinal field studies and experiments that use densely re peated measures of emotions and relevant outcomes” (p. 820). This remark raises the question of why, if Fredrickson un derstands that a longitudinal (within-per son across-time) study is needed to test the theory, this was not acknowledged in the initial article (Fredrickson & Losada, 2005), in the book that highlights the research of that article (Fredrickson, 2009, Chapter 7), or in the correction to the original article (Fredrickson & Losada, 2013), the latter of which in sisted on the validity of the results of Fredrickson and Losada’s (2005) withintime across-persons study. Indeed, why wasn’t the appropriate study design used in the first place? After all, it has been pointed out many times over many years that the data and the analysis used to test a theory should correspond to that theory and that a test of a within-person theory nearly always requires within-person data and analysis. Doing research the wrong way, while delaying doing it the right way “until later,” is not acceptable after so many years of discussions of this issue.
627
REFERENCES
Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013), The complex dynamics of wishful thinking: The critical positivity ratio. Ameri can Psychologist, 68, 801-813. doi:10.1037/ a0032850 Fredrickson, B. L. (2009). Positivity: Ground breaking research reveals how to embrace the hidden strength of positive emotions, over come negativity, and thrive. New York, NY: Crown. Fredrickson, B. L. (2013). Updated thinking on positivity ratios. American Psychologist, 68, 814-822. doi: 10.1037/a0033584 Fredrickson, B. L., & Losada, M. F. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60, 678-686. doi:10.1037/0003-066X.60.7 .678 Fredrickson, B. L., & Losada, M. F. (2013). Correction to Fredrickson and Losada (2005). American Psychologist, 68, 822. doi: 10.1037/ a0034435
Correspondence concerning this comment should be addressed to Carol A. Nickerson, 203 North Lynn Street #35, Champaign, IL 61820. E-mail: [email protected] http://dx.doi.org/10. K»7/a0036980
N o n lin e a r D y n a m ic a l M o d e ls in P sych olo gy A re W id e s p re a d a n d Testable Stephen J. Guastello Marquette University Fredrickson and Losada’s (2005) claimed connection between positivity ratios in hu man interactions and the Lorenz chaotic attractor and the extensive critique thereof (Brown, Sokal, & Friedman, December 2013) lead me to offer several remarks about nonlinear dynamical systems (NDS) models in psychology that would-be re searchers should understand if the goal is to move the science forward. D ifferent M odels Share Com mon Features
Frederickson and Losada (2005) hypothe sized that a critical point in a research variable existed at which the temporal pat terns of events were qualitatively different on either side of the critical point. This hypothesis seemed to be the only link to the Lorenz attractor; there are actually many nonlinear models that could contain similar critical points. Thus the validity of a can didate model requires testing it as a whole; one cannot infer the global dynamics from a single local feature such as the presence of a critical point. Inasmuch as no tests of
628
either variety were performed with real data, Frederickson and Losada were left with no more than a weak metaphor. Chaotic movements of variables within the phase space or time series of observations comprise another hallmark of the Lorenz attractor, which were not as sessed. The problem of the uniqueness of the model deepens, however, because many chaotic systems are already known. Researchers who have suspected chaotic behavior in their data are more apt to mea sure turbulence or other metrics of deter ministic variability and associate them with other precursors and outcomes (Guastello, 2009). Literal Testing Is Possible
Although some of the foregoing ideas were captured in Brown et al.’s (2013) recom mendations for VA1-VA5 for testing dy namical models, VA1-VA5 did not go far enough to explain that it is possible to test models by using time series data; analyses of different types are possible depending on how one wants to frame the problem (Guastello & Gregson, 2011). For models similar to Equations 1 and 2 below (the equations of the Lorenz model would qual ify), data are prepared by setting a zero point and a standard calibration of scale for all variables in the model. Any error asso ciated with approximating a differential function from a difference function be comes part of [1 — R2]:
Az = p 0 + p,Z[ + p 2Szi +
(1)
z2 = 0[Bzi * exp(02zi) + 03
(2)
