Mathematical models are useful insofar as they enable us to simplify and subsequently better understand complex processes and phenomena. Suppose, for example, that an experimental psychologist who, after admin istering a treatment to a subject, is able to measure anger levels on a 21-point rating scale (ranging from —10, indexing “no an ger,” to 10, indexing “extreme anger”) at eight discrete time intervals (f = 0 , 1 , . . . , 7) observes the pattern summarized below: Time
t = 0 f —1 t = 2 t = 3
Level o f Anger 1 0 —8 Time
6
—5
t = 4 t = 5 t = 6 t —1
Level o f Anger 4
—3
2
—1
Furthermore, suppose that other experi mental psychologists independently repli cate this experiment and find similar pat terns of damped alterations in the levels of anger of their subjects. Could we write down a mathematical model that describes the observed changes in anger over time? Indeed, the first-order difference equation Anger, = Anger0( - c ) ‘
(1)
adequately describes how a subject’s anger levels evolve over time for some constant Icl < 1, where Angert is the observed level of anger at time t and Anger0 is the initial level of anger. Instead of presenting large sets of data every time we discuss the effect of the treatment on an arbitrary subject’s anger levels, we could simply state that the process follows Equation 1. Therefore, an important property of mathematical modeling in a psychology context is its ability to summarize a com plex process (e.g., the temporal evolution of an emotion) into a single equation. Of course, one can suggest other mathematical models that capture the data presented in our example. The researcher’s goal is thus to choose the model that best represents the underlying psychological process under study. Myung (2000) argued that this is not necessarily the model that provides the best fit for a given set of data, since a highly complex model can provide a good fit with out necessarily bearing any interpretable relationship with the true process (p. 190). Other than complexity, a second criterion that governs model selection is the extent to which its assumptions and predictions conform to reality. In economics, where the use of mathematical modeling is more prevalent than in psychology, there has been a long and interesting debate on this issue with no particular consensus in the literature. On the one hand, a number of econ omists believe that mathematical models
should describe real-world phenomena. George Ainslie, for example, has recently criticized the quasi-hyperbolic delay dis count function that is used extensively by behavioral economists as the standard model of impulsiveness, noting that its popularity stems more from “a desire to preserve the tractability of classical eco nomic discount functions than from either parsimony or a need to fit experience” (Ainslie, 2012, p. 4). His critique highlights an apparent willingness on the part of so cial scientists to trade off the empirical validity of their mathematical models for the sake of achieving sharp but often flawed predictions. Other economists, such as Ariel Rubinstein, argue that mathemati cal models in economic theory are not meant to have any predictive power. In his account of dilemmas of an economic theo rist, Rubinstein (2006) noted that “the word ‘model’ sounds more scientific than ‘fable’ or ‘fairy tale’ although I do not see much difference between them” (p. 881). Nonetheless, despite no unified con sensus on the role of mathematical models in the social sciences, I believe that psy chologists should not necessarily shy away from incorporating these models in their toolkit because they offer a powerful method for capturing the underlying behav ioral and cognitive processes that they study. However, as Brown et al. (2013, p. 801) cautioned, they additionally should verify that the primary conditions for thenvalid application have been met. REFERENCES
Ainslie, G. (2012). Pure hyperbolic discount curves predict “eyes open” self-control. The ory and Decision, 73, 3-34. doi:10.1007/ si 1238-011-9272-5 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. (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 Myung, I. J. (2000). The importance of complex ity in model selection. Journal o f Mathemat ical Psychology, 44, 190-204. doi:10.1006/ jmps. 1999.1283 Rubinstein, A. (2006). Dilemmas of an eco nomic theorist. Econometrica, 74, 865- 883. doi:10.11ll/j.l468-0262.2006.00689.x
September 2014 • American Psychologist
