1
Finding Order in the Noise:
A Conceptual Overview on the Shift to Complex Dynamical Models
in Psychology
By Gaby Lunansky, student number 10206345 Bachelor Thesis, 2016
Psychological Methods, University of Amsterdam Supervisor: Lisa Wijsen, MSc
2 Abstract
Since the emergence of the new complexity sciences in the 1970s (e.g. Zeeman, 1979; Gleick, 1987; Stewart, 1989), researchers in the psychological field have been applying complex dynamical systems theory into psychological models. Dynamical complexity models study the behaviour of a system over time and bring new possibilities for modelling qualitative changes and complex interactions. Dynamical complexity models thus bring a new perspective on the possibilities and functions of models in psychology. However, many suggestions for new complexity models in psychology have failed to build formal, testable complexity models and have only presented verbal representations of possible complex behaviour in psychological attributes (Wagenmakers, van der Maas & Farrell, 2012). In order to build new complexity models in psychology, the fundamental differences between linear regression models and complexity models have to be clear. This thesis focuses on the shift in the function of psychological models from traditional, linear models to complex dynamical models. I give a conceptual overview of the main differences between traditional, linear models and new complex dynamical models in psychology. The most important shift in the function of complex dynamical models is that the focus is on the system’s process over time, not on the underlying, linear relationship between variables at a certain point of time. This has to do with three important aspects of complex dynamical models that will be discussed: the concept of irreversible time, the concept of equilibrium and the concept of qualitative change. Finally, it is argued that complex dynamical systems also have an influence on the discussion on inter-individual variance versus intra-inter-individual variance, as both kinds of variances should be taken into account for many complex dynamical models in psychology.
3 Introduction
The psychological field applied mostly linear regression models since its entrance into formal science. The main focus has been on finding underlying linear relationships between variables and behaviour structures. This excluded the possibility to do research on changing processes, where the underlying structures change over time and are not always linear. Even though there was an interest in the process of change in psychology, there was not an accepted methodology to do research of this kind. As Cronbach and Furby stated in their essay “How we should measure" change": Or should we?” (1970): “It appears that investigators who ask questions regarding gain scores [change] would ordinarily be better advised to frame their questions in other ways.” (p 80).
However, in the physical field, there was an emergence of new complexity sciences in the 1970’s (e.g. Zeeman, 1979; Gleick, 1987; Stewart, 1989). These new complexity sciences implied a shift of focus from traditional linear systems following strict laws to complex dynamical systems. As these new complexity sciences brought new ways of modelling and methods for doing research, psychological researchers have applied complex dynamical systems theory into their own models for the past decades. Psychological attributes are being modelled as dynamical systems in various psychological fields, e.g. reaction time
(Wagenmakers, Farrell & Ratcliff, 2004; Wagenmakers, van der Maas & Farrell, 2011), psychopathological disorders (Cramer, Waldorp, van der Maas & Borsboom, 2010; Kendler, Zachar & Craver, 2010; Kendler, 2012; Borsboom & Cramer, 2013), personality (Cramer et al., 2012; Costantini et al., 2015) and developmental psychology (van der Maas & Molenaar, 1992; van der Maas & Molenaar, 1996; Keller, 2005). Research on dynamical systems in psychology is growing and developing advanced methods to build formal psychological complexity models (e.g. Grasman, van der Maas & Wagenmakers, 2009; van Borkulo et al., 2014). As a result, there is an increasing focus on interdisciplinary research programs, combining insights from biology, neuroscience, physics and mathematics with psychology. Although fully accepted and developed dynamical complexity models are yet absent in
psychology, the enthusiasm for complexity is growing. This can be shown by for example the special Complexity programme by the Netherlands Organisation for Scientific Research (NWO) which funded seven million euros to academic research on complexity in 20091.