where z is the dependent variable (order parameter) that is measured at two or more points in time, A and B are control (inde pendent) variables, |3,- are linear regression weights, and 0,- are nonlinear regression weights. The results of the nonlinear mod els should be compared against alternative theoretical models, which are often linear but not always so. Of further interest, Equations 1 and 2 represent two different dynamical structures containing critical points. The K n o w led g e Base Is G ro w in g
Fredrickson (December 2013) wrote, Most valuable to the maturation of this work will be longitudinal field studies and experiments that use densely repeated measures of emotions and relevant outcomes alongside pioneering dynamic mathematical and statistical models . . . Al though physics, chemistry, and engineering have more experience modeling complex systems than does psychological science, human emo tions are clearly dynamic, multicomponent sys
tems that show self-sustaining upward and downward spirals sensitive to changing circum stances . . . [which] are likely to become ever more relevant to affective science and positive psychology, as they have for biology, econom ics, and public health, (p. 820)
The quoted passage does not reflect that applications of NDS are known in most major areas of psychology. Many concepts are well developed and supported by real data (Guastello, Koopmans, & Pincus, 2009). They have been progressing for more than 30 years (Guastello, 2009), in cluding the temporal dynamics of emotion (Dishion, 2012; Isenhower, Frank, & Kay, 2012) and idea generation by creative prob lem solving groups (Guastello, 1998). A theme that is coalescing here is one of blithe disregard for the extant literature in NDS methodology and applications in psychology by those who purport to con tribute something new. I encounter this ap proach to science occasionally in manu scripts I have been asked to review for various journals and in print after publica tion. Readers can become misled to ascribe more originality than warranted where it does not belong, to miss important connec tions, and to ignore the precedent sources that could further the underlying research objective. Granted, it often happens that pertinent prior publications are often splat tered about many books and journals, sometimes in places where one would not readily think to look, and their connections to the target are not always apparent. This is where reviewers can play a constructive rather than a merely judgmental role in the publication process. Leaks in the collective effort, however, can produce intellectual wreckage such as the example examined here. REFERENCES
Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013). The complex dynamics of wishful thinking: The critical positivity ratio. Ameri can Psychologist, 68, 801-813. doi:10.1037/ a0032850 Dishion, T. (2012). Relationship dynamics in development of psychopathology: Introduc tion to the special issue. Nonlinear Dynamics, Psychology, and Life Sciences, 16, 237-241. Fredrickson, B. L. (2013). Updated thinking on positivity ratios. American Psychologist, 68, 814-822. doi: 10.1037/a0033584 Fredrickson, B. L., & Losada, M. T. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60, 678-686. doi:10.1037/0003-066X.60.7 .678 Guastello, S. J. (1998). Creative problem solving groups at the edge of chaos. The Journal of Creative Behavior, 32, 38—57. doi: 10.1002/j .2162-6057.1998.tb00805.x
September 2014 • American Psychologist
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Nelson on Peace Psychology special issue ....................................................626 On Brown et al. (2013) and Fredrickson (2013) Nickerson .......................................... 626 Guastello ........................................... 628 Brown et al.......................................... 629 Musau ................................................ 632 Hamalainen et al................................. 633 Lefebvre & Schwartz ......................634 Brown et al. reply ............................. 636 http://dx.doi.org/10.1037/a0037415
