Hick's law: Difference between revisions

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{{Short description|Time to make a decision as a result of the possible choices}}

'''Hick's law''', or the '''Hick-Hyman law''', is a [[human-computer interaction]] [[model (abstract)|model]] that describes the time it takes for a [[user (computing)|user]] to make a decision as a function of the possible choices he or she has. Given ''n'' equally probable choices, the average reaction time ''T'' required to choose among them is approximately

{{more footnotes|date=October 2013}}

'''Hick's law''', or the '''Hick–Hyman law''', named after British and American psychologists [[W. E. Hick|William Edmund Hick]] and [[Ray Hyman]], describes the time it takes for a person to make a decision as a result of the possible choices: increasing the number of choices will increase the decision time [[logarithm]]ically. The Hick–Hyman law assesses cognitive information capacity in choice reaction experiments. The amount of time taken to process a certain amount of bits in the Hick–Hyman law is known as the "rate of gain of information". The plain language implication of the finding is that increasing the number of choices does not directly increase the time to choose. In other words, twice as many choices does not result in twice as long to choose. Also, because the relationship is logarithmic, the increase in time it takes to choose becomes less and less as the number of choices increases.

==Background==

:<math>T = b \log_{2}(n + 1)</math>

In 1868, [[Franciscus Donders]] reported the relationship between having multiple [[stimulus (psychology)|stimuli]] and choice reaction time. In 1885, J. Merkel discovered that the response time is longer when a stimulus belongs to a larger set of stimuli. Psychologists began to see similarities between this phenomenon and [[Information Theory|information theory]].{{Who|date=August 2017}}

Hick first began experimenting with this theory in 1951.<ref>{{Cite journal|last=Hick|first=W.E.|year=1952|title=On the rate of gain of information|journal=Quarterly Journal of Experimental Psychology|volume=4|issue=1|pages=11–26|doi=10.1080/17470215208416600|s2cid=39060506 |url=http://www2.psychology.uiowa.edu/faculty/mordkoff/InfoProc/pdfs/Hick%201952.pdf}}</ref> In his first experiment, there were 10 lamps arranged circularly around the subject. There were 10 [[Telegraph key|Morse keys]] for each of his fingers that corresponded to these lamps. A running pre-punched tape roll activated a random lamp every 5 seconds; 4 electric pens recorded this lamp activation on moving paper in [[4-bit computing#:~:text=With 4 bits, it is,digit represented by four bits.|4-bit binary]]. When the subject tapped the corresponding key, the 4 pens recorded the response, using the same system. Although Hicks notes his experimental design using a 4-bit binary recording process was capable of showing up to 15 positions and "all clear", in his experiment he required the device to give an accurate record of reaction time between 10 options after a stimulus for the experiment.

where ''b'' is a constant that can be determined empirically by fitting a line to measured data. According to Card, Moran, and Newell (1983), the +1 is "because there is uncertainty about whether to respond or not, as well as about which response to make." The law can be generalized in the case of choices with unequal probabilities ''p_i'' of occurring, to

Hick performed a second experiment using the same task, while keeping the number of alternatives at 10. The participant performed the task the first two times with the instruction to perform the task as accurately as possible. For the last task, the participant was asked to perform the task as quickly as possible.

While Hick was stating that the relationship between reaction time and the number of choices was logarithmic, Hyman wanted to better understand the relationship between the reaction time and the mean number of choices. In Hyman’s experiment, he had eight different lights arranged in a 6x6 matrix.<ref>{{cite journal|last=Hyman|first=R|title=Stimulus information as a determinant of reaction time|journal=Journal of Experimental Psychology|date=March 1953|volume=45|issue=3|pages=188–96|pmid=13052851|doi=10.1037/h0056940|s2cid=17559281}}</ref> Each of these different lights was given a name, so the participant was timed in the time it took to say the name of the light after it was lit. Further experiments changed the number of each different type of light. Hyman was responsible for determining a [[linear relation]] between reaction time and the information transmitted.

