Prediction based on TSLP

Abstract

Bronchial asthma is a heterogeneous disease affecting more than 300 million people, and its occurrence increases every year. Severity of disease is characterized by interaction of genetic factors and numerous environmental factors. The technique of regression model construction is used to determine significance and dependence between factors. The data of 70 children with diagnosed bronchial asthma and 20 children of a control group were used in the study. 142 factors were studied and level of thymic stromal lymphopoietin (TSLP) in blood serum was taken as a value of a referable variable to construct regression models. Prediction model parameters were assessed. Errors distribution law was defined on the basis of a ten-factor regression model analysis. In the absence of large deviations from normal distribution law, validity coefficients are close to the errors under normal distribution law. A comparative study of histograms is provided to distribute model's regressors values. Bronchial asthma, child, regression model, residual plot, TSLP Bronchial asthma is a chronic pulmonary inflammatory disease [1]. Development of approaches in the disease treatment as well as new medications creation did not allow consolidating control over bronchial asthma. For this reason, the search of inflammation biomarkers is the most pressing challenge Thymic stromal lymphopoietin can be used as one of inflammatory markers [4, 5]; that is hematopoetin cytokine isolated from mouse thymus epithelium culture [6, 7]. When studying the role of thymic stromal lymphopoietin its direct involvement in initiation and support of allergic inflammation in bronchial asthma is confirmed [8, 9]. Asthma phenotyping is based on diversity of disease clinical signs and heterogeneity [10, 11]. Special attention is paid to study of severe uncontrolled asthma in children [12, 13].

1. Introduction

2023 Copyright for this paper by its authors.

Regression models are a common way to predict a course of bronchial asthma in children [19, 20].

Regression model defines a relationship between the observed value Yi and factors X K ( k = 1...K ) characterizing bronchial asthma. It can be represented as follows

Yi = F (X1i , X 2i ,..., X ki,..., X Ki ) + i , ( 1 ) here Yi is the value of a referable variable for the i -th test, i = 1...n ; X ki is the predictor’s (regressor’s) value for the i –th test; i is the prediction error for the value of Yi of the referable variable in the i –th test having the following properties:

E(i ) = 0, 2 (i , i ) = 2 , 2 (i ,  j ) = 0, i  j. ( 2 )

The study purpose is research of the regression model features of thymic stromal lymphopoietin level in the blood serum.

The regression model (1) shows the trend of the referable variable change

Yˆ = F (X1, X 2 ,..., X K ),

E(Y ) = Yˆ, ( 3 ) depending on the change of predictors’ X k values, which characterize bronchial asthma, and the range of Yi values around the value of F (X1, X 2 ,..., X K ) . The range of values for each level of predictors X k follows probability distribution with mathematical expectation of E(i ) = 0 and mean square deviation of (i ) =  .

Definition of a set of predictors X k for the referable variable Y analyzing is the central task of

regression model construction, which defines connection of factors characterizing bronchial asthma. Consistent increase of the number of predictors in the model is one of the methods ( 1 ). It is supposed that an addition of a new predictor to an existing set of predictors helps to decrease the 2 (i ) value. Selection of predictors X k starts in the virtue of the regressor as a cause factor importance. The referable variable of Y depends on it. Selection of the type of functional dependence ( 3 ), as well as selection of the X k factors, is one of the main problems of severe bronchial asthma study. In the most of the cases the dependence is defined as a result of a definite number of successive approximations. Model of linear regression is used as an initial approximation during functional dependence constructing [21]. Also special attention is to be paid to causal relationships of Y and X k , given that statistical relationship of the referable variable Y and predicate X k doesn’t mean that the value of the referable variable Y causally depends on the value of predicate X k . All the above questions will be studied in details in this paper. 2. Materials and methods