The author acknowledges financial support from the Competence Development Fund of Southern Norway through a grant to the Faculty of Eco nomics and Social Sciences, University of Agder. Correspondence concerning this comment should be addressed to Andrew Musau, Faculty of Economics and Social Sciences, University of Agder, Gimlemoen 51, 4604 Kristiansand, Nor way. E-mail: [email protected] http://dx.ctoi.org/10.1037/a0037048
M a th e m a tic a l M o d e lin g Is M o re Than Fitting Equations Raimo P. Hamalainen, Jukka Luoma, and Esa Saarinen Aalto University The article by Brown, Sokal, and Friedman (December 2013) regarding the Fredrick son and Losada (2005) article discussed the use of differential equations in science and repeated our earlier observation (Luoma, Hamalainen, & Saarinen, 2008) that there is lack of justification for the use of the Lorenz equations in the latter article. In this comment we want to point out that Brown et al. presented a very narrow view on mathematical modeling in behavioral re search. We describe how the conceptual use of mathematical models is essential in many fields. Fredrickson and Losada (2005) used mathematical modeling to suggest a hy pothesis that at least three positive emo tional experiences to every negative emo tional experience is a bifurcation point separating people who flourish from those who languish. The hypothesis was based on an analysis of Losada’s (1999) model of team interaction, which in turn was based on the famous Lorenz (1963) equations. We (Luoma et al., 2008) had previously pointed out that the “reasoning behind the model equations” (p. 760) and “the predic tive validity of the model” (p. 757) is ques tionable. Brown et al. (2013) addressed these problems again and claimed that the Fredrickson and Losada (2005) article “contains numerous fundamental concep tual and mathematical errors” (p. 801). This is an overstatement, as there are no clear mathematical errors, but the problems lie in the justification of the use of the model in this context, as we had already shown before. Brown et al. based their ar ticle on general skepticism toward mathe matical modeling in the behavioral sci ences. Their set of requirements for using mathematical modeling, in general, and differential equations, in particular, is too restrictive. The view that models should be used as in physics by fitting equations to
633
empirical data is too narrow. In the behav ioral sciences, phenomena are enormously complex. In this context, very few prob lems can be described by differential equa tions so that these could be fitted to empir ical data successfully. However, we believe that these challenges are a rationale to em brace mathematical modeling, not to dis card it. We argue that mathematical model ing constitutes a useful tool for the behav ioral sciences in a more general way. Math ematical modeling constitutes a tool and a language to help in the process of scientific reasoning. Modeling should be considered not only as finding equations to describe the reality but as a process of giving struc ture to theoretical knowledge and empirical observations. For example, Losada (1999) had strong empirical data about the ratio of positivity and negativity in teams, and thus it was quite natural to try to find a simple mathematical model which could give in sight into the possible dynamics of the phe nomenon. There are many examples where mathematical models have helped generate useful insight and even create entirely new fields of research despite the fact that these models have not been fitted to empirical observations. For example, models may be used in a metaphorical way, to ease under standing of a complex phenomenon. Smith and Thelen (2003) used the mathematical theory of nonlinear dynamic systems to understand questions concerning how the human mind emerges during the early years of a child’s life. Very simple mathe matical models have been used effectively to aid reasoning about the complex system of evolution. Hamilton (1964) introduced an equation to help understand when genes that promote altruistic behaviors are likely to spread in a population. This “Hamilton’s rule” has been very influential in evolution ary theory. Game-theoretical models have been used extensively to explain individual and group behavior in competitive and co operative settings. These models have been influential even though they have not been fitted to empirical observations. Modem economic theory is based on mathematical models with very strong assumptions about human rationality. These assumptions have been the basis of economics despite the known fact that the assumptions are not in agreement with the actual behavior of peo ple. We argue that researchers should increase, not decrease, the use of models but be sensitive to the way in which they are used. When models are used to facil itate scientific reasoning, the most impor tant issue is not the adequacy of fit with