Peace P sych o lo g y Should Include th e S tu d y o f P eaceful In d iv id u a ls Linden L. Nelson California Polytechnic State University The selection of topics for the special issue on peace psychology (October 2013) prob ably gave readers the impression that peace psychology should be defined as the study of conflict and peace at intergroup, societal, and global levels. The titles for each of the first five articles included the words war or intergroup conflict, and the titles for each of the other three articles included the words terrorism, group violence, or global well-being. It seems odd that the psycho logical discipline focused on peace would be more concerned with societal and global issues, normally addressed by sociology and political science, than with intraper sonal and interpersonal peacefulness. Although it is true that the literature commonly identified as “peace psychol ogy” has focused mostly on intergroup conflict, there are reasons for defining peace psychology more broadly to include intrapersonal and interpersonal peaceful ness. One reason for studying peacefulness at all levels is that peacefulness at one level affects peacefulness at other levels. The final article in the special issue (Cohrs, Christie, White, & Das, 2013) suggested that positive emotions may increase peace fulness at the intrapersonal, interpersonal, and intergroup levels. To the extent that 626
positive emotions are peaceful inner states, it could be said that intrapersonal peace increases peacefulness at the interpersonal and intergroup levels. In addition to positive emotions, there are other components of inner peace such as self-acceptance, congruence between as pects of self, and nonviolence toward self. Many studies using both diverse and com prehensive measures of inner peace have found a positive correlation between inner peace and the interpersonal peacefulness of individuals (for a list and description of such studies, see Nelson, 2014). Also, mea sures of happiness and life satisfaction cor related positively with holding values of universalism (e.g., equality, social justice, a world at peace, protecting the environ ment) in a study with large representative samples of people from 26 countries (Schwartz, 2010). Furthermore, there is considerable evidence showing that interpersonally peaceful individuals tend to hold relatively peaceful attitudes about in tergroup and international relations (Nel son, 2014). The peace psychology literature sup ports my contention that the study of both intrapersonal and interpersonal peaceful ness should be included in the definition of peace psychology even though the litera ture shows more evidence of peace psy chologists’ interest in interpersonal peace fulness than in intrapersonal peacefulness. For example, 41% of the 297 essays in the Encyclopedia o f Peace Psychology (Chris tie, 2012) address interpersonal conflict or peacefulness to at least some extent, and 12% of the entries include discussion about intrapersonal peacefulness or distress. Among the 21 books in the Peace Psychol ogy Book Series published by Springer, 14 address interpersonal peacefulness to some extent, and nine of the books in the series include discussion concerning intraper sonal peacefulness. Creating a peaceful world would seem to entail the development of peaceful individuals, and peace psychology can most effectively contribute toward that ob jective by studying and promoting peace fulness at all levels, including intraper sonal, interpersonal, and intergroup.
REFERENCES
Christie, D. J. (Ed.). (2012). Encyclopedia of peace psychology. Malden, MA: WileyBlackwell. Cohrs, J. C., Christie, D. J., White, M. P., & Das, C. (2013). Contributions of positive psychol ogy to peace: Toward global well-being and resilience. American Psychologist, 68, 590600. doi: 10.1037/a0032089 Nelson, L. L. (2014). Peacefulness as a person ality trait. In G. K. Sims, L. L. Nelson, & M. R. Puopolo (Eds.), Personal peacefulness: Psychological perspectives (pp. 7-43). New York, NY: Springer, doi: 10.1007/978-1-46149366-2_2 Schwartz, S. H. (2010). Basic values: How they motivate and inhibit prosocial behavior. In M. Mikulincer & P. R. Shaver (Eds.), Prosocial motives, emotions, and behaviors: The better angels of our nature (pp. 221-241). Washing ton, DC: American Psychological Associa tion. Correspondence concerning this comment should be addressed to Linden L. Nelson, 290 Ramona Drive, San Luis Obispo, CA 93405. E-mail: [email protected] http://dx.doi.org/10.1037/a0036961
N o Em pirical Evidence fo r Critical P o sitivity Ratios Carol A. Nickerson Champaign, Illinois Fredrickson and Losada (2005) theorized that a ratio of positive affect to negative affect (positivity ratio) of 2.9013 acts as a critical minimum or “tipping point” for well-being; a person must meet or surpass this ratio to flourish. Fredrickson and Losada (2005, pp. 683-684) tested this the ory with two samples of college students, reporting that flourishing and nonflourish ing students had mean positivity ratios of 3.2 and 2.3, and 3.4 and 2.1, respectively, in the two samples; they considered their results empirical evidence for their theory because the two means for each sample flanked the theorized critical minimum positivity ratio. Fredrickson and Losada also theorized, but did not test, that flour ishing disintegrates at a critical maximum positivity ratio of 11.6346.
