Hick's law

Law

Given n equally probable choices, the average reaction time T required to choose among the choices is approximately:

${\displaystyle T=b\cdot {\log }_2(n+1)}$

where b is a constant that can be determined empirically by fitting a line to measured data. The logarithm expresses depth of "choice tree" hierarchy – log₂ indicates binary search was performed. Addition of 1 to n takes into account the "uncertainty about whether to respond or not, as well as about which response to make."

In the case of choices with unequal probabilities, the law can be generalized as:

${\displaystyle T=bH}$

where H is strongly related to the information-theoretic entropy of the decision, defined as

${\displaystyle H=\sum _i^n{p}_i{\log }_2(1/p_i+1)}$

where p_i refers to the probability of the ith alternative yielding the information-theoretic entropy.

Hick's law is similar in form to Fitts's law. Hick's law has a logarithmic form because people subdivide the total collection of choices into categories, eliminating about half of the remaining choices at each step, rather than considering each and every choice one-by-one, which would require linear time.

Relation to IQ

E. Roth (1964) demonstrated a correlation between IQ and information processing speed, which is the reciprocal of the slope of the function:

${\displaystyle \text{Reaction Time}=\text{Movement Time}+\frac{{\log }_2(n)}{\text{Processing Speed}}}$

where n is the number of choices. The time it takes to come to a decision is proportional to:

${\displaystyle \frac{{\log }_2(n)}{\text{Processing Speed}}}$

Stimulus–response compatibility

The stimulus–response compatibility is known to also affect the choice reaction time for the Hick–Hyman law. This means that the response should be similar to the stimulus itself (such as turning a steering wheel to turn the wheels of the car). The action the user performs is similar to the response the driver receives from the car.

Exceptions

Studies suggest that the search for a word within a randomly ordered list—in which the reaction time increases linearly according to the number of items—does not allow for the generalization of the scientific law, considering that, in other conditions, the reaction time may not be linearly associated to the logarithm of the number of elements or even show other variations of the basic plane.

Exceptions to Hick's law have been identified in studies of verbal response to familiar stimuli, where there is no relationship or only a subtle increase in the reaction time associated with an increased number of elements, and saccade responses, where it was shown that there is either no relationship, or a decrease in the saccadic time with the increase of the number of elements, thus an antagonistic effect to that postulated by Hick's law.

The generalization of Hick's law was also tested in studies on the predictability of transitions associated with the reaction time of elements that appeared in a structured sequence. This process was first described as being in accordance to Hick's law, but more recently it was shown that the relationship between predictability and reaction time is sigmoid, not linear associated with different modes of action.

Hick's law is sometimes cited to justify menu design decisions. For example, to find a given word (e.g. the name of a command) in a randomly ordered word list (e.g. a menu), scanning of each word in the list is required, consuming linear time, so Hick's law does not apply. However, if the list is alphabetical and the user knows the name of the command, he or she may be able to use a subdividing strategy that works in logarithmic time.