Introduction
The question of whether mental disorders should be thought of as discrete categories or as continua represents an important issue in clinical psychology and psychiatry. The current setup of diagnostic systems such as the Diagnostic and Statistical Manual of Mental Disorders, 5th edition (DSM-5; American Psychiatric Association, 2013) and the 10th revision of the International Classification of Diseases (ICD-10; World Health Organization, 1992) typically adheres to a categorical model, in which discrete diagnoses are based on patterns of symptoms.
This approach is rooted in psychiatric traditions that go back to the work of Emil Kraepelin (e.g. see Kraepelin & Dierendorf, Reference Kraepelin and Dierendorf1915), who laid the foundation for a psychiatric categorization system that views the science and diagnosis of mental disorders as a branch of medicine. In medicine, tracing observable symptoms (e.g. foggy eyesight, headaches) to specific diseases (e.g. a tumour in the brain) plays a central role (Hyland, Reference Hyland2011). In fact, the successes of modern medicine are predicated on the insight that, in many cases, treatment should be directed at diseases (e.g. removing the tumour) rather than, for instance, at the observable symptoms themselves, because in medicine diseases function as root causes (Borsboom & Cramer, Reference Borsboom and Cramer2013). In this scheme of thinking, the decision of which treatment to assign to an individual depends on which disease that person has, rather than on the observable symptoms. Thus, the task of the physician is to identify diseases through diagnosis, after which an appropriate treatment can be selected. This idea functions as a template for health care systems around the world, and mental health care is no exception.
The current health care model assumes that psychiatric categorizations ‘carve nature at its joints’, as Plato puts it. However, such categorizations often involve apparently arbitrary conventions. For instance, while the DSM-5 diagnosis of major depression requires five or more symptoms to be present, it is unclear whether the resulting categorization is empirically superior to one that would require four or six symptoms for a diagnosis. If such categorizations do not have parallels in reality, for instance because they are essentially arbitrary cut-scores on a continuum, current practices may frustrate rather than facilitate scientific progress. For example, artificially dichotomizing a continuous variable can lead to dramatic drops in statistical power to detect effects on that variable (MacCallum et al. Reference MacCallum, Zhang, Preacher and Rucker2002; Van der Sluis et al. Reference Van der Sluis, Posthuma, Nivard, Verhage and Dolan2013). Also, such practices may lead to suboptimal treatment (or no treatment at all); both because evidence for diagnosis–treatment combinations is compromised, and because individuals seeking treatment may not fit the distinctions imposed by the system.
Thus, the question of whether psychopathology constructs are kinds or continua is of central importance to psychiatry and clinical psychology. In the past two decades, researchers have started to approach this question empirically (Meehl, Reference Meehl1992, Reference Meehl1995; Waller & Meehl, Reference Waller and Meehl1998). Collectively known as taxometrics, empirical approaches are based on the idea that multivariate distributions of psychometric indicators can be used to infer the structure of the underlying construct (Ruscio et al. Reference Ruscio, Haslam and Ruscio2006). This research programme has grown rapidly in recent years: Haslam et al.'s (2012) review discusses no fewer than 177 taxometric studies.
The present paper reviews such modelling approaches from a psychometric perspective. First, we discuss classic psychometric conceptualizations of latent variables and approaches to investigate them. Second, we examine psychometric models that integrate categorical and continuous features of latent constructs. Third, we review alternative approaches based on network modelling, which allow constructs to be continuous for some individuals but categorical for others. We conclude that current methodologies and data are insufficient to conclusively decide whether psychopathology constructs are continuous or categorical.
Measurement theoretical definitions of kinds and continua
The question of how to represent empirical phenomena (such as depression) using mathematical structures (such as continuous or categorical systems) is dealt with in measurement theoryFootnote 1 Footnote † (e.g. Suppes & Zinnes, Reference Suppes, Zinnes, Luce, Bush and Galanter1963; Krantz et al. Reference Krantz, Luce, Suppes and Tversky1971). Importantly, all measurement starts with categorization; namely with the formation of equivalence classes. Equivalence classes are sets of individuals who are exchangeable with respect to the attribute of interest (e.g. a psychiatric disorder). Thus, we can imagine that, according to some ideal method of observation, all depressed individuals would be assigned the same abstract symbol, say ‘D’, while the rest of the population would be assigned the symbol ‘H’. If all of the individuals in the population can be assigned these labels unambiguously, and individuals with the same label are indistinguishable with respect to the attribute of interest (just like two electrons are indistinguishable with respect to their charge), then we end up with two categories, as in the leftmost panel of Fig. 1. This is a nominal measurement scale (Stevens, Reference Stevens1946), because even if the symbols may be numerals (e.g. 0 and 1), they do not carry numeric information but only serve to convey class membership.
