Use of the complex of models of regression for analysis of the factors that determine the severity of bronchial asthma

Background: According to an International Study of Asthma and Allergies in Childhood (ISAAC), the prevalence of asthma in children of 6-7 years old has increased by 10%, and at the age of 13-14 years by 16% over the last decade. Determining the factors that are keys to the occurrence of the disease and its severity is important in explaining the pathogenesis of bronchial asthma. Methods: Analyzed 142 indicators of clinical and paraclinical examination of 70 children with asthma. To select factors that could be significant in the formation of severe asthma, applied the method of logistic regression with step-by-step inclusion of predictors. Both quantitative and qualitative characteristics were selected. Each qualitative attribute was coded “1” if the child had this characteristic, or “0” if this characteristic had not been established. The formation of a severe asthma course was accepted according to (1) and the absence of a severe asthma flow formation as (0). Results: Analyzed the model of paired regression, the boundary value of thymic stromal lymphopoietin was established, exceeding which indicates the high probability of the presence of severe bronchial asthma. Increasing the value of thymic stromal lymphopoietin by 10 pg/mL suggests an increase in the likelihood of severe asthma by 10%. Conclusions: A complex of steam regression models has been developed to determine the factors characterizing the severity of bronchial asthma. The risk of developing severe bronchial asthma in children has been determined and 15 factors have been identified that affect severe asthma.


Introduction
Nowadays, there are 339 million people suffering from bronchial asthma in the world [1]. Despite many years of research, asthma is still the most common chronic disease within the children population in different countries of the world, and the incidence increases with every passing year [2]. Asthma remains one of the most common causes of disability in pediatric ages and takes the 4th place in the structure of common disabilities among 10-14 years old children [3][4][5]. Patients' quality of life significantly decreases and is followed by notable economic expenditures both for the family and for the society [6]. The first symptoms of the disease often appear in childhood and it is still difficult to prognosticate further course of the disease. Prognostication of asthma course and disease in children is still a difficult problem, because of the multifactority of the disease [7]. Numerous clinical studies of uncontrolled asthma course approve the necessity to analyze the factors influencing severe forms of the disease [8][9][10]. One such factor can be thymic stromal lymphopoietin (TSLP). There is a relationship between TSLP dysregulation and its role in the pathogenesis of atopic diseases, such as atopic dermatitis, asthma, allergic rhinitis and eosinophilic esophagitis [11,12]. Detection of cases at early stages with the risk of severe asthma development in children is one of the main problems. This gives us the opportunity to use individual methods of therapy and observation in these cases.
Selection and substantiation of regression models which define the characterizing severity of bronchial asthma are the aims of the study.

Methods
Let's take a look at the method of construction of the complex of regressive models containing K number of inputs (regressors) , ( ) and M number of outputs , ( ).
Let's present the dependence of the inputs and outputs in this form where -is a random error. The random error is determined by the fact that measured experimental values of inputs and outputs deviate from its initial values due to random errors. We will construct the regressive model for the case when the random error in the equation of regression is subject to the normal distribution law.
Let's assume that after examining of i-patient parameter will equal , ( ). Value of equals to .
which is distributed by the normal law with the mathematical expectation and standard deviation , that characterizes the error of measurement. Let's assume that values of the factors obtained after examining every particular i-patient (clinical examining data) were received with the same accuracy that allows us to write .
Then probability distribution of can be presented as . (2) The probability that the value of output for i-patient is in this range can be calculated with the following equation .
Since the events under which factor equals are independent then the possibility that values of the explained factor for the first patient lays within , for the second patient lays within ,…, for N-patient lays within is the multiplication of possibilities.
It follows that the possibility (4) will be maximal if the sum of the squares of errors is minimal .
Let's show the received system of equations in the following form .
Unknown coefficients are determined by the solution of the system of linear equations (8) with the defined system of functions which is in many cases an orthogonal system of functions on the interval of changes of values of the parameters . In that regard let's present the dependence as an orthogonal range of Wiener [13]: (9) , , Differentiate the function respectively on , obtain , , .
Substitute the values of partial derivatives (11) of the function into the system of equations (10), we obtain the system of equations for identification of unknown rates : , , , Engage the initial moment, expressed through the frequencies of appearances of covariates , , , , .
Values for initial moments calculated with the formula (13) will be the same possible as the relevant values for the general population as we increase the number of experiments N. Considering (13) we obtain a linear system of equations z a z 1 2 In case of selection of linear relationship for parameters expression (9) become with the value of coefficients a and , that are determined by the system of equations (14): Using the definition of dispersion and covariance of a random variable , Increasing of covariance rate via the absolute value leads to enhancement of the dependence between and parameters. If we turn to dimensionless variables , , we obtain a dimensionless equation of model of couple regression , , , that looks more compact comparing to (16) x , y ( ) Unprocessed data of experimental research are presented in [14]. The study respects for human rights in accordance with the current legislation of Ukraine, follows international ethical requirements and does not violate any scientific ethical norms as well as standards of biomedical research. Individual data of the patients are encoded. Before the experiment conduction, factors that could affect the severity of bronchial asthma are identified. The following 142 factors are categorized [ Table 1], allowing to unify the data analysis and structure the relationships between them [15] [ Figure 1-6].  TSLP], [ Figure 1-4] [14]. The number before the name of the factor determines the category to which this factor relates to, (Table 1: TLSP factor, category №13) [15].
Factors that were identified after clinical examination in not more than two patients were excluded from the input unprocessed data [ Table   2]. The list of excluded factors is in Table 2. It's expected that excluded factors don't affect the result of the research too much so they can be omitted. It allowed shortening the number of factors for analysis by ~10%. So, 130 factors were used for the regressive model. The next step of the research was dividing the data into two different parts. The first part was used for the construction of the regressive model [16], the second part -the testing one, for verification of the regressive model adequacy.

