Population-natural-kinds

15The first generalization we must make is to consider our particular child, represented by a particular part of its lifeline, as a member of some population-natural-kind, that is to say, a natural kind represented by a particular population of individuals. For instance, we can generalize this particular child has a member of the natural kind “children with ADHD”, “boys”, “three-year-olds” etc. These natural kinds are nonexclusive, meaning that we can treat the particular child as a representative of the natural kind “three year old boys with ADHD”. The importance of this generalization is that we assume that there is something natural and specific about this class, represented by the population of three-year-old boys with ADHD. That is, there exists a set of properties that defines this class as an instance of each of these particular natural kinds, in which case it must be a combination of the defining properties of the combined natural kinds.

16 One of the central aims of science is to discover and specify the classes of natural kinds that define the subject of this particular science, e.g. developmental psychology. Take an example from elementary chemistry: if “water” is a chemical natural kind, chemistry must be able to define water by its essential chemical properties (its defining properties). In this particular case, it is that water is defined by its molecules, characterized by the bond between two hydrogen and one oxygen atoms, with outcomes being other types of natural kinds. These defining properties are absolutely generalizable, in the sense that they are true of every instance of water wherever or whenever it occurs (water on Mars and water on Earth are both chemically speaking H₂O). In the colloquial use of the word water, however, we often refer to a particular amount of liquid that is here-and-now before us, consisting of H2O plus any amount of contaminations such as NaCl, oxygen, bacteria, calcium carbonate etc. These contaminations that are almost always present in any particular, observed amount of liquid that we call water can in a sense be treated as if it were chemical random and non-essential variance occurring in virtually all instances of observed quantities of water. In principle, chemists are able to also physically remove these contaminants from the water, and keep “pure water”, chemically defined by H₂O and nothing else.

Psychological-natural-kinds

17As chemical elements are defining features of chemical natural kinds consisting of various types of molecules, mental and behavioral natural kinds are defining features of psychological natural kinds represented by specific populations of individuals. For instance, if ADHD is a natural kind, which corresponds with a particular population of individuals with ADHD, ADHD must be definable by a set of properties in the form of psychological-natural-kinds (or more precisely mental- and behavioral-natural-kinds). These defining natural kinds take the form of psychological functions, such as executive functioning, working memory, delay gratification and so forth. These psychological functions are examples of classifications of specific sorts of observable events (e.g. a child waiting for a bigger present instead of actually choosing an immediate that smaller one). They classify these events as members of an open set, and this set is defined as a propensity of a particular person or child. A typical assumption of standard psychological theory is that this propensity is identified as a single causal factor within the person, that is to say that it is viewed as an identifiable internal cause of all the events (behaviors, choices, etc.) that constitute a particular natural kind.

18 This standard view stands in sharp contrast with a view often advocated in the theory of complex dynamic systems, which is that a particular class of events that is interpreted as a propensity of a particular system, such as particular forms of behaviors in a specific child, is causally explained as an emergent result of the real-time intertwining of a great many causal processes. These processes involve properties of the agent’s body as well as properties of the agent’s direct environment, and these properties are produced by a constant loop of reciprocal causal relationships. In that sense, there is no identification of a particular class of propensities with a single underlying causal factor in the person. In the standard models, a particular class of behaviors that are subsumed under a natural kind—e.g. behaviors and events counting as depressive symptoms—is often defined by a single, underlying causal factor, namely depression (Schmittmann, Cramer, Waldorp, Epskamp, Kievit, & Borsboom, 2013). Later in this paper, we will explain the difference between the causal theories underlying the standard models of observable behavioral phenomena and those of the models based on the approach of complex dynamic systems.

19As a population-natural-kind—such as the population consisting of people with ADHD, or the population consisting of three-year-olds, or the population consisting of human beings—is represented by sets of individuals, a behavioral or mental natural kind is represented by sets of mental and physical activities or operations, carried out by human beings. For instance, the natural kind “executive functioning” is assumed to correspond with a set of activities that require executive functioning, such as inhibiting one’s immediate response in order to obtain a greater reward afterwards. Put differently, there is a “population” or more precisely an open set of behaviors and actions that act as representatives of a particular behavioral or mental natural kind.