Fig. 1. How different theories on the structure of attributes (top) imply distinct latent variable models (bottom) in the case of dichotomous indicator variables. The top of the figure progresses from a theoretical structure with two kinds of people, via an ordered structure, to a continuum. The bottom of the figure shows the parallel psychometric progression from a latent class analysis (LCA) model (left) to a parametric item response theory (IRT) model (right) via ordered latent class models (or, equivalently, non-parametric IRT models).
Now, we may not succeed in finding an observational procedure (or even in agreeing on a hypothetical one) that in fact yields the desired equivalence classes. For instance, we may – and typically do – find that individuals who have been assigned the same label are not indistinguishable with respect to the attribute of interest: there appear to be significant differences between, say, cases of depression that feature the full range of symptoms and those that barely meet the diagnostic criteria. An intuitive way to accommodate this is to create more than two categories; e.g. ‘no depression’ (H), ‘mild depression’ (M) and ‘severe depression’ (S).
Because there are now three classes rather than two, next to the relation between individuals within classes (equivalence) we may also represent systematic relations between members of different classes. One way to do this is by invoking the concept of order, as represented by assigning numbers to the individuals in the classes (e.g. assign a 0 to all H-individuals, a 1 to all M-individuals, and a 2 to all S-individuals). Note that any numerical assignment that represents this order will do: the numbers still have no quantitative meaning (Michell, Reference Michell1997, Reference Michell1999). This representation, which is shown in the middle panel of Fig. 1, is known as an ordinal scale in measurement theory (Stevens, Reference Stevens1946) and as an ordered categorical structure in statistics (e.g. Agresti, Reference Agresti2013).
This procedure may, however, fail too. For example, we may find that even within the H, M and S-classes, there are non-trivial differences between individuals that we wish to represent. In this case we can break up these classes into even more subclasses; say M is split into M1, M2 and M3, and S is split into S1, S2 and S3. If (and only if) these additional subclasses also conform to order relations, such that H < M1 < M2 < M3 < S1 < S2 < S3, then we may represent them with a scale that starts to approach continuity: a continuous variable can be seen as an extension of this line of reasoning to infinitely many (possible) subclasses. The continuity hypothesis formally implies that (a) in between any two positions lies a third that can be empirically instantiatedFootnote 2 (just like for any two people who are 1.8 and 1.9 m tall, we might find a third who is 1.85 m tall), and (b) that there are no gaps in the continuum (just like, within the normal spectrum, there are no impossible body heights). This situation is represented in the rightmost panel of Fig. 1.
In psychological terms, categorical representations line up naturally with an interpretation of disorders as discrete disease entities (e.g. tumours, which are present or not), while continuum hypotheses are most naturally consistent with the idea that a construct varies continuously in a population (e.g. bodily height, which everybody has in some quantity). In a continuous interpretation, the distinction between individuals with and without, say, a DSM-5 diagnosis would thus be analogous to the distinction between tall people and people of average height: it depends on the imposition of a cut-off score that does not reflect a gap that is inherent in the attribute itself. Note that this does not mean that the delineated categories are arbitrary in a general sense or are ‘just’ social conventions: the delineation of extremely tall people may not correspond to any clear cut-off, but surely extremely tall people possess qualities and problems that the rest of us do not have, and the concept of being extremely tall certainly does not rest primarily on a social convention.
Kinds and continua as psychometric entities
If mental disorders were directly observable, the task of categorizing them as continuous or categorical would be relatively straightforward, and could proceed exactly as illustrated before. In this case, we could actually create equivalence classes of individuals with the same disorder status, see how many of these suffice, and test whether they conform to order relations. All this is standard procedure in the natural sciences, where the basic equivalence relations (being equally heavy, being equally tall) can be determined directly through experiment (Trendler, Reference Trendler2009; Markus & Borsboom, Reference Markus and Borsboom2012). With a sufficiently large number of neatly ordered categories (say, 30 levels of depression, which would differ exclusively in degree), few would object to treating depression as continuous, at least for most practical and research purposesFootnote 3 .