634
empirical data, but whether the model fits the purpose of the model at hand (see Sterman 2002). Models of psychological phenomena can be contributive even if they contain problematic assumptions. Likewise, we can use mathematical mod els of social phenomena even if their psychological assumptions are controver sial. In a recent article we have shown how psychological phenomena related to communicating about models can lead to unjustified conclusions (Hamalainen, Luoma & Saarinen 2013). The history of science has shown that even geniuses like Einstein have published results with mis takes (Ohanian, 2008). These are typi cally fixed later by the authors them selves or the scientific community, usually by presenting new constructive ideas without playing down the profes sional competence of the original au thors. R EFER ENC ES
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(9), 801-813. doi: 10.1037/a0032850 Fredrickson, B. L., & Losada, M. F. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60(1), 678-686. doi:10.1037/0003-066X.60.7 .678 Hamalainen, R. P., Luoma, J., & Saarinen, E. (2013). On the importance of behavioral op erational research: The case of understanding and communicating about dynamic systems. European Journal o f Operational Research, 228(3), 623-634. doi:10.1016/j.ejor.2013.02 .001
Hamilton, W. D. (1964). The genetical evolu tion of social behaviour. Journal o f Theo retical Biology, 7(1), 1-16. doi: 10.1016/ 0022-5193(64)90038-4 Lorenz, E. N. (1963). Deterministic nonperi odic flow. Journal o f the Atmospheric Sci ences, 20(2), 130-141. doi:10.1175/15200469(1963)0202.0.CO;2 Losada, M. (1999). The complex dynamics of high performance teams. Mathematical and Computer Modelling, 50(9-10), 179-192. doi: 10.1016/S0895-7177(99)00189-2 Luoma, J., Hamalainen, R. P., & Saarinen, E. (2008). Perspectives on team dynamics: Meta learning and systems intelligence. Systems Re search and Behavioral Science, 25(6), 757767. doi: 10.1002/sres.905 Ohanian, H. C. (2008). Einstein’s mistakes: The human failings of genius. New York, NY: W. W. Norton. Smith, L. B., & Thelen, E. (2003). Development as a dynamic system. Trends in Cognitive Sciences, 7(8), 343—348. doi: 10.1016/S13646613(03)00156-6 Sterman, J. D. (2002). All models are wrong: Reflections on becoming a systems scientist.
System Dynamics Review, 18(4), 501-531. doi: 10.1002/sdr.261
Correspondence concerning this comment should be addressed to Raimo P. Hamalainen, Depart ment of Mathematics and Systems Analysis, School of Science, Aalto University, P.O. Box 11100, 00076 Aalto, Finland. E-mail: raimo [email protected] http://dx.doi.org/10.1037/a0036949
A n Em pirical R atio in Search o f a T h eo ry Vladimir A. Lefebvre University of California at Irvine Robert M. Schwartz University of Pittsburgh Medical Center An interesting problem appears in positive psychology. Two related experiments with a large number of subjects found an empir ical ratio of positive to negative emotional balance consistent with previous research. In the first study by Losada (1999), trained coders rated as positive or negative the utterances during a team meeting for each member of “flourishing” and “languishing” business teams. In the second experiment, college students described their psycholog ical state daily for 28 days by endorsing positive or negative adjectives (Fredrick son & Losada, 2005). The results of these two experiments are connected as follows: For flourishing subjects the ratio of positive to negative emotion is greater than 3, and for languishing subjects this ratio is less than 3. These results have been supported by considerable previous research over several decades (Fredrickson, 2013). A natural question arises: Why does “3” play such a role? Fredrickson and Losada (2005), who reported this phenom enon, constructed a mathematical model based on the theory of nonlinear dynamic systems, which generated the constant close to number 3, specifically 2.9. This model caused a compelling objection by Brown, Sokal, and Friedman (December 2013) that led to the model’s being re tracted by Fredrickson (December 2013). Thus, the reason for 3:1 being an important ratio has not been found, as yet. We have, in effect, a verified empirical ratio in search of a theory. In this comment, we show how to account for the phenomenon of the 3:1 ratio using a mathematical model of reflex ive awareness (Lefebvre, 1977, 1992; Schwartz, 1997) that captures the property of the mind to model self and other at increasing levels of awareness. The model
