=Paper=
{{Paper
|id=Vol-2861/paper_43
|storemode=property
|title=A Merging Method to Discretizing and Grouping the Input Factors of ANOVA Model while Research of Time Dynamic of the Students Intelligence Quotient
|pdfUrl=https://ceur-ws.org/Vol-2861/paper_43.pdf
|volume=Vol-2861
|authors=Anastasiia Timofeeva,Tatiana Avdeenko,Olga Razumnikova
}}
==A Merging Method to Discretizing and Grouping the Input Factors of ANOVA Model while Research of Time Dynamic of the Students Intelligence Quotient==
https://ceur-ws.org/Vol-2861/paper_43.pdf
A Merging Method to Discretizing and Grouping the Input
Factors of ANOVA Model while Research of Time Dynamic of
the Students Intelligence Quotient
Anastasiia Timofeevaa, Tatiana Avdeenkoa and Olga Razumnikovaa
a
Novosibirsk State Technical University, 20, Karla Marksa ave., Novosibirsk, 630073, Russia
Abstract
In present work we study, with use of multivariate ANOVA model, the influence of
independent factors such as year, faculty, gender, on the indicators of students' general
intelligence (IQ) with a sample collected in 1991-2013 at the Novosibirsk State Technical
University. The peculiarity of models of this type is that the response is a quantitative
variable, and the input features must be qualitative. Therefore, first, the problem of
converting quantitative features into categorical ones (discretization) arises, second, with a
large number of levels of input qualitative features their grouping is required. If the variables
are strongly correlated, then both tasks should be solved simultaneously. In this case, the
optimal quality of the model should be ensured in accordance with a certain criterion.
Existing methods for the features type conversion are limited to one of the tasks
(discretization or grouping) and often do not take into account the relationships between the
features. Therefore, an original approach is proposed that allows solving the problem and
interpreting the results obtained.
Keywords 11
intelligence, Flynn effect, analysis of variance, discretization, grouping, interaction effect
1. Introduction
The intelligence quotient (IQ) is associated with the quality of people’s life and its duration. Thus,
a study carried out in Scotland, and presented in [1], showed that the probability of surviving to 76
years depends significantly on the IQ level detected at the age of 11 years. The studies carried out
were based on IQ measurements of 2792 children in 1932 in Scotland, born in 1921, the fate of 79.9%
(2230) of which was subsequently tracked. One possible explanation for these findings is that
intelligence enhances people's health care by helping them to acquire problem-solving skills that are
useful for preventing chronic diseases, accidental injuries, and for adhering to complex treatment
schemes.
There are other reasons for the influence of IQ on the quality of life, and, as a consequence, on its
duration. Thus, in the article [2], based on a survey of 6870 participants living in England in 2007, a
positive correlation was found between the level of verbal IQ and the feeling of happiness. People
with lower IQ were found to be less happy than people with higher IQ.
On the other hand, recent studies show that high intelligence is associated with increased anxiety
and stress, and can also cause chronic depression [3]. It is also noted that gifted people are more likely
than others to suffer from asthma and allergies [4], and are also susceptible to autoimmune diseases
[5].
SLET-2020: International Scientific Conference on Innovative Approaches to the Application of Digital Technologies in Education,
November 12-13, 2020, Stavropol, Russia
EMAIL: a.timofeeva@corp.nstu.ru (Anastasiia Timofeeva); avdeenko@corp.nstu.ru (Tatiana Avdeenko); razoum@mail.ru (Olga
Razumnikova)
ORCID: 0000-0001-9900-026X (Anastasiia Timofeeva); 0000-0002-8614-5934 (Tatiana Avdeenko); 0000-0002-7831-9404(Olga
Razumnikova)
©️ 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
370
All of the above indicates the relevance of conducting research based on the accumulation and
multivariate statistical analysis of intelligence indicators and its relationship with various time and
demographic factors. A special place in these studies is played by the phenomenon of a gradual
increase of IQ in the 20th century, known as the "Flynn effect". The effect was observed in different
countries and for different categories of test subjects [6]. For example, in [7] it was concluded that a
representative sample of Americans from 1932 to 1978 every year coped better and better with IQ
tests, while the overall increase in average IQ over 46 years was 13.8 points. However, since the end
of the 20th century, the reverse temporal dynamics of IQ (or the anti-Flynn effect) began to be
observed, the reasons for which remain unclear [8, 9, 10].
