Use of the complex of models of regression for analysis of the factors that determine the severity of bronchial 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 [3-5]. 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 [8-10]. 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) xk , ( k = 1, K) and M number of outputs ym , ( m = 1, M).

Let’s present the dependence of the inputs and outputs in this form

ym = Fm (x1, x2 ,..., xK-1, xK ) + em

where e m - 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 e m is subject to the normal distribution law.

Let’s assume that after examining of i-patient parameter xk will equal xk i , ( i = 1, N). Value of ym equals to ym i .

ym i = Fm (x1i , x2 i ,..., x(K-1)i , xK i ) + emi

which is distributed by the normal law with the mathematical expectation

Fm i = Fm (x1i , x2 i ,..., x(K-1)i , xK i )

and standard deviation

sm i = sm (x1i , x2 i ,..., x(K-1)i , xK i ) ,

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

sm1 = sm 2 = ... = sm N = sm = const .

Then probability distribution of ym i can be presented as

ym for the first patient lays within (ym1, ym1 + dym ), for the second patient lays within (ym 2 , ym 2 + dym ),…, for N-patient lays within (ym N , ym N + dym ) is the multiplication of possibilities.

Let’s determine the average Fm (x1i , x2 i ,..., x(K-1)i , xK i )so that expression (4) would comply with the maximum value. In view of the fact that

It follows that the possibility (4) will be maximal if the sum of the squares of errors em i is minimal

å(ym i - Fm (x1i , x2 i ,..., x(K-1)i , xK i ))

Expression (6) complies with the fact that there are two dependencies Fm (x1i , x2 i ,..., x(K-1)i , xK i ) , for which the set of

experimental values ym i of the output parameter ym is the most possible. Emphasize that the values of factors determined after

examination of every particular i-patient were received with the same accuracy. In another case expression (6) should be written in this form

The function Fm (x1, x2 ,..., xK-1, xK ) can be depicted as a superposition Z of known functions

Z

Fm (x1, x2 ,..., xK-1, xK ) = åazgz (x1, x2 ,..., xK-1, xK )

z=1

with az coefficients, that are derived from the case (6)

^N e ¶em i = - N æ - Z ö × =

Let’s show the received system of equations in the following form

Z N N

åav ågv i × gz i =åym i × gz i .

v=1 i=1 i=1

Unknown coefficients az are determined by the solution of the system of linear equations (8) with the defined system of functions

gz (x1, x2 ,..., xK-1, xK ) which is in many cases an orthogonal system of functions on the interval of changes of values of the parameters xk . In that regard let’s present the dependence Fm (x1, x2 ,..., xK-1, xK ) as an orthogonal range of Wiener [13]:

Fm (x1, x2 ,..., xK-1, xK ) = a + åbkxk + å å ckk2xkxk2 + å å å dkk2k3xkxk2xk3 +...

k=1 k=1 k2=1 k=1 k2=1 k3=1

This submission allows expressing coefficients az at functions gz (x1, x2 ,..., xK-1, xK ) through initial moments of random errors Xk , Ym . Define the values a, bk , ckk2 , dkk2k3..., at which the left side of the expression (6) accepts the minimal value. For this we find extreme value for expression (6) regarding the rates a, bk , ckk2 , dkk2k3... with the dependence of input and output factors (9):

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].

Table 1. Classification of factors [2]

№	Name	Common information
1	Course type	Severe persistent, moderate persistent, mild persistent, intermittent
2	Principal diagnosis (case taking)	Allergic rhinitis, atopic dermatitis, deflection of the nasal septum, secondary cardiomyopathy
3	Anamnesis of disease	Number of years from the first symptoms
4	Anamnesis of life	Bronchial asthma in mother, allergic rhinitis in mother, bronchial asthma in father, allergic rhinitis in father, bronchial asthma in relatives of the second generation
5	Complete blood count	RBC, HBC, WBC, eosinophils%, basophils%, band neutrophils%, segmented neutrophils%, lymphocytes%, monocytes%, ESR
6	Urinalysis test	Color, appearance, specific gravity, pH, leukocytes, transitional epithelial cells, mucus
7	Skin allergy test (pollen allergens)	Chenopodium, birch, sagebrush, sunflower, timothy, cock's-foot, rye, elytrigia, ryegrass, alder, fescue grass, walnut, linden, plantain, nettle, dandelion, sumpweed, foxtail, ragweed, chestnut, cypress, maple
8	Skin allergy test (household allergens)	Domestic dust, pillow feather, daphnia, rabbit hair, cat hair, dog hair, sheep wool
9	Food allergy test	Egg white, carp fish, pollack fish, milk, lemon, apple, raspberries, beetroot, cabbage, beef, pork, chicken, banana, watermelon, orange, tangerine, grape, cocoa, soybeans, black tea, tomato, beans, rice groats, buckwheat grits, wheat groats, oat grits, corn grits, rye grits
10	Ig E	Serum immunoglobulin E
11	Spirogram	VC, FVC, PEF, FEF(MEF) 25%, FEF(MEF) 50%, FEF(MEF) 75%, FEV 25-75%, FEV1, FEV/FVC %, MVV.
12	Immunological status	Leukocytes, neutrophils (%;103 cells), lymphocytes (%;103 cells), CD3 (%;103 cells), CD4 (%;103 cells), CD8 (%;103 cells), CD16 (%;103 cells), CD22 (%;103 cells), CD25 (%;103 cells), IgA, IgM, IgG, phagocytosis of latex, %, phagocytic number, total complement (СН 50), CIC with 3.5% PEG, units., spontaneous NBT tests, %, spontaneous IAN tests, units., stimulated NBT test, %, stimulated IAN test, units, lysosomal cationic proteins, units
13	TSLP	The level of serum thymic stromal lymphopoietin
14	Age	Age of the examined children from 6 to 17 years
15	Gender	Surveyed 56 boys and 14 girls

A large number of factors that determine the presence of a certain sign are coded with 0 or 1 (e.g. [4. Bronchial asthma in mother]=1, if this sign characterizes the patient and [4. Bronchial asthma in mother]=0, if the sign is absent). There are factors in the analysis above that can accept values related to several situations. For instance, factor [8. Pillow feather], Figure 6, characterizing the degree of intensity of a diagnostic skin test is coded so every degree of intensity relates to a certain number. ([8. Pillow feather]=0 if the sign is absent, [8. Pillow feather]=1 or 2 if the reaction is weakly positive, 3 and 4 if it is positive. Factor [8. Domestic dust] also relates to such kind of factors (0- if there is no reaction to dust, 1 and 2 – mild reaction, 3 and 4 – positive, Figure 2). Factors that can turn continuous variables after measurement are characterized by the unit of measurement ([3. Number of years from the first symptoms], [12. CD8 10*3 cells]), [13.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.

Table 2. List of excluded factors [15]

№	Name	The number of patients with the present factor	Description of the category of factor [Table 1]
1	Vasomotor rhinitis	2	2. Principal diagnosis (case taking)
2	Allergic rhinosinusitis	2	2. Principal diagnosis (case taking)
3	Corn	2	7. Skin allergy test (pollen allergens)
4	Acacia	2
5	Willow	1
6	Sorrel	2
7	Hazel	2
8	Egg yolk	2	9. Food allergy test
9	Hake	2
10	Cucumber	2
11	Carrot	2
12	Potatoes	2

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- 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. Severe persistent]. We have already analyzed the factor “thymic stromal lymphopoietin” [13. TSLP] [21-24], which is used for assessment of the severity of bronchial asthma. Its high numbers are usually for severe bronchial asthma [25-27]. Researchers Soumelis [28] and Lee [29] proved the importance of TSLP in the pathogenesis of atopic dermatitis.

The relationship between the researched factors is determined by the correlation coefficient [16], which complies with the level of the linear relationship between factors. As a consequence of that, it's advisable to choose a combination of factors for the construction of the complex of the models of matched regression those to have both the highest and the lowest values of the correlation coefficient via absolute value. The group of factors with the highest value of the correlation coefficient is supposed to have significant relations between factors in the model of matched regression. By contrast, a group of factors with the lowest value of the correlation coefficient is supposed 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

rx1y1 ³ 0.21. (19)

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

rx1y1 < 0.21. (20)

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 (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│ rx1y1│< 0.02 is given in Table 4. The extended list of factors satisfying the restriction (20) is in [16].

Table 3. Complex of the pair regression models

№		Explainable factor, y1	Regressor, x1	Regression model (16)
1	- 0.26	13. TSLP	6. Color	𝑦₁ = 33.36 − 15.16𝑥₁
2	-0.26		12. CD8 10*3 cells	𝑦₁ = 60.27 − 44.6𝑥₁
3	-0.24		5. ESR	𝑦₁ = 39.4 − 3.61𝑥₁
4	0.22		4. Bronchial asthma in relatives of the second generation	𝑦₁ = 23.75 + 24.67𝑥₁
5	0.25		14. Age	𝑦₁ = 3.83 + 2.0𝑥₁
6	0.26		2. Allergic rhinitis	𝑦₁ = 17.9 + 15.16𝑥₁
7	0.28		8. Domestic dust	𝑦₁ = 9.37 + 7.23𝑥₁
8	0.30		8. Pillow feather	𝑦₁ = 16.73 + 10.8𝑥₁
9	0.37		2. Atopic dermatitis	𝑦₁ = 23.03 + 49.04𝑥₁
10	0.40		8. Sheep wool	𝑦₁ = 15.76 + 17.67𝑥₁
11	0.45		8. Rabbit hair	𝑦₁ = 16.94 + 15.26𝑥₁
12	0.77		1. Severe persistent	𝑦₁ = 16.63 + 68.04𝑥₁
1	-0.26	1. Severe persistent	12. CD8 10*3 cells	𝑦₁ = 0.52 − 0.5𝑥₁
2	-0.25		12. CD 25%	𝑦₁ = 0.40 − 0.01𝑥₁
3	-0.23		12. CD 25 10*3 cells	𝑦₁ = 0.29 − 0.24𝑥₁
4	-0.21		10. Ig E	𝑦₁ = −0.03 + 0.19𝑥₁
5	0.24		8. Rabbit hair	𝑦₁ = 0.08 + 0.09𝑥₁
6	0.25		5. Eosinophils	𝑦₁ = 0.03 + 0.02𝑥₁
7	0.32		8. Sheep wool	𝑦₁ = 0.04 + 0.16𝑥₁
8	0.32		8. Domestic dust	𝑦₁ = −0.08 + 0.09𝑥₁
9	0.27		7. Foxtail	𝑦₁ = 0.08 + 0.19𝑥₁
10	0.30		7. Cypress	𝑦₁ = 0.07 + 0.15𝑥₁
11	0.30		2. Allergic rhinitis	𝑦₁ = 0.03 + 0.2𝑥₁
12	0.32		3. Number of years from the first symptoms	𝑦₁ = −0.01 + 0.02𝑥₁
13	0.34		8. Pillow feather	𝑦₁ = 0.02 + 0.14𝑥₁
14	0.43		2. Atopic dermatitis	𝑦₁ = 0.1 + 0.65𝑥₁
15	0.77		13. TSLP	𝑦₁ = −0.09 + 0.01𝑥₁

Table 4. Regressors for which lack of connection with the explained factor is assumed

№	Regressor, x1	№	Regressor, x1
1.	9. Black tea	9.	6. Specific gravity
2.	6. Appearance	10.	9. Apple
3.	7. Walnut	11.	7. Sunflower
4.	12. Total complement (СН 50)	12.	6. Transitional epithelial cells
5.	7. Sagebrush	13.	12. IgA
6.	12. Lymphocytes, 10*3 cells	14.	8. Daphnia
7.	2. Deflection of the nasal septum	15.	5. Segmented neutrophils, %
8.	4. Bronchial asthma in mother

In Figure 1, there is a model of matched regression that defines the relationship between the explained factor [13. TSLP] and the regressor [2. Atopic dermatitis]. The straight line defines the relationship between the explained factor and the regressor. The points designate the experimental data, received after clinical research. As it was mentioned above, enhancement of the severity of the disease leads to an increase in TLSP level. In the works of Ziegler and Artis, the relationship between [13. TSLP] factor and [2. 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. Atopic dermatitis] and the explained factor is 0.37 [Table 3] which implies a weak relation. The average value of [13. TSLP] in case of the absence of the sign [2. Atopic dermatitis] is 23 pg/mL and 72 pg/mL ([2. Atopic dermatitis]=1) in case of its presence. Despite the fact that the presence of the sign [2. Atopic dermatitis] leads to an increase of the factor [13. TSLP] by 40, a low number of patients who have this sign allows to create, in the real work, only approximate model of matched regression that confirms the statement about stochastic increase of the factor [13. TSLP] in case of the presence of the sign [2. Atopic dermatitis], seen in [30].