September 2014 • American Psychologist
Copyright of American Psychologist is the property of American Psychological Association and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
t = 0 f —1 t = 2 t = 3
Level o f Anger 1 0 —8 Time
6
—5
t = 4 t = 5 t = 6 t —1
Level o f Anger 4
—3
2
—1
Furthermore, suppose that other experi mental psychologists independently repli cate this experiment and find similar pat terns of damped alterations in the levels of anger of their subjects. Could we write down a mathematical model that describes the observed changes in anger over time? Indeed, the first-order difference equation Anger, = Anger0( - c ) ‘
(1)
adequately describes how a subject’s anger levels evolve over time for some constant Icl < 1, where Angert is the observed level of anger at time t and Anger0 is the initial level of anger. Instead of presenting large sets of data every time we discuss the effect of the treatment on an arbitrary subject’s anger levels, we could simply state that the process follows Equation 1. Therefore, an important property of mathematical modeling in a psychology context is its ability to summarize a com plex process (e.g., the temporal evolution of an emotion) into a single equation. Of course, one can suggest other mathematical models that capture the data presented in our example. The researcher’s goal is thus to choose the model that best represents the underlying psychological process under study. Myung (2000) argued that this is not necessarily the model that provides the best fit for a given set of data, since a highly complex model can provide a good fit with out necessarily bearing any interpretable relationship with the true process (p. 190). Other than complexity, a second criterion that governs model selection is the extent to which its assumptions and predictions conform to reality. In economics, where the use of mathematical modeling is more prevalent than in psychology, there has been a long and interesting debate on this issue with no particular consensus in the literature. On the one hand, a number of econ omists believe that mathematical models
should describe real-world phenomena. George Ainslie, for example, has recently criticized the quasi-hyperbolic delay dis count function that is used extensively by behavioral economists as the standard model of impulsiveness, noting that its popularity stems more from “a desire to preserve the tractability of classical eco nomic discount functions than from either parsimony or a need to fit experience” (Ainslie, 2012, p. 4). His critique highlights an apparent willingness on the part of so cial scientists to trade off the empirical validity of their mathematical models for the sake of achieving sharp but often flawed predictions. Other economists, such as Ariel Rubinstein, argue that mathemati cal models in economic theory are not meant to have any predictive power. In his account of dilemmas of an economic theo rist, Rubinstein (2006) noted that “the word ‘model’ sounds more scientific than ‘fable’ or ‘fairy tale’ although I do not see much difference between them” (p. 881). Nonetheless, despite no unified con sensus on the role of mathematical models in the social sciences, I believe that psy chologists should not necessarily shy away from incorporating these models in their toolkit because they offer a powerful method for capturing the underlying behav ioral and cognitive processes that they study. However, as Brown et al. (2013, p. 801) cautioned, they additionally should verify that the primary conditions for thenvalid application have been met. REFERENCES
Ainslie, G. (2012). Pure hyperbolic discount curves predict “eyes open” self-control. The ory and Decision, 73, 3-34. doi:10.1007/ si 1238-011-9272-5 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. (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 Myung, I. J. (2000). The importance of complex ity in model selection. Journal o f Mathemat ical Psychology, 44, 190-204. doi:10.1006/ jmps. 1999.1283 Rubinstein, A. (2006). Dilemmas of an eco nomic theorist. Econometrica, 74, 865- 883. doi:10.11ll/j.l468-0262.2006.00689.x
September 2014 • American Psychologist
The author acknowledges financial support from the Competence Development Fund of Southern Norway through a grant to the Faculty of Eco nomics and Social Sciences, University of Agder. Correspondence concerning this comment should be addressed to Andrew Musau, Faculty of Economics and Social Sciences, University of Agder, Gimlemoen 51, 4604 Kristiansand, Nor way. E-mail: [email protected] http://dx.ctoi.org/10.1037/a0037048