At the Novosibirsk State Technical University for 23 years from 1991 to 2013, the intelligence of
1st year students was tested according to the Amthauer method. The sample consisted of 3,677
students of both sexes from various departments of the university in the natural science, technical, and
humanitarian fields of knowledge. As a result of the analysis of these data, it becomes possible to
establish the influence of factors such as gender and faculty on the IQ of students, as well as to study
the temporal dynamics of changes in the IQ of students studying at a Russian university.
For the research, a multivariate analysis of variance was chosen, in which the response (dependent
variable) is the final IQ of students, measured on a scale of relationships. Categorical independent
features, measured on a nominal scale, are student gender, faculty, and year of study. The aim of the
study is to identify the influence of independent factors on the dependent variable - a quantitative
indicator of IQ. It is important to assess not only the impact of factors separately, but also their
interactions.
When conducting long-term studies of intelligence, it is not always possible to develop an
experimental design that makes it possible to obtain optimal estimates of the effects in the ANOVA
model, since it is difficult to ensure such conditions under which a similar sample population of
individuals would be surveyed every year. In this regard, the analyzed sample is characterized by an
uneven distribution of students across faculties and survey years, i.e. in one year, students from one
subset of faculties were surveyed, and in the next year, students from another subset. To construct an
acceptable analysis of variance model under these conditions, in present paper the method of
agglomerative discretization and grouping of input features was developed, investigated and applied.
The article has the following structure. Section 2 provides an overview of the existing
discretization methods, substantiates the development of a new method. Section 3 presents the quality
criteria investigated in the article for constructing optimal discretization. In Section 4, we describe the
ANOVA model used. Section 5 describes the developed discretization algorithm. Section 6 contains
the results of the studies of the proposed approach, and section 7 contains their interpretation for
solving the multifactor task of studying the IQ of students. In section 8 we provide a conclusion on
the work.
2. Overview of discretization methods
A good overview of the current state of research on discretization methods is presented in [11, 12].
If the transformation of a quantitative attribute into a qualitative one is carried out in such a way as to
ensure the best agreement with the response, then we are talking about supervised discretization. This
task can be solved using top-down (divisive) discretization techniques or bottom-up (agglomerative)
techniques. In the first case, a gradual division into intervals occurs, and in the second, the intervals
are merged. At each step of such algorithms, an evaluation function is calculated that characterizes the
quality of the division into intervals. In addition, the stop criterion is important, which determines that
further partition (merging) does not make sense.
For example, an efficient recursive partitioning algorithm MDLP [13] evaluates the quality based
on information gain based on entropy, and the stopping criterion is derived from the principle of
minimum length description. The chi-square statistic is popular in the agglomerative merging
problem. Algorithms such as ChiMerge [14], Chi2 [15] were built on its basis. Both approaches are
designed for classification tasks, that is, they assume that the response is categorical. Therefore, their
application to transform a set of input variables in the construction of ANOVA models requires
discretizing the response, which can lead to the loss of significant information.
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Another group of discretization methods, the so-called wrapping methods, focuses on the quality
of the estimated model. Thus, these methods simultaneously solve the learning problem. The existing
algorithms are built for classifiers, for example, such simple ones as a majority class voting classifier
[16], or more general classifiers such as Naive Bayes [17].
Compared to the problem of discretization, the grouping problem has not been studied so deeply in
the literature. A fairly complete overview of grouping methods is presented in [18]. Many commercial
data mining packages suggest excluding variables that have too many categories. This approach,
however, cannot be considered acceptable in cases where the research interest is to assess the effects
of just such variables. Effective grouping methods allow for fewer, more informative categories. This
can be done by Sequential Forward Selection method [19]. It is a greedy that initializes a group with
the best category and then iteratively adds new categories to this first group. Decision tree algorithms
often solve the grouping problem with a greedy heuristic based on bottom-up categorization. The
CHAID algorithm [20] uses this greedy approach with a criterion close to the ChiMerge criterion
[14]. In [18], a new method of grouping MODL based on the Bayesian approach was proposed, as
well as the discretization method MODL [21]. It searches for the most likely grouping model for the
given dataset. Optimization is done using a greedy bottom-up algorithm.