In Figure 2, there is a model of matched regression, defining the relation between the explained factor [13. TSLP] and the regressor [8. Domestic dust]. The correlation coefficient between the explained factor [13. TSLP] and the regressor [8. Domestic dust] is 0,28 [Table 3] that assumes a weak relation. Learning the dependence of the values of factor [13. TSLP] on the value of the regressor [8. Domestic dust], the values of the explained factor [13. TSLP] were determined (7,0-21,0 pg/mL) at which the skin test in the conducted clinical research is considered weakly positive ([8. Domestic dust]=1 or [8. Domestic dust]=2), in case of significant allergic reaction to this allergen ([8.Domestic dust]=3 or [8.Domestic dust]=4) with maximal TSLP index 38,3 pg/mL. Conducted experimental research say about the tendency to increasing of [13. TSLP] depending on the severity of allergic reaction of factor [8. Domestic dust]. Shifting from one degree of severity of the factor [8. Domestic dust] to another leads to increasing of the prognostic rate of [13. TSLP] proportionally to the value of the coefficient of matched regression b1 = 7, 23. The absence of the sign is defined by the value of the coefficient a = 9,37 .

In Figure 3, there is a model of matched regression that defines the dependence of explained factor [13. TSLP] on the regressor [4. Bronchial asthma in relatives of the second generation]. The correlation coefficient between the explained factor [13. TSLP] and the regressor [4. Bronchial asthma in relatives of the second generation] is 0,22 [Table 3] that assumes the presence of weak relation. On the basis of the analysis of clinical research, an assumption is made that the presence of the sign of the regressor for the considered sample leads to the value of the factor [13. TSLP] that is higher than 50 pg/mL. In case of absence of the sign that characterizes this regressor a prognostic value for [13. TSLP] is 21 pg/mL.

In Figure 4, there is a model of matched regression defining the dependence of the explained factor [1. Severe persistent] on the regressor [13. TSLP]. The correlation coefficient between the explained factor and the regressor is 0,73 [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 [2. Atopic dermatitis], [8. Domestic dust], [4. Bronchial asthma in relatives of the second generation], that is an important result of the real work. The presence of a strong relationship allows us to say about the general qualitative tendency of the explained factors depending on the similar regressors. The analysis of the model of a matched regression allows defining a break-off value of the [13. TSLP] factor, at which we can claim about a high possibility of the presence of severe bronchial asthma. Brake-off value of the factor [13. TSLP] is 63 pg/mL. This model of a matched regression can be, at some point, interpreted as a probabilistic model for prognostication of the severity of bronchial asthma, thus the distribution function of random variable determining the value of the [13. TSLP] factor can be roughly written as

ìF(x) = 0, [13.TSLP] < 9.0;

On the basis of the distribution function, F(x) = 0, there was an important assumption made within the clinical experiment, that

increase of [13. TSLP] by 10 pg/mL leads to an increase in the possibility of severe bronchial asthma by 10%.

In 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 the equation of regression is submitted to the normal distribution law, and the values of the factors determined after examination of every patient were received with the same accuracy. It made it possible to elaborate the way of assessment of the risk for the development of severe bronchial asthma in children and identify 15 factors that affect severe bronchial asthma. It was shown that the regressors chosen for the analysis are weakly linked to the explained factors, which makes it possible to determine only qualitative dependence between them. The analysis of factor thymic stromal lymphopoietin [13. TSLP] confirms the conclusions [21-23, 29], that it’s the most prominent indicator for the assessment of the severity of bronchial asthma. The correlation coefficient between [1. Severe persistent] and the regressor [13. TSLP] is 0,77 that implies a strong dependence between them. The analysis of the model of matched regression made it possible to identify a border value of the factor [13. TSLP], surpassing of which is considered as a high possibility for the presence of severe bronchial asthma. The increase of the factor [13. TSLP] by 10 pg/mL is considered as 10% higher possibility for severe bronchial asthma.

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

Article Information:

Abstract:

Keywords:

Article :

Introduction:

Methods:

Z

z=1

Z N N

v=1 i=1 i=1

k=1 k=1 k2=1 k=1 k2=1 k3=1

Results and Discussion:

Conclusion

References:

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