To understand the enthusiasm for complex dynamical models in psychology one has to grasp what new possibilities they bring for doing psychological research. Dynamical systems
4 theory focuses on the behaviour of systems and their change over time. It is primarily a
mathematical and physicist field, describing the system in several mathematical functions and calculating its behaviour with different input values2 (Gleick, 1983). Complex dynamical systems have some specific characteristics making them fundamentally different from
traditional linear models. An important aspect is the different perception on time. Traditional linear psychological models assume individuals as time invariant constants and analyse the variation between people (the inter-individual variance). Time invariance means time doesn't explicitly have an influence on the observed variables, i.e. a process is time invariant when an outcome is not affected by earlier outcomes and when it does not matter at what point in time the measurement occurs (Molenaar, 2004). Molenaar (2009) already argued that most
psychological attributes – and people in general – do not meet this time invariance condition. Some examples of time variant processes in psychology are the development of the
embryonic brain (Edelman, 1987 via Molenaar, 2004), infant motor development (Thelen, Kelso & Foger, 1987), memory (Barton, 1994), perception in the visual network (Linsker, 1988) and early lexical development in children (Li, Zhao, & Mac Whinney, 2007), since there are indicators of the underlying (growth) structure of the processes changing over time. Complex dynamical models make it possible to build models that do not meet the time invariance assumption, by modelling the process of the interaction of variables over time.
Another important difference between complex dynamical models and traditional linear models is that complex dynamical systems can have phase transitions, meaning they can (suddenly) change radically in their behaviour (Stewart & Peregoy, 1983). These changes are qualitatively in nature, i.e. the system does not change in a stable and linear manner. Of course, the ability of modelling qualitative change rises many new possibilities for
psychological models (e.g. phase transitions in developmental psychology (van der Maas & Molenaar, 1992), however, it also brings difficulties; complexity systems can be
fundamentally unpredictable over a long time since the smallest change in initial values might cause a completely different development (the butterfly effect) (Gleick, 1983).
These difficulties are a real problem for building psychological complexity models. Despite the latest investigations on complex psychological behaviour, there is also critique on how psychological researchers use dynamic systems theory to model complex behaviour. It has been argued that many proposed complex models are only verbal representations of
2 In this thesis I give a small overview of only the most relevant aspects for this thesis’ subject, more
comprehensive introductions to the field are for example by Glendinning (1994), Tschacher and Scheier (1997) and Luenberger (1979).
5 possible complex behaviour in psychological attributes, instead of testable and formal
(mathematical) models (Wagenmakers, van der Maas & Farrell, 2012). In order to build new complexity models in psychology, the fundamental differences between linear regression models and complexity models have to be clear. As the research on complexity in psychology is growing, it is important to understand what this shift really entails and to grasp what new possibilities it brings for building models and theories. In this thesis, I investigate the conceptual shift in the function of psychological models from linear to complex dynamical models. I give a conceptual overview of the most fundamental differences between
traditional, linear models and new complex dynamical models in psychology. First, I discuss the linear regression model and its assumptions. To illustrate the theoretical implications of using a linear regression model, the factor model will be shortly explained. Second, I discuss complex dynamical models, their assumptions and the associated difficulties. As an example of the new possibilities brought by complex dynamical models, the network perspective of psychopathological disorders will be discussed. Finally, I argue that using complex dynamical models has new implications for the distinction of inter-individual and intra-individual
variance.
Linear regression models
Traditionally, psychological models are based on linear regression equations. Here follows a brief outline of linear regression and modelling. The basic idea is that there is a linear
regression equation that can formally represent the relationship between two or more variables. The simple linear regression function is:
𝑦𝑦𝑖𝑖 = 𝑎𝑎 + 𝛽𝛽𝛽𝛽𝑖𝑖+ 𝜀𝜀𝑖𝑖
Where there are 𝑛𝑛 datapoints of the dataset ��𝛽𝛽𝑖𝑖, 𝑦𝑦𝑖𝑖�, 𝑖𝑖 = 1, … , 𝑛𝑛�. There is a dependent or outcome variable 𝑦𝑦𝑖𝑖, an independent variable 𝛽𝛽𝑖𝑖, an intercept 𝑎𝑎 (the expected value of 𝑦𝑦𝑖𝑖 when 𝛽𝛽𝑖𝑖 = 0), a regression coefficient 𝛽𝛽 (indicating the expected increment of the dependent
variable 𝑦𝑦𝑖𝑖 per unit change of the independent variable 𝛽𝛽𝑖𝑖) and an independent error variable 𝜀𝜀𝑖𝑖. Of course, this simple regression function can be made more complex by adding more
dependent variables and/or adding interacting regression functions.
