1 - Generating expectations

Let v_k, j denote the associative strength from stimulus k to stimulus j. On any given trial, the expectation of stimulus j, e_j, is given by:

$$ \tag{Eq.1} e_j = \sum_{k}^{K}x_k v_{k,j} $$

x_k denotes the presence (1) or absence (0) of stimulus k, and the set K represents all stimuli in the design.

2 - Learning associations

Changes to the association from stimulus i to j, v_i, j, are given by:

Δv_i, j = x_iα_iβ_j(λ_j − v_i, j)

where α_i is the associability of (or attention devoted to) stimulus i, β_j is a learning rate parameter determined by the properties of j, and λ_j is a the maximum association strength supported by j (the asymptote).

3 - Learning to attend

The parameter α_i changes as a function of learning, proportionally to the difference between the absolute errors conveyed by i and all the other predictors¹, via:

$$ \tag{Eq.3} \Delta \alpha_{i} = x_i\theta_i \sum_{j}^{K}\gamma_j(|\lambda_j - \sum_{k \ne i}^{K}v_{k,j}|-|\lambda_j - v_{i,j}|) $$ where θ_i is an attentional learning rate parameter for stimulus i (usually fixed across all stimuli). Although Mackintosh (1975) did not extend their model to account for the predictive power of within-compound associations, the implementation of the model in this package does. This can sometimes result in unexpected behavior, and as such, Eq. 3 above includes an extra parameter γ_j (defaulting to 1/K) that denotes whether the expectation of stimulus j contributes to attentional learning. As such, the user can set these parameters manually to reflect the contribution of the different experimental stimuli. For example, in a simple “AB>(US)” design, setting γ_US = 1 and γ_A = γ_B = 0 leads to the behavior of the original model.

4 - Generating responses

There is no specification of response-generating mechanisms in MAC1975. However, the simplest response function that can be adopted is the identity function on stimulus expectations. If so, the responses reflecting the nature of j, r_j, are given by:

r_j = e_j