The severity of the course of bronchial asthma depends on many factors. Clinical and laboratory studies were carried out on 90 children aged 6 to 18. 70 children with bronchial asthma of various degrees of severity as well as 20 healthy school-aged children were included into the main group. 142 predictors were studied, 11 factors were selected from the bottom in accordance with the selection method. Multivariate linear regression models have been developed and analyzed to predict the severity of bronchial asthma disease. The dependence of the forecast quality of the observed value on the number of model regressors is analyzed. The MSE value was used as a characteristic of forecast quality. An estimate of the number of regressors required for a significant increase in the forecast quality is presented. The law of distribution of the error in predicting the severity of bronchial asthma disease in a multifactorial linear regression model has been substantiated. The visual representation of multivariate models is made using the residual plot.
Studying not only the factors, but also determining their relationship with each other, is an important step in understanding the course of the disease in each individual case. Analysis of the multifactorial nature of bronchial asthma underlies the prediction of the disease and its course [12]. Numerous studies have examined various categories of factors. Commonly used factors include age, gender of the child, whistling breath, allergic sensitization, Ig E [13, 14].
To assess the prognosis of a severe course of bronchial asthma, both linear and nonlinear multifactorial models are used, containing a different number of regressors (Table 1), aimed at determining predictors that are unambiguously significant in determining the severity of the course of bronchial asthma disease [15].
In this case, the values of the regressors are determined by both quantitative and qualitative values. Table 1 presents data on some common types of models for predicting the severity of bronchial asthma and on the number of predictors in these models.
It should be noted that the models presented in Table 1 with the same number of regressors are used to analyze the prediction of the severity of bronchial asthma by different initial factors.
The presence of a large number of models with different numbers of regressors makes the issue of choosing both the type of model and the number of regressors in the model relevant. In this research, we analyze the dependence of the forecast quality on the number of model regressors.
The process of building a linear regression model with a large number of regressors is quite laborious. The computational complexity of the algorithm for constructing a regression model grows in proportion to the square of the number of regressors in the model. Therefore, when analyzing the severity of bronchial asthma, linear models are usually used with the number of regressors, the number of which does not exceed 5-7 (Table 1). Linear models with a small number of regressors can be considered as a tool for preliminary analysis of a set of experimental data. A deeper analysis requires an increase in the number of regressors in the model. Due to the fact that the results of predicting the severity of the course of bronchial asthma depend on a sufficiently large number of weakly dependent factors with approximately the same scale of formation of the explained value, an increase in the number of regressors in the model leads to a slight increase in the quality of the prediction of the observed value. On the other hand, the presence of a large number of weakly dependent factors with approximately the same scale of formation of the value of the explained quantity leads to the fact that linear regression models are a good tool for predicting the severity of the progression of bronchial asthma disease due to the fact that the distribution of the prediction error for the quantity of the regressors K 10 satisfies the normal distribution law. However, the issue arises, how many regressors the model should contain and how much the prediction accuracy of the model will be estimated to increase with an increase in the number of regressors. This work is devoted to the analysis of this problem.
The regression model that allows you to determine the severity of the course of the bronchial asthma can be summarized as follows:
Yi F X1, X 2 ,...,X k i
(
To analyze the influence of the number of regressors on the quality of predicting the severity of the disease of bronchial asthma, we will use the data set formed during the examination of 90 children with a diagnosis of bronchial asthma aged 6 to 18 years. The investigation contains data from the anamnesis of life and diseases of patients, laboratory and diagnostic indicators of the examination. 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. To analyze the dependence of the parameters on the quality of prediction, 142 factors were selected, which were encoded. As a result of the studies, for each examined patient, the values of 142 factors were recorded, on which, it is assumed, the severity of the course of the bronchial asthma disease may depend. As a result of preliminary analysis, invalid data were excluded from this set. The resulting dataset in the form of a 90x142 matrix [26] was used to build a regression model. As a result of the phased elimination, 11 factors were identified that match the criterion
ry xm max , rxmxv min . (
The selected factors are used in this work to construct linear regression models for predicting the
severity of bronchial asthma disease. The type of model is determined by criteria (
Due to the fact that to assess the severity of the course of bronchial asthma disease, a large number of weakly dependent factors with approximately the same resulting contribution of the predicted observable value, proportional ryx , were selected, choice for prediction a linear model for prediction suggests that the error has a normal distribution with distribution characteristics: E( i ) 0 , 2 ( i ) 2 , 2 ( i , j ) 0 , j i .
(
This feature, characteristic of models for predicting the severity of bronchial asthma disease, will be used to compare the prediction accuracy of linear models with different numbers of regressors.
It should also be added that the spread of error values i for each range of predictor values X m obeys a probability distribution with mathematical expectation E( i ) 0 and standard deviation ( i ) .
The first step of the study after the choice of factors (Table 2) is to build a set of linear regression models for predicting the severity of bronchial asthma disease and comparing the quality of the prediction of the observed value for a different number of model regressors. As a criterion for comparing models, we will use the MSE value n MSE i2 .