The goal of the researcher is to find the linear function that best represents the relationship of the variables of interest. This linear function is what is called the model, describing the most important variables affecting the psychological attribute and describing
6 how variables act singly and in concert (Heiby, 1995). Thus, the researcher has to find the most important independent variables that affect the dependent variable. This is normally done by controlling the effects of the independent variables through experimental
manipulation, by studying them in isolation, and seeing what effect they have on the dependent variables (Hoyert, 1992).
Linear functions satisfy the superposition principle, which states that the function meets the homogeneity and additivity property (Barton, 1994). The additivity property states that for every function 𝐹𝐹(𝛽𝛽 + 𝑦𝑦) = 𝐹𝐹(𝛽𝛽) + 𝐹𝐹(𝑦𝑦), i.e. that the common effect of two
independent variables at a given value and measurement point on a dependent variable is the same as the sum effect of the two independent variables individually (Barton, 1994). The homogeneity property states that for every function 𝐹𝐹(𝑎𝑎𝛽𝛽) = 𝑎𝑎 𝐹𝐹(𝛽𝛽), i.e. the strength of the input function is proportional to the strength of the output function (Barton, 1994).Since linear functions satisfy the superposition principle, a small change in the (initial) value of the independent variable produces a proportionally small change in the predicted dependent variable. The superposition principle makes it convenient to use linear equations for modelling: linear equations can be represented with a straight line on a graph which makes them easy to interpret, relatively easy solvable and most important, they can be taken apart and put together again (Heiby, 1995). Thus, the researcher can experimentally manipulate the individual independent variables and add their effect to the general linear equation.
Linear models can be tested by fitting them to the data. There are many different methods, but the basic idea is that the predicted values by the model have to be close to the observed values. If a large part of the variance of the dependent variables can be explained by the independent variables, the researcher knows he or she has found important variables. The deviation between the expected values and the observed values is called error. There are different types of error. Random error, or noise, is error one cannot control for. Systematic error is error by bias, i.e. some systematic error of the researcher (either in measurement or modelling). The variance of the data that is unexplained by the independent variables is called residual variance. This residual variance should be as small as possible, but without the model overfitting the data. Overfitting happens when the model parameters fit the specific dataset too well, lowering the generalizability of the model for new data.
There are some important assumptions in linear regression (Heiby, 1995). First of all, naturally, the assumption is that the data are linear. A second assumption is statistical
independence of errors, thus only random error is assumed. Additionally, homoscedasticity is assumed. This means that the error in the data has constant variance and is independent of the
7 variance of the independent variables. Finally, the error distribution is assumed to be normally distributed (Heiby, 1995). Thus, many assumptions in linear regression are linked to the conception that the error is random. However, sometimes this is not the case. There are possible transformations for when these assumptions haven’t been met, for example
transforming the data to its log function. Another possibility is that the data that is assumed to be random error is not in fact random error. This possibility will be elaborated in the non-linear paragraph.
Linear regression models in psychology: Factor analysis
To illustrate the use of linear regression models in psychology, a widely-used linear model in psychology will be briefly discussed: the factor model. The factor model is a latent variable model. A latent variable is a variable which is impossible to observe directly, but which acts as a common cause for manifest (i.e. observable) variables. Thus, latent variable models assume that the relationship between two or more manifest variables can be explained by an underlying latent variable, causing the variation in the manifest variables (Borsboom, Mellenbergh & Van Heerden, 2003). Controlling for the latent variable (i.e. assuming the existence of a latent variable causing the variation in the manifest variables, and holding this latent variable constant), the correlation between the manifest variables disappears, which is called local independence (Bollen, 2002).
In a factor model, the latent variables are called factors. The factor model tests if the factors have a causal influence on the manifest variables, by analysing the variance of the manifest variables. The more equally the variance of the manifest variables is (the
covariance), the more plausible that the manifest variables are caused by the same underlying factor (Bollen, 2002). The residual variance in a factor analysis is the variance of the manifest variable which is not caused by the factor, i.e. the unique variance of the manifest variable. The residual variances of the manifest variables are assumed to be independent of each other and should be as small as possible, representing random error fluctuations of the manifest variables. If the residual variance of an manifest variable is too high, there is not much variance caused by the factor in the model, indicating a poor fit of the model (Bollen, 2002). This fitting to data is normally done with a likelihood function: a function that calculates the most likely expected parameters from the model, and compares these expected values with the observed values in the data (Bollen, 2002). The fit of the model depends on how big the difference is between the expected values by the model and the observed values in the data.
