Critical thinking
Article: Borsboom, Rhemtulla, Cramer, van der Maas, Scheffer and Dolan (2016)
Kinds versus continua: a review of psychometric approaches to uncover the structure of psychiatric constructs

The present paper reviews psychometric modelling approaches that can be used to investigate the question whether psychopathology constructs are discrete or continuous dimensions through application of statistical models.

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 DSM-V typically adheres to a categorical model, in which discrete diagnoses are based on patterns of symptoms.

But, such categorizations often involve apparently arbitrary conventions.

Measurement theoretical definitions of kinds and continua

All measurement starts with categorization, the formation of equivalence classes.
Equivalence classes: sets of individuals who are exchangeable with respect to the attribute of interest.
We may not succeed in finding an observational procedure that in fact yields the desired equivalence classes.

• We may find that individuals who have been assigned the same label are not indistinguishable with respect to the attribute of interest.
  Because there are now three classes rather than two, next to the relation between individuals within cases (equivalence), we may also represent systematic relations between members of different cases.
• One may do so by invoking the concept of order.
  But, we may find that within these classes, there are non-trivial differences between individuals that we wish to represent.

If we break down the classes further, we may represent them with a scale that starts to approach continuity.

The continuity hypothesis formally implies that:

• in between any two positions lies a third that can be empirically instantiated
• there are no gaps in the continuum.

In psychological terms, categorical representations line up naturally with an interpretation of disorders as discrete disease entities, while continuum hypotheses are most naturally consistent with the idea that a construct varies continuously in a population.

• in a continuous interpretation, the distinction between individuals depends on the imposition of a cut-off score that does not reflect a gap that is inherent in the attribute itself.

Kinds and continua as psychometric entities

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.
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 underlay the thoughts, feelings and behaviours we do observe.

• 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.

Models assume that, given a specific level of a latent variable, the indicators are uncorrelated.
This feature, local independence, is consistent with a causal interpretation of the effects of the latent on the observed variables.

• 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.
One may apply models in an attempt to determine the form of the latent structure.
This can be done in two ways:

• Taxometrics: 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
• 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.

The logic underlying taxometric analysis.
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.
If the underlying variable is a binary variable, then the covariance between any two indicators is expected to vary with the value of the proxy.

• at low values of the proxy, most individuals are healthy (they are in the healthy class)
  Within this class the covariance among the items is expected to be zero.
• 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 hypothesis.

To carry out a taxometric analysis:

•  one arbitrarily chosen variable is denoted as the ‘index’ variable, and is assumed to be a proxy of the underlying construct.
•  then, over a moving window of values on the index variable, the covariance between the other two variables 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 form 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 continuous latent structure. This is called MAXCOV
  • MAXEIG plots the eigenvalue of a matrix of item covariances instead of simple bivariate covariance, allowing more variables to be used in analysis.

The taxometric approach is not uncontroversial in psychometrics.

• it has long rested purely on the visual inspection of a plotted function instead of on a formal hypothesis test.
• one of its core assumptions (categorical latent structures will produce peaked covariance functions) is not necessarily true.
• 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.
• latent variable modelling approaches do rest on a firm mathematical basis.

Latent variable approaches are not without problems

• many continuous variable models have statistically equivalent categorical or mixture counterparts.
  • 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

McGrath and Walters (2012) 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 in the structure is determined to be categorical.

The hypothesis of kinds and continua do not exhaust the space of possibilities, so that evidence against one hypothesis is not necessary evidence for the other.

Alternative latent variable models

Factor mixture models

Finite mixture models partition the population into distinct latent classes, but allow for continuous variation within these classes.
If that variation is itself measured through a number of indicator variables, then we obtain a factor mixture model.

• the factor mixture model can this be understood as latent class model in which each latent class is characterized by its own common factor model.
• conditioning on the latent classes no longer renders the observed variables independent, as their conditional distribution is characterized by the factor model.
• 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 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.

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.
Mixture modelling allows us the connect factor model and latent class models by means of intermediate models and associated constrains.

GoM models

In GoM models, one can also depart form a simple latent class model to integrate continuous features.

• Here, the continuous variation concerns group membership itself.
• GoM model allows individuals to be members of multiple classes at the same time.
  GoM model holds that an individual belongs to both classes at the same time, but to different degrees.
• In the GoM model, the degree of membership is expressed in terms of a set of probabilities that sum to unity.

But, the GoM model is not widely used in psychometric applications, probably due to a lack of readily assessable statistical software to apply the GoM model.

Network models and dynamical systems

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.

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 indicators.
Correlations between variables commonly seen as ‘indicators’ then arise from a network of causal effects among these variables themselves (they form mechanistic property clusters).

• as such, in networks, there are no latent variables that function as psychologically meaningful common causes, even through individual nodes and connections may stand under the influence of factors that are not directly observed.

Individual differences in network structure may lead to different patterns of symptom dynamics.

• for a person with a weakly connected network, external stressors may lead to an increase in the number of symptoms that are activated but, when the external stressors are removed, the person will spontaneously and smoothly return to equilibrium.
• strongly connected networks may show strongly non-linear behaviour with sudden jumps from one state to another.
  • in strongly connected networks, symptom activation may be increased through feedback loops.

Differences in dynamics across different network structures are important to the kinds vs continua discussion.

• they show that disorders may be discrete kinds for some people (with strongly connected networks), and continuous for others (with weakly connected networks).
• individual differences may look like a continuous distribution even if, intra-individually, the transition from a healthy to a disordered state is discontinuous.

If present, discontinuous transitions have direct measurable consequences that may be exploited in further research.
Transitions from a healthy state to a disordered state are typically preceded by early warning signals that indicate that the system is close to a tipping point for a transition.