Unfortunately, in psychology, we have no way to decide conclusively whether two individuals are ‘equally depressed’. This means we cannot form the equivalence classes necessary for measurement theory to operate (Krantz et al. Reference Krantz, Luce, Suppes and Tversky1971; see also Trendler, Reference Trendler2009; Markus & Borsboom, Reference Markus and Borsboom2013). The standard approach to dealing with this situation in psychology is to presume that, even though equivalence classes for theoretical entities like depression and anxiety are not subject to direct empirical determination, we may still entertain them as hypothetical entities purported to underlie the thoughts, feelings and behaviours that we do observe (Borsboom, Reference Borsboom2005). Under this assumption, we may investigate these theoretical constructs indirectly, by conceptualizing them as the common cause of a set of symptoms or item responses (e.g. Cronbach & Meehl, Reference Cronbach and Meehl1955; Bollen, Reference Bollen1989; Reise & Waller, Reference Reise and Waller2009).
For example, a psychometric model could specify the hypothesis that there are just two kinds of people (e.g. depressed and non-depressed individuals), and that these two kinds of people respond differently to the questions in a clinical interview; this would yield a formal representation of the hypothesis that disorders are discrete kinds (e.g. a latent class model; Lazarsfeld & Henry, Reference Lazarsfeld and Henry1968; McCutcheon, Reference McCutcheon1987). Alternatively, a model could hold that there are not just two discrete categories of individuals, but rather that individuals differ from each other in degree (e.g. everybody, including healthy individuals, has a certain level of depression). This would yield a formal representation of the hypothesis that depression is not a kind but a continuumFootnote 4 . If symptoms were recorded dichotomously, the resulting mathematical structure would be an item response theory (IRT) model (Reise & Waller, Reference Reise and Waller2009). An IRT model states that the probability of endorsing an item depends is a monotonic function of a person's position on the measured latent variable, so that a higher position on that latent variable is associated with a higher probability of endorsing the item. The precise form of the item characteristic curve (ICC; which relates the item response probability to the latent variable) can vary across IRT models, but in standard applications is often taken to be a logistic curve.
Crossing these conceptualizations of latent structure with the structure of observations yields Bartholomew's (Reference Bartholomew1987) classic taxonomy of measurement models as represented in Table 1.
Table 1. Cross-classification of latent variable models for discrete v. continuous latent and observed variables
The models in Table 1 are identical in that the latent and observed variables feature as independent and dependent variables, respectively (see also Mellenbergh, Reference Mellenbergh1994). Also, in each of the latent variable models represented in Table 1, the observed variables are assumed to be statistically independent, conditional on the latent variables. Thus, the models assume that, given a specific level of a latent variable (e.g. depression), the indicators (e.g. ‘feelings of guilt’ and ‘suicidal ideation’) are uncorrelated. This feature, which is known as local independence, is consistent with a causal interpretation of the effects of the latent on the observed variables (Borsboom et al. Reference Borsboom, Mellenbergh and Van Heerden2003). In such a causal interpretation, variation in the latent variable is not merely associated with, but in fact causally responsible for, variation on the observed variables.
The distribution of observed variables is typically taken as a given in psychometric modelling, as it is dictated by the response format used in questionnaires or interviews. The structure and distribution of the latent variable, however, may feature as a research question, rather than a known. This is often the case in psychiatric nosology, because we do not have strong independent evidence to resolve the question of whether psychiatric disorders vary continuously or categorically in the population. In this case, one may apply the models in Table 1 in an attempt to determine the form of the latent structure. This can be done in two ways. First, by inspecting particular consequences of the model for specific statistical properties of (subsets of) items, such as the patterns of bivariate correlations expected to hold in the data (Waller & Meehl, Reference Waller and Meehl1998). Second, on the basis of global fit measures that allow one to compare whether a model with a categorical latent structure fits the observed data better than a model with a continuous latent structure (De Boeck et al. Reference De Boeck, Wilson and Acton2005; Lubke & Neale, Reference Lubke and Neale2006, Reference Lubke and Neale2008; Lubke & Miller, Reference Lubke and Miller2015). The former of these approaches is typically denoted by the name ‘taxometrics’, while the latter is not, and we will follow this terminology here; however, it should be noted that taxometrics rests on exactly the same psychometric model as general latent variable modelling, and in this sense the approaches are complementary (see also Schmitt et al. Reference Schmitt, Aggen, Mehta, Kubarych and Neale2006; McGrath & Walters, Reference McGrath and Walters2012).