20 For simplicity, let us assume that we have a population-natural-kind described as “three-year-olds with ADHD”, and this natural kind corresponds with a particular human population. It has already been hinted at that the sets we are dealing with should in principle be open sets. For instance, we note that the actual members of the population of three-year-olds with ADHD are constantly shifting, because three-year-olds are growing up and becoming for-year-olds, but new three-year-olds enter the population. And thus, we may assume that this population-natural-kind remains more or less stable as far as its representing population is concerned, but this is in that sense also a kind of “latent” variable. Let us for the sake of the argument also assume that the essential or defining properties of this population-natural-kind can be defined in terms of two mental or behavioral-natural-that kinds, namely executive functioning and working memory. There is of course an ongoing discussion of whether executive functioning is indeed a real natural kind, but for the sake of the current argument—and in line with a great number of actual empirical studies—we shall assume executive functioning and working memory can indeed be treated as natural kinds. We also know of course that in order to actually characterize ADHD as a natural kind, we need much more than these two properties, but the basic reasoning remains the same irrespective of how many properties one needs to carry out a particular classification.

Finding relationships between variables as a core issue in scientific research

26The question of what explanation exactly entails amounts to a very difficult, if not murky, philosophical and theoretical issue that we shall not delve into in this article. Instead, I believe that eventual problems can be clarified if we look at what hopefully the majority of the readers will recognize as standard or widespread practice in our discipline, and take that practice as the operationalization of what researchers and behavioral sciences and the development considered to be an explanation. I hope that readers will agree with me that what most researchers attempt to find are relationships between variables. For instance, they want to know the relationship between working memory and math achievement in primary school children, or the relationship between verbal IQ and theory of mind, or verbal IQ and a child’s score on a test for executive functioning. Researchers mostly express this relationship in the form of regression models or indicators such as correlations.

27 Given the widespread and almost automatic use of these models and indicators, we may assume that they provide an accepted, and largely uncontested, operationalization of the nature of scientific relationships in our discipline. Researchers also want these relationships to be generalizable, which, as we have seen earlier, implies that they want them to be true for a particular population of interest. With this aim in mind, researchers will try to discover this relationship based on data that are representative for their population of interest. That is, they will define a particular relationship between two variables as the contingency between the value of one variable and the value of the other variable as it exists in every individual member of their representative sample of unintended population. For instance, in a representative sample of primary school children, every child has a particular value on the variable executive functioning, and a particular value on the variable math performance (we shall refrain here from all sorts of problems regarding the measurement of such values and assume that the measurement is as correct as can be). So, the relationship between these two variables is actually represented by a geometric object, which is a cloud of points, with each point defined by two coordinates, one of which is the value on the first, the other the value of the second variable. However, researchers will not treat this representation as if it were a direct representation of the real relationship between the variables, because they see this cloud of points as an error-laden picture of the real relationship.

Figure1. x-y graph of data points in the form of a two-dimensional cloud

Figure1. x-y graph of data points in the form of a two-dimensional cloud

The properties of relationships featuring in data models

28But what—in the practice of behavioral and developmental research—counts as a representation of the real relationship then? Given the ubiquity of regression models (I shall discuss the use of correlations later), a relationship must be something that creates a directed, unique and uniform mapping of one variable onto the other.

29 First, a relationship is something “directed” that gives you a value of the second variable, given a value of the first. The first variable is then taken to be the independent variable, in the sense that the researcher can basically take any value of the independent variable he or she wants, and the second variable is the dependent variable, in the sense that its value depends on the chosen value of the independent variable (all this is of course extremely basic and almost trivial knowledge, but sometimes it is very interesting to go back to one’s axioms in an attempt to see what it eventual limitations are). The interesting thing is that the choice of these terms—independent and dependent—creates intended or unintended connotations with statements relating to causal control. For instance, if I change something in the independent variable, which basically means that I now select another value, something will change in the dependent variable. The word change has a purely (and very simple) mathematical meaning, in the sense of mathematically changing the value of the number, but it is easily misinterpreted as implying causal consequences. For instance, if I have a regression model showing a strong linear relationship between executive functioning and math performance, any mathematical change in the value of the independent variable executive functioning will correspond with a particular mathematical change in the value of the dependent variable math performance. Thus, with executive functioning being the independent variable, it is very easy to assume that a real increase in a child’s executive functioning must thus lead to a real increase in this child’s math performance. I bet that most articles showing such a relationship will conclude with recommendations urging the applied reader to consider a training program for executive functioning in order to improve math performance. Everyone knows that one should be very cautious with such consequences—correlation is not causation—but in spite of this rational consideration, beliefs very easily diverge into the domain of causal connections. In this particular case, the reason for the causal interpretation is the belief that executive functioning is a causal component necessary for performing a composite sort of activity such as solving math problems.