90 children aged 6 to 18 years were involved into the study of bronchial asthma in children in view of the phenotype associated with thymic stromal lymphopoietin. The main group amounted to 70 children with diagnosed bronchial asthma and the control group amounted to 20 children. The average age of children with bronchial asthma was 11 years. For every patient, information on 142 factors that could be the cause of bronchial asthma was gathered, processed and analyzed. The study was conducted with respect for human rights and in accordance with international ethical requirements; it doesn't violate any scientific ethical standards and standards of biomedical research [22]. Parents were questioned about symptoms characteristic to bronchial asthma in patients as well as the patients’disease anamnesis. The results of the questioning were added to patient’s materials. Clinical features of disease and results of laboratory tests were studied. Level of thymic stromal lymphopoietin in patients’ blood serum was defined with enzyme immunoassay using a commercial test system manufactured by Biotechne (ELISA, USA) on the immune-enzyme analyzer “Labline-90” (Austria). Enzyme immunoassay is based on sandwich-type technique which is characterized by cobinding of biotin-labeled antibodies to the analyzed analyte. Amount of thymic stromal lymphopoietin in a batch is defined on an analytical curve in accordance with routine practice for such experiments. Amount of thymic stromal lymphopoietin is measured in picograms per 1 ml of serum (pcg / ml) [23]. Indicators “Severe” and “TSLP” were taken as a prediction parameter Y that defines severity of bronchial asthma in children. Negative effect of thymic stromal lymphopoietin cytokine on the course of disease was proved in [24, 25]. All the above questions will be studied in details in this paper.

3. Basic materials. Prediction model of bronchial asthma severity

3.1. Assessment of prediction model parameters

For linear model of multiple regression with model parameters of 0 , k :

constant additive Yˆi and random additive i values. As defined for an error i ( 2 ), the observed variable Yi assessment follows:

 K  K K E(Yi ) = E0 + k X ki + i  = 0 + k X ki + E(i ) = 0 + k X ki,

 k=1  k=1 k=1 Model parameters of 0 , k have an assessment:

E(b0 ) = 0,

E(bk ) = k ,

mxk =

K b0 = my − mxk bk , k=1 1 n

 X ki, n i=1 bk =

Kx y

k , Kxk xk my = 1 n

Yi , n i=1 Kxk xk = 1 n(X ki − mxk )2, n i=1

Kxk y = 1 n (X ki − m k )(Yi − my ).

x n i=1

Since, when predicting the value of the observed Yi , the values of regressors X ki are known, unknown values of 0 , k with assessment of their mathematical expectation ( 6 ) are defined by a linear combination of random values of Yi of the observed variable. To estimate the mean square deviation of (Yi ) let’s use the definition of i error ( 2 )

 K  2 0 + k X ki + i  = 2 (i ), 2 (i ,  j ) = 0. ( 8 )

 k=1 

In the  = const assumption, the value of a random variable of Y is defined by probability distribution with mathematical expectation ( 5 ) and the mean square deviation of  = const which does not depend on predictor X value. Assessment of 2 value can be defined as follows: n 2 = E(MSE ), MSE = , SSE = (Yi − Yˆi )2,

SSE n − m i=1 here m = (K + 1) is the number of constraints, it can be defined by a number of equations of ( 7 ) type. Retaining connections of b0 = B0 (Y1,Y2 ,...,Yn ) , bk = Bk (Y1,Y2 ,...,Yn ) ( 7 ) correspond to the regression model ( 4 ). Regardless, (n − m) of independent values Yi can be defined. The rest m of Yi values are defined from the system of equations ( 7 ). Expressions ( 6 ) and ( 9 ) give assessment for coefficients of the regression model and mean square deviation of  . Defining of b0 , bk coefficients according to the ordinary least squares technique, regardless the function of error rate distribution i , makes it possible ( 5 ) ( 6 ) ( 7 ) ( 9 ) to calculate the unbiased point estimations for b0 , bk , that have minimal dispersion. However, to define the space of b0 , bk parameters, assumption on the error rate i distribution law is required. 3.2.

Assessment of prediction error distribution law

When studying the factors characterizing bronchial asthma severity, let us introduce an assumption of a normal error rate distribution i and define when the conditions of the assumption are met. For a normal law of error rate distribution i with the conditions ( 2 ), designation N (0, 2 ) is used. It is known that bronchial asthma severity is determined by a large number of factors that are weakly interconnected. When investigating severity of bronchial asthma in paper [26], more than 100 factors are being analyzed with correlation index rxk xv : rxk xv =

Kxk xv Dxk Dxv , xk = Dxk ,

Dxk = n1 i=n1 (X k i − mxk )2 , k, v = 1...K , between factors X k , X v that satisfies inequality rxk xv  0.15 . A weak relation is observed not only between factors that are used as the linear model regressors, but also between the referable Y value, characterizing bronchial asthma severity, and the regressor X k . Since value of correlation index ry xk : ry xk =

Kxk y , ,  y =

Dy ,

Dy =

1 n(Yi − my )2 , k , v = 1...K ,

Dy Dxk n i=1 between the referable value Y and factor X k is small, it is necessary for the model ( 1 ) to have enough number of regressors to provide required accuracy of referable value prediction. If the error rate  of the referable value Y is formed under the influence of a large number of X k factors, which are weakly dependent against each other and have similar rates of referable value Y formation, then the error rate distribution  can be moved into proximity with normal distribution. If the number of factors which are weakly dependent on each other is more than ten, we can assume that the error rate distribution i of the referable value for the linear regression model ( 4 ) is not different from the normal law of errors. Thus, error rate distribution i follows the normal law of errors for the regression model ( 4 ). Indeed, if the model ( 4 ) has summands whose effect on the sum (16) dispersion is overbearingly large compared to other summands, then we can ignore the summands whose effect is vanishingly small. If the number of factors in resulting model is small, then the assumption of the normal law of errors i should be discarded. Thus, due to the fact that the result of predicting of bronchial asthma severity depends on sufficiently large number of weakly dependent on each other X k factors, having approximately similar rates of the referable variable Y value forming, prediction error i distribution for a number of regressors K  10 tends to the normal law of errors. In this study, let us use the model of linear regression ( 4 ) with ten regressors to justify our assumption about the normal law of errors. Additionally, let us mention that procedures of model parameters estimation based on the t-distribution are as a rule sensitive only to large deviations of errors from normal distribution. If there are no large deviations from normal law of errors, validity coefficients for any error are close to an error following the normal law of errors. ( 10 ) ( 11 ) 3.3.

Assessment of factors using distribution histogram

Schematic prediction of regression factor of X k is one the main criteria of model regressors selection. This analysis will allow us to obtain diagnostic information about a predictor variable by determining outlying values of X k factor that has an effect on selection of regression function factors. Diagnostic information about a range and concentration of X levels in the study helps to adjust the range of regression analysis certainty. For schematic prediction, when choosing model parameters, we use a distribution histogram of model regressors qualitative values. Distribution histograms of model regressors qualitative values are given in Fig.1.

This analysis will allow us to correct the experimental data by excluding the data containing measurement errors. Comparative study of distribution histograms of model regressors values presented in various research papers is used to estimate the quality of experiment. The value for each regressor is defined by the value of a qualification factor, which characterizes the degree of regressor’s effect on bronchial asthma severity (0 – no effect, and 1-,2-,3-,4- degree of regressor’s effect on bronchial asthma manifestation in a child). The regressors “Atopic dermatitis”, “Allergic rhinitis” and “BA in relatives of second generation” reproduces the results in papers [27]. The results of the regressors “Sheep wool”, “Rabbit hair”, “Dog hair”, and “Pillow feather” are correspondent to the paper [28]. The rest of the factors correspond to statistical average data for the disease. 3.4.

Model construction

Р1 – comparison between the clinical sign presence and absence groups’ Р2 – comparison with the control group

Analysis of difference of thymic stromal lymphopoietin levels depending on the presence or absence

of asthma clinical signs sets only one possible difference – the value of negative heredity for allergy but not separately for asthma (table 2). The findings confirm the study of Gilda Varicchi, which shows that the gene of thymic stromal lymphopoietin is located on the 5q22.1 chromosome next to the “atopic cytokine” cluster on 5q31 and is responsible for allergy manifestations [29]. CD25 10*3 cells 100.739 here the calculation of ry xm , rxmxv is done in accordance with the formulas ( 10 ), ( 11 ). Raw experimental data for the factors selection are given in [30]. Numerical characteristics of the set of factors in accordance with the criterion (12) are given in Table 2. The selected factors are used to construct a linear regression model in this study ( 4 ).

The values of correlation coefficients between the factors of the rxm xv model, as well as between the model factors and the observed value ry xm are given in Table 3.

In Table 3, there are twelve factors for construction of a linear regression model, containing ten

factors. The number of techniques which are used to select 10 regressors for the model (16) out of twelve factors is defined by the following expression

12! 2!(12 − 2)!

Thus, 132 models are studied in this paper: 66 models with the above mentioned regressors for the observed value of Severe and 66 models with the above mentioned regressors for the observed value of TSLP. In Tables 4 and 5 four models with ten regressors for the Severe and TSLP values prediction are (13)

given. The value of

MSE ( 9 ) was the selection criterion of models for TSLP prediction. The models are placed in the descending order of the

MSE value. The models to predict the Severe value contain the same regressors as the model to predict the TSLP. The value of MSE for the 66 TSLP prediction models corresponds to the inequality 22,97  MSE  26,44 . It follows the assumtion that the regressors given in Table 2 have approximately similar rates of the referable value Y formation. If there is a predominant factor in a set of regressors, the value of MSE in the TSLP prediction models under is present. - 6 the absence of this factor would be significantly larger than in the model where this predominant factor

4.1. Diagnostic assessment of the model using Residual Plot

Let us consider the residual ei = Yi − Yˆi (14) as an observed value during diagnostic assessment of linear regression model. The observed residuals e show the properties of the  error ( 4 ) that is fundamental for linear regression model assessment i i 7 6 8 2 , 0 8 3 1 3 , 0 7 3 7 3 , 0 8 8 7 3 , 0 9 0 , 6 4 3 , 9 1 2 3 6 0 , 0 5 3 0 , 0 4 1 3 0 , 0 9 6 , 4 8 6 , 3 2 , 1 8 5 , 1 - 9

- 9 me = n1 i=n1 ei = 0, s2 = n −1 m i=n1 (ei − me )2 = n −1 m i=n1 ei2 = MSE. (15) The number of freedom degrees for the linear regression model with ten regressors is m = 11. The value of the dependences of residuals e = f (zvalue ) and e = f (Yˆ ) (Fig. 2) for the model diagnostic assessment.

MSE acts as an assessment of mean square deviation  . We will use a residual plot showing The value of zvalue j is defined by equation if z j = zvalue j . To calculate cumulative probability Pj for the sorted residual of e j , ( j = 1..n p ) let us use the formula

Pj = 1 2 z j −

 x22 dx,  exp −   Pj = (2 j − 1) / 2n p. (16) (17)

The emphasis should be on assessment of experimental data outliers. In Fig. 2, two outliers in

experimental data used for prediction of TSLP in the linear model No 101 (Table 2) with ten regressors are circled. This model contains the factors given in Table 2, excluding X 4 , X 6 . Model No 101 with the specified ten factors has the minimum value of s = 22.9728 among analyzed 66 models of linear regression for the observed TSLP.

An outlier for the residual e j can be connected with the measurement error of both the factor being predicted and one of its regressors. Let us construct M models with (M-1) regressors to define the source of the outlier in the M-factor model. If the outlier of e j remains in each of M (M-1)-factor models constructed, hence the Yi value is the source of the outlier. If for one of M models the outlier e j disappears, then the factor which is absent in the model is the source of the outlier. This technique will be called M-regressors technique. On each step of the M-regressors technique application we remove several outliers specifying the source of their appearance (Table 6). The values of data for definite sources of outliers will be replaced with empty values. The criterion for outlier definition is as follows: experimental selections with minimum and maximum deviation of the actual value of e j from the expected value of e j (deviation of the actual value of e j from the straight line the expected values are placed on) are taken. If absolute value of one of the values is significantly larger than of the other one, the iteration should be made for the maximum absolute value of e j . Only the experiments with existing model regressors values were used to define coefficients of the linear regression model. In Table 6 iterative approximants to define the outliers for the selection with number of elements n p = 61 with the total number of experiments of 90 are given. In 29 cases of experimental data, the model regressors contain missed values. As the first iteration, residuals e j for patients number 22 and 26 (Fig. 3-6) were assessed. The sources of outliers ( Yi , X m i ) were defined based on the outliers M in the models (M-1) study. are the source for the outlier 81, and the seven factors X 1 , X 2 , X 3 , X 5 , X 7 , X 8 , TSLP are the source for the outlier 39. If there is no factor X 7 in a model, then there is no outlier as well. Along with this, factor X 7 is the source for both outlier 81 and 39.

Since several factors are the source for each outlier, then it is supposed that the TSLP factor also

takes an effect on the e j value of the outlier. The consecutive removal of outliers (Table 6, Fig. 4, Fig. 5) resulted to the change of MSE value for the ten-factor model.

Table 4 shows the consecutive change of MSE value on each step of the outliers removal. After the outliers removal, the MSE value decreased by 30%: from MSE =22.97 to MSE =15.89 under 11 restrictions applied by linear regression model's coefficients. For the four values of the error e j in Fig. 6 zvalue  −1,5 and for the four values of the error e j when zvalue  1,5 . Error e j probability can be defined from expression (16), where z j = e j / MSE . Calculations show that z j  1,5 . This corresponds to the error value probability of P( z  1,5) = Pj  0,1 . In Fig. 7 residual plot of e = f (Yˆ ) depending on the predicted value of Yˆ for the selection values of zvalue  1,5 up to the value of MSE =10.51 for 45 patients in the selection and under 11 restrictions applied by linear regression model’s coefficients is shown. 4.2.