The logic underlying taxometric analysis is, at first sight, straightforward (but see Maraun et al. Reference Maraun, Slaney and Goddyn2003; Maraun & Slaney, Reference Maraun and Slaney2005). If the underlying construct is continuous, then the covariance between any two indicators conditional on a given range of a proxy of the construct should be same regardless of the exact range. For example, if the variable underlying depression is a (single) continuous dimension, then the relation between the scores on, say, an insomnia item and a suicidal ideation item should be about the same in people with low, intermediate or high scores on a third item that plausibly acts as a proxy to the depression construct (e.g. a sad mood item). In contrast, if the underlying variable is a binary variable (comprising two classes, i.e. healthy v. depressed), then the covariance between any two indicators conditional on a given range of the proxy of the construct is expected to vary with the value of the proxy. Specifically, at low (high) values of the proxy, most individuals are healthy (depressed), i.e. they are in the healthy (depressed) class. Within this class the covariance among the items is expected to be zero (as per local independence). In contrast, at intermediate levels of the proxy, we will find a mixture of individuals that belong to either class. Here, the conditional covariance between the indicators is expected to be larger, because between-class differences contribute to the covariance. Taxometric analysis capitalizes on such implications of latent structure hypotheses.
To carry out a taxometric analysis, one arbitrarily chosen variable (e.g. mood) is denoted the ‘index’ variable, and is assumed to be a proxy for the underlying construct. Then, over a moving window of values on the index variable, the covariance between the other two variables (i.e. suicidal ideation and feelings of worthlessness) is plotted. If the underlying construct is categorical, the resulting covariance curve will be peaked at the point where the selected groups contain equally many individuals from each latent class: because groups with very low or very high scores on the index variable will be composed almost entirely of individuals from one latent class, in which we have local independence, the correlation between the plotted variables will be lower for very high or very low scores of the index variable. As a result, a peaked covariance curve suggests a categorical latent structure, while a flat curve suggests a continuous latent structure. This particular method is called MAXCOV (for MAXimum COVariance). In a similar vein, MAXEIG (MAXimum EIGenvalue) plots the eigenvalue of a matrix of item covariances instead of a simple bivariate covariance, allowing more variables to be used in the analysis. Several other methodologies have been constructed over the years, which use the same idea of identifying divergent predictions from different hypotheses concerning the structure of latent variables (Ruscio et al. Reference Ruscio, Haslam and Ruscio2006; McGrath & Walters, Reference McGrath and Walters2012).
Despite its popularity, it should be noted that the taxometric approach is not uncontroversial in psychometrics. The reason is that it has long rested purely on the visual inspection of a plotted function instead of on a formal hypothesis test (Haslam et al. Reference Haslam, Holland and Kuppens2012). Another reason is that one of its core assumptions (categorical latent structures will produce peaked covariance functions) is not necessarily true (Maraun et al. Reference Maraun, Slaney and Goddyn2003; Maraun & Slaney, Reference Maraun and Slaney2005); for example, violations of distributional assumptions concerning measurement error may lead to peaks and valleys in the covariance function even if the latent variable is continuous (Molenaar et al. Reference Molenaar, Dolan and Verhelst2010). Some of these concerns may be less of a problem in methods based on simulation (Ruscio et al. Reference Ruscio, Ruscio and Meron2007), and McGrath & Walters (Reference McGrath and Walters2012) suggest that, despite the above problems, taxometric procedures do perform reasonably well in systematic simulations. Nevertheless, the fact that taxometrics lacks a comprehensive mathematical foundation is a considerable weakness, because it implies that the validity of taxometric techniques must be judged on a case-by-case basis.
Complementary to taxometric analysis, one may use latent variable modelling as a framework in which to query the structure of psychiatric constructs. In such approaches, one can compare the fit of a model in which the latent variable is represented as being categorical with that of a model in which the latent variable is represented as a continuous dimension to decide which model is superior (Lubke & Miller, Reference Lubke and Miller2015). This counters at least some of the above concerns, as latent variable modelling approaches do rest on a firm mathematical basis (Lubke & Muthén, Reference Lubke and Muthén2005). However, latent variable approaches are not without problems either. For instance, it is well known that many continuous variable models have statistically equivalent categorical or mixture counterparts: that is, a fitting model with a categorical latent variable does not imply that the construct itself is categorical, because a continuous model might fit the same data equally well (e.g. Molenaar & Von Eye, Reference Molenaar, Von Eye, Von Eye and Clogg1994; Halpin et al. Reference Halpin, Dolan, Grasman and De Boeck2011; see also Erosheva, Reference Erosheva2005).
McGrath & Walters (Reference McGrath and Walters2012) have systematically evaluated the performance of latent variable models and taxometric procedures, and propose a combination of modelling approaches, in which taxometric strategies are used to detect categorical structures, whereas latent class or profile models are used to select the optimal number of classes if the structure is determined to be categorical. A thoughtful combination of different methodologies indeed appears the most sensible currently available strategy for investigating the issue. However, it is remarkable that no systematic and principled methodological procedure appears to have emerged from the psychometric work on this issue. One possible reason for this situation is that the hypotheses of kinds and continua do not exhaust the space of possibilities, so that evidence against one hypothesis is not necessary evidence for the other (as seems to be the implicit assumption in taxometrics). In the next sections, we discuss models that may indeed substantiate this idea, because they have both categorical and continuous aspects.
