The person as channel: entropy, choice reaction time and the limits of information theory

A design review produces a sentence such as "eight commands, so the choice costs three bits". The arithmetic is exact. The sentence that usually follows is not. It says a person is a channel with a fixed capacity, and that time is information over capacity. This unit argues that entropy measures the uncertainty an interface presents, given a distribution over outcomes, and that the fixed-capacity claim is unsupported. The rates that would define the capacity disagree, each comes from one narrow task, and the relation they summarise breaks outside the range measured. Unit 6's law survived its own derivation; the fixed-capacity reading of this unit's law does not.

Entropy is exact once the distribution is named

Shannon (1948) defined entropy as the sum, over outcomes, of each probability times the logarithm of one over that probability: zero when the outcome is certain and greatest when the outcomes are equally likely. MacKay (2003) splits the definition in two, the information content of an outcome being the logarithm of one over its probability and the entropy of an ensemble averaging that quantity. Entropy is therefore largest for equiprobable outcomes and additive for independent ones. English is roughly 50 per cent redundant over blocks of about eight letters (Shannon, 1948).

A constructed case shows why the distribution matters. A dialog with five equally likely actions carries 2.32 bits; if one action takes 60 per cent of the traffic and the rest split, the same five actions carry 1.77 bits. The same arithmetic gives 2.00 bits for four equally likely trial types and 1.36 bits for one at .70 and three at .10, the skews of probability that Mordkoff (2017) manipulates. Entropy compares two distributions over the same outcomes; it does not report how long the decision takes.

Hick measured a slope, not a capacity

Hick (1952) took the definition seriously. In choice-reaction experiments with up to ten alternatives he found the rate of gain of information constant on average within the duration of one perceptual-motor act, at about five bits per second, and a ten-choice experiment in which participants traded errors for speed gave the same result. Reaction times across his ten stimuli related to the objective uncertainty about the response.

Two qualifications are easy to lose: Hick confined the constancy to one act, not to the person, and five bits per second is a slope through a set of conditions rather than a capacity of a channel. Hyman (1953) supplied the general form, in which reaction time grows with the number of equally probable alternatives and depends on the information content of the stimulus, written as mean reaction time equals a plus b times H (Mordkoff, 2017).

The rate is not one number, even inside one paradigm

If b were a capacity, b would be stable. Longstreth, El-Zahhar and Alcorn (1985) ran five experiments with a response-duration measure, in which the start of the response is the same for every stimulus and its termination differs. They replicated an anomaly, a slope of about 15 milliseconds per bit against the usual 150; their procedure excluded the three conditions that carried the standard explanations, and their replacement makes time a function of the number of alternatives.

Proctor and Schneider (2018) list what the law does not handle: compatibility, practice, very large set sizes and sequential dependencies. Mordkoff (2017) adds that the block-level relation says nothing about the trial types inside the block, where times differ, and that trial-frequency predictors do not fit those differences: his hierarchical model keeps average uncertainty first and adds relative trial probability, and it fits well. A relation that needs a second layer is a description of behaviour, not a capacity.

A capacity needs a channel, and a task is not one

Cover and Thomas (2006) define the information capacity of a discrete memoryless channel as the maximum mutual information between input and output over all input distributions, and prove the channel coding theorem: rates below capacity are achievable with codes whose error probability tends to zero, and rates above are not. The definition presupposes a stated input alphabet, output and noise process, and the theorem is asymptotic.

A worked instance shows what the definition demands. Take a four-alternative choice, with the four outcomes equally likely and independent from trial to trial. The entropy is exactly 2 bits and the mean reaction time is a number one can measure, but the capacity is the maximum mutual information over input distributions, and the input alphabet here is the experimenter's choice of which alternative to present. Nothing in the screen or the task fixes the noise process or the code the person uses, and the coding theorem's promise of error-free transmission applies to long blocks of symbols rather than to one press. Reading a capacity out of the slope requires setting those three things by assumption, and none of them is a property of the display.

An interface supplies the input alphabet, not the code, the output or the noise. MacKay (2003) adds that an information content belongs to an outcome under a distribution rather than to a screen.