8 intelligence, personality and psychopathological disorders like depression. It is the basis of the psychopathological diagnostic system and the DSM. The theoretical implication is that there is an assumed underlying common cause of the manifest indicator variables (Borsboom, Mellenbergh & Van Heerden, 2003). For the psychopathological diagnostic system, this means that depression is viewed as the common cause of the depression symptoms. In other words, the patient suffers from depression symptoms because he or she is depressed. The one factor model implies that the clinician should focus on this latent variable itself to cure the patient, since the depression latent variable is the common cause of the symptoms (Cramer & Borsboom, 2015). If the common cause has been treated, the symptoms should also disappear, like the symptoms of a cancer would disappear if the tumor is removed (Cramer & Borsboom, 2015).
Complex Dynamical Models
Contrary to assuming an underlying, constant linear structure like the traditional regression models, dynamical systems theory studies the way in which systems behave and change over time. As stated before, dynamical systems theory is a broad field and an extensive
introduction is beyond the scope of this thesis. Therefore, the most relevant concepts for a conceptual comparison of traditional linear models and complex dynamical models in
psychology will be discussed. The most important shift in the function of models in non-linear dynamical models is that the focus of the model is on its process over time, not with the relationship between the most important variables at one certain point in time. This has to do with three important aspects of complex dynamical models that will be discussed in this paragraph: the concept of irreversible time, the concept of equilibrium and the concept of qualitative change or phase transitions.
Complexity models take a different perspective on the influence of time than traditional, linear models. This different perspective has to do with the distinction between open and closed systems. Traditional linear models are normally modelled as closed systems, meaning they only operate internally and have no exchange with their environment. In contrast, open systems do exchange elements (e.g. energy, matter) and have feedback loops with their environment (Prigogine & Stengers, 1984). Closed systems treat time as if it were reversible. Reversible time is endless, cyclical and ongoing, without an ending point. A moment (in past or future) is assumed to be like any other moment. However, a new perspective on the relationship between time and energy was proposed from the field of thermodynamics, stating that there is ‘an inescapable loss of energy in the universe’ (Toffler,
9 1983, p 19). This statement became the Second Law of Thermodynamics in the 19th century and challenged the perspective that time is reversible. It became clear that the universe, like all other natural and biological systems, is aging and that this is not a reversible process (Prigogine & Stengers, 1984). For open systems there actually is a clear arrow in the direction of time, thus, these systems are not time invariant. This means they do not meet the
stationarity principle, which entails that the mean of a variable is constant over time. Thus, to adequately model an open system, its process over time has to be taken into account (Green et al,. 1999).
To model a dynamic system over time, the model normally has an underlying
differential equation. In differential equations the next state of the system (next result of the equation) is always dependent of the former state (former result of the equation) (Green et al,. 1999). Mathematically, dynamics can be linear or non-linear. Linear dynamics assume that systems are most effectively modelled with linear equations, whose solutions can combine to another solution. Because of the superposition principle, a small change in the (initial) value of the independent variable produces a proportionally small change in the predicted dependent variable. In contrast, non-linear systems use non-linear equations that don’t meet the
superposition principle. Thus, the output of the non-linear equation is not directly proportional to its input. There are discontinuities or sudden changes (Stewart & Peregoy, 1983). In a non-linear model each controlling variable affects all other variables. If the value of one variable changes, the effects of all other variables subsequently change. Thus, the effects of each variable depend upon the simultaneous states of all other variables (Barton, 1994).
Running the data through a system of equations in which the results feed back into the same equations themselves is a way of finding the pattern of solutions for non-linear dynamic equations. This process is called iteration and every new iteration represents a new time point (Barton, 1994). A classic non-linear equation is:
𝛽𝛽′= 𝛽𝛽 + 𝑟𝑟𝛽𝛽(1 − 𝛽𝛽)
where there is a constant feedback of the result back into the equation (Barton, 1994). The 𝛽𝛽′ is the result of former iteration of the equation, and becomes the 𝛽𝛽 value for the next iteration of the equation (the next time point), creating a pattern of results. The 𝑟𝑟 is the parametric factor of the equation. To illustrate why this equation is non-linear, figure 1 shows the trajectory of a system with the underlying equation 𝛽𝛽′= 𝛽𝛽 + 𝑟𝑟𝛽𝛽(1 − 𝛽𝛽), a starting value of
10 𝛽𝛽 = 0.5 and 𝑟𝑟 varying from 1.5 – 33
. For the first values of 𝑟𝑟, the system shows stability, but as 𝑟𝑟 increases there is a sudden change, and there is a bifurcation into two equilibrium points. The equilibrium is the state of the system where there is only one solution to the dynamic equation and the trajectory of the system is stable (Barton, 1994).