M a th e m a tic a l M o d e lin g Is M o re Than Fitting Equations Raimo P. Hamalainen, Jukka Luoma, and Esa Saarinen Aalto University The article by Brown, Sokal, and Friedman (December 2013) regarding the Fredrick son and Losada (2005) article discussed the use of differential equations in science and repeated our earlier observation (Luoma, Hamalainen, & Saarinen, 2008) that there is lack of justification for the use of the Lorenz equations in the latter article. In this comment we want to point out that Brown et al. presented a very narrow view on mathematical modeling in behavioral re search. We describe how the conceptual use of mathematical models is essential in many fields. Fredrickson and Losada (2005) used mathematical modeling to suggest a hy pothesis that at least three positive emo tional experiences to every negative emo tional experience is a bifurcation point separating people who flourish from those who languish. The hypothesis was based on an analysis of Losada’s (1999) model of team interaction, which in turn was based on the famous Lorenz (1963) equations. We (Luoma et al., 2008) had previously pointed out that the “reasoning behind the model equations” (p. 760) and “the predic tive validity of the model” (p. 757) is ques tionable. Brown et al. (2013) addressed these problems again and claimed that the Fredrickson and Losada (2005) article “contains numerous fundamental concep tual and mathematical errors” (p. 801). This is an overstatement, as there are no clear mathematical errors, but the problems lie in the justification of the use of the model in this context, as we had already shown before. Brown et al. based their ar ticle on general skepticism toward mathe matical modeling in the behavioral sci ences. Their set of requirements for using mathematical modeling, in general, and differential equations, in particular, is too restrictive. The view that models should be used as in physics by fitting equations to
633
empirical data is too narrow. In the behav ioral sciences, phenomena are enormously complex. In this context, very few prob lems can be described by differential equa tions so that these could be fitted to empir ical data successfully. However, we believe that these challenges are a rationale to em brace mathematical modeling, not to dis card it. We argue that mathematical model ing constitutes a useful tool for the behav ioral sciences in a more general way. Math ematical modeling constitutes a tool and a language to help in the process of scientific reasoning. Modeling should be considered not only as finding equations to describe the reality but as a process of giving struc ture to theoretical knowledge and empirical observations. For example, Losada (1999) had strong empirical data about the ratio of positivity and negativity in teams, and thus it was quite natural to try to find a simple mathematical model which could give in sight into the possible dynamics of the phe nomenon. There are many examples where mathematical models have helped generate useful insight and even create entirely new fields of research despite the fact that these models have not been fitted to empirical observations. For example, models may be used in a metaphorical way, to ease under standing of a complex phenomenon. Smith and Thelen (2003) used the mathematical theory of nonlinear dynamic systems to understand questions concerning how the human mind emerges during the early years of a child’s life. Very simple mathe matical models have been used effectively to aid reasoning about the complex system of evolution. Hamilton (1964) introduced an equation to help understand when genes that promote altruistic behaviors are likely to spread in a population. This “Hamilton’s rule” has been very influential in evolution ary theory. Game-theoretical models have been used extensively to explain individual and group behavior in competitive and co operative settings. These models have been influential even though they have not been fitted to empirical observations. Modem economic theory is based on mathematical models with very strong assumptions about human rationality. These assumptions have been the basis of economics despite the known fact that the assumptions are not in agreement with the actual behavior of peo ple. We argue that researchers should increase, not decrease, the use of models but be sensitive to the way in which they are used. When models are used to facil itate scientific reasoning, the most impor tant issue is not the adequacy of fit with
634