==Law==

[[Image:Hick's law plotted data.svg|thumb|right|300px|Data from W. E. Hick (1952) demonstrating Hick's Law: The relationship between reaction time and number of response options across two participants (red and blue).]]

Given ''n'' equally probable choices, the average reaction time ''T'' required to choose among the choices is approximately:

:<math>T = b \cdot \log_{2}(n + 1)</math>

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&nbsp;– 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."<ref>{{cite book|last1=Card|first1=Stuart K.|last2=Moran|first2=Thomas P.|last3=Newell|first3=A.|author-link3=Allen Newell|year=1983|title=The Psychology of Human–Computer Interaction|publisher=Lawrence Erlbaum|___location=Hilldale, London}}</ref>

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

:<math>T = b H</math>

where ''H'' is strongly related to the [[information theory|information-theoretic]] entropy of the decision, defined as

:<math>H = \sum_i^n p_i \log_{2}(1/p_i + 1)</math>

where ''p_i'' refers to the probability of the ''i''th alternative yielding the information-theoretic entropy.

Hick's law is similar in form to [[Fitts' law]]. Intuitively, one can reason that 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, requiring linear time.

Hick's law is similar in form to [[Fitts's law]]. Intuitively, one can reason that 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, requiringwhich would require linear time.

===Relation to IQ===

[[File:EN HICK.png|right|thumb|Roth's application of Hick's law<br>"Bit" is the unit of log₂(n).]]

E. Roth (1964) demonstrated a correlation between IQ and information processing speed, which is the [[Multiplicative inverse|reciprocal]] of the slope of the function:<ref name="roth1964">{{cite journal|last=Roth|first=Erwin|year=1964|title=Die Geschwindigkeit der Verarbeitung von Information und ihr Zusammenhang mit Intelligenz|trans-title=The speed of processing information and its connection with intelligence|journal=Zeitschrift für Experimentelle und Angewandte Psychologie|language=de|volume=11|pages=616–622}}</ref>

:<math>\text{Reaction Time} = \text{Movement Time} + \frac{ \log_{2}(n) } {\text{Processing Speed} }</math>

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

proportional to :<math>\frac{ \log_{2}(n) } {\text{Processing Speed}} </math>

==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==

[[File:Sigmoid2.jpg|right|thumb|180px|The sigmoid function remains a better representation than the linear function of the relationship between predictability and reaction time.]]

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,<ref name = Longstreth1985>{{cite journal | last1 = Longstreth | first1 = L. E. | last2 = El-Zahhar | first2 = N. | last3 = Alcorn | first3 = M. B. | year = 1985 | title = Exceptions to Hick's Law: Explorations With a Response Duration Measure | journal = Journal of Experimental Psychology: General | volume = 114 | issue = 4 | pages = 417–434 | language = en | doi = 10.1037/0096-3445.114.4.417| pmid = 2934496 }}</ref> and saccade responses, where it was shown that there is either no relationship,<ref name = Kveraga2002>{{cite journal | last1 = Kveraga | first1 = K. | last2 = Boucher | first2 = L. | last3 = Hughes | first3 = H. C. | year = 2002 | title = Saccades operate in Violation of Hick's law | journal = Experimental Brain Research | volume = 146 | issue = 3 | pages = 307–314 | language = en | doi = 10.1007/s00221-002-1168-8| pmid = 12232687 | s2cid = 21845688 }}</ref> 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.<ref name = Lawrence2008>{{ cite journal | last1 = Lawrence | first1 = B. M. | last2 = St. John | first2 = A. | last3 = Abrams | first3 = R. A. | last4 = Snyder | first4 = L. H. | year = 2008 | title = An anti-Hick's effect in monkey and human saccade reaction times | journal = Journal of Vision | volume = 8 | issue = 26 | pages = 26.1–7 | language = en | doi = 10.1167/8.3.26 | pmid = 18484832 | doi-access = free }}</ref>

Hick's law has been shown to apply in experiments where the user is presented with ''n'' buttons, each having a light bulb beside them. One light bulb is randomly lit up, after which the user must press the corresponding button as quickly as possible. Obviously, the decision to be made here is very simple, requiring little conscious thought.

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.<ref name = Stadler1992>{{cite journal | last1 = Stadler | first1 = M. A. | year = 1992 | title = Statistical Structure and Implicit Learning Serial | journal = Journal of Experimental Psychology: Learning, Memory, and Cognition | volume = 18 | issue = 2 | pages = 318–327 | language = en | doi = 10.1037/0278-7393.18.2.318}}</ref><ref name = Remillard2001>{{cite journal | last1 = Remillard | first1 = G. | last2 = Clark | year = 2001 | title = Implicit Learning of First-, second-, and Third-Order Transition Probabilities | journal = Journal of Experimental Psychology: Learning, Memory, and Cognition | volume = 27 | issue = 2 | pages = 483–498 | language = en | doi = 10.1037/0278-7393.27.2.483| pmid = 11294445 }}</ref> This process was first described as being in accordance to Hick's law,<ref name = Jamieson2009>{{cite journal | last1 = Jamieson | first1 = R. K. | last2 = Mewhort | year = 2009 | title = Applying an exemplary model to the serial reaction-time task: Anticipating from experience | journal = The Quarterly Journal of Experimental Psychology | volume = 62 | issue = 9 | pages = 1757–1783 | language = en | doi = 10.1080/17470210802557637| pmid = 19219752 | s2cid = 24593123 }}</ref> but more recently it was shown that the relationship between predictability and reaction time is [[sigmoid function|sigmoid]], not [[linear function|linear]] associated with different modes of action.<ref name = Pavão2016>{{cite journal | last1 = Pavão | first1 = R. | last2 = Savietto | first2 = J.P. | last3 = Sato | first3 = J.R. | last4 = Xavier | first4 = G. F. | last5 = Helene | first5 = A. F. | year = 2016 | title = On Sequence Learning Models: Open-loop Control Not Strictly Guided by Hick's Law | journal = Scientific Reports | volume = 6 | language = en | doi = 10.1038/srep23018 | pmid = 26975409 | page=23018| pmc = 4792158 | bibcode = 2016NatSR...623018P }}</ref>

Hick's law is sometimes cited to justify [[menu (computing)|menu]] design decisions (for an example, see [http://www.catb.org/~esr/writings/taouu/html/ch04s03.html]). However, applying the model to menus must be done with care. 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, the user will likely be able to use a subdividing strategy that may well require logarithmic time.

Hick's law is sometimes cited to justify [[menu (computing)|menu]] design decisions (for an example, see [http://www.catb.org/~esr/writings/taouu/html/ch04s03.html]). However, applying the model to menus must be done with care. 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 willknows likelythe name of the command, he or she may be able to use a subdividing strategy that mayworks well requirein logarithmic time.<ref>{{Cite book | last1 = Landauer | first1 = T. K. | last2 = Nachbar | first2 = D. W. | chapter = Selection from alphabetic and numeric menu trees using a touch screen | title = Proceedings of the SIGCHI conference on Human factors in computing systems - CHI '85 | pages = 73 | year = 1985 | isbn = 978-0897911498 | doi = 10.1145/317456.317470| s2cid = 17669570 }}</ref>

Yet another situation is when the user does not know the exact name of the command they seek in a menu, but would likely recognize it if they saw it. In this case, the user may or may not be able to use a subdividing search strategy, depending in part on how menu items are categorized and how well the user can use categories to speed their search.

==See also==

For Hick's law and [[Fitts' law]] considerations in the context of [[menu (computing)|menu]] and submenu design, see Landauer and Nachbar (1985).

* [[Power Lawlaw of Practicepractice]]

* ''[[The Paradox of Choice]]''

*[[Fitts's law|Fitts's Law]]

== See also Notes==

{{reflist}}

* [[The Magical Number Seven, Plus or Minus Two]]

* [[Power Law of Practice]]

==References==

*{{cite book|last1=Cockburn|first1=Andy|last2=Gutwin|first2=Carl|last3=Greenberg|first3=Saul|title=Proceedings of the SIGCHI Conference on Human Factors in Computing Systems |chapter=A predictive model of menu performance |date=April 28 – May 3, 2007|pages=627–636|___location=San Jose, California|doi=10.1145/1240624.1240723|isbn=9781595935939|chapter-url=http://ir.canterbury.ac.nz/bitstream/10092/662/1/12602883_paper191-cockburn.pdf|hdl=10092/662|s2cid=7340315 |hdl-access=free}}

* Original work

**{{cite [[journal|last=Hick|first=W. E. Hick]]. |title=On the rate of gain of information. |journal=Quarterly Journal of Experimental Psychology,|date=1 4:11-26,March 1952|volume=4|issue=1|pages=11–26|doi=10.1080/17470215208416600|s2cid=39060506 |url=http://www2.psychology.uiowa.edu/faculty/mordkoff/InfoProc/pdfs/Hick%201952.pdf}}

**{{cite journal|last=Hyman|first=R. Hyman. |title=Stimulus information as a determinant of reaction time. |journal=Journal of Experimental Psychology,|date=March 1953|volume=45:188-196, 1953|issue=3|pages=188–96|pmid=13052851|doi=10.1037/h0056940|s2cid=17559281}}

*{{cite book|last=Rosati|first=L.|chapter=How to design interfaces for choice: Hick-Hyman law and classification for information architecture|editor1=Slavic, A.|editor2=Salah, A.|editor3=Davies, C.|title=Classification and visualization: interfaces to knowledge: proceedings of the International UDC Seminar|date=October 24–25, 2013|___location=The Hague, The Netherlands|pages=125–138|chapter-url=https://www.researchgate.net/publication/270890224}}

* Selected subsequent work

*{{cite book|last1=Roy|first1=Q.|last2=Malacria|first2=S.|last3=Lecolinet|first3=E.|last4=Guiard|first4=Y.|last5=Eagan|first5=J.|title=Proceedings of the SIGCHI Conference on Human Factors in Computing Systems |chapter=Augmented letters: Mnemonic gesture-based shortcuts |date=April 27 – May 2, 2013|pages=2325–2328 |___location=Paris, France|doi=10.1145/2470654.2481321|isbn=9781450318990 |s2cid=15928158 |url=https://hal.archives-ouvertes.fr/hal-01164207/file/aletters-author-version.pdf |chapter-url=http://biblio.telecom-paristech.fr/cgi-bin/download.cgi?id=12999}}

** T. K. Landauer and D. W. Nachbar. Selection from alphabetic and numeric menu trees using a touch screen: Breadth, depth, and width. In Proceedings of [[Association for Computing Machinery|ACM]] CHI 1985 Conference on Human Factors in Computing Systems, pages 73--78, 1985. http://doi.acm.org/10.1145/317456.317470

*{{cite journal|last=Seow|first=Steven C.|year=2005|title=Information Theoretic Models of HCI: A Comparison of the Hick–Hyman Law and Fitts' Law|journal= Human-Computer Interaction|volume=20|issue=3|pages=315–352|doi=10.1207/s15327051hci2003_3|citeseerx = 10.1.1.86.4509 |s2cid=14436546 }}

* Overviews

*{{cite book|last=Welford|first=Alan T.|title=Fundamentals of Skill|___location=Methuen, Massachusetts|year=1968|pages=61–65}}

** Stuart K. Card, Thomas P. Moran, [[Allen Newell]] (1983). The Psychology of Human-Computer Interaction.

** A. T. Welford. Fundamentals of Skill. Methuen, 1968. Pages 61-65.

==External links==

* [http://www.usabilityfirst.com/glossary/main.cgi?function=display_term&term_id=266hicks-law/ Usability Glossary: Hick's Law]

[[Category:Human-computerExperimental interactionpsychology]]

[[Category:Human–computer interaction]]