Figure 1
The trajectory of a dynamical system can be thought to exist in the hypothetical phase space. This phase space shows the position of the system at any point in time (Hoyert, 1992). A simple phase space can be depicted in a two or three dimensional graph, with the state variables (the most important independent variables that specify the instantaneous state of the system) represented on the axes. By plotting the outcome pattern of the iterations over the (assumed) underlying equation of the dynamical system, the trajectory of the system over time becomes visible (Hoyert, 1992). The trajectory is dependent on the attractors of the system. The attractor is the force that determines important features of the system, like the size and shape. There are numerous types of attractors; a system can be drawn to point
attractors, limit cycles or so-called strange attractors. In the simpler attractors (point and limit cycles), the system will approach the specific point or curve, independent from the starting value (Hoyert, 1992).. However, with the strange attractors, this is not the case. The initial values determine all future trajectories and there is a high sensitivity for input values, meaning that a small difference in the initial values of the independent variables can lead to a
completely different outcome process of the system, a phenomenon called the butterfly effect (discovered by Lorenz in 1961; Gleick, 1983). The system becomes unstable, which is what
3
11 happens in figure 1.
Another aspect of non-linear dynamical models is that they can have qualitative changes. Physics has very clear examples of qualitative changes, e.g. water that changes from a liquid to a solid when it freezes. In dynamical system theory there are different models with different qualitative changes, like phase transitions and sudden jumps. A sudden jump is not a simple acceleration, or caused by a large change in the independent variables. A sudden jump is caused by a smooth continuous change in the independent variables, which can be caused when the attractor of the system changes, becoming unstable itself (Glendinning,
1994). However, these sudden jumps in the dependent variables are not smooth, continue changes but radical and qualitative changes.
An example of a model with sudden jumps is the cusp model (Grasman, van der Maas & Wagenmakers, 2009). The cusp model is a model that can be in two different states, having a radical change from one state into the other. Figure 2 shows a cusp-model for data that shows a sudden jump. The data are collected by measuring 103 points of a simple catastrophe machine (for examples, see Zeeman, 1979). The three dimensional surface represents the behaviour of the system for different values of the independent variables. Two of the three modes are stable, but the one in the middle (the fold) is unstable, which means that the system jumps from one stable state to the other (Grasman, van der Maas & Wagenmakers, 2009). Thus, for certain values of the independent variables, the system suddenly radically jumps from one state to the other. Figure 2 shows no data points in the fold, since the data-points can only be in one of either stable states but not in the bifurcation of the sudden jump.
12 Thus, in complex dynamical models, it is important to model the process of
development of a system over time because the model is assumed to be time variant, the system can have different states (depending on how far away it is from its equilibrium) and there are possibilities of qualitative changes like sudden jumps. These aspects are not visible if the system would be modelled measuring at one point in time. However, the discussed characteristics also bring technical and conceptual difficulties for modelling complex dynamical models.
The already discussed sensitivity of input values, or butterfly effect, implies that it might be useless to study each independent variable in isolation in a non-linear dynamical system, which makes it a difficult task for the researcher to find the important independent variables and to understand their simultaneous effects all together (Hoyert, 1992). Another implication of not meeting the superposition principle is that it is fundamentally impossible to predict the behaviour of a complex dynamic system over a long period of time, since every current state of the independent variables affects the future output of the other independent variables through a feedback mechanism. Thus, a very small measurement error in the initial estimation of the model parameters can increase exponentially in every iteration or feedback loop (Hoyert, 1992). A final difficulty of dynamic complexity models is distinguishing noise from relevant data. Non-linear dynamical models do not follow a simple linear regression line and can have sudden perturbations or jumps. Given that every small error in prediction can lead to huge error in the further estimation of the model, erroneously distinguishing noise from relevant data can have big implications for the correctness of the model. The ‘noise’ might actually be indicating a characteristic of the complexity system, being relevant for the correct representation of the model.