Septem ber 2014 • A m erican P sychologist © 2014 American Psychological Association 0003-066X/14/$ 12.00 Vol. 69, No. 6, 6 26-637
Recently, Brown, Sokal, and Fried man (December 2013) convincingly demonstrated that the mathematical work underlying Fredrickson and Losada’s (2005) critical minimum positivity ratio was both flawed and misapplied, con cluding that their “claim to have demon strated the existence of a critical mini mum positivity ratio of 2.9013 is entirely unfounded” (p. 801). Fredrickson (De cember 2013) replied that, although the mathematical work might be flawed, sub stantial empirical evidence exists for crit ical minimum and maximum positivity ratios and, more generally, for an (un specified) nonlinear relation between the positivity ratio and well-being. In fact, neither Fredrickson and Losada’s study nor the other studies cited by Fredrickson (2013, pp. 818-819) provided empirical evidence for either tipping points or a more general nonlinear relation between the positivity ratio and well-being, for one or the other or both of two reasons. First, even if there existed a critical minimum positivity ratio of 2.9013, the fact that the two mean positivity ratios obtained in each of the two samples in Fredrickson and Losada’s (2005) study flanked it would not provide empirical evidence for the theorized tipping point between nonflourishing and flourishing. The design of this study made it impos sible to test for the existence of the tip ping point because the well-being scores were dichotomized to create two groups of respondents, those who were flourish ing and those who were not. Dichotomiz ing well-being completely obscured the (possible) existence of a tipping point or any other nonlinearity; two points deter mine a line. Therefore, the results of Fredrickson and Losada’s study and those of the studies they cited that also used two well-being groups (p. 683) were compatible with either a linear or a non linear relation between the positivity ra tio and well-being. Second, there was a mismatch be tween Fredrickson and Losada’s (2005) theory and the data they used to test it. The theory clearly described a psycho logical process that occurs within person across time. Testing a within-person across-time theory requires observations on two (or more) variables for one person over many time points; the analysis can be repeated for any number of persons. Fredrickson and Losada’s data and anal yses, however, tested a within-time across-persons theory. Testing a withintime across-persons theory involves ob servations on two (or more) variables over many persons for one time point; the
analysis can be repeated for any number of time points. Despite the common belief that the result of a within-time across-persons analysis is some sort of an “on average” equivalent of the result of a within-per son across-time analysis, there is in fact no necessary relation between the results of the two different types of analyses even if performed on the same “persons by time points” data set. Because a with in-time across-persons analysis can, and probably will, give a result different from that of a within-person across-time anal ysis, it should not be used to test a with in-person across-time theory unless it can be shown (a) that persons are essentially replicates of one another or “exchange able,” with observations that differ being due to random variation only, and (b) that the structure of the across-persons varia tion between the variables under consid eration is the same as the structure of the within-person variation, in which case the two analyses should give similar re sults. This situation is very unlikely in psychological studies involving process, development, or change. Unfortunately, all of the studies cited by Fredrickson and Losada (2005, p. 683) and Fredrickson (2013, pp. 818-819) used within-time across-persons data and analyses, and so, like Fredrickson and Losada’s study, did not provide empirical evidence for Fredrickson and Losada’s within-person across-time theory, contrary to Fredrick son’s claim. An appropriate test of one or the other or both of Fredrickson and Losada’s (2005) theorized critical positivity ratios requires that each person’s positivity ratio and well being be assessed at many time points. The analysis should begin with a visual inspection for each person of the relation between the positivity ratio and well-being over these time points; this might include creating a three-dimensional (time by positivity ratio by well-being) scatterplot. Given the strong pre dictions of the theory, visual inspection might be sufficient to determine whether critical minimum and maximum positivity ratios ex ist, and if so, whether they equal 2.9013 and 11.6346, respectively, or some other values (which might not be the same for each per son). If the data are so noisy that it is difficult to identify critical positivity ratios via visual inspection, or if a more formal analysis is needed or desired, the statistical methods de veloped for identifying “change points” in regression and time-series models should be useful. The results can then be summarized across all persons. O f course, the appropriate withinperson across-time research strategy is
