## 2 Materials and Methods
### Applied Statistical Methods and Analysis of Success
The grand mean term (\(\lambda_{0}\)) is a normalising constant estimated to make cell probabilities sum to 1, according to the following formula [39] (Equation (3)):  
\[\lambda_{0}=-ln\sum_{ijk}\exp\left(\lambda_{ijk}^{ABC}+\lambda_{jk}^{BC}+ \lambda_{k}^{C}\times\theta_{ijk}^{D}\right) \tag{3}\]  
The \(\lambda\)-terms can easily be converted to odds ratio as follows [41] (Equation (4)):  
\[\lambda_{111}^{ABC}=\frac{1}{8}\times\ln\left(\frac{m_{111}\ \times\ m_{221}}{m_{121}\ \times\ m_{211}}\times\frac{m_{122}\ \times\ m_{212}}{m_{112}\ \times\ m_{222}}\right)\text{ and }\lambda_{11}^{BC}=\frac{1}{8} \times\ln\left(\frac{m_{121}\ \times\ m_{211}}{m_{111}\ \times\ m_{221}}\times\frac{m_{122}\ \times\ m_{212}}{m_{112}\ \times\ m_{222}}\right) \tag{4}\]  
In matrix terms, model (1) can be described by the following matrix equation (Equation (5)):  
\[ln\left(\overrightarrow{m}\right)=X\times\overset{\rightharpoonup}{b} \tag{5}\]  
where \(\overset{\rightharpoonup}{b}=(\beta_{0},\beta_{1},\beta_{2},\beta_{3})\) is a \(4\times 1\) parameter vector for the terms in model (1), \(\overrightarrow{m}\) is a \(8\times 1\) vector of the cell probabilities and X is the \(8\times 4\) design matrix. The first element in the parameter vector is identical to the grand mean; the 4 th column of the design matrix includes the covariate means. Model (1) can be written in the following form using formula (Equation (6)):  
\[\ln\left( \begin{array}{c}m_{111}\\ m_{112}\\ m_{121}\\ m_{122}\\ m_{211}\\ m_{212}\\ m_{221}\\ m_{222}\end{array} \right)=\left[ \begin{array}{cccc}1&1&1&23.43\\ 1&-1&-1&-20.3\\ 1&-1&-1&13.37\\ 1&1&1&-8.65\\ 1&-1&1&24.6\\ 1&1&-1&-20.76\\ 1&1&-1&9.8\\ 1&-1&1&-5.76\end{array} \right].\left[ \begin{array}{c}\beta_{0}\\ \beta_{1}\\ \beta_{2}\\ \beta_{3}\end{array} \right] \tag{6}\]  
Scheirer-Ray-Hare (SRH) test is an equivalent of parametric two-way ANOVA test including the interaction terms as well. It is based on ranks and also suitable for any situations where ordinal data are used. On the other hand, the test makes no assumption about the normal distribution of the data. The procedure of the SRH test is as follows [42]:  
1. Carry out a standard parametric ANOVA on the ranked data
2. Calculate the mean square (MS) by dividing the total sum of squares by the total degree of freedom
3. Take the relevant sum of squares (SS) of the factors and divide by the value of MS and calculate the SS/MS value for each factor
4. Take the relevant SS/MS value and the degree of freedom (df) of the relevant factor and compute the \(\chi^{2}\) distribution with the parameters of SS/MS and degrees of freedom in order to gain the modified \(p\)-values  
We used SRH test to determine the significant differences in the eye-tracker variables using the significant interaction terms of the loglinear model. These variables were also tested in the loglinear model but did not have any direct influence to the cell frequencies. The significant combinations of variables (subject \(\times\) success \(\times\) image and image \(\times\) success) were also analyzed in detail by Mann-Whitney test for the parameters of the Eye-Tracker. Survival analysis was developed for cases where the task is the analysis of the elapsed time until an event occurs. This elapsed time is usually called as survival time [43; 44]. In this research the analysis of the survival time requires special methods since the decision times of anomalous and normal trichromats cannot be directly compared because of the lack of normal distribution. The aim of the method is to estimate the survival function and the execution of related significance analyses. The survival function gives that how does the rate of the survivals decrease approximately from the starting point in the function of time [45]. The comparison of the survival diagrams was executed with the generalized Wilcoxon-test [46]. For the survival analysis, Statistica 12.0 software-package (Statsoft Inc. Tulsa, OH, USA) was used.