However, these difficulties do not always apply in every (stage of a) complex
dynamical model and can also be partly evaded. Different equations might be the best ones to describe the different stages of the system. Normally, when a system is said to be in a stable state, meaning the system is close to its equilibrium, linear equations are used to describe the system (Gleick, 1987). Thus, not every stage of the dynamical system needs to be modelled with non-linear equations. Also, sometimes there are possible transformations for non-linear equations to linear equations, thereby avoiding some of the discussed difficulties (see for example Lo & Andrews (2015) about using Generalized Linear Mixed Models for transforming non-linear reaction time data). Another important method are ‘dimension reduction techniques’ (Fontanella, Fontanella, Ippoliti & Valentini, 2015), which try to find underlying attractors of a high-dimensional system (i.e. the joint probability model over all
13 the variables), thus trying to simplify the model. The reduced dimensionality model should represent the minimum needed parameters to find the intrinsic dimensionality of the data. One example is assuming a known, underlying linear attractor field (e.g. a Markov Random Field), which means that although the system is assumed to be complex and dynamic, its underlying attractors are already being assumed in order to make it easier to build a model.
Complex dynamical models in psychology: Network models for psychopathology The difficulties for modelling complex dynamical models did not stop researchers from investigating the possibilities of complexity in psychology. Although these investigations sometimes lack formal models and testable hypotheses (Wagenmakers, van der Maas, & Farrell, 2012), there are proposals of new complexity models in psychology. A recently new and growing theory is the network perspective (Borsboom & Cramer, 2013), which applies complex dynamical models to psychopathological disorders like depression. The network perspective (Borsboom & Cramer, 2013) is the first theory to present a formalized model on psychological disorders without taking a latent variable perspective (like the factor model previously discussed). The network model perspective claims that there is no latent variable – the disorder – causing the symptoms, but that the network of interactions between symptoms is the disorder (Borsboom & Cramer, 2013). This means that depression symptoms are no longer thought of as dependent variables of the latent common cause variable named
‘depression’, but instead the symptoms and their interactions themselves are the depression. The nodes in the network represent the symptoms, and the edges between the symptoms represent the causal relationship between the symptoms (Borsboom & Cramer, 2013). The activated symptoms have a causal influence on their connected other symptoms. There is no need for an underlying, latent variable to explain the relationship between depression symptoms (Borsboom & Cramer, 2013). The symptoms themselves interact, and it is this interaction of symptoms that accounts for the depression.
The network model theory clearly calls for a different clinical approach towards therapy (Cramer & Borsboom, 2015). There is no longer one underlying factor causing the symptomology, which means that there is no underlying disorder that needs to be treated. The symptomology itself needs to be treated, focusing on the most central symptoms. These central symptoms have the most connections and most important causal connections with other symptoms, implying that if these central symptoms can be treated, their connected symptoms will also lower in activation. This leads to an important difference between the network perspective and the latent variable model of depression: their view on the
14 exchangeability of symptoms (Cramer & Borsboom, 2015). DSM criteria state that in order to get the diagnosis of depression, the patient has to get a score of for example 5 out of 8 major depression symptoms. Two patients diagnosed with major depression disorder can have a different symptomology, but get an identical diagnosis. A latent variable model would treat symptoms as exchangeable, since the symptoms all have the same common cause (Cramer & Borsboom, 2015). Contrary, the network perspective takes the individual symptomatology of the patient into account and presents a formal way to give a different importance to depression symptoms, namely, their centrality in the network.
The network perspective of depression is an example of how new dynamical models bring new theoretical insights into psychology. Currently, researchers are still investigating the depression networks develop over time, using longitudinal repeated measures data. However, there are indications of complex behaviour in depression networks; van de
Leemput et al., (2014) found indications of critical slowing down before depression networks reach a tipping point and the mood depression symptoms activate. Critical slowing down is specific behaviour a dynamic system can show before reaching a tipping point into a phase transition. Van de Leemput et al., (2014) took this phenomenon from dynamical systems theory and found criteria indicating critical slowing down in depression network by showing that there is an elevated correlation and variance of the depression symptoms, thus more fluctuation, indicating a loss of resilience of the depression network before the mood depression symptoms activate. Thus, even though the whole trajectory of the depression network is not yet fully known, looking at specific indications of complex behaviour can also bring insights into the dynamic behaviour of systems.