Thus, most of the existing supervised discretization algorithms are designed to solve classification
problems, that is, for categorical response. They are mainly aimed at improving the quality of
predicting the response (quality of classification) [[22],[23]]. Moreover, they are usually univariate. In
this regard, it seems relevant to develop an algorithm for the optimal categorization of input features,
taking into account their interrelationships, to build a model of analysis of variance. Here
categorization includes two tasks: discretization of quantitative variables and grouping of nominal
features. Due to the specifics of the practical task, the construction of response predictions is
secondary, therefore, the use of criteria such as cross-validation in order to assess the quality of the
model and avoid overfitting is limited. The main task was to obtain and interpret estimates of the
effects of influencing factors. As a result, we had to resort to goodness-of-fit criteria.
3. Goodness of fit criteria
Most often, the quality of a regression model is judged by the coefficient of determination,
calculated as
ESS
R2 1 ,
TSS
where ESS is residual sum of squares of the model, TSS is total sum of squares of the model.
However, this indicator has an obvious drawback. With increasing complexity of the model
(including new variables), it is possible to better describe the response, thereby decreasing ESS and
increasing R 2 . However, the number of the degrees of freedom decrease, which is in no way taken
into account when calculating the coefficient of determination.
To check the significance of the model, the F-statistic is used, calculated as
R2 N m
F ,
1 R2 m 1
where N is the number of observations, m is the number of estimated parameters. It takes degrees of
freedom into account, so the increase in model complexity must be offset by a sufficient decrease in
the residual sum of squares.
Akaike information criterion is often used in the problem of feature selection, for example, in the
stepwise regression procedure. It provides a trade-off between goodness of fit and complexity of the
model (number of parameters). The Akaike criterion is calculated as follows.
AIC 2m N log ESS .
It should be borne in mind that with a very large number of categories, building good groupings is
difficult because of the risk of overfitting the model. In the extreme case, to avoid overfitting,
efficient grouping methods can combine all values into one group, thereby excluding the variable
from consideration. In order to prevent such a situation, the stopping criterion must include a
condition for the minimum number of categories (for example, two).
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4. ANOVA model
For research, the following model of analysis of variance was formulated:
yktji k t j kt kj tj ktj ktji , (1)
where yktji – i -th observed value corresponding to the IQ level for a student of the k -th sex of the j -
th faculty in the year t , k is the effect of the k -th sex ( k 1 for male, k 2 for female), t is the
effect of the year t , t 1991,...,2013 , j is the effect of the j -th faculty, j 1,...,11 , kt is the
interaction effect of the k -th sex and t -th year, kj is the effect of the interaction of the k -th year
and the j -th faculty, tj is the effect of the interaction of the j -th faculty and the t -th year,
ktj is the effect of the interaction of the k -th sex, t -th year and j -th faculty, ktji is a random
error.
It is impossible to estimate all the effects in model (1). Usually they resort to reduction. This
estimates paired comparisons with some baseline, for example, 2 1 is the influence of female
versus male. The first levels of factors are taken as the baseline levels.
The distribution of the studied students is uneven over the years (see table 1). There is a close
relationship between the variables Faculty and Year. The chi-square statistic is 8092.9, which
indicates a significant correlation at 0.1% significance level. Nevertheless, it should be borne in mind
that the original contingency table has a very large dimension (220 degrees of freedom), and, as a
consequence, cells with a small number of observations, which negatively affects the correctness of
the chi-square test. For confirmation, the correlation ratio was calculated, showing the influence of the
faculty for the year. It is 0.192 (F-statistic is equal to 86.9), which also speaks of a significant
connection at 0.1% significance level.
