A memory-based model of Hick's law

Abstract

We propose and evaluate a memory-based model of Hick's law, the approximately linear increase in choice reaction time with the logarithm of set size (the number of stimulus-response alternatives). According to the model, Hick's law reflects a combination of associative interference during retrieval from declarative memory and occasional savings for stimulus-response repetitions due to non-retrieval. Fits to existing data sets show that the model accounts for the basic set-size effect, changes in the set-size effect with practice, and stimulus-response repetition effects that challenge the information-theoretic view of Hick's law. We derive the model's prediction of an interaction between set size, stimulus fan (the number of responses associated with a particular stimulus), and stimulus-response transition, which is subsequently tested and confirmed in two experiments. Collectively, the results support the core structure of the model and its explanation of Hick's law in terms of basic memory effects.

Keywords: Hick’s law; choice reaction time; cognitive modeling; fan effect; memory.

Figure 1

Data sets illustrating Hick’s law, the approximately linear increase in choice reaction time with the logarithm of set size (n). Points represent data, gray lines represent linear fits (i.e., fits of Equation 1), and black lines represent memory-based model fits. Merkel’s (1885) data are from a table in Woodworth and Schlosberg (1954). The other data were estimated from figures in their respective sources. Hyman’s (1953) data come from conditions with equiprobable alternatives. Venables’ (1958) data come from his normal subjects.

Figure 2

Hick’s law as a function of practice. Left panel: Data from Hale (1968) for correct responses only, estimated from his figure. Right panel: Memory-based model fit.

Figure 3

Stimulus–response repetition effects that challenge the information-theoretic view of Hick’s law. Left panel: Data from Kornblum (1969), estimated from his figure. Right panel: Memory-based model fit. Gray lines represent linear fits to the data for comparison with the memory-based model fit.

Figure 4

Schematic illustration of stimulus (S) and set sources and their associations (denoted by arrows) with stimulus–response (S–R) chunks in declarative memory. Black and gray outlines indicate active and inactive elements, respectively, for a situation in which stimulus 1 from set size 2 is presented.

Figure 5

Parameter space partitioning results for modeling Hick’s law with variation in total source activation (W) and the proportion of the total source activation allocated to the set source (p_w).

Figure 6

Schematic illustration of the stimulus–response alternatives used in Experiments 1 and 2. Letters were stimuli and number keys were responses. Fan-1 stimuli (outlined in gray) appeared in only one set whereas fan-2 stimuli (outlined in black) appeared in two sets but mapped onto different responses.

Figure 7

Left panels: Data from Experiments 1 and 2 (top and bottom panels, respectively) as a function of stimulus–response transition (repetition or switch), stimulus fan (1 or 2), and the logarithm of set size (n). For Experiment 2, the data are for set repetitions only. Right panels: Memory-based model fits for Experiments 1 and 2 (top and bottom panels, respectively). Note that the y-axis scales differ between Experiments 1 and 2 to enable better visualization of the three-way interaction in each experiment.

Figure 8

Memory-based model predictions for correct reaction time and accuracy (left and right panels, respectively) as a function of the logarithm of set size (n) for different levels of noise (s).

Figure A1

Data from Experiment 2 for set switches as a function of response transition (repetition or switch), stimulus fan (1 or 2), and the logarithm of set size (n).