Longitudinal Data in Dynamical Complexity Models The previous sections discussed conceptual differences between traditional linear
psychological models and complexity models and the new theoretical possibilities brought by complexity models, like the new network perspective of depression. However, there is another important aspect of using complexity models in psychology. A much debated issue is the fact that inter-individual variance is not always equivalent to intra-individual variance. Dynamic complexity models bring new possibilities for making correct inferences based on both kinds of variance.
To understand the difference of inter-individual and intra-individual variance, the hypothetical behaviour space was introduced. This hypothetical behaviour space contains all scientific relevant information of individuals (De Groot, 1954, via Molenaar, 2004). All life
15 histories of a population can be represented by a collection of life trajectories in this
behaviour space. Inter-individual variance is the variation between people at certain measurement moment(s), and intra-individual variance is the variation of one person over time. Thus, for the collection of life-trajectories in the behaviour space, inter-individual variance is collected by (Molenaar & Ram, 2010):
- Selecting a fixed subset of relevant variables for the research - Selecting one or more time points for the measurement occasions
- Calculating the variance by pooling across subjects at the selected time points and for the selected variables
In contrast, intra-individual variance is collected by (Molenaar & Ram, 2010):
- Selecting a fixed subset of relevant variables for the research - Selecting a fixed subject
- Calculated the variance is by pooling across time points for the selected subject and for the selected variables
The important point of this distinction is that these two types of variation are not always equivalent, thus only under specific conditions the intra-individual variation will give the same results as the inter-individual variation. The equivalence between these two structures only exists in ergodic processes. An ergodic process is one in which the ensemble average (the average of the population) is the same as the time average (the average of one person over a certain timespan). The two necessary conditions for a process to be ergodic are stationarity (the mean of the variable has to be constant over time, i.e. time invariant) and homogeneity of the population (Molenaar 2004).
However, psychologic processes normally do not meet this ergodic criteria. Thus, intra-individual variation and inter-individual variation are not equivalent in most cases. This implies that one cannot take intra-individual inferences based on inter-individual variation and vice-versa (Molenaar, 2004). In regular factor analysis, the used variance is inter-individual variance. This means that one cannot say anything about the score of an individual on itself, other than its score relative to others. However, it is a fairly common use in psychology to let an individual take some ability-test, analyse the results with regular factor analysis and make intra-individual inferences of the ability of the individual.
There are many situations where psychologists are interested in the individual score of a person, e.g. for an IQ test or in a diagnostic setting. For this, the intra-individual variance of
16 that person is needed. Thus, since most psychological attributes don’t meet the ergodicity principle, psychologists needs longitudinal, repeated-measures data to make intra-individual inferences. However, collecting repeated-measures data is more time-intensive for both the researcher as for the participants. Although there still is a main focus on collecting inter-individual data and doing inter-inter-individual analyses, there is an increase in awareness of the importance of collecting intra-individual data. With new techniques such as Experience Sampling Methodology (ESM) where people are asked to fill in several questions during the day via an app on their telephone, the first investigations on longitudinal networks of
depression have started (Bringmann et al., 2013).
However, complex dynamical systems bring a new point of interest in this matter. Since complexity models normally do not meet the stationarity principle (e.g. do not have a constant mean value over time), but are now starting to being measured over time, it is possible for the inter-individual variance to change over time. Stated in terms of the behavioural space: if there are clear trends of different individual life trajectories, such as certain sub-populations having significant different trajectories, the inter-individual variance should be calculated over the whole range of individual trajectories and not at only one time-point, or important information will be missed. Thus, if the data do not meet the ergodicity principle, because of non-stationarity and heterogeneity of the population, the variation between all individual trajectories should be calculated to obtain the correct inter-individual variance. This means that both intra-individual variance and inter-individual variance should be taken into account. Otherwise, the inter-individual variance will miss relevant data-points to be calculated correctly. If the dynamic system is not ergodic, one cannot just simply ‘pool across subjects’ to calculate the inter-individual variance and model a system on the
population level, since the subjects’ trajectories could show trends of sub-populations which should be taken into account.