Alternative latent variable models
Recent developments in statistical modelling have produced various models that blur the kinds–continua distinction, in the sense that they accommodate both categorical and continuous latent structures at the same time. Fig. 2 illustrates how our basic model with two latent classes can be extended in two other directions than the standard order used in the foregoing [i.e. the transition from latent class analysis (LCA) to IRT as represented in Fig. 1]: by including continuous latent variables within classes (factor mixture models), and by making group membership itself a matter of degree [grade-of-membership (GoM) models]. In addition, network models allow the very same structure to be continuous and categorical, depending on the parameters of the network.
Fig. 2. Alternative psychometric models for representing kinds and continua. The left top panel show the standard progression from a two-class latent class analysis (LCA) to an item response theory (IRT) model, as in Fig. 1. The left bottom panel shows the inclusion of continuous variation within classes (factor mixture model). The right top shows a fuzzy set representation, in which individuals belong to multiple classes at the same time (grade-of-membership model). The right bottom panel shows network models, which can have categorical or continuous dynamics depending on the structure of the network (see Fig. 4).
Factor mixture models
Finite mixture models partition the population into distinct latent classes, but allow for continuous variation within these classes (McLachlan & Peel, Reference McLachlan and Peel2000). If that variation is itself measured through a number of indicator variables, then we obtain a factor mixture model (Yung, Reference Yung1997; Dolan & Van der Maas, Reference Dolan and Van der Maas1998; Arminger et al. Reference Arminger, Stein and Wittenberg1999; Lubke & Muthén, Reference Lubke and Muthén2005; Muthén, Reference Muthén, Hancock and Samuelsen2008). The factor mixture model can thus be understood as a latent class model in which each latent class is characterized by its own common factor model (see bottom left panel of Fig. 2). Importantly, conditioning on the latent classes no longer renders the observed variables independent, as their conditional distribution is characterized by the factor model. However, observed variables are assumed to be conditionally independent given both the latent class and the latent continuous factor. Alternatively, the factor mixture model can be understood as a multi-group common factor model in which group membership is unknown. The latent class variable then takes the place of an observed grouping variable. The factor mixture model may distinguish between healthy and affected individuals, but also allows for quantitative individual differences within the two classes. For instance, such a model may propose a categorical distinction between people with and without depression, but at the same time may allow for continuous variation in the degree to which depressive symptomatology is present within each of these classes.
Factor mixture models provide a useful framework for formalizing the distinction between categorical and continuous latent variables in terms of distributional assumptions and model constraints (Muthén, Reference Muthén2006; Masyn et al. Reference Masyn, Henderson and Greenbaum2010; Von Davier et al. Reference Von Davier, Naemi and Roberts2012; Clark et al. Reference Clark, Muthén, Kaprio, D'Onofrio, Viken and Rose2013). For instance, consider Fig. 3, in which we display the latent distribution above the ICCs for three binary items, which specify how the expected value of the items (e.g. the expected value of the symptom ‘suicidal ideation’) varies with positions on the latent variable (e.g. major depression). The top-left figure represents an IRT model with the assumption of that the latent variable follows a normal distribution. One's position on the latent trait determines one's probability of endorsing the items through the ICCs in the figure directly below the normal density: e.g. the more depressed, the higher the probability of endorsing the item ‘suicidal ideation’. This is a standard IRT model as discussed earlier.
Fig. 3. From an item response theory model with a latent normal density (top left), via a two-component discrete factor mixture model subject to measurement invariance (top right) and the same model without measurement invariance (bottom left) to an unconstrained latent class model (bottom right).
Now we can gradually move from this dimensional model towards a categorical model by changing the model constraints. For example, the top-right figure represents the same model as the top-left figure, but now in two distinct groups which follow the same ICCs (so that measurement invariance holds; Mellenbergh, Reference Mellenbergh1989; Meredith, Reference Meredith1993). The bottom-left figure represents this same two-group model, but here the groups are allowed to follow different ICCs. From this model, we can subsequently derive a latent class model by simply shrinking the variances in each group towards zero; here, the latent classes can be viewed as specific values on the continuous factor (under additional ordering restrictions, this model is equivalent to a model with ordered latent classes, as represented in the middle panel of Fig. 1; see also Croon, Reference Croon1990; Vermunt, Reference Vermunt2001; Masyn et al. Reference Masyn, Henderson and Greenbaum2010).