30 Second, we want a relationship to be unique, which means that we want to map a particular value of the independent variable onto one and only one particular value of the dependent variable. This uniqueness is what distinguishes the “true” representation of the relationship from the observed representation of the relationship in the form of a cloud of points. In the data cloud, correspondences are not unique. A particular value of the independent variable corresponds with a range of the variables of the dependent one, but this is customarily attributed to the fact that the observed variables depend on more independent variables than the one we are presently controlling (see figure 2).

Mapping a particular X value onto a range of Y values depending on the form of the data cloud

Mapping a particular X value onto a range of Y values depending on the form of the data cloud

31Third, we want a relationship to be uniform. For instance, a relationship between executive functioning and math performance should in principle hold for all possible values of executive functioning and all possible values of math performance. There should not be “this” kind of relationship for low values, yet another for somewhat higher values, and still another one for even higher values. Of course, sometimes we discover that the relationship diverges from a simple linear relationship, and seems to take the form of an inverted U-shape relationship for instance. If this is so, we will nevertheless conserve the uniformity of the relationship, namely by invoking a quadratic regression model (or any other polynomial function of the independent variable that does the job). The regression model – linear, quadratic or even higher order—holds for each and every possible value of the independent and dependent variable. In some cases, however, the set of observations is believed to consist of observations coming from different subpopulations, with different sorts of relationships between the dependent and independent variables. In that case, we can conceive of the distribution as a mixture model, but for each of the postulated subsamples, a uniform relationship between the independent and dependent variable is nevertheless assumed.

32 The limit of a mixture model is that every individual observation represents a unique subpopulation. There exist no intrinsic reasons to automatically reject the possibility or interest of such uniqueness, although the thought of it will probably horrify most researchers in psychology and developmental psychology. However, if this limit case applies, then every subpopulation is represented by only a single data point, and a single data point does not allow us to represent a relationship (a single point may belong to an infinite number of straight lines). How can we obtain a set of observations that characterize such a unique case? We can do so by sampling repetitive observations within the individual, e.g. repeated math performances, repeated performances of executive functioning and so forth. These repetitive observations must, by necessity, come from a real time series, a real time serial succession of events. But then, the statistical association between the observations corresponding with a particular relationship between the variables measured, is not governed by any form of, mostly unknown, dynamic causal relationship, i.e. and eventually reciprocal causal effect of one variable onto another as it unfolds and eventually changes over the course of time.

33 There is no a priori reason to assume that the structure of statistical relationships expressed by independent observations of cases belonging to a particular category (individuals from a particular group such as children with ADHD) is similar to the structure of statistical relationships expressed by dependent observations of cases belonging to a particular time series or real-time process. The lack of such a priori reason relates to the issue of ergodicity, which is now receiving increasing attention in developmental science (references).

Relationships over time and relationships with time

42Suppose now that we have a ruler measuring an important dimension of the state in which we are interested, which for instance is the child’s numerical knowledge at a particular time (Van Geert & Fischer, 2009). In fact, this ruler is everything that assigns a one-dimensional numerical value to the complex property “number knowledge” of the child at a particular moment. An example of such a ruler might be a test of numerical knowledge, but such rulers are in fact extremely crude, in the sense that they are not capable of capturing the details of what the child knows, in a particular context and at a particular moment. But let us, for the sake of the argument assume that we have such a virtually perfect ruler, that the current state of the child’s numerical knowledge is represented by a number A, and the next state by a number B. Hence, the change we observe is a causal change from a system state described as A into a system state described as B, and this description is a kind of shorthand for the underlying causal processes that we do not exactly know. So, whenever we draw a description such as

the → arrow sign explicitly implies a causal process, transforming a state described by the label A into a state described by the label B over the course of time.

