# What Makes a UI Simple? Difficulty and Complexity in Tasks Engaging Visual-Spatial Working Memory
## 4 Results
### Index of Difficulty  
In our formulation for the index of difficulty in accordance with (11), we considered three possible options for the memorized information items \(N_{I}=\{S_{P},F_{N},F_{M}\}\), corresponding to the significant factors we previously found in (13):  
\[IDM_{SP}=S_{P}\log_{2}S_{0}. \tag{15}\]  
\[IDM_{FN}=F_{N}\log_{2}S_{0}. \tag{16}\]  
\[IDM_{FM}=F_{M}\log_{2}S_{0}. \tag{17}\]  
In Table 3 we present the candidate indexes' ranges, means, SDs and correlations (all of them significant, \(p<0.001\)) with the dependent variables. As expected, due to the designation stage, \(T_{M}\) had the highest correlation (r\({}_{4546}\) = 0.390) with \(IDM_{SP}\), which was also the strongest correlate for _Complexity_ (r\({}_{4546}\) = 0.421). However, for \(E_{M}\), the highest correlation (r\({}_{4546}\) = 0.340) was found with \(IDM_{FN}\), which was even higher than with \(F_{N}\) (see in Table 2 above), which supports Hypothesis 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **r (\(E_{M}\))** & **\(\tau\) (\(Complexity\))** \\ \hline \(S_{0}\) & 0.176 & 0.13 \\ \(S_{P}\) & 0.231 & **0.424** \\ \(F_{N}\) & **0.332** & 0.329 \\ \(F_{M}\) & 0.239 & 0.115 \\ \(L_{RLE}\) & 0.302 & 0.410 \\ \(L_{DEF}\) & 0.215 & 0.342 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Correlations of the independent variables with \(E_{M}\) (Pearson) and _Complexity_ (Kendall tau-b).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Index of Difficulty** & **Range** & **Mean (SD)** & **r (\(E_{M}\))** & **r (\(T_{M}\))** & **\(\tau\) (\(Complexity\))** \\ \hline \(IDM_{SP}\) & 18.6–67.2 & 36.5 (10.2) & 0.259 & **0.390** & **0.421** \\ \(IDM_{FN}\) & 4.6–51.7 & 22.6 (7.5) & **0.340** & 0.211 & 0.314 \\ \(IDM_{FM}\) & 4.6–41.4 & 15.9 (6.3) & 0.251 & 0.068 & 0.124 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Descriptive statistics and correlations for the different formulations of the index of difficulty.  
In the regression analysis for \(E_{M}\), we first calculated its average values per levels of each of the index of difficulty formulations (unlike in Fitts' law, we don't do it per trials, since we predict not time, but accuracy), as well as per each level of _Complexity_. We obtained the following regression equations with the respective statistics:  
\[E_{M}=-0.026+0.003 IDM_{SP}, \tag{18}\]  
\[\text{R}^{2}=0.597,\,\text{F}_{1,15}=22.2,\,p<0.001\]  
\[E_{M}=-0.067+0.007 IDM_{FN}, \tag{19}\]  
\[\text{R}^{2}=0.893,\,\text{F}_{1,16}=133.5,p<0.001\]  
\[E_{M}=-0.021+0.007 IDM_{FM}, \tag{20}\]  
\[\text{R}^{2}=0.827,\,\text{F}_{1,13}=62.2,\,p<0.001\]  
\[E_{M}=-0.031+0.039 Complexity, \tag{21}\]  
\[\text{R}^{2}=0.572,\,\text{F}_{1,20}=26.7,\,p<0.001\]  
The model (21) with _Complexity_ had the lowest \(\text{R}^{2}\), while the model with \(IDM_{FN}\) (19) explained the most variation in \(E_{M}\), and we will be using this formulation in further analysis. The regression model for performance time due to this factor had \(\text{R}^{2}=0.474\) (\(\text{F}_{1,16}=14.4,\,p=0.002\)):  
\[T_{M}=4.8+0.049 IDM_{FN}. \tag{22}\]  
The lower degree of variation explained by the index of difficulty in \(T_{M}\) compared to \(E_{M}\) is natural, since the former is affected not only by the working memory capabilities, but also by skill with a computer mouse that was used in the designation stage in the experiment.