Thus, while the factor model and the latent class model are distinct models in Table 1, mixture modelling allows us to connect them by means of intermediate models and associated constraints. Importantly, the existence of these models implies that the choice between fully discrete and fully continuous latent structures does not exhaust the possibilities. As De Boeck et al. (Reference De Boeck, Wilson and Acton2005) note, within this psychometric framework, the distinction between categories and continua is itself a matter of degree (see also Von Davier et al. Reference Von Davier, Naemi and Roberts2012).
GoM models
In GoM models, one can also depart from a simple latent class model to integrate continuous features. Here, however, the continuous variation concerns group membership itself. Where the latent class model assumes that every person belongs to one and only one latent class, the GoM model allows individuals to be members of multiple classes at the same time (Erosheva, Reference Erosheva2005). Thus, instead of proposing that an individual belongs either to, say, the class of typical or to the class of atypical depression (the focal hypothesis in LCA), the GoM model holds that an individual belongs to both classes at the same time, but to different degrees. The model thus is a psychometric instantiation of the logical concept of a ‘fuzzy set’: a set that has no clear boundaries, and to which different objects belong to different degrees (Verkuilen et al. Reference Verkuilen, Kievit, Zand Scholten, Belohavek and Klir2011). In the GoM model, the degree of membership is expressed in terms of a set of probabilities that sum to unity (so, in a two-class model, if one's GoM for one class equals 60%, GoM to the other class must equal 40%). Erosheva (Reference Erosheva2005) and Asparouhov & Muthén (Reference Asparouhov, Muthén, Hancock and Samuelsen2008) provide treatments of different statistical representations of the GoM modelFootnote 5 . The move from a latent class to a GoM model is illustrated in the top-right panel of Fig. 2.
The GoM model has been applied to schizophrenia (Manton et al. Reference Manton, Korten, Woodbury, Anker and Jablensky1994; Jablensky, Reference Jablensky2006, Reference Jablensky2010), depression (Woodbury & Manton, Reference Woodbury and Manton1989) and personality disorders (Nurnberg et al. Reference Nurnberg, Woodbury and Bogenschutz1999) and, in these applications, meaningful classes could indeed be defined. However the GoM model is not widely used in psychometric applications, probably due to the lack of readily accessible statistical software to apply the GoM model. However, this has recently changed with the appearance of Robitzsch's (Reference Robitzsch2014) package for the statistical software environment R. The GoM model can also be fitted in Mplus by representing it as a multilevel latent variable model (Asparouhov & Muthén, Reference Asparouhov, Muthén, Hancock and Samuelsen2008). These software advances invite the further application of the model to clinical psychology and psychiatry.
Network models and dynamical systems
In traditional models discussed so far, theoretical constructs are assumed to be either categorical or continuous for all elements of the population. However, it is possible that the transition to and from a psychiatric disorder proceeds as a categorical sudden transition for some individuals, whereas it is a smooth process of change for others. Recently developed network models accommodate this possibility, and thus shed a new light on the question of whether disorders should be thought of as categories or as continua.
Psychometric latent variable models represent differences in the structure of psychiatric constructs as differences in the distributional form of a latent variable, which acts as a common cause of the indicatorsFootnote 6 . Thus, when one considers ‘insomnia’ and ‘fatigue’, two symptoms of depression, a latent variable model assumes that when insomnia and fatigue covary, this is the result of their common dependence on depression (insomnia←depression→fatigue). Recently developed network models of psychiatric disorders (Cramer et al. Reference Cramer, Waldorp, van der Maas and Borsboom2010, Reference Cramer, Borsboom, Aggen and Kendler2012a , Reference Cramer, van der Sluis, Noordhof, Wichers, Geschwind, Aggen, Kendler and Borsboomb ; Borsboom & Cramer, Reference Borsboom and Cramer2013), in contrast, assume that insomnia and fatigue covary because they are causally related: if one does not sleep, one will get tired eventually (insomnia → fatigue). Correlations between variables commonly seen as ‘indicators’ then arise from a network of causal effects among these variables themselves (i.e. they form so-called mechanistic property clusters; Kendler et al. Reference Kendler, Zachar and Craver2011); as such, in networks, there are no latent variables that function as psychologically meaningful common causes, even though individual nodes and connections may of course stand under the influence of factors (e.g. genetic effects or life events) that are not directly observed.