43We can also describe a sequence of such transformations, and such a sequence will always and explicitly correspond with a time sequence, namely a sequence of events that occurs in the child that we are describing in this particular example

44Hence, what we have here is a description, for instance in terms of our numerical “ruler” of a real-time process. Being a real-time process, it must occur in a real-time system, for instance a particular child. If we have some sort of idea of the form or content of the arrow symbol, we basically have a dynamic system.

45 It might be so that this sequence of numerical values, indicating a process unfolding over time, shows some sort of mathematical structure. This sequence is determined by the succession of data points, and is thus dependent on what lies behind the arrow symbol, which connects to successive points in time. If A, B, C, D etc. actually represent values of a single variable, as measured by a particular “ruler”, and X is a name of that variable, then what we actually represent is

46 Hence, the relationship that we seek is given by the duplets (X_t, X_{t +1}), (X_{t +1}, X_{t +2}), (X_{t +2}, X_{t +3}), etc. These duplets are ordered in time, they define a sequence. The relationship between these duplets may change over time, and there is no assumption of necessary uniformity here. That is to say, if we also indexed the arrows, namely

47we would get a sequence, which like any other temporal sequence, can vary over the course of time.

48 These duplets concern one and only one variable, namely X. Now compare this with the situation in which we have two variables, for instance the variable X such as working memory and a variable Y, such as time or age. A classical association model would then be defined by duplets of the following type (X_i, Y_i), (X_j, Y_j), (X_k, Y_k), and so forth. The relationship between X and Y would be considered to be uniform across all possible values of X and Y (remember that this uniformity also holds in the case of quadratic or polynomial models, where the uniform relationship between X and Y is described in the form of a polynomial, which might result in various kinds of U-shaped or curved patterns). The one-dimensional sequence of X values and arrows (one-dimensional because it concerns only X, which is just one variable) is the outcome of an underlying causal structure and process, a causal structure describing a particular complex dynamic system, such as a particular child.

49What does this sequence tell us about the properties of this underlying system? That of course depends on the properties of the sequence. If the observed sequence is relatively simple, we might eventually discover that the sequence represented by →_{t +1}, →_{t +2}, →_{t +3}, →_{t +4}, does not change, implying that the causal principle connecting a preceding state with a successive state in a time series is a constant for the entire time stretch that we have studied (in this particular case the sequence spanning a duration from 1 to 4). But it is most likely that many observed sequences will be quite complicated themselves, which is what you expect to get from a complex dynamic system anyway. In fact, the series only informs us about one single dimension or variable of the system, X, which, if the system is complex, is but one of the incredibly many variables that characterize the system. However, there exists a very interesting mathematical theorem, the Takens-Ruelle theorem, that tells us that important information about the underlying dynamic system consisting of a great many interacting variables, is conserved in the temporal structure of any, relatively arbitrarily variable that the system consists of (Takens, 1981; Eckman, Kamphorst, & Ruelle, 1987).

50To actually understand what kind of information this is, we must first appreciate the fact that a complex system consists of eventually a very, very great many causally connected or, as it is usually stated coupled, variables. A coupling means some sort of causal influence between one variable and another, or one component and another, in which the actual change in one component causally affects some other component. An example of a causal coupling is when the change in one component causes a particular change in another. Yet another example of a causal coupling is when the change in one component causes the other component to actually counteract the change in such a way that it (the other component) may remain stable.

51Now suppose that we have a system which is very strongly coupled, i.e. very strongly internally coordinated. That is, any change in a particular variable will affect a particular change in all other variables and vice versa. In this case, such a system may very rapidly evolve towards a state in which all variables become completely coordinated, and in which the property of anyone variable perfectly predicts the property of all others. This is a system that can be specified by only one descriptive variable. Another way of stating this is to say that it is a system characterized by a one-dimensional state space. Yet another way of stating this is to say that it is a system in which the degrees of freedom have been reduced to only one.