Both inter-individual variance and intra-individual variance should thus be calculated to see how a system behaves over time on a population level. Bringmann et al. (2013)
developed a method to do this by analysing a multilevel vector autoregression (VAR) model on the data. With this analysis, it is possible to see the development of the population
depression network over time. The VAR model can test for trends in intra-individual depression network trajectories, thus check if there are sub-populations having a different structure of depression. Hopefully, methods for analysing both inter-individual variance and intra-individual variance will be more frequently used as the research on complex dynamical psychological systems increases.
17 Discussion
This thesis investigated the conceptual shift in the function of models from traditional linear models in psychology to complex dynamical models. Important changes in the function of models are complex dynamical models taking ‘time’ into account, not meeting the
superposition principle (thus, the change in input values not being proportional to the change in output values of the model) and having the possibility of qualitative changes in the model (i.e. chaos, the butterfly effect). A consequence of the shift to complex dynamical models is that models no longer try to find the product of the interaction of some variables at a certain point in time, but instead have to model the process of the interactions of these variables over time.
I discussed theoretical implications of the shift to complex dynamical models in psychology. First, time is now perceived and modelled as irreversible instead of reversible, which is more plausible for living and changing and open organisms like human beings. Complexity models cannot be simply assumed to be time-invariant. Secondly, complex dynamical models make it possible to model qualitative changes and phase transitions in psychological attributes, for example in developmental psychology. Thirdly, complex dynamical models open the possibility for modelling complex and causal interactions of various variables without having to assume latent variables, i.e. the discussed network perspective of psychopathology. Finally, dynamical models call for more longitudinal data collection. For this last subject, I argued that complex dynamical models also bring a new aspect into the inter-intra individual variance discussion, since non-stationary dynamic models should take both inter- and intra-individual variance into account for their model.
There is much more to say about every topic I have only briefly touched, and the discussed assumptions, methods or models were simplified to some extent. However, the goal of this thesis was to give a conceptual overview of the most important (theoretical)
implications of the shift from traditional linear to complex dynamical models in psychology. Given the enormous research area of complex dynamical systems, the most relevant subjects have been discussed at their most fundamental level. Further readings for more extensive discussions or specific subjects are suggested in the text, in footnotes or in the reference section.
As Perna and Masterpasqua (1997, p 7) state: “Physicists, chemists, and biologists are finally speaking a language that offers a new way to apprehend the complexity and
18 linear models”. With a real understanding of the fundamental issues underlying the shift to complexity models, comprehending both the new possibilities complexity brings for
modelling psychological attributes as also the difficulties it entails, the psychological field can work towards formal, dynamic complexity models to gain a better understanding on its
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23 APPENDIX A
Code for creating figure 1
# Bifurcation for-loop # Gaby Lunansky
# Based on values & function in Barton, S. (1994). Chaos, self-organization , and psychology. American Psychologist, 49(1), page 6
rmax = 3
plot(-1, -1, xlim = c(1.5, rmax), ylim = c(0, 1.4), xlab = "r", ylab = "Y",
main="Bifurcation Arising from Iterations of Non-Linear Equation
x'=x+rx(1-x)") r <- seq(1.5, rmax, by = 0.0001) n <- 150 for (z in 1:length(r)) { xl <- vector() xl[1] <- 0.5 for (i in 2:n) {
xl[i] <- xl[i - 1] + r[z] *xl[i - 1]*(1- xl[i -1])
}
uval <- unique(xl[30:n])
points(rep(r[z], length(uval)), uval, cex = 0.1, pch = 19)
24 APPENDIX B
Code for creating figure 2 (data available upon request)
# Cusp model # Gaby Lunansky
# Code from UvA course 'Constructing Psychological Theories' by D. Borsboom & H.J.L. van der Maas
library(cusp)
data <- read.table('zeeman gaby.txt',header=TRUE, dec=',') #data available upon request x1= data[,'A'] x2=data[,'B'] x1 <- as.numeric(x1) x2 <- as.numeric(x2) z=as.numeric(data[,'X']) data<-data.frame(x1,x2,z)
fit<-cusp(y~z,alpha~x1+x2,beta~x1+x2,data)
cusp3d(fit, main="Cusp Model With Data from Zeeman Catastrophe Machine (N=1