Seow (2005) sets the two surviving information-theoretic laws side by side, and records that HCI took up Fitts's law and left the choice law behind. He treats the pointing rate as homologous to Hick's rate of gain of information: so reading one capacity into both is an inference from the units rather than a result.

A task's time is a sum of stage costs

Sternberg (1969) offers the alternative shape. His additive-factor method treats factors whose effects on mean reaction time are additive as acting on separate processing stages, and it produced staged models of classification tasks. Stage durations add without being stochastically independent, so a task's time is a sum of stage costs and one rate dividing its entropy across the whole task has the wrong shape.

Card, Moran and Newell (1980) price uncertainty in operators instead, observing 1,280 user-system-task interactions over 28 users, 10 systems and 14 tasks and modelling unit-task time as a linear function of analyst-chosen operators, including a mental operator estimated from their data. Expert time was predicted to within about 20 per cent.

Miller (1956) gives the reason the sum cannot be read off the screen. The spans of absolute judgement and immediate memory limit how much information can be received and remembered, and recoding into chunks stretches the bottleneck. When a user's chunks are not the interface's items, the entropy of the display is not the entropy of the decision. Whose chunks those are is the other half. Recoding is a property of the person's knowledge rather than of the display, so a user working in a second language and a screen-reader user hearing the items in sequence all chunk differently from the practised participant the slope was fitted on.

Contested question: bits, alternatives, or neither?

The first position treats the law as a description of a processing stage within a range rather than a capacity: Hick (1952) states the range himself, Proctor and Schneider (2018) catalogue the effects omitted, and Longstreth and colleagues (1985) report a slope an order of magnitude from the canonical one, with a replacement law in the number of alternatives.

The second holds that entropy is the right variable and the slope a real rate. Its strongest evidence is Hyman's own. Hyman (1953) manipulated average uncertainty three ways, by the number of alternatives, by unequal frequencies and by sequential dependency, and the regression of reaction time on information was the same whichever method was used. A relation that survives three manipulations of one quantity is not an artefact of one procedure, and Proctor and Schneider (2018) call it one of the few acknowledged laws in psychology. The three positions are unequal in evidence: Hick's and Hyman's are single-paper laboratory results with up to ten alternatives. Longstreth's anomaly comes from a response-duration paradigm whose measurement has itself been disputed, and Mordkoff's is a model fitted to existing data rather than a new experiment.

The third denies that a single function of whole-set entropy will do: Mordkoff (2017) shows the block-level relation leaves trial-level differences unexplained, and that the model fitting them takes relative trial probability as a second predictor.

What would settle it is a design that separates the candidates. Set size, probability skew and sequential dependency would be manipulated orthogonally, so that H and the number of alternatives move independently, and a model in H, Longstreth's function of the number of alternatives and Mordkoff's two-layer model would be compared on the same data with the criterion fixed in advance. The winner would then be tested on prediction outside the fitted range. A slope that changes with the method used to manipulate H would count against the entropy reading, because Hyman's within-subject constancy is its strongest support. Meanwhile, compute the entropy of a choice set only when you have a defensible distribution over its members, use it to compare two arrangements, and never divide by a rate to obtain a time. Price the decision with an operator sequence instead.

The comparison is the part an interface can use

The usable part is the comparison. Card (1982) treats menu selection as recognition rather than recall, which is why menus suit untrained users. Tullis (1985) used a sorting technique with hierarchical cluster analysis to map users' view of a system's functions, then compared a broad menu hierarchy with a deep one, where users made significantly fewer inappropriate selections. The number of choices was not the whole story; their arrangement changed the error rate.

Where the argument is heading

The direction of travel is from a single rate to a layered model. Proctor and Schneider (2018) describe modelling in which the relation is embedded in the decision process, and Mordkoff (2017) extends the law to the trial level. Seow (2005) points the same way: HCI adopted the law whose parameter attached to a device and a task, and left the one whose parameter did not.

The sources this unit rests on

• Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379-423. doi:10.1002/j.1538-7305.1948.tb01338.x Defines entropy as the uncertainty of a set of probabilities and reports ordinary English as roughly 50 per cent redundant over blocks of about eight letters. It supplies the quantity computed here.
• Hick, W. E. (1952). On the rate of gain of information. Quarterly Journal of Experimental Psychology, 4(1), 11-26. doi:10.1080/17470215208416600 Ran choice-reaction experiments with up to ten alternatives and found the rate of gain of information constant at about five bits per second. Hick confined that constancy to one perceptual-motor act.
• Hyman, R. (1953). Stimulus information as a determinant of reaction time. Journal of Experimental Psychology, 45(3), 188-196. doi:10.1037/h0056940 Showed that reaction time grows with the number of equally probable alternatives and expressed the result as a function of the stimulus information content: the second half of the Hick-Hyman law.
• Miller, G. A. (1956). The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review, 63(2), 81-97. doi:10.1037/h0043158 Argues that the spans of absolute judgement and immediate memory limit how much information can be received and remembered, and that recoding into chunks stretches the bottleneck.
• Sternberg, S. (1969). The discovery of processing stages: extensions of Donders' method. Acta Psychologica, 30, 276-315. doi:10.1016/0001-6918(69)90055-9 Introduces the additive-factor method, in which factors with additive effects on mean reaction time act on separate processing stages; stage durations add, so a task's time is a sum of stage costs.
• Card, S. K., Moran, T. P. and Newell, A. (1980). The keystroke-level model for user performance time with interactive systems. Communications of the ACM, 23(7), 396-410. doi:10.1145/358886.358895 Observed 1,280 user-system-task interactions over 28 users, 10 systems and 14 tasks, and modelled unit-task time as a linear sum of analyst-chosen operators, predicting expert time to within about 20 per cent.
• Card, S. K. (1982). User perceptual mechanisms in the search of computer command menus. Proceedings of the 1982 Conference on Human Factors in Computing Systems (CHI). doi:10.1145/800049.801779 Analyses how users search a command menu and treats selection as recognition rather than recall, which is why menus suit untrained users.
• Tullis, T. S. (1985). Designing a menu-based interface to an operating system. ACM SIGCHI Bulletin, 16(4), 79-84. doi:10.1145/1165385.317471 Used a sorting technique with hierarchical cluster analysis to learn users' view of a system's functions, then compared a broad menu hierarchy with a deep one, where users made significantly fewer inappropriate selections.
• Longstreth, L. E., El-Zahhar, N. and Alcorn, M. B. (1985). Exceptions to Hick's law: explorations with a response duration measure. Journal of Experimental Psychology: General, 114(4), 417-434. doi:10.1037/0096-3445.114.4.417 Five experiments with a response-duration measure replicated a slope of about 15 milliseconds per bit against the usual 150, proposing a replacement in the number of alternatives rather than in entropy.
• Seow, S. C. (2005). Information theoretic models of HCI: a comparison of the Hick-Hyman law and Fitts' law. Human-Computer Interaction, 20(3), 315-352. doi:10.1207/s15327051hci2003_3 Compares the two surviving information-theoretic laws of human performance and asks why HCI took up Fitts's law and left the Hick-Hyman law behind. It treats the pointing rate as homologous to Hick's.
• Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory. Wiley, 2nd edition. doi:10.1002/047174882X Develops entropy, mutual information and channel capacity, defining capacity as the maximum mutual information over input distributions and proving the coding theorem. It makes capacity conditional on a stated channel.
• MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. https://www.inference.org.uk/mackay/itila/ Defines the information content of an outcome and the entropy of an ensemble as the average of that quantity, greatest for equiprobable outcomes and additive for independent variables. It requires the distribution to be named first.
• Mordkoff, J. T. (2017). Effects of average uncertainty and trial-type frequency on choice response time. Psychonomic Bulletin and Review, 24(6), 2012-2020. doi:10.3758/s13423-017-1263-2 Notes that the law relates average uncertainty to block mean response time and leaves trial-type differences inside the block unexplained, and fits a hierarchical model adding relative trial probability.
• Proctor, R. W. and Schneider, D. W. (2018). Hick's law for choice reaction time: a review. Quarterly Journal of Experimental Psychology, 71(6), 1281-1299. doi:10.1080/17470218.2017.1322622 Reviews Hick's experiments and analyses and sets out the challenges the law does not handle: stimulus-response compatibility, practice, very large set sizes and sequential dependencies.