52Now consider a system in which there are no systematic causal couplings, or no couplings at all between any variable of the system. That is, the variation in one variable is completely independent of the variation in another. To describe the states of the system, one meets is state space that has as many dimensions as a number of variables or components that the system consists of. It is highly likely, and in fact quite certain from the information we have about natural systems, that complex adaptive systems such as children can be characterized by state spaces that are somewhat between complete one dimensionality and complete multidimensionality (see Wijnants et al. 2012, for an example regarding dyslexia).

53In fact, the size of the state space that we need to describe the possible behavior of the system and the way that behavior varies over time, tells us something about the degree of organization (coupling, coordination, etc.) of the system. At current, we see an emergent field of statistical methods that use the information coming from the temporal variability in a particular indicator variable (which might be as simple as reaction time in the visual identification of words on a computer screen) as a means to develop quantitative measures for the degree of internal organization or coordination of the system. These techniques are known under names such as recurrence quantification analysis, detrended fractal analysis, and the types of variability analyzed are known under highly colorful names such as white noise, pink noise or brown noise (see Kello et al., 2010).

Types of relationships over time

54In summary, a sequence of events, represented by numbers or eventually categories corresponding with a particular measurable variable, can have a statistical structure that allows us to estimate important properties of the underlying causal structure, pertaining to the level of coordination or coupling among the variables that characterizes a particular system. Are there any other forms of regularity, also pertaining to the underlying causal structure in the system, which we can extract out of the observed time serial variability of a system’s behavior?

55To begin with, the time serial variability might reveal a very simple sort of dynamics. For instance, the next state may be equal to the preceding state plus some constant value (which might stochastically vary). That is to say, the sequence can be described as an iteration of a mathematical equation, for instance the function + a and we can then write the sequence as

56 or, in the form of the change formula

57 (the change in y,Δy, corresponding with a particular change over time, Δt, equals a constant, a). This is actually what we might expect to find if the linear regression models are true of individual processes (always plus or minus some sort of unsystematic stochastic factor). However, the time serial data we have on individual change processes hardly if ever show this sort of overall simple linear increase. These processes are characterized by various patterns of change, such as S-shaped change, inverted J-shaped change, discontinuous change, stepwise changes, inverted U-shaped change, overlapping waves, changing variability, temporary regressions, or stepwise changes each of which is preceded by temporary regression (Van Geert, 1998). Of course, each of these patterns can be reconstructed on the basis of some sort of ad hoc combination of linear additions and subtractions. This would actually mean that there exists no common underlying mathematical expression for the structure of change in time serial developmental datasets, and by default, that every pattern is likely to have its highly particular type of mechanism.

58 Another approach is to try to define a mathematical expression for the iterative change pattern that corresponds with theoretically feasible causal principles of developmental change. If such expression is capable of producing the plethora of developmental time series that we can find in the data, we have both theoretical and empirical justification for the assumption that the underlying complex causal process behaves in some sort of systematic way, which is captured by our mathematical equation.

59 In earlier publications I have tried to show that a number of general theoretical principles of growth may account for a wide variety of individual growth patterns (Van Geert, 1991, 1994; Van Geert & Steenbeek, 2005). The first principle that such growth models entail is that the amount of change of a variable representing some sort of complex system property, is proportional to what is “already there”, i.e. is proportional to the current level of the variable. The simplest mathematical function expressing this idea is exponential change

60or in the form of the change formula

61 (the dot represents a multiplication). Another very general principle of growth is that it depends on resources that are intrinsically limited. That is, as a variable approaches the limit set by the limited resources, growth will decline in a way that is proportional to the level already attained. Both principles can be combined in one single equation, which in fact amounts to the logistic growth equation already discovered in the 19^th century as a general model of growth in a wide variety of phenomena (it was first discussed in the context of population growth by the 19^th century Belgian mathematician Verhulst)

62 (for K being a limiting factor corresponding with the set of available resources). The equation should be read as follows: the next state of the variable is equal to the preceding state y_t plus a certain amount of change, which is expressed by y_t . r . ( 1 – y_t/K). This change is determined by the multiplication of three components: the preceding state of the variable, y_t, the rate of growth, r, and a damping factor ( 1 – y_t/K). It is easy to see that as the level of the variable approaches the value K, y_t/K approaches the value 0, which means that the amount of change per unit time also approaches 0. As soon as y_t = K, the amount of change y_t . r . ( 1 – y_t/K) equals y_t . r . 0, which means that change is 0 and that the variable has reached its stable level.

