## 2 Preliminary Research
### ANOVA Model  
As it was stated earlier, our purpose is to identify an influence of variables, such as year, department, and gender, on the response variable, i.e., the IQ test (or one of its nine subtests). Since these variables are considered categorical, it seems reasonable to use the ANOVA model to solve the problem. The exploratory analysis showed that the explanatory variables were highly correlated, especially the year and department factors. The ANOVA model allows us to estimate not only the impact of the above factors on the IQ values separately, but also their interactions.  
\begin{table}
\begin{tabular}{c c c} \hline
**Department** & **Abbreviation** & **Year** \\ \hline Automation and computer engineering & ACEF & 1994, 1995, 1997–1999, 2001, 2002, 2004 \\ Mechanical engineering and technologies & MTF & 1993, 1995–2001, 2013 \\ Radio engineering and electronics & REEF & 1992–2002, 2006, 2008–2010 \\ Business & FB & 1997–1999, 2001–2005 \\ Humanity education & HEF & 1994, 1998–2004, 2006–2008, 2010–2012 \\ Aircraft engineering & AEF & 1993, 1994, 1997, 2000, 2002–2006 \\ Mechatronics and automation & MAF & 1994, 1995, 1997, 1998 \\ Applied mathematics and computer science & AMCSF & 1995–2004 \\ Physical engineering & PhEF & 1991, 1994–1996, 1998–2000, 2003, 2004, 2006, 2009 \\ Power engineering & PEF & 1994, 1995, 1997–1999, 2001, 2002, 2004 \\ \hline \end{tabular}
\end{table}
Table 1: Natural distribution of tested students by departments and years.  
Figure 1: Violin plot of the IQ variable distributed by the years.  
In order to solve the above problem, the ANOVA model is formulated, as follows,  
\[y_{k\
eq ij}=\mu+\alpha_{k}+\beta_{t}+\gamma_{j}+(\alpha\beta)_{kt}+(\alpha \gamma)_{kj}+(\beta\gamma)_{tj}+(\alpha\beta\gamma)_{ktj}+\varepsilon_{ktji}, \tag{1}\]  
where \(y_{k\
eq ij}\) is the \(i\)-th observed value corresponding to the IQ level for a student of the \(k\)-th gender of the \(j\)-th department in the year \(t\), \(\alpha_{k}\) is the effect of the \(k\)-th gender (\(k=1\) for male, \(k=2\) for female), \(\beta_{t}\) is the effect of the year \(t\), \(t=1991,\ldots,2013\), \(\gamma_{j}\) is the effect of the \(j\)-th department, \(j=1,\ldots,10\), \((\alpha\beta)_{kt}\) is the interaction effect of the \(k\)-th gender and \(t\)-th year, \((\alpha\gamma)_{kj}\) is the effect of the interaction of the \(j\)-th department and the \(t\)-th year, \((\alpha\beta\gamma)_{ktj}\) is the effect of the interaction of the \(k\)-th gender, \(t\)-th year and \(j\)-th department, \(\varepsilon_{ktji}\) is a random error.  
It is impossible to estimate all the effects of the model (1). Usually, they resort to reducing the parameters estimating not the parameters themselves, but paired comparisons with some baseline levels. For example, the difference \((\alpha_{2}-\alpha_{1})\) is the influence on the IQ of the female factor versus the male one. In the present research, we have taken the first levels of the factors as the baseline ones.  
A critical requirement for using the ANOVA model is the condition of homoscedasticity of the dependent numerical variable for the selected levels of factors. As shown by the exploratory analysis, the prerequisites for applying the model are not achieved with natural discretization and grouping of factors. Therefore, the problem arises of aggregating the explanatory variables is combining their values into larger categories. For the ordinal variable year, it is advisable to use discretization methods, and for the categorical variable department to use grouping methods. A method that implements a combination of these approaches in application to the problem being solved is presented in the next section.