empirical data, but whether the model fits the purpose of the model at hand (see Sterman 2002). Models of psychological phenomena can be contributive even if they contain problematic assumptions. Likewise, we can use mathematical mod els of social phenomena even if their psychological assumptions are controver sial. In a recent article we have shown how psychological phenomena related to communicating about models can lead to unjustified conclusions (Hamalainen, Luoma & Saarinen 2013). The history of science has shown that even geniuses like Einstein have published results with mis takes (Ohanian, 2008). These are typi cally fixed later by the authors them selves or the scientific community, usually by presenting new constructive ideas without playing down the profes sional competence of the original au thors. R EFER ENC ES
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(9), 801-813. doi: 10.1037/a0032850 Fredrickson, B. L., & Losada, M. F. (2005). Positive affect and the complex dynamics of human flourishing. American Psychologist, 60(1), 678-686. doi:10.1037/0003-066X.60.7 .678 Hamalainen, R. P., Luoma, J., & Saarinen, E. (2013). On the importance of behavioral op erational research: The case of understanding and communicating about dynamic systems. European Journal o f Operational Research, 228(3), 623-634. doi:10.1016/j.ejor.2013.02 .001
Hamilton, W. D. (1964). The genetical evolu tion of social behaviour. Journal o f Theo retical Biology, 7(1), 1-16. doi: 10.1016/ 0022-5193(64)90038-4 Lorenz, E. N. (1963). Deterministic nonperi odic flow. Journal o f the Atmospheric Sci ences, 20(2), 130-141. doi:10.1175/15200469(1963)0202.0.CO;2 Losada, M. (1999). The complex dynamics of high performance teams. Mathematical and Computer Modelling, 50(9-10), 179-192. doi: 10.1016/S0895-7177(99)00189-2 Luoma, J., Hamalainen, R. P., & Saarinen, E. (2008). Perspectives on team dynamics: Meta learning and systems intelligence. Systems Re search and Behavioral Science, 25(6), 757767. doi: 10.1002/sres.905 Ohanian, H. C. (2008). Einstein’s mistakes: The human failings of genius. New York, NY: W. W. Norton. Smith, L. B., & Thelen, E. (2003). Development as a dynamic system. Trends in Cognitive Sciences, 7(8), 343—348. doi: 10.1016/S13646613(03)00156-6 Sterman, J. D. (2002). All models are wrong: Reflections on becoming a systems scientist.
System Dynamics Review, 18(4), 501-531. doi: 10.1002/sdr.261
Correspondence concerning this comment should be addressed to Raimo P. Hamalainen, Depart ment of Mathematics and Systems Analysis, School of Science, Aalto University, P.O. Box 11100, 00076 Aalto, Finland. E-mail: raimo [email protected] http://dx.doi.org/10.1037/a0036949
A n Em pirical R atio in Search o f a T h eo ry Vladimir A. Lefebvre University of California at Irvine Robert M. Schwartz University of Pittsburgh Medical Center An interesting problem appears in positive psychology. Two related experiments with a large number of subjects found an empir ical ratio of positive to negative emotional balance consistent with previous research. In the first study by Losada (1999), trained coders rated as positive or negative the utterances during a team meeting for each member of “flourishing” and “languishing” business teams. In the second experiment, college students described their psycholog ical state daily for 28 days by endorsing positive or negative adjectives (Fredrick son & Losada, 2005). The results of these two experiments are connected as follows: For flourishing subjects the ratio of positive to negative emotion is greater than 3, and for languishing subjects this ratio is less than 3. These results have been supported by considerable previous research over several decades (Fredrickson, 2013). A natural question arises: Why does “3” play such a role? Fredrickson and Losada (2005), who reported this phenom enon, constructed a mathematical model based on the theory of nonlinear dynamic systems, which generated the constant close to number 3, specifically 2.9. This model caused a compelling objection by Brown, Sokal, and Friedman (December 2013) that led to the model’s being re tracted by Fredrickson (December 2013). Thus, the reason for 3:1 being an important ratio has not been found, as yet. We have, in effect, a verified empirical ratio in search of a theory. In this comment, we show how to account for the phenomenon of the 3:1 ratio using a mathematical model of reflex ive awareness (Lefebvre, 1977, 1992; Schwartz, 1997) that captures the property of the mind to model self and other at increasing levels of awareness. The model
September 2014 • American Psychologist
Copyright of American Psychologist is the property of American Psychological Association and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