63 We also know that the change in a variable can be dependent on the influence it undergoes from other variables, and which is supposed to be proportional to the level or strength of these other variables. In line with the general assumption that growth is dependent on the level already attained, the effect of other variables, which we shall denote by v_{i, j, k…} can be represented by a simple mathematical equation namely

64(for s being an influencing factor which can be positive, in which case we call the variable v_i a supportive variable, and a competitive variable in case the value is negative).

65 In a growth process, a variable may affect another one not by its level but by the amount it changes, i.e. its increase or decrease. For instance, the increase of a particular variable, such as a child’s knowledge of multiplication and division, may consume shared resources such as practice time or effort at the time this particular knowledge is explicitly taught or practiced by the child. In this case, the effect of the variable v on the variable y occurs via its first derivative, and can be written as

Figure3. Examples of resulting temporal patterns

Figure3. Examples of resulting temporal patterns

66(for w a parameter that can again be supportive or competitive i.e. positive or negative).

67Finally, in a system subject to growth, variables can be connected in various ways. For instance, the level of a particular skill can be positively affected by skill-specific motivation, but on the other hand the skill specific motivation can be positively or negatively affected by the level already attained. The relationships can in principle also be indirect, as when a particular skill positively affects the increase in motivation that affects the growth of another skill, which then positively or negatively affects the first skill. Such connections can in principle be highly idiosyncratic, i.e. typical of a particular individual, for instance during the development of specific interests in children. The general point, however, is that the variables describing a complex developing system are, in principle, connected in a wide variety of ways. That is, such a complex system can be described by a network of interacting variables, the interactions of which are described in the form of theoretically generalized growth equations.

68 An example of a mathematical model of such a developing system is:

69 The equations express the change in the number of coupled variables, named as A, B, C, D, etc. (only the first two are actually shown in the above equation). The change is expressed as the multiplication of two change factors, which can be found within the main parentheses. The factor 1 – L_A/C_A defines a physical limit on the change of the variable, which is expressed by the parameter C_A. For instance, Olympic athletes running the 100 m outperform the overwhelming majority of the population. However, even if Olympic records are continuously broken, the maximum speed is governed by physical limitations. The first factor, described between the large parentheses is the core factor and expresses the network structure of the model. It consists of two sub-factors left and right of the plus-sign. Left of the +, we find the logistic change factor, which we already discussed under equation 11. The limiting parameter K_LA is determined by all the constant factors that affect the change in the variable at issue, A in this case. For instance, genetic factors, including their epigenetic effects, might be part of this specific parameter. Right of the +, we find the sum of all the supporting or competing factors, which we discussed under equation 12. The influence of a variable, for instance B, on the variable A, is expressed by the multiplication of the support factor s, which might also be negative in the case of a competitive relationship between the two variables, the level of the A-variable, and the level of the supporting or competing variable, which in the present example would be variable B. In the equation, the supporting or competing variables are represented by the generic name V. If the network represented by the above equation is a so-called small world network, every variable is directly connected to a limited number of other variables. However, almost all variables are interconnected by means of indirect connections: for instance, A is directly connected to B, and B is directly connected to C, implying that A is indirectly connected to C.

70 It is interesting to observe that, depending on parameter values, the network model governed by the above equations naturally produces a wide variety of temporal trajectories that show many of the qualitative phenomena that have been described in the literature (Fischer and Van Geert, 2014; Van Geert et al., submitted). It generates sequences of S-shaped growth, stepwise growth, temporal regressions, inverted U-shaped growth, long-term couplings between variables, and sequences of overlapping waves.

