A Joint Modelling Approach to Analyze Risky Decisions by Means of Diffusion Tensor Imaging and Behavioural Data

Understanding dependencies between brain functioning and cognition is a challenging task which might require more than applying standard statistical models to neural and behavioural measures to be accomplished. Recent developments in computational modelling have demonstrated the advantage to formally account for reciprocal relations between mathematical models of cognition and brain functional, or structural, characteristics to relate neural and cognitive parameters on a model-based perspective. This would allow to account for both neural and behavioural data simultaneously by providing a joint probabilistic model for the two sources of information. In the present work we proposed an architecture for jointly modelling the reciprocal relation between behavioural and neural information in the context of risky decision-making. More precisely, we offered a way to relate Diffusion Tensor Imaging data to cognitive parameters of a computational model accounting for behavioural outcomes in the popular Balloon Analogue Risk Task (BART). Results show that the proposed architecture has the potential to account for individual differences in task performances and brain structural features by letting individual-level parameters to be modelled by a joint distribution connecting both sources of information. Such a joint modelling framework can offer interesting insights in the development of computational models able to investigate correspondence between decision-making and brain structural connectivity.

As can be noticed, Model 4 has the lowest DIC, andR approaching 1, and can be selected as the best model.

Multivariate Model tuning in JAGS
In this section we discuss some computational details related to the tuning of the Multivariate Student's t-distribution. More precisely, we focus on the estimation of the correlation parameters in the covariance matrix of the multivariate model.
In our model representation, we assumed a sparse covariance matrix in which some correlation parameters were fixed to zero since our confirmatory approach focused on testing specific meaningful relations between cognitive and neural parameters. Thus, prior distributions should be considered for each correlation coefficient in the decomposed covariance matrix.
The JAGS probabilistic programming framework allows to embed prior distributions in the 4 hierarchical model by considering the Multivariate Student's t-distribution as modelled according to a precision matrix instead of a covariance matrix. Matrix inversion is then needed when correlation coefficients have to be obtained. However, matrix inversion problems may arise when the assumption of positive-definite matrix is violated, and it is often the case in which this happens.
Differently, prior distributions for the covariance matrix parameters can be specified when the Multivariate Normal distribution is considered. This allows to estimate correlation coefficients when the Multivariate Student's t-distribution is taken into account by overcoming the limitations related to matrix inversion.
Consider the vector-valued random variable X * = [γ * , β * , α * 0 , α * 1 ]. We assume X * to have a Standard Multivariate Normal distribution: with 4-dimensional zero mean vector and covariance matrix Σ as structured according to the main text. Consider now the vector-valued random variable X = [γ, β, α 0 , α 1 ] containing the individual level cognitive and neural parameters. We model X as a Multivariate Student's t distributed random variable as follows: where µ = [µ γ , µ β , µ δ1 , µ δ2 ] and V is a Chi-square distributed random variable with parameter ρ denoting degrees of freedom. Note that parameters µ and Σ are estimated in a hierarchical bayesian framework in the main text.
Here, ζ was fixed to a default value such that ζ = 5. In general, values of ζ in the range  do not compromise posterior sampled chains mixing in our application. However, when ρ is treated as a parameter to estimate and left free to vary within a broader range, chains mixing is not ensured and parameter estimates are unreliable.

3 Simulation Study
In this section we provide a simulation study aimed to explore model performance in meaningful scenarios. In particular, a Monte Carlo 3-factorial design is employed to recover Effective Sample Size of posterior MCMC samples and Computation Time across levels of three factors, namely, Number of subjects, Number of ROI-to-ROI connections (which we refer to as ROIs), and the tuning scenarios.
In particular, in each cell of the factorial design, parameters are sampled from the prior, synthetic neural and behavioural data are simulated based on the sampled parameters, and the simulated data pattern is used to fit the joint model. For each cell, the process of data simulation and model fitting is repeated for 10 times. Number of subjects are allowed to vary across three levels, that is, [10,30,50], whilst the Number of ROIs across two levels, that is, [5,10]. Simulating neural data consists in directly sampling an FA measure related to a specific ROI-to-ROI connection. The tuning factor consists in two levels in which the degrees of freedom of Multivariate t-distribution is fixed to a default value such that ζ = 5, or is treated as a free parameter with an exponential prior such that ζ ∼ Exp(1/30). As can be noticed, computation time increases slightly linearly, based on the number of individuals.
Increasing the number of ROIs seems to contribute to extend the computation time especially for higher sample sizes. Scenarios in which ζ is treated as a free parameter (scenario 3 and 4) are the most computationally expensive. It is worth noticing that, in general, computations are rather cheap and this is due to the simplified joint model adopted. The increase of the computation time might not be linear in case the full model is employed. Figure 2 shows the distribution of the Effective Sample Size across the MCMC joint posterior samples for each cell of the factorial design. Such a metric helps assessing the convergence of the MCMC sampels, but it also quantifies how much independent information there is in autocorrelated chains (Krushke, 2010). Having non-autocorrelated chains ensure to decrease the uncertainty of the estimation of posterior quantities of interest, such as credible intervals, which are useful for empirical research. As a substantiated heuristic, the higher the Effective Sample size the better.
It is computed according to: where N is the number of samples for a given chain, and ACF(·) is the autocorrelation function at lag k. In general, scenarios in which ζ is fixed to a default value show a higher Effective Sample Size compared to that of the scenarios in which ζ is sampled and included in the joint posterior. Enriching information yielded by the data, by increasing both sample size and ROIs, seems to ensure a more reliable posterior distributions. This provides an advantage when more data are available, due the fact that computation time seems to be affected by larger data structure to a lesser extent. In a similar way, reliability of posterior samples when ζ is assumed as a free parameter seems to be affected by the amount of data available. On the other hand, ζ seems to be generally unreliable even when larger datasets are 8 considered. In our case, mean Effective Sample Size estimates for ζ samples were 143(SD=18) (resp. 272(SD=34)) for 50 synthetic subjects and 5 ROIs (resp. 15 ROIs).

Sensitivity Analysis
In this section we provide a Sensitivity Analysis to assess the influence of several class of prior distributions on model's posterior. Due to the complexity of the hierarchical model we propose an informal approach (Roos et al., 2015) based on a factorial design in which repetitive fits of the model are performed with ad hoc modified prior inputs. We provide 7 scenarios in which several classes of prior distributions are employed for the main parameter of interest: µ γ , µ β , µ α0 , µ α1 , ρ = (ρ 1 , ρ 2 , ρ 3 , ρ 4 ).
The choice of relying on such a subset is justified by the fact that the high number of parameters in the model would make an exhaustive treatment of each (hyper) prior infeasible. Table 1

JAGS Code
Here, the JAGS code of the generative model is provided: