# What Makes a UI Simple? Difficulty and Complexity in Tasks Engaging Visual-Spatial Working Memory
## 2 Methods and Related Work
### Information Theory-Based Measures  
The enormous effect of Information Theory rapidly caused attempts to apply [PERSON]'s approaches to describe human behavior. One of the regularities in cognitive psychology that stood the test of time was proposed in 1952 and was subsequently named [PERSON]'s law:  
\[RT\sim b\cdot\log_{2}(N+1), \tag{1}\]  
where \(RT\) is reaction time when choosing from equally probable alternatives, \(N\) is the number of alternatives, and \(b\) is a constant that [PERSON] named the rate of gain of information. When proposing the final formulation for his model, [PERSON] noted that it corresponds to the number of steps in hierarchical sequence of binary classification decisions [23]--hence the logarithm to the base two. Indeed, today we have convincing evidence that human information processing is based on binary search (dichotomies).  
Further, the generalization of (1) to non-equiprobable alternatives was proposed by [PERSON]:  
\[RT=a_{H}+b_{H}\cdot H, \tag{2}\]  
where \(H\) is the entropy (information quantity) of the stimuli set, \(a_{H}\) and \(b_{H}\) are empirically defined coefficients.  
Another akin regularity, inspired by the Shannon-Hartley theorem on maximum transmission rate in a noisy channel, was put forward in 1954 and consolidated as [PERSON] law:  
\[MT=a+b\log_{2}(A/W+1)=a+b\cdot ID, \tag{3}\]  
where \(MT\) is time needed to perform a rapid aimed movement to a target, \(A\) is movement distance, \(W\) is size of the target, \(a\) and \(b\) are coefficients usually found from regression. [PERSON] goal was to calculate the \"transmission rate\" of human motor system, so he formulated the index of difficulty (_ID_) as can be seen in (3) and used it to express the index or performance (_IP_):  
\[IP=ID/MT. \tag{4}\]  
An important subsequent advance was introduction of the means to account for the movement accuracy: the \"effective\" index of difficulty (_IDe_). In each actual movement task performed, effective distance (_Ae_) can be calculated as the distance actually travelled. Effective target width (_We_) is the normalized area where movements for each particular combination of the independent variables terminate (details can be found e.g., in [30]). Thus, the classic [PERSON] model (7) with the _IDe_ becomes:  
\[MT=a^{\prime}+b^{\prime}\cdot ID_{e}=a^{\prime}+b^{\prime}\cdot\log_{2}(A_{e}/ W_{e}+1). \tag{5}\]  
This allowed obtaining the motor system throughput (_TP_) measure that has substituted [PERSON] IP and that is calculated with some experimental data as follows:  
\[TP=\
icefrac{{1}}{{P}}\sum_{i=1}^{P}\big{(}\
icefrac{{1}}{{M}}\sum_{j=1}^{M} \frac{ID_{eij}}{MT_{ij}}\big{)}, \tag{6}\]  
where \(P\) is the number of participants and \(M\) is the number of combinations for the independent variables (_A_ and _W_) in the experiment. _TP_ is an objective and full measure of both speed and accuracy in certain experimental conditions, independent of a particular task's setup.  
Of the two above regularities, [PERSON] law sees much wider usage in HCI, which is particularly due to high practicality of the throughput measure, which can be used to effectively compare different versions of UIs or interface devices. For instance, [PERSON] law studies that employ standardized methodology reported throughput values in the ranges 3.7-4.9 bit/s for mouse, 1.6-2.6 for joystick, 1.0-2.9 for trackball, etc. [30] (p. 784). On the contrary, applicability of [PERSON]'s law (2) is rather limited, since it is unclear how to calculate the information quantity in stimuli in most practical tasks [31]. Examples of successful applications of the law in HCI include design of specially organized menus [32], describing behavior in visual search [33; 34] and perception of visual complexity [12]. Still, objective measurement of \"cognitive throughput\" remains much more complicated compared to the motor system's one, as we detail in the next sub-section.