Results and Discussion
Conducted literature review characterizes bronchial asthma as a heterogeneous disease with an increasing number of severe cases [17][18][19]. Numerous clinical research of low-controlled severe bronchial asthma confirms the necessity of studying new biological markers for appropriate therapy onset [20]. For understanding the pathophysiology of severe bronchial asthma, the analysis of factors affecting the beginning of severe forms is needed. Severe bronchial asthma will be characterized by explained factors [1.
The relationship between the researched factors is determined by the correlation coefficient [16] to have no relations between factors. Both groups are needed for the research. The first group allows defining the set of factors that can be used to identify the severity of bronchial asthma. In our research, we assume that there is a relationship between explained factor and regressor (the weak relationship is also taken into account), if the correlation coefficient is The full list of factors with the value of the correlation coefficient (19), used for the construction of the model of matched regression is given in [16]. The second group of factors is characterized by the absolute value of the correlation coefficient .
It's assumed that for this group there is no relationship between the explained factor and the regressor. Such an approach allows us to reduce the dimensionality of the model. In our case, it's only 15 factors out of 130 (including excluded ones) for which the condition is met x1y1 r 0.21 ³ x1y1 r 0.21 < (19). Equations of the regressive models for the factors defined by restriction (19) are given in Table 3. The list of factors for which the lowest value of the correlation coefficient│ │< 0.02 is given in Table 4. The extended list of factors satisfying the restriction (20) is in [16].   Table 4. Regressors for which lack of connection with the explained factor is assumed Atopic dermatitis] regressor is defined, however, quantitative characteristics are not considered [30]. The real work complements the research conducted in [30], by quantitative relation. Presence of the sign [2.
Atopic dermatitis] increases the possibility of bronchial asthma occurrence. This relation is portrayed in Figure 1. The correlation coefficient of the factor [2.
In Figure 2, there is a model of matched regression, defining the relation between the explained factor [13. TSLP] and the regressor [8.   Table 3] which implies a strong relation and confirms conclusions made in [16]. The correlation coefficient, in this case, is a few times greater than for regressors mentioned above   Figure 5, there is a model regression that determines the dependence of the explained factor [1. Severe persistent] on the regressor [2. Allergic rhinitis]. The correlation coefficient between the explained factor and the regressor is 0.3 (Table 3), which tells us about a low dependence and confirms conclusions made in [16]. In case of the absence of this factor in the history of the disease (the disease ([2. Allergic rhinitis]=1)) an assumption based on the real work can be made that only 15% of the children may develop severe bronchial asthma. The absence of the factor ([2. Allergic rhinitis]=0) doesn't exclude the possibility of severe bronchial asthma development.
In conclusion, we will analyze the regressive model that determines the dependence of the explained factor [1. Severe persistent] on the regressor [8. Pillow feather]. The regressor is coded with numbers: 0-absence of the sign, 1 and 2 -weakly positive, 3 and 4 -positive.
In the conducted research, the regressor accepted values in the interval [0;4]. If the value of the regressor [8. Pillow feather]=3, then the prognostic value for the explained factor is 0.45. It's assumed that if [8. Pillow feather]=4 then the value of the explained factor will probably surpass 0.5.

Conclusion
In this work, the complex of models pair regression for determining the severity of bronchial asthma was presented. The main provisions used for creating the complex of models regression were determined and substantiated. The premises were used, that a random error in