Asessment of regression model coefficients

Let us consider the test for m coefficients

H 0 : m = 0, H  : m  0. (23) The case of m = 0 corresponds to assessment for Y

E(Y ) = 0 + 1 X 1 + ... + m−1 X m−1 + 0  X m + m+1 X m+1 + ... +  M X M .

For the error i normal distribution this condition means not only the fact that there is no linear relationship between Y and X m . In fact, it means that there is no relationship between Y and X m at all, because for the model under studying only linear relationship between Y and X m is supposed. The value of the bm coefficient can be represented as

n bm = ryxm xym + 0(rxv xm )  KKxxmmxym = i=1 (X m i − mxm )(Yi − m y ), mxm = n1 i=n1 X m i.

n  (X mi − mxm i=1 )2 For regression model with normal error i distribution n n n  (X m i − mxm )Yi −  (X m i − mxm )m y  (X m i − mxm )Yi n bm  i=1 i=n1 (X mi −i=m1xm )2 = i =i=n11 (X mi − mxm )2 = i=1 km iYi .

Coefficient km i is a determinate function of X m i regressor having the following properties n nkm i = in=1 (X mi − mxm ) = 0, km i = n X m i − mxm , i=1  (X mi − mxm )2  (X mi − mxm )2 i=1 i=1 (24) (25) n n i=n1 km i X mi = i=i1=n(1X(Xmim−i−mmxmxm)X)2mi = 1, i=n1 km2 i =  i=ni1=1( X(Xmmii−−mmxmxm)2)2 = i=n1 (X mi1− mxm )2 .

Coefficient bm normal distribution law results from the fact that coefficient bm is a linear

combination of random value of Yi . To assess the regression model coefficient bm , let us consider the following expressions:

 n  n n  M  n n M E(bm )  E km iYi  = km i E(Yi ) = km i 0 + v X v i  = 0 km i + km i v X v i =  i=1  i=1 i=1  v=1  i=1 i=1 v=1 = n km i Mv X v i = Mv n km i X v i = m + Mv n km i X v i = m + 0(rxv xm ), (26) i=1 v=1 v=1 i=1 vv=1m, i=1 2 (bm )  2  nkm iYi  = nk m2 i2 (Yi ) = 2 nk m2 i + 0(rxv xm ) = n 2  i=1  i=1 i=1 (X mi − mxm )2 i=1 + 0(rxv xm ).

We will use them to calculate the value of Student’s t-test [31] tm value = s(bbmm ) , s(bm ) = (bm )  n (X mi − mxm )2 (27) i=1 for coefficient bm of the model when the number of items in selection is n=45 (Fig. 7) and the number of restrictions is M = 11. Critical value of t=0.05,r=34  2.032 corresponds to the statistics (27) when signification factor is  = 0.05 and the number of freedom degrees is r = (45 − 11) = 34 . Since distribution of bm / s(bm ) is the t-distribution, then confidence interval for bm coefficients can be defined by relationship

P(bm − s(bm ) t=0.05,r=34  m  bm + s(bm ) t=0.05,r=34 ) = 1 − , (28)  , hence

m = bm  bm , bm = s(bm ) t=0.05,r=34.

Results of the confidence interval calculations for several factors are given in Table 7. (29)

5. Conclusions

The technique of multifactor regression model construction is developed to predict bronchial asthma severity in children, when thymic stromal lymphopoietin level in blood serum is taken as a referable variable. Whereas disease severity is defined by a large number of factors that are weakly dependent on each other, assumption on normal error distribution was suggested. In preparation of experimental data comparative analysis of distribution histograms for model regressors values was used. The comparative study of experimental data and the data given in research papers on bronchial asthma severity was done to assess the quality of raw data. The results of histograms analysis were used to correct initial values of raw experimental data. During the analysis the data containing measurement error were excluded. The technique of multifactor linear regression model construction was developed to analyze bronchial asthma severity in children. The model of linear regression with ten regressors was studied in details. More than 100 factors effecting on bronchial asthma severity in children were analyzed to form a set of model’s regressors. The criteria which allow forming an optimal set of model’s regressors were developed. As a result of statistical analysis it was revealed that a confidence interval for coefficients of linear regression model is limited rather widely. It proves the necessity of the model further improvement. Further research perspectives involve model construction techniques based on nonlinear regression equations or neural network application. 6. References [12] I. Pavord, N. Hanania, Controversies in Allergy: Should Severe Asthma with Eosinophilic

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