Table 1
The ratio of faculties and survey years in the sample
Faculty Abbreviation Survey years
automation and computer
ACEF 1994, 1995, 1997-1999, 2001, 2002, 2004
engineering
mechanical engineering and
MTF 1993, 1995-2001, 2013
technologies
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 enginiiring AEF 1993, 1994, 1997, 2000, 2002-2006
mechatronics and automation MAF 1994, 1995, 1997, 1998
applied mathematics and computer
AMCSF 1995-2004
science
1991, 1994-1996, 1998-2000, 2003, 2004,
physical engineering PEF
2006, 2009
power engineering PEF 1994, 1995, 1997-1999, 2001, 2002, 2004
natural sciences NSF 2007
Consequently, it is impossible to assess all the effects of faculty and year interactions in order to
separate the effect of student specialization from the time trend. Therefore, it is necessary to discretize
the Faculty and Year variables in such a way as to ensure the optimal quality of estimation of the
model, which includes interaction effects.
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5. The developed algorithm
The algorithm is developed for the case when it is required to discretize one quantitative variable
and group one categorical variable, and the variables are highly correlated. It can be extended to the
case when there are more than two variables, but with a large number of variables and levels the curse
of dimension arises.
The pseudocode of the algorithm for the optimal categorization of input features, taking into
account their interrelationships for constructing an analysis of variance model, is shown in Figure 1.
Q quality x1 , x2
repeat
for k 0 to T0 1 do
if k 0 & T0 2 then x1 merge x1 , k , k 1 else x1 x1
if k 0 then Q1 0,0, k quality x1, x2
for i 1 to K0 1 do
for j i 1 to K 0 do
if K0 2 then x2 merge x2 , i, j else x2 x2
Q1 i, j, k quality x1, x2
end for
end for
end for
Q opt Q1
i , j ,k
if Q p Q | Q Q then break
Q : Q
i , j , k arg opt Q
* * *
i , j ,k
1
if k * 0 then x1 : merge x1 , k * , k * 1 , T0 : T0 1
if i* 0 then x2 : merge x2 , i* , j * , K0 : K0 1
end repeat
return x1 , x2
Figure 1: Pseudocode of the developed algorithm
Input: raw data including response values, quantitative factor x1 with T0 levels, and qualitative
factor x2 with K 0 levels.
The thresholds were selected simultaneously for two variables by the agglomerative merging
method. The initial model was built taking into account all available levels of factors. Further, one
boundary between the levels was successively removed. For a categorical variable, all possible pairs
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of factor levels were considered, for a quantitative variable, only adjacent values. In addition, such an
option was considered when the levels were not combined. It was assigned an index 0 according to
the variable for which the levels were not combined. This is done in case the optimal solution is to
combine levels in only one of the variables. If the best value of the quality index corresponding to the
optimal solution was achieved, the levels were combined. Then the procedure was repeated until an
improvement was obtained.
The function quality x1 , x2 returns an indicator of the quality of fitting an ANOVA model of the
form (1) (determination coefficient, F-statistic, AIC) depending on the input data.
The function merge x, i, j combines the levels i, j of a variable x so that the number of levels is
reduced by one. If the input variable included K levels, then the function returns the transformed
variable with K 1 levels numbered from 1 to K 1 .
Since the optimization of the goodness-of-fit criteria can go in different directions (for the
determination coefficient and F-statistic it is maximization, for AIC it is minimization), we denoted
the optimal value as opt . Wherein Q p Q | Q Q means that Q is no better than Q .
6. Results
The choice of the determination coefficient as an evaluation criterion did not give any results,
since the original partition provided the minimum residual sum of squares. As expected, any merging
of intervals led to a decrease in the determination coefficient.
The use of the F-statistic, on the contrary, led to the fact that at each step there was an
improvement in the values of the evaluation function. Thus, the work of the algorithm ended only
when the intervals could no longer be combined, that is, when there were two categories left for each
feature. The faculty of AMCSF stood out in a separate group, as well as 1991. The results of
evaluating such a model indicate one significant effect - on the AMCSF, compared to the rest of the
faculty, IQ is 6.3 points higher (significant at the 1% level).
The use of the Akaike information criterion made it possible to obtain more interesting results.