1 n i1 For comparison, consider 1, 2, 3, 5 and 10 factor linear regression models, for the construction of which we used the factors from Table 2.
For the eleven factors presented in Table 2, we construct 11 models with ten factors (Table 3).
The columns labeled «Xm» show the values of the coefficients before the regressor code Xm, which
can be identified using Table 2. The free term of an equation are presented in the table in column "A".
Linear regression model №1 and № 10 are designated as Y1 , Y10 has the form:
Y7 . By virtue of the above-justified assumption, that error has a normal distribution, it follows that
the points characterizing the value of residuals ei for the value of prediction errors i , should lie on a
small neighborhood one straight line. Anomalous values are shown in circles on the graphs. The outliers
were probably due to errors in the operation of the equipment used to change the values of quantitative
factors or to the carelessness of the personnel in the preparation of raw survey data. Also, abnormal
values can be associated with the presence of incorrect answers of patients in the personal survey sheet
submitted for the study.
(
The presence of outliers can lead to a significant distortion of the form of the regression model, and, accordingly, to an increase in the error. In this regard, the anomalous values of the regressors in the prepared dataset should be changed or excluded from the set that will be used to build a linear regression model. Table 3 shows the MSE value for each of the models Y1 Y10 . Models Y1 , Y10 correspond to the lowest and highest MSE values. Each of the models in Table 3 has a significant number of anomalous values. The MSE value for each of the model Y1 Y10 is approximately the same (Table 3).
To improve the prediction accuracy of the regression model, exclude outliers from the dataset and
rebuild the models Y1 Y10 based on the changed data. There are a number of methods for correcting
the anomalous values that are presented in the dataset. We will take advantage of excluding rows from
the dataset that correspond to patients with abnormal values of one or more regressors. After excluding
six rows from the dataset, each of which corresponds to the outlier in Figure 1, the coefficients for the
linear regression models were recalculated. Linear regression models Y1 , Y10 (
Y1 0.19 0.11X1 0.31X2 0.01X3 0.02X4 0.31X5 0.02X6 0.01X7 (
The final results of the analysis of the models after excluding outliers are presented in Table 4. For
each model, the MSE value was determined before and after excluding anomalous values. There is a
decrease in the MSE value by several times. For the model Y1 the MSE value decreased from 0.071 to
0.027, and for the model Y10 the MSE value decreased from 0.088 to 0.037. As before excluding
outliers, the model Y1 has the best indicator according to the MSE criterion (
Analysis of the results presented in the table shows that the models Y1 , Y5 , Y8 contain almost the same outliers with coded patient numbers 66, 39, 49, 35, 20, 26. Models Y2 , Y4 also contain almost the same outliers with coded patients 66, 39, 35, 20, 26 and 66, 49, 35, 20, 26. Models Y6 , Y9 , Y11 contain outliers with coded patient numbers 66, 49, 39, 35, 20, 26. coefficients for some of them are presented in table 5.
Models are selected from Table 5 according to the smallest and largest values of MSE,Y78 (MSE=0.025) и Y182 =(MSE=0.061):
Y78 0.18 0.09X1 0.27X 2 0.02X 4 0.35X5 0.03X6 0.04X9 0.1X10 ,
Y182 0.15 0.13X1 0.01X3 0.04X7 0.04X8 0.07X9 0.08X10 0.03X11 .
(
The values of these indicators will be used to analyze the quality of prediction for models with a different number of regressors.
The residual plots shown in Figure 4 correspond to the linear regression models Y78 , Y182 . We see that the residual plot for the model Y182 (Figure 4) (Figure 4) contains a sufficiently large number of anomalous values for the residuals ei , which led to a significant increase in the MSE value of the model
Models Y7 , Y10 also contain almost the same outliers with coded patient numbers 69, 48 ,49, 36, 20, 39, 25, 26 and 66, 39, 19, 69, 25, 20, 49, 26. The sequence is indicated in ascending order of the residual error value. Analysis of the results of Table 4 allows us to form a set of rows to which outliers correspond and which are candidates for exclusion of the dataset. 3.2.
The next step of the study is the construction of 7-factor linear regression models. To build the models, we use the factors from Table 2. From the eleven factors, 330 seven-factor models were built and analyzed
C171
11! 7!(11 7)! . 63 63 63
When we construct five-factor models, we use the same approach that was used to build seven-factor models. To build the models, we use the factors from Table 2. From the eleven factors, 462 five-factor models were built and analyzed
C151 coefficients for some of them are presented in table 6.
To graphically represent the analysis results, residual plots were selected for the model Y309 , with
the lowest MSE value and for the model Y133 with the lowest MSE value and for the model (Figure 5).
Model Y309 with the lowest MSE and the model Y133 with the highest MSE have an analytic view:
Y309 0.15 0.29X 2 0.36X 5 0.03X 6 0.05X 9 0.09X10 , (
Three-factor models are not widely used in the analysis of the severity of bronchial asthma disease.
