Understanding Motivation with the Progressive Ratio Task: a Hierarchical Bayesian Model

Abstract

The progressive ratio task (e.g., Wolf et al., Schizophrenia Bulletin, 40(6):1328–1337, 2014) is often used to assess motivational deficits of individuals with mental health conditions, yet the number of studies investigating its underlying mechanisms is limited. In this paper, we present a hierarchical Bayesian model for the cognitive structure of the progressive ratio task. This model may be used to investigate the underlying mechanisms of human behavior in progressive ratio tasks, which can identify the factors contributing to participants’ performance. A simulation study shows satisfactory parameter recovery results for this model. We apply the model to a progressive ratio data set involving people with schizophrenia, first-degree relatives of people with schizophrenia, and people without schizophrenia. Our analysis reveals that people with schizophrenia are more affected by elapsed time than people without schizophrenia, tending to lose motivation to exert effort as they spend more time and effort in the task, regardless of the effort-reward ratio. The first-degree relatives show intermediate effects of time and effort-reward optimization between people with and without schizophrenia, which indicates that first-degree relatives might share some deficits with people with schizophrenia, only not as severe.

Appendix : Model comparison

We conducted model comparisons to justify the choice to leave out effects of the reward level V_ic,n in Eq. (3) for the probability to quit within initialized sets, and to justify having individual levels in the intercept and main effect parameters.

We compared the model stated in Eqs. (2) and 3, and the following model which contains effects of the reward level,

$$ p_{ic,n} = \frac{1}{1+\exp(-Z^{p}_{ic,n})}, $$

$$ \begin{array}{ll} Z^{p}_{ic,n} & = \mu_{ic} + \mu_{ic}^{10} \hat{V}_{ic,n}^{10} + \mu_{ic}^{01} \hat{V}_{ic,n}^{01} \\ & + \eta^{p}_{ic} \hat{M}_{ic,n} + \eta_{c}^{10} \hat{V}_{ic,n}^{10} \hat{M}_{ic,n} + \eta_{c}^{01} \hat{V}_{ic,n}^{01} \hat{M}_{ic,n} \\ & + \gamma^{p}_{ic} \hat{t}_{ic,n} + \gamma_{c}^{10} \hat{V}_{ic,n}^{10} \hat{t}_{ic,n} + \gamma_{c}^{01} \hat{V}_{ic,n}^{01} \hat{t}_{ic,n} \\ & + \psi_{c} \hat{M}_{ic,n} \hat{t}_{ic,n}, \end{array} $$

(6)

and

$$ q_{ic,n} = \frac{1}{1+\exp(-Z^{q}_{ic,n})}, $$

$$ \begin{array}{ll} Z^{q}_{ic,n} & = \mu_{ic} + \mu_{ic}^{10} \hat{V}_{ic,n}^{10} + \mu_{ic}^{01} \hat{V}_{ic,n}^{01} \\ & + \eta^{q}_{ic} \hat{M}_{ic,n} + \eta_{c}^{10} \hat{V}_{ic,n}^{10} \hat{M}_{ic,n} + \eta_{c}^{01} \hat{V}_{ic,n}^{01} \hat{M}_{ic,n} \\ & + \gamma^{q}_{ic} \hat{t}_{ic,n} + \gamma_{c}^{10} \hat{V}_{ic,n}^{10} \hat{t}_{ic,n} + \gamma_{c}^{01} \hat{V}_{ic,n}^{01} \hat{t}_{ic,n} \\ & + \psi_{c} \hat{M}_{ic,n} \hat{t}_{ic,n}, \end{array} $$

(7)

We compared their model fit by way of the deviance information criterion (DIC), using both Spiegelhalter et al.’s (2002) and Gelman et al.’s (2013) methods to compute the effective number of parameters. A smaller DIC indicates relatively better fit.

We obtained a chain for each model containing 5000 burn-in samples and 30000 total iterations. To avoid autocorrelations, we thinned the chain by keeping every 6th iteration, resulting in 6000 samples from the posteriors to compute the model comparison statistics. The hierarchical Bayesian model from Eqs. (2) and 3 has a DIC of 615.28 according to Spiegelhalter et al.’s (2002) method, and 797.06 according to Gelman et al.’s (2013) method. The alternative model from Eqs. (6) and 7 has a DIC of 619.24 according to Spiegelhalter et al.’s (2002) method, and 838.61 according to Gelman et al.’s (2013) method. The log Bayes factor of the hierarchical Bayesian model (Eqs. (2) and (3)) over the alternative model is 19.76, according to the generalized harmonic mean estimator method (Gronau et al., 2017). Both DIC and the Bayes factor indicate a better fit of the hierarchical Bayesian model, which justifies leaving out the reward level effects for the probability to quit within a set.

To justify the inclusion of individual-level parameters in Eq. (2), we compared the hierarchical Bayesian model from Eqs. (2) and 3 (denoted as “full”) and truncated models excluding each one of the individual-level parameters. We generated 5000 burn-in samples and 30000 total iterations for each model, and kept every 6th iteration to compute the model statistics. Table 2 shows the DICs and log Bayes factors: all model comparison statistics favor the full model.

Table 2 Model comparison statistics