## 3 Proposed Method
### Results of Discretization and Grouping  
#### 4.2.1 Partitioning for General IQ  
First, we apply the Algorithm 1 to find the optimal partitioning of the explanatory variable values for the general IQ as a response. The result of partitioning depends a lot on a constant \(c\), determining an intermediate option of the goodness function between the _AIC_, which pays more attention to the accuracy \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & \(c=7\) & \(c=7.5\) & \(c=7.75\); \(8\) & \(c=\)**log\(N\) (\(BIC\))** \\ \hline Year & **38.28** & 28.18 & 26.38 & 37.54 \\ Department & 160.27 & 202.02 & **206.49** & 204.51 \\ Gender & 23.04 & **45.64** & 38.53 & 39.72 \\ Year:Department & **35.41** & 22.90 & 32.64 & 31.06 \\ Year:Gender & 1.57 & 5.70 & 6.17 & **7.58** \\ DepartmentGender & 0.20 & 3.58 & **4.41** & 3.24 \\ Year: DepartmentGender & 2.61 & 2.75 & **3.81** & 3.38 \\ sumF & 261.38 & 310.77 & 318.42 & **327.04** \\ Opt number of periods \(T^{*}\) & 3 & 5 & 4 & 3 \\ Opt number of groups \(J^{*}\) & 4 & 3 & 3 & 3 \\ The number \(m^{*}\) of best solutions & 12 & 2 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: F-statistics for the ANOVA model depending on the parameter \(c\). Figure 2: The proportion of cases at which the minimum value of \(Q\) was achieved. of the model, and \(BIC\), which penalizes more for the complexity of the model. To make a choice between the \(AIC\) and the \(BIC\) (a choice between 26 values of the constant \(c\)), it is necessary to involve some additional criteria. As such a criterion, it was decided to use F-statistic for each explanatory variable and their interactions, obtained from ANOVA tables with the analysis results. The choice in favor of a particular solution can be made based on the greater significance of the parameter of the most interest to the researcher, since the greater value of the F-statistics indicates the greater significance of the corresponding effect. Table 2 presents the results only for those \(c\) values for which the maximum F-statistics was achieved for either one variable, or interaction of variables, or for the sum of F-statistics (sumF). The resulting optimal number of periods is denoted by \(T^{*}\), the optimal number of department groups is denoted by \(J^{*}\). For \(c=7\), the year variable and the effect of interaction between the year and the department are of the greatest importance in F-statistics. The influence of the gender variable is the strongest for the partition obtained at \(c=7.5\). The department variable, the effects of the department-gender interaction, as well as the interaction of all three factors, have the highest F-statistics at \(c=7.75\), 8. The maximum influence of the interaction effect between the gender and the year, as well as the total influence of all variables, is found for the model built based on the \(BIC\) criterion (\(c=\log N\)). From Table 2, we see possible partitions by year into 3, 4, and 5 periods, depending on the value of \(c\) constant:  
* Three periods: 1991-2002, 2003-2005, and 2006-2013;
* Four periods: 1991-2002, 2003-2005, 2006-2012, and 2013;
* Five periods: 1991-2000, 2001-2002, 2003-2005, 2006-2012, and 2013. The solution corresponding to five periods is not typical, it occurs only once among the best partitions and corresponds to \(c=7.5\) with highest F-statistics for gender. Therefore, this solution with too many periods should be abandoned, unless the variable gender takes precedence over other explanatory variables. Moreover, the solutions corresponding to four and five periods contain the latest period consisting of only one 2013. However, in 2013, only 16 students were tested, so it would be considered inappropriate to separate this year into a distinct period. Therefore, it is reasonable to choose a partition into three periods by the variable year. The decision to limit ourselves to three periods by the variable year is also confirmed by the analysis of F-statistics.