Model fitting and selection

The same data, averaged by group and condition, were then used to optimize a model of precision- weighted predictive processing in the covert orienting paradigm.

Speed of processing estimation

Speed of processing (SoP) components, estimated for the whole sample, are shown in Table 8.2. Table 8.2: Study 6 estimated sum of perceptual and motor speed of processing RT components by target modality and SOA.

Perceptual and motor SoP components (ms)

SOA Auditory target Somatic target Visual target

100 478 452 287

300 432 394 244

500 430 357 222

Model variations

A series of model variations was developed to investigate the most probable source, given overall model assumptions, of the group difference in validity identified in the classical analysis (Figure 8.2). All model variations were nested within the overall model described above.

1. Salience model: group difference in γ

In the first model, SoP estimates were based on the sample as a whole; confidence was fixed at 80%; and a single estimate of basal precision parameter τ was derived for the whole sample. Salience parameters γ were estimated independently for each group.

2. Basal precision model: group difference in τ

In the second model, SoP estimates were based on the sample as a whole, confidence was fixed at 80%, and salience parameters γ were estimated for the whole sample. Basal precision parameter τ was estimated independently for each group.

92 3. Confidence model: group difference in α

The preceding models both posit group differences in mechanisms contributing to perceptual precision. An alternative hypothesis suggests that depressed participants differ from healthy controls not in terms of perception itself, but in requiring a higher level of confidence (i.e., more time for evidence accumulation) before being willing to commit to a response. In the third model, therefore, a confidence parameter α was estimated for each group as follows:

𝛼 ~ 𝑈𝑛𝑖𝑓𝑜𝑟𝑚(0.5,1) (80)

The confidence parameter α overlapped substantially with basal precision parameter τ, due to their similarity in roles. An increase in τ increases the precision of the conditional confidence in the target, pulling the lower bound of the confidence estimate away from 0. Similarly, a reduction in α, by definition, pulls the lower bound of the confidence estimate in the same direction, reducing the amount of precision required to attain the cut-off. Only the unique contribution of τ to the prediction of cue and target differentiates the two parameters. Due to this overlap, attempts to estimate the two parameters simultaneously were found to lead to difficulties with convergence, and so τ was fixed at 2 for both groups. SoP estimates and salience parameters γ were estimated for the sample as a whole.

4. Speed of processing model: group difference in SoP

The fourth model assumed that confidence and precision parameters did not differ by group, and that group differences were solely accounted for by differences in perceptual and motor speed of processing. SoP and its consequences for data y were estimated independently for the two groups. Confidence was fixed at 80%, and precision parameters γ and τ were estimated for the whole sample. Combination models

All possible combinations of these models were also considered (a total of 15 models). Models in which both τ and α varied by group were discarded following initial runs which demonstrated that these functionally similar parameters traded off each other when both were allowed to vary within the same model, causing difficulties with convergence. The remaining 11 models were compared in terms of fit (Appendix A.3., Figure A.6).

It was observed that the addition of a group difference in SoP to any other group difference or combination of differences worsened the fit of the model, without any compensatory gains in terms of reduced model complexity. Five models which included group differences in SoP and at least one other parameter were discarded for this reason. Six models remained: the four index models described above, and two combination models as follows:

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5. Salience and precision model: group differences in γ and τ

This model assumes that group differences are present in both state-dependent and basal precision parameters (i.e., salience parameters γ and basal precision parameter τ). These parameters were allowed to vary by group, while confidence was held constant at 80% and SoP was estimated for the whole sample.

6. Salience and confidence model: group differences in γ and α

This model locates group differences in the salience parameters γ and the confidence parameter α. Basal precision was held constant at 2 and SoP was estimated for the whole sample.

Bayesian model comparison

All six models were entered into a Bayesian model comparison using the product space method (Carlin & Chib, 1995). The model comparison proceeded by fitting each model to the data y averaged by group, cue modality, cue validity and SOA; i.e., pooled over target modality as in the ANOVA (and for the same reasons). Posterior model probabilities are shown in Table 8.3. Bayesian model comparison penalises model complexity in addition to rewarding good fit, and so the basal precision model has a higher posterior probability than the more complex salience model, despite the latter’s better fit (Appendix A.3., Figure A.6). However, the best-fitting model was also the most probable one, providing a better description of the data by a Bayes factor of 13.02 relative to its nearest competitor. This model was the salience and precision model, in which both the salience parameter γ and the basal sensory precision parameter τ varied by group.

Table 8.3: Study 6 posterior model probabilities for predictive processing model of covert orienting

Model Location of group difference Posterior probability

5 Salience and precision model γ and τ 0.93

6 Salience and confidence model γ and α 0.07

2 Basal precision model τ 2.70 x 10-12

1 Salience model γ 1.28 x 10-14

3 Confidence model α 8.63 x 10-18

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