September 2014 • American Psychologist
more costly, labor-intensive, and timeconsuming than that used by Fredrickson and Losada (2005). Positive affect and negative affect would need to be assessed daily for many months to enable the com putation of multiple monthly positivity ratios; well-being would need to be as sessed once at the beginning of each month. These assessments are not as ar duous as they first seem, however, be cause recent advances in technology (e.g., mobile phones, electronically acti vated voice recorders, Internet websites) make recording many observations over a long time period much easier than has been the case in the past. A more serious problem is that some respondents— per haps most— might never achieve tippingpoint values of the positivity ratios (if they exist, and whatever they might be), or never move from one state of well being to the other, or both, so that a very large sample of respondents would be needed for a long period of time to be able to collect enough data for an ade quate test of the theory, necessitating consideration of ways to maintain re spondent motivation and limit respondent attrition. Such practical problems do not justify collecting within-time across-per sons data to test a within-person acrosstime theory, however. Fredrickson (2013) herself, in reply to Brown et al. (2013), remarked that “most valuable to the maturation of this work will be longitudinal field studies and experiments that use densely re peated measures of emotions and relevant outcomes” (p. 820). This remark raises the question of why, if Fredrickson un derstands that a longitudinal (within-per son across-time) study is needed to test the theory, this was not acknowledged in the initial article (Fredrickson & Losada, 2005), in the book that highlights the research of that article (Fredrickson, 2009, Chapter 7), or in the correction to the original article (Fredrickson & Losada, 2013), the latter of which in sisted on the validity of the results of Fredrickson and Losada’s (2005) withintime across-persons study. Indeed, why wasn’t the appropriate study design used in the first place? After all, it has been pointed out many times over many years that the data and the analysis used to test a theory should correspond to that theory and that a test of a within-person theory nearly always requires within-person data and analysis. Doing research the wrong way, while delaying doing it the right way “until later,” is not acceptable after so many years of discussions of this issue.
627
REFERENCES
Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013), The complex dynamics of wishful thinking: The critical positivity ratio. Ameri can Psychologist, 68, 801-813. doi:10.1037/ a0032850 Fredrickson, B. L. (2009). Positivity: Ground breaking research reveals how to embrace the hidden strength of positive emotions, over come negativity, and thrive. New York, NY: Crown. Fredrickson, B. L. (2013). Updated thinking on positivity ratios. American Psychologist, 68, 814-822. doi: 10.1037/a0033584 Fredrickson, B. L., & Losada, M. F. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60, 678-686. doi:10.1037/0003-066X.60.7 .678 Fredrickson, B. L., & Losada, M. F. (2013). Correction to Fredrickson and Losada (2005). American Psychologist, 68, 822. doi: 10.1037/ a0034435
Correspondence concerning this comment should be addressed to Carol A. Nickerson, 203 North Lynn Street #35, Champaign, IL 61820. E-mail: [email protected] http://dx.doi.org/10. K»7/a0036980
N o n lin e a r D y n a m ic a l M o d e ls in P sych olo gy A re W id e s p re a d a n d Testable Stephen J. Guastello Marquette University Fredrickson and Losada’s (2005) claimed connection between positivity ratios in hu man interactions and the Lorenz chaotic attractor and the extensive critique thereof (Brown, Sokal, & Friedman, December 2013) lead me to offer several remarks about nonlinear dynamical systems (NDS) models in psychology that would-be re searchers should understand if the goal is to move the science forward. D ifferent M odels Share Com mon Features
Frederickson and Losada (2005) hypothe sized that a critical point in a research variable existed at which the temporal pat terns of events were qualitatively different on either side of the critical point. This hypothesis seemed to be the only link to the Lorenz attractor; there are actually many nonlinear models that could contain similar critical points. Thus the validity of a can didate model requires testing it as a whole; one cannot infer the global dynamics from a single local feature such as the presence of a critical point. Inasmuch as no tests of