71 In addition, the model generates predictions for the distribution of exceptional performance levels, which are not symmetrically distributed that are in fact highly skewed. The model generates developmental trajectories in which the predictability of the final level of a particular developmental variable or ability increases with age. It also predicts that heritability, defined as a correlation with genetic endowment, increases with increasing age, as is demonstrated by the empirical data. Generating these patterns and predictions does not require any particular fine-tuning of the network model parameters. In fact these patterns and predictions more or less automatically result from random variation in the model parameters.

72 Hence, the patterns of intra-individual variability that these dynamic network models generate comply with the patterns and phenomena found in empirical research on individual change patterns. Neither the equations of the underlying developmental model, nor the patterns of intra-individual variability generated by the model resemble the equations and variability patterns from the standard linear models that were believed to be generalizable to all possible cases, for all possible cases being defined by variability between individuals constituting a particular population.

A misconception about generalization

73Our conclusion is that one cannot study the internal causal structure of a particular system, e.g. a particular child, by studying the structure of associations between variables as they differ between individuals, i.e. between children or between systems of a particular kind (children this particular case). That is to say, there is no logical connection between the numerical or statistical structure of associations between variables across individuals, and the internal causal structure of an individual system, e.g. an individual child, that belongs to the sample that you have studied. By logical connection I mean that statements from one domain—for instance the domain of inter-individual differences and the statistical models of such differences—by logical necessity apply to the other domain, for instance the domain of the internal causal structure of a particular system such as a child.

74Note however that this assumption of logical connection or logical necessity is often, albeit tacitly, made. In fact, this assumption forms the major justification for the fact that most of the research carried out in the behavioral sciences concerns studies of associations between variables in preferably big samples of individuals. This assumption is also what prevents the behavioral sciences, including developmental psychology, from becoming theoretically and empirically mature. The reasoning behind this assumption goes roughly as follows.

75Suppose that we have found a relationship between two (or any other relevant number of) variables in a very big sample representative of the entire population (e.g. the entire population of children with ADHD). If the relationship holds for the entire population, then it must be a lawful relationship, or something that at least approximates a lawful relationship very closely. By definition, a lawful relationship holds for every possible specific case, including patterns of intra-individual variability.

76However, the tricky thing is that there are two different meanings of the statement “holding for the entire population”. One is that it holds for the population as an entitiy. I shall call it a type I statement. For instance, when I say that this population counts 1,000,000 members, I make a statement about the cardinality of the population on the entity level (the population as an entity with 1,000,000 as property). Such type of statement does of course not hold for every individual member of the population: counting 1 million members is of course not a property of every single member. Another statement is one that is true for every member of the population. I shall call it a type II statement. For instance, “this population consists of male persons” is true for the entire population (all persons are male) and for every individual (every member of the population is a male person). However, we also often say that “this is the male population”, but a population is only male or female in a transitive sense (failing to appreciate to describe the difference would among others lead to paradoxes, as described in Bertrand Russell’s type theory).

77 Under standard practice, if a relationship between two variables across the population is found and to defined, it is automatically considered as a type II statement, and not a type I statement. The justification for this is most likely that every individual in a sense “lies on” the relationship, that every individual is a point in the relationship as it is regularly represented by means of a regression line. But that, I believe, is a categorical error. A relationship between two variables over an entire population is a type I statement holding for the population as an entity. Every individual member of the population constitutes as if it were one single point on the line that describes the relationship. This is similar to each individual constituting one unit in the cardinality property of the population as a whole. No one would ever make a type I – type II confusion error if the statement regards the cardinality of the population, but the error is very commonly made in the context of relationships between variables. Examples of type I properties of a population are its cardinality (number of members) and its history. Typical type II statements concern the defining features of the individual members that define them as members of the set represented by a particular population, such as having ADHD, or being between five and 12 years old. Another example might concerned the geographic localization of the population. For instance if it is characterized by the fact that as an entity it occurs in a particular region, then living in that region must be true of all its individual members, and thus be a type II statement.