Individual differences in network structure may lead to different patterns of symptom dynamics. For instance, insomnia may quickly result in fatigue in Bob's network (because of a strong connection between insomnia and fatigue), but less quickly in Alice's network (in which this connection is weaker; see Fig. 4). For a person with a weakly connected network, external stressors (like losing one's job) may lead to an increase in the number of symptoms that are activated but, importantly, when the external stressors are removed, the person will spontaneously and smoothly return to equilibrium. Strongly connected networks, however, can behave differently: they may show strongly non-linear behaviour with sudden jumps from one state to another (Thom, Reference Thom1975; Zeeman, Reference Zeeman1977; Van der Maas & Molenaar, Reference Van der Maas and Molenaar1992; Cramer, Reference Cramer2013). For example, for the case of depression this means that in a person with a strongly connected network, a given perturbation (e.g. an adverse life event) has the potential to trigger a full and persistent depressive episode, while that same perturbation may only yield a temporary elevation of depressed symptoms in a weakly connected network (see also Pe et al. Reference Pe, Kircanski, Thompson, Bringmann, Tuerlinckx, Mestdagh, Mata, Jaeggi, Buschkuehl, Jonides, Kuppens and Gotlib2015). The reason for this difference is that, in strongly connected networks, symptom activation may be increased through feedback loops; e.g. a person who suffers from lack of interest may avoid social contacts, which may lead to further diminished interest. Thus, in a strongly connected network, the spreading of activation through the symptom network can cause a depressive episode, which may then be maintained by the feedback loops in the network.
Fig. 4. Alice's network, which has a weak connectivity profile (left top) gives rise to a single basin of attraction (right top), corresponding to a healthy state. Continued stress (perturbation; force being exerted on the ball from the left) may cause prolonged changes in state (elevated ball), but upon removal of stress the network will return to the healthy state. Bob's more strongly connected network (bottom left) features alternative stable states (bottom right) and a tipping point. If external stress exceeds a given value, then the system collapses to the disordered state, which is itself stable. Thus, the system will maintain the disordered state, even if the stressor is removed.
These differences in dynamics across different network structures are important to the kinds v. continua discussion, because they show that disorders may be discrete kinds for some people (e.g. people with strongly connected networks) and continuous structures for others (e.g. people with weakly connected networks). Thus, individual differences data may look like a continuous distribution even if, intra-individually, the transition from a healthy to a disordered state is discontinuous (see also Borsboom et al. Reference Borsboom, Mellenbergh and Van Heerden2003; Molenaar, Reference Molenaar2004; Molenaar & Campbell, Reference Molenaar and Campbell2009; Adolf et al. Reference Adolf, Schuurman, Borkenau, Borsboom and Dolan2014). This may explain why taxometric investigations of individual differences data often claim evidence for continuity, even in cases where the within-person phenomenology would seem to suggest a discontinuous shift from health to disorder, like post-traumatic stress (Forbes et al. Reference Forbes, Haslam, Williams and Creamer2005) and psychosis (Ahmed et al. Reference Ahmed, Buckley and Mabe2012), and that evidence is often mixed, depending strongly on the specific sample that was investigated.
If present, discontinuous transitions have direct measureable consequences that may be exploited in further research, because transitions from a healthy state to a disordered state are typically preceded by early warning signals (EWS; Scheffer et al. Reference Scheffer, Bascompte, Brock, Brovkin, Carpenter, Dakos, Held, van Nes, Rietkerk and Sugihara2009, Reference Scheffer, Carpenter, Lenton, Bascompte, Brock, Dakos, van de Koppel, van de Leemput, Levin, van Nes, Pascual and Vandermeer2012) that indicate that the system is close to a tipping point for a transition. An important EWS is critical slowing down, which means that a system near a transition will take longer to recover from random perturbations of its state. In the context of psychiatric disorders, this would mean that a person at the brink of developing an episode of depression will recover more slowly than a healthy person from a relatively minor daily stressors such as an unpleasant phone call with their mother-in-law. As a result, the system's state at a given time point becomes more predictable from previous time points. Studies on mood fluctuations indeed suggest that people who feature such increased predictability (emotional inertia; Kuppens et al. Reference Kuppens, Allen and Sheeber2010) are at elevated risk for mood disorders, in line with the idea that mood shifts may be preceded by EWS (Van de Leemput et al. Reference Van de Leemput, Wichers, Cramer, Borsboom, Tuerlinckx, Kuppens, Van Nes, Viechtbauer, Giltay, Aggen, Derom, Jacobs, Kendler, Van der Maas, Neale, Peeters, Thiery, Zachar and Scheffer2014) as the system becomes less resilient (see also Montpetit et al. Reference Montpetit, Bergeman, Deboeck, Tiberio and Boker2010).