When applying the algorithm, three groups of faculties were distinguished. From table 1 it is clearly
seen that there are years in which some faculties were not covered by the study. This problem was
partially solved by discretizing the variable Year. Table 2 shows the proportions of students of
faculties of three groups studied in a given range of years. For example, for the first group, there were
no periods left when the faculties of this group were not covered by the study. Nevertheless, there is a
gap for the second group of faculties in 2009, and for the third - in 2008 and 2010-2013. Therefore, it
was not possible to estimate the corresponding effects.
Table 2
Shares of students of faculty groups in the total number of students studied in a given range of years
Survey years 1st group 2nd group 3rd group
1991-1996 0.697 0.073 0.230
1997 0.556 0.148 0.296
1998-1999 0.661 0.195 0.144
2000 0.203 0.228 0.568
2001 0.296 0.245 0.460
2002 0.579 0.274 0.147
2003-2005 0.306 0.320 0.373
2006-2007 0.394 0.518 0.088
2008 0.600 0.400 0
2009 0.759 0 0.241
2010-2013 0.556 0.444 0
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After discretizing the variables, a model was estimated describing the dependence of IQ on gender,
faculty, and year and on their interactions. It turned out that gender has an insignificant effect on the
level of intelligence. Therefore, the gender factor was eliminated and the model was re-estimated.
Table 3 provides a summary table with the values of the F-statistic and p-value for 1% significance
level. Almost all the effects of the variable Year turned out to be significant at the 5% or 10% level.
For the base year, the effect of the faculties of the second group compared to the first was –5.3 and is
significant at the 1% level. The effect of faculties of the third group compared to the first for the base
year is estimated as 1.2 and is significant at the 10% level. Most of the interactions between the year
and the faculty were significant. The general average is estimated at 112.3.
Table 3
The significance of factors
Factor Degrees of freedom F-statistics Critical F-value p-value
Year 10 14.88 0.55 <210-16
Faculty 2 244.41 0.63 <210-16
Faculty:Year 17 11.38 0.53 <210-16
Figure shows the predicted IQ values by year and depending on the group of faculties.
1
120
2
3
predicted IQ
115
110
1991-1996 1997 1998-1999 2000 2001 2002 2003-2005 2006-2007 2008 2009 2010-2013
Figure 2: Model estimation results
7. Interpreting the Results
From the point of view of specialization, the distinguished groups of faculties can be divided as
follows. The first group is technical and economic faculties, the second is humanitarian and applied
faculties, and the third is physics and mathematics. The latter group, on average, is characterized by
the highest level of intelligence. Although since 2006 the IQ has dropped and has become comparable
to the level of intelligence of students in other faculties. But during this period, a group of students
with a physical and mathematical specialization was observed very little (see Table 1): only PEF 2006
(22 students) and 2009 (14 students). Therefore, the decline in IQ may be due to the non-
representativeness of the sample.
In the 2000s, there was instability of IQ indicators among students of technical and economic
specialization. Growth period 2006-2007 can be explained by the fact that in 2007 only the NSF was
observed from this group, which was characterized by higher IQ indices.
For students of humanitarian and applied specialties from 2000 to 2005 in general, there was an
increase in intelligence indicators, and then a sharp decline began in 2006-2008. In 2009, the faculties
376
of this group were not studied, so the interaction effect could not be estimated, and the IQ forecast is
based only on the main effects. This explains the sharp increase in the IQ forecast in 2009, which
cannot be considered reasonable.
8. Conclusion
Thus, in this work, an analysis of variance model was constructed to study the influence of input
factors on the IQ of students. To build a qualitative model, taking into account the specifics of the
collected data, a new agglomerative method for discretizing and grouping input features was
developed and tested. The interpretation of the obtained estimation results is carried out. In practice,
the obtained results of interpretation can be used in the construction of individual educational
trajectories, which is one of the key problems of the modern digital educational environment [24].
Future work involves the improvement of the developed algorithm in terms of finding the optimal
solution, as well as the development of alternative models for the study of students' IQ with
subsequent comparison of the results.
9. Acknowledgements
The research is supported by Ministry of Science and Higher Education of Russian Federation
(project No. FSUN-2020-0009).
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