These models are used as an indicator for superficially determining the severity. However, we believe
that three-factor models are worth considering for a general understanding of the issue of how much
the forecast accuracy increases when moving from a three-factor linear regression model to a five-factor
or seven-factor linear regression model. To build three-factor models, the factors from Table 2 were
used. Of the eleven factors, 165 three-factor models were constructed and analyzed
11!
C131 165, (
3!(11 3)! coefficients for some of them are presented in table 7. To demonstrate the analysis results, residual plots were selected for the model Y3 and Y103 (Figure 6).
As in previous multivariate model analyzes, the first model Y3 corresponds to the lowest MSE, and the second model corresponds to the highest MSE of the analyzed number from three-factor models. The analytical presentation of the models has the form:
Y3 0.11 0.12X1 0.36X 2 0.48X 5 , Y103 0.02 0.004X3 0.06X7 0.09X11 , (
The jump-like dependence of the values for the residuals ei from zvalue is explained by the fact that
for the forecast regressors are used, which are represented by qualitative values (for example, there is
the presence of a feature or there is no presence of a feature).
Indeed, [Allergic rhinitis], [Atopic dermatitis] and [Bronchial asthma in relatives of second
generation] were chosen as regressors for predicting the severity of bronchial asthma disease for the
model corresponding to the best result in terms of the quality of fit (
These factors are decisive in the superficial diagnosis of the observed value. In contrast to the model Y3 (Figure 6.а), for the model Y103 the points characterizing the values of the residuals lie on one straight line, with the exception of several outliers ei which contains each of the models considered above. Model Y103 is presented by regressors [Number of years from the first symptoms], [Domestic dust], [CD25 10*3 cells], among which the values of the two regressors are given by quantitative
In conclusion, we will consider two-factor (Table 8) and one-factor (Table 9) linear regression models. As a result of the analysis, 55 two-factor models and 11 paired regression models were considered:
11! 11! C121 55, C111
2!(11 2)! 1!(111)! coefficients for some of them are presented in tables 8 and 9.
Analytical representation of the two-factor model Y21 , Y23 is: 11 ,
Y21 0.01 0.01X 4 0.5X5 , Y23 0.1 0.01X3 0.06X7 .
The model Y21 corresponds to the lowest MSE and the model Y23 corresponds to the highest MSE. Note to MSE that the paired regression model contains the same factor that is present in the two-factor model:
Y5 0.05 0.47X5 , Y7 0.07 0.06X7 . (
Thus, a two-factor linear regression model is a refinement of a paired regression model. An important note is that the paired regression model with the minimum MSE contains the [Bronchial asthma in relatives of second generation] regressor, which in the two-parameter model is supplemented by the [Bronchial asthma in father] factor, and in the three-parameter model [Allergic rhinitis], [Atopic dermatitis]. The prediction accuracy of the two-parameter regression model is quite close to the prediction accuracy of the three-parameter regression model. 4. Analysis of results In the previous section, a detailed analysis of multivariate linear regression models, consisting of ten, seven, five, three, two factors, as well as an analysis of the paired regression model was carried out. For each category of models, the model with the lowest and highest MSE values was found. The obtained MSE values are used to compare the quality of predicting the observed value, presented in Figure 7. The dotted line in the graph shows the average value MSE, which is half the sum of the smallest and largest values. The results obtained clearly show that for linear regression models dependent on two to five factors, the value MSE has almost the same value. Improving the forecasting quality is achieved by increasing the number of regressors. With an increase in the number of regressors, the range of variation of the value is significantly narrowed MSE MSEmin; MSEmax . In this case MSEmin the value changes slowly with an increase in the number of regressors. When moving from a five-factor linear regression model to a ten-factor linear regression model, the the average value MSE (the dotted line) decreased by an amount not exceeding 20%. Approximately the same value MSEmin is explained by the fact that when the number of factors decreases, outliers are excluded.
Indeed, on the one hand, a decrease in the number of factors should lead to an increase in the error. On the other hand, a model with fewer factors contains only outliers that correspond to the model factors, which accordingly improves the model's accuracy. In this regard, an important conclusion should be made about the need for preliminary data processing. The presence of outliers can lead to a decrease in accuracy with an increase in the number of regressors.
As shown in this work, due to the fact that for models with ten or more factors, the error has a
normal distribution with distribution characteristics (
In this work, we performed a comparative analysis of the quality of predicting the severity of the course of bronchial asthma depending on the number of regressors in the model. For comparative analysis, a multivariate linear regression model was used. The substantiation of the distribution law for the forecasting error is given. The comparative analysis of values MSE for multivariate linear regression models using the example of the considered dataset shows that the use of models with less than six factors is inappropriate. The results obtained indicate that linear regression models with a small number of factors have approximately the same value MSE . As a result of performing this study, an important conclusion was obtained that the value MSE slowly decreases with an increase in the number of model regressors. This raises the relevance of the search for new methods for predicting the severity of bronchial asthma disease, including the use of Machine learning. A prospect for further research is to analyze the quality of fit the observed value depending on the number of regressors for different types of nonlinear regression models.
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