The article suggests methods for estimating the parameters of logistic regression for different conditions. In the case of a single polytomic input variable with a minimum number of categories - a method for assessing chances and probabilities. In this case, the quality of classification can be evaluated separately for each input variable: the assessment does not depend on the connectedness of the input variables, which allows not to check the correlation and preliminary selection of significant variables. For several variables, it is proposed to use a Bayesian classifier, which, if there is no correlation between the attributes, assigns specific individuals of the population to a certain class for health reasons. If there is a correlation of factor attributes and complex dependencies between input variables, it is proposed to use the maximum likelihood estimation. As a result of the analysis, a ready-made mathematical apparatus will be obtained, which makes it possible in practice to obtain the values of the the probabilities of diseases under various initial data..
Diagnostic methods [1-2] in medicine play a crucial role. The accuracy of the diagnosis and the speed with which it can be made depends on many factors: the condition of the patient, the available data on the symptoms and signs of the disease, the results of laboratory tests, but most importantly, the qualifications of the doctor himself [3-6]. An accurately diagnosed diagnosis as soon as possible allows increasing the chance of curing the patient [7]. Based on all these considerations, it is natural to try to determine the conditions under which the diagnosis can be made as quickly and accurately as possible.
For many centuries, doctors have been trying to solve this problem with varