They reflect two distinct approaches to understanding connectivity. One approach—dynamic causal modelling DCM —tries to model how activity in one brain area is affected by activity in another using models of effective connectivity , while the other—Granger causal modelling GCM —tests for the signature of these influences by looking for correlations in the activity of two or more regions using models of functional connectivity.

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Previously, the relative accuracies of these methods, in disclosing patterns of communication among brain regions, were unknown. Here, we consider the motivation behind the two techniques, their underlying assumptions, and the implications of David et al. Most human brain mapping studies appeal to one of two principles of functional brain organisation: functional segregation and integration. Functional segregation posits a regionally specific selectivity for neuronal computations; for example, certain brain areas e. Functional integration, on the other hand, speaks to distributed interactions among functionally segregated regions.

Studies of functional integration seek to understand how regional responses are mediated by connections between brain areas and how these connections change with experimental manipulations or disease. Functional integration is usually analysed in terms of functional or effective connectivity.

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Images of these haemodynamic responses are typically acquired every few seconds, producing a time-series of fMRI data at each point in the brain. Functional connectivity is defined as a statistical dependency between these regional responses over time e. Analyses of functional connectivity are concerned with the spatial deployment of these dependencies; in other words, which areas correlate with which other areas. On the other hand, effective connectivity is concerned with the directed influence one brain region exerts on another.

This approach, unlike functional connectivity, tries to understand how one brain region affects another. To measure effective connectivity, one has to have a model of how this influence is mediated. Analyses of effective connectivity then try to quantify coupling in terms of the parameters of the connectivity model. In what follows, we will consider DCM and GCM in light of the above distinction between functional and effective connectivity. In , two techniques were introduced that addressed temporal dependencies and directed influences among distributed brain responses.

However, beyond this, they differ radically in their ambitions and domains of application. We will look at these differences from the point of view of their underlying models, the inferences they afford, their implicit notion of causality, and their history. These models invoke hidden neuronal and biophysical states that generate data.

In contrast, GCM rests upon a phenomenological model of temporal dependencies among the data themselves [ 5 ], without reference to how those dependencies were caused see Figure 1. This distinction becomes crucial for fMRI, because fMRI signals are haemodynamic convolutions of underlying neuronal signals. In other words, the fMRI signals are the products of a complicated chain of physiological events that are initiated by changes in neuronal activity.

This means that the observed fMRI response to a neuronal activation can be delayed and dispersed by several seconds. The convolution or impulse response function, mapping from underlying neuronal activity to observed fMRI responses, is called a haemodynamic response function and typically peaks at about four seconds see Figure 2.

DCM assumes haemodynamic signals are caused by changes in local neuronal activity, mediated by experimental inputs e. DCM is based on a model of this distributed processing and is parameterised by the strength of coupling among the neuronal regions. This neuronal model is then supplemented with a haemodynamic model that converts the neuronal activity into predicted haemodynamic signals.

For fMRI, the neuronal models are usually fairly simple and are based upon low-order approximations to otherwise complicated equations describing the evolution of neuronal states see Figure 1. In contrast, the haemodynamic model is rather complicated see Figure 2. Both the neuronal and haemodynamic parts of the DCM are specified in terms of non-linear differential equations in continuous time hence dynamic. The parameters of these equations encode the strength of connections and how they change with experimental factors.

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It is these parameters DCM tries to estimate. In DCM for fMRI, bilinear differential equations describe the changes in neuronal activity x t i in terms of linearly separable components that reflect the influence of other regional state variables. Known deterministic inputs u t elicit a change in neuronal states directly though c i or increase the coupling parameters a ij in proportion to the bilinear coupling parameters b ij. The neuronal states enter a region-specific haemodynamic model to produce the outputs y t i.

GCM tries to model the ensuing dependencies among the outputs with a time-lagged linear regression of the current response on previous responses up to an order denoted by p. The DCM is effectively a state-space model formulated in continuous time; whereas the GCM is a vector autoregression model in discrete time. See Figure 2 for a fuller explanation of the haemodynamic part of the model. A This schematic shows the architecture of a haemodynamic model for a single region. Neuronal activity induces a vasodilatory and activity-dependent signal s that increases blood flow f.

Flow causes changes in volume and deoxyhemoglobin v and q. These two haemodynamic states enter an output non-linearity to give the observed fMRI signal y. B This transformation from neuronal states x t i to haemodynamic response y t i is encoded graphically by the boxes in the previous figure and corresponds to a convolution. The implicit convolution kernel or haemodynamic response function is shown in the insert for typical values of the haemodynamic model's parameters.

Conversely, the model used by GCM is formulated in discrete time and usually rests upon the assumption that any statistical dependencies among brain regions can be approximated by a linear mapping over time-lags although more sophisticated non-linear models can be used; e. GCM has no notion of experimental inputs or evoked responses and assumes the fMRI signals are stationary and are driven by random fluctuations; see [ 7—9 ]. The parameters of their underlying regression models encode the degree of statistical dependence between regions and are simple regression coefficients.

In summary, the models employed by DCM are complicated and domain-specific, in relation to the simple and generic models used in GCM. So why go to the trouble of creating realistic models of brain processes? Basically, because it allows one to compare different models or hypotheses about distributed neuronal computations.

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In DCM one fits or inverts the models by optimising the distribution of their parameters i. Put simply, one finds the distribution of parameters that renders the data the most likely, under the DCM considered. This optimisation furnishes two things. It provides the most likely parameters for any given model, and the model evidence itself. This evidence is simply the probability of observing the data under a particular model.

The model evidence is a very important quantity because it allows one to compare different models and adjudicate among them [ 10 ]. In other words, it allows one to explore model space and find the best model that explains the data in a parsimonious way. If one equates each model with a hypothesis about the neuronal architectures subtending observed data, the model evidence provides a quantitative and principled measure for evaluating beliefs about different hypotheses.

If the model with the mapping has more evidence than the model without, one can conclude that the mapping or dependency exists. This inference is usually the end-point of the GCM, because the parameters regression coefficients per se have no biophysical meaning. In summary, DCM enables model comparison over a number of competing hypotheses or models and inference on the biophysical parameters of the model selected.

In GCM there are only two models, with and without a particular functional connection, and the object is to infer that this dependency exists. In both cases, establishing evidence for one model, in relation to another, allows one to declare some causal relationship; but is the nature of this causality the same? Source: Web of Science. Effective connectivity within the distributed cortical network for face perception.

Cereb Cortex — Interhemispheric integration of visual processing during task-driven lateralization. J Neurosci — Cerebral pathways in processing of affective prosody: A dynamic causal modeling study. Neuroimage — Analyzing and shaping human attentional networks.

Neural Netw — Synaptic plasticity and dysconnection in schizophrenia. Biol Psychiatry — Shifts of effective connectivity within a language network during rhyming and spelling. Dissociating reading processes on the basis of neuronal interactions. J Cogn Neurosci — Investigating the functional role of callosal connections with dynamic causal models. Ann N Y Acad Sci 16— Where bottom-up meets top-down: Neuronal interactions during perception and imagery.

A dynamic causal modeling study on category effects: Bottom-up or top-down mediation? This is why it is referred to as Granger causality or G-causality.

Conversely, causality in DCM is used in a control theory sense and means that, under the model, activity in one brain area causes dynamics in another, and that these dynamics cause the observations. Standard intent-to-treat analyses examine only the Z association with Y and so are estimating the effect of treatment assignment , rather than a physiologic effect of received treatment X. Can we also estimate the latter effect?

The answer is yes, provided we can make further not necessarily unique quantitative assumptions. The graph makes clear that we should not expect the crude X-Y association to equal the X-Y effect, because of confounding by U. These facts alone can allow one to put bounds on the X-Y effect, 3 Sec. Suppose we go beyond Figure 1c by assuming the linear structural relations. Second, we can substitute 4a into 4b to get. This ratio is an example of an instrumental-variables estimate of effect; 3 Sec.

For instrumental variables, algebraic modelling led to discovery of assumptions plausible in some settings that are sufficient for estimating the effects of interest from the given data. Nonetheless, by focussing our attention on basic qualitative relations, graphs can help identify fallacies in causal inference. Some examples were given in our discussion of Figures 1a and 1b ; as another example, some epidemiologists still believe mistakenly that an extraneous factor cannot induce selection bias unless it is a risk factor for disease. Consider a case-control study of magnetic-field exposure X and childhood leukaemia Y, with U representing socioeconomic factors and S selection.

It has been argued though disputed that socioeconomic factors have little or no effect on childhood-leukaemia risk as opposed to diagnosis or mortality ; there is evidence, however, that those factors are associated with magnetic fields and with participation. Figure 1d summarizes this background. Consequently, U would have to be controlled in order to ensure an unbiased estimate of the X-Y effect. Such control could not be accomplished if U were unmeasured or poorly measured. Note however that if X itself affected selection, there would be no way to remove the resulting selection bias through control of a covariate.

When using models in data analysis, it is essential to consider the distribution of exposure and confounders in the combined study population of all treatment or exposure groups that are under comparison, not in some specific target group of policy interest. Furthermore, in a population-based case-control study this population will be the source population of cases and controls, not just the subjects selected into the study. A controversial issue in all theories of causation is whether a variable must be manipulable to be considered potentially causal.

Even when technology advances enough to allow alteration of a previously immutable characteristic e. In potential-outcome models, the levels of immutable variables may be represented by strata i. In graphical and structural models, immutable variables may appear as exogenous variables, and so are not distinguished from manipulable exogenous variables. A more severe problem arises when variables that are not interventions are treated as interventions for planning purposes. This effect is quite dependent on how the disease is eliminated; for example, if it is eliminated by chemoprevention or vaccination, there may be occasional fatal side effects, or there may be causal or preventive effects on other potentially fatal diseases.

Of the four causal modelling methods reviewed here, SCC models the only ones originating in epidemiology stand apart in requiring specification of mechanisms within the individual units under study. There are rarely data to support such detailed specification, which may explain why SCC models have seen little use beyond teaching examples. Structural equations have seen extensive analytic application especially in the social sciences 10, 12, 13, 31 , and potential-outcome models have been used to derive permutation tests for randomized trials for 80 years.

Furthermore, the most recent innovations based on potential outcomes g-estimation 38, 39 and marginal structural modelling 40 are designed for longitudinal data on time-varying exposures and confounders, which precludes their use in many if not most studies; the techniques also require special programming. Due to their qualitative form, graphical models have not led to as many analytic techniques as have algebraic models.

On the other hand, they can be easily applied in any study to display assumptions of causal analyses, and to check whether covariates or sets of covariates are insufficient, excessive, or inappropriate to control given those assumptions. There are now at least four major classes of causal models in the health-sciences literature: Causal diagrams graphical causal models , potential-outcome models, structural-equations models, and sufficient-component cause models.

Causal diagrams can provide an easily understood depiction of qualitative assumptions behind a causal analysis, while potential-outcome and structural-equations models can depict more detailed quantitative assumptions about responses of units comprising the study population. Sufficient-component cause models differ from the other models in that they depict more elaborate qualitative assumptions about causal mechanisms within population units.

Four causal diagrams used in examples. In all four, X and Y are the exposure and outcome variables under study. Two distinct sufficient-component cause SCC models for the set of mechanisms within an individual; each leads to the same potential-outcome model. The authors would like to thank Charles Poole, Judea Pearl, Katherine Hoggatt, and a referee for helpful comments on this paper. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide.

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Graphical models. Potential-outcome counterfactual models. Multifactorial causation and the sufficient-component cause model. Structural-equations models. Graphical versus algebraic representations. An overview of relations among causal modelling methods Sander Greenland. Oxford Academic. Google Scholar. Babette Brumback.

Cite Citation. Permissions Icon Permissions. Abstract This paper provides a brief overview to four major types of causal models for health-sciences research: Graphical models causal diagrams , potential-outcome counterfactual models, sufficient-component cause models, and structural-equations models. Bias , causal diagrams , causality , confounding , data analysis , direct effects , epidemiological methods , graphical models , inference , instrumental variables , risk analysis , sufficient-component cause models , structural equations.

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## Structural equation modeling

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