Thus, network models provide a fresh way of thinking about the problem of kinds v. continua, and suggest new avenues for research. Psychometrically, network models can be fitted as so-called Markov random fields (Kindermann & Snell, Reference Kindermann and Snell1980; Epskamp et al. Reference Epskamp, Cramer, Waldorp, Schmittmann and Borsboom2012). The R-packages IsingFit (Van Borkulo et al. Reference Van Borkulo, Borsboom, Epskamp, Blanken, Boschloo, Schoevers and Waldorp2014) and qgraph (Epskamp et al. Reference Epskamp, Cramer, Waldorp, Schmittmann and Borsboom2012) implement data-analytic approaches that may be used to this end. Individual differences in network structures can be studied by using individual differences in time series, with a multilevel modelling framework that allows for inter-individual variation in intra-individual networks (Bringmann et al. Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters, Borsboom and Tuerlinckx2013, Reference Bringmann, Lemmens, Huibers, Borsboom and Tuerlinckx2015).
Discussion
The current evaluation of psychometric conceptualizations and models shows that the distinction between continua and kinds is considerably more subtle than has been presupposed in the literature. Several psychometric models explicitly integrate continuous and categorical aspects of the construct studied, and in network models the very same construct can show continuous or discontinuous behaviour, depending on the network structure. Novel psychometric models conceptualize the distinction between kinds and continua in interesting ways, and suggest new ways of thinking about the issue.
In particular, these approaches may serve to elucidate why, so far, few research lines have led to a univocal verdict on the structure of studied constructs. As Haslam et al. (Reference Haslam, Holland and Kuppens2012) note, results vary over contexts, types of data, and substantive domains in ways that are not easily interpreted (see also Lubke & Miller, Reference Lubke and Miller2015). As a result, it is possible to find evidence both for and against taxa in the taxometric literature of almost any disorder. We have suggested here that one reason for this paucity of strong results may be that the models considered represent a limited and perhaps inappropriate subset of theoretical positions, which portray the kinds and continua hypotheses as mutually exclusive and exhaustive. A fair evaluation of the psychometric literature does not support this portrayal.
An important novel approach that may serve to increase our resolution concerning the structure of psychopathology involves the study of intra-individual transitions from and to disorder states (Boker et al. Reference Boker, Molenaar and Nesselroade2009; Molenaar et al. Reference Molenaar, Lerner and Newell2013). Ecological momentary assessment procedures provide a fruitful way of studying transitions (Hamaker et al. Reference Hamaker, Nesselroade and Molenaar2007; Van de Leemput et al. Reference Van de Leemput, Wichers, Cramer, Borsboom, Tuerlinckx, Kuppens, Van Nes, Viechtbauer, Giltay, Aggen, Derom, Jacobs, Kendler, Van der Maas, Neale, Peeters, Thiery, Zachar and Scheffer2014). Wichers (Reference Wichers2014) has described a theoretical program that connects the macro-level of mental disorders to the micro-dynamics of mood states as they fluctuate and influence each other over short time periods. The combination of this program with advanced psychometric models for time series is likely to shed light on the conditions that would lead to continuous v. categorical change.
A final extension that may further our understanding of categorical and continuous aspects of disorders concerns the domain of experimental manipulations. It is well known in the literature on phase transitions in other areas of psychology (Van der Maas et al. Reference Van der Maas and Molenaar1992) that the study of transitions requires a careful experimental setup, in which variables that drive transitions can be isolated and manipulated – ideally, the system is moved into and out of a number of distinct states within a single experiment (e.g. Dutilh et al. Reference Dutilh, Wagenmakers, Visser and Van der Maas2010). Of course, such manipulations are ethically questionable in clinical work; however, one could imagine clever combinations of quasi-experimental research (e.g. prospective studies involving adverse life events), experimental manipulations (e.g. treatment studies), and model-based simulations that might be used as proxies to a full experimentally controlled procedure. Incorporating such experimental features may considerably increase the resolution of psychometric attempts to uncover the structure of psychopathology.
In conclusion, both the models and the data used in current research on the structure of psychopathology are limited and often suboptimal. As a result, we should hesitate to draw strong conclusions from the current literature. Fortunately, extensions into models for within-person dynamics, experimental setups, and multivariate systems with continuous and categorical variables are now within reach. These developments are likely to shed more light on the structure of psychiatric constructs in the years to come.
Acknowledgements
D.B. is supported by ERC Consolidator Grant no. 647209. M.R. is supported by ERC Career Integration Grant no. 631145. A.O.J.C. is supported by Veni grant no. 016.155.083 from the Netherlands Organization for Scientific Research. M.S. is supported by a Spinoza Prize awarded by the Netherlands Organization for Scientific Research and ERC Advanced Grant no. 5120755-01.
Declaration of Interest
None.