The notion of ergodicity and the study of change

78The reasoning that a theory holding for a representative set of all possible observations of a particular phenomenon, also holds for a specific subsets of those observations, for instance in the form of repeated observations in a single person over a sufficient amount of time, appears so obvious that it is almost not worth discussing. The reasoning is certainly true in the general abstract sense, but the way it is applied to theories and observations in the behavioral sciences relies upon an unproven and probably also false assumption that the statistical structure of relationships between variables holding between subjects—which is what the observation of all possible observations of that relationship actually entails—, is statistically similar to the structure of relationships between those variables holding within subjects. There exists a particular term referring to the condition in which the statistical structure of some property of interest, derived from the ensemble of subjects, is similar to the statistical structure of this property of interest, derivable from any individual from this ensemble, which is ergodicity.

79A short article by Tarko (2005) on the Softpedia website  [1] provides a particularly illuminating example of this principle:

80Suppose you are concerned with determining what the most visited parks in a city are. One idea is to take a momentary snapshot: to see how many people are this moment in park A, how many are in park B and so on. Another idea is to look at one individual (or few of them) and to follow him for a certain period of time, e.g. a year. Then, you observe how often the individual is going to park A, how often he is going to park B and so on. Thus, you obtain two different results: one statistical analysis over the entire ensemble of people at a certain moment in time, and one statistical analysis for one person over a certain period of time. … The idea is that an ensemble is ergodic if the two types of statistics give the same result.

81In this particular case, the statistical structure of the inter-individual data will most likely differ from the great majority of statistical structures that one obtains by studying individual patterns, i.e. intra-individual variability. Nevertheless, all the different patterns of intra-individual variability will produce a particular pattern of inter-individual variability. It is not necessary that the majority of intra-individual patterns should be relatively similar to one another and similar to the overall pattern. The structure of the intraindividual models, describing individual processes, does not bear any necessary resemblance to the structure of the inter-individual model.

82The difference between the methods is that they provide a different kind of information. The first method tells you something about parks, namely how popular they are with people, with “people” being the “unidentified” ensemble of all people in the city. The second method tells you something about people, namely how fond they are of parks and how they differ in their preferences (e.g. some people don’t like parks at all, others like very particular parks, and still others like parks that attract many people).

83If we apply this example to the method of finding relations between variables by sampling across the ensemble of all possible persons, we can conclude that the first statistical method—sampling across individuals—tells us something about variables, namely how they are distributed across people. The method of finding relations between variables by sampling across consecutive events in a particular person tells us something about processes, namely how particular variables behave during the course of people’s activities, in the short or in the long run. Processes are sequences ordered in time that are specific to specific timelines.

84Specific timelines occur in individuals, such as a particular person, a particular family, and so forth. Populations are in fact sets of independent timelines (independent in the sense that they are methodologically treated as not coming about as a result of interactions between members in the population). The average of the processes that occur in individuals specifies neither a type I property, nor a type to property of the population. That is to say, it does not specify a process as it occurs to the population as an entity, and it does not specify the process as it occurs in every individual constituting that population. A population can of course be characterized by a particular process, for instance the process of its history, consisting of its growth, stabilization, change of properties and so forth. But, as we stated earlier, this sort of historical process does not in any way describe the processes as they occur in the population’s individual members.

85It has often be stated that process observations lead to idiosyncratic theories, i.e. theories that apply to individual cases (Molenaar, 2004; Molenaar & Campbell; 2009; Molenaar, Sinclair, Rovine, Ram, & Corneal, 2009; Hamaker, 2012). However, idiosyncratic theories can in principle be subsumed under nomothetic laws, i.e. laws that apply to every individual process (Van Geert, 1991; Molenaar, 2013; Steenbeek, Janssen & Van Geert, 2012; Van der Steen, Steenbeek, Van Dijk, & Van Geert, 2014). For instance, the principle of resource dependent exponential growth might serve as one of those “laws” of development, at least to the extent to which a concept such as “law” is applicable to behavioral sciences. Another such “law” may be that developmental systems consist of networks of relatively sparsely connected components affecting each other’s short-term and long-term fluctuations and changes (see for instance the recent developments and the dynamic network structure of the brain, Bullmore & Sporns, 2009; Bullmore & Bassett, 2011; Rubinov & Sporns, 2010).