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either variety were performed with real data, Frederickson and Losada were left with no more than a weak metaphor. Chaotic movements of variables within the phase space or time series of observations comprise another hallmark of the Lorenz attractor, which were not as sessed. The problem of the uniqueness of the model deepens, however, because many chaotic systems are already known. Researchers who have suspected chaotic behavior in their data are more apt to mea sure turbulence or other metrics of deter ministic variability and associate them with other precursors and outcomes (Guastello, 2009). Literal Testing Is Possible
Although some of the foregoing ideas were captured in Brown et al.’s (2013) recom mendations for VA1-VA5 for testing dy namical models, VA1-VA5 did not go far enough to explain that it is possible to test models by using time series data; analyses of different types are possible depending on how one wants to frame the problem (Guastello & Gregson, 2011). For models similar to Equations 1 and 2 below (the equations of the Lorenz model would qual ify), data are prepared by setting a zero point and a standard calibration of scale for all variables in the model. Any error asso ciated with approximating a differential function from a difference function be comes part of [1 — R2]:
Az = p 0 + p,Z[ + p 2Szi +
(1)
z2 = 0[Bzi * exp(02zi) + 03
(2)
where z is the dependent variable (order parameter) that is measured at two or more points in time, A and B are control (inde pendent) variables, |3,- are linear regression weights, and 0,- are nonlinear regression weights. The results of the nonlinear mod els should be compared against alternative theoretical models, which are often linear but not always so. Of further interest, Equations 1 and 2 represent two different dynamical structures containing critical points. The K n o w led g e Base Is G ro w in g
Fredrickson (December 2013) wrote, Most valuable to the maturation of this work will be longitudinal field studies and experiments that use densely repeated measures of emotions and relevant outcomes alongside pioneering dynamic mathematical and statistical models . . . Al though physics, chemistry, and engineering have more experience modeling complex systems than does psychological science, human emo tions are clearly dynamic, multicomponent sys
tems that show self-sustaining upward and downward spirals sensitive to changing circum stances . . . [which] are likely to become ever more relevant to affective science and positive psychology, as they have for biology, econom ics, and public health, (p. 820)
The quoted passage does not reflect that applications of NDS are known in most major areas of psychology. Many concepts are well developed and supported by real data (Guastello, Koopmans, & Pincus, 2009). They have been progressing for more than 30 years (Guastello, 2009), in cluding the temporal dynamics of emotion (Dishion, 2012; Isenhower, Frank, & Kay, 2012) and idea generation by creative prob lem solving groups (Guastello, 1998). A theme that is coalescing here is one of blithe disregard for the extant literature in NDS methodology and applications in psychology by those who purport to con tribute something new. I encounter this ap proach to science occasionally in manu scripts I have been asked to review for various journals and in print after publica tion. Readers can become misled to ascribe more originality than warranted where it does not belong, to miss important connec tions, and to ignore the precedent sources that could further the underlying research objective. Granted, it often happens that pertinent prior publications are often splat tered about many books and journals, sometimes in places where one would not readily think to look, and their connections to the target are not always apparent. This is where reviewers can play a constructive rather than a merely judgmental role in the publication process. Leaks in the collective effort, however, can produce intellectual wreckage such as the example examined here. REFERENCES
Brown, N. J. L., Sokal, A. D., & Friedman, H. L. (2013). The complex dynamics of wishful thinking: The critical positivity ratio. Ameri can Psychologist, 68, 801-813. doi:10.1037/ a0032850 Dishion, T. (2012). Relationship dynamics in development of psychopathology: Introduc tion to the special issue. Nonlinear Dynamics, Psychology, and Life Sciences, 16, 237-241. Fredrickson, B. L. (2013). Updated thinking on positivity ratios. American Psychologist, 68, 814-822. doi: 10.1037/a0033584 Fredrickson, B. L., & Losada, M. T. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60, 678-686. doi:10.1037/0003-066X.60.7 .678 Guastello, S. J. (1998). Creative problem solving groups at the edge of chaos. The Journal of Creative Behavior, 32, 38—57. doi: 10.1002/j .2162-6057.1998.tb00805.x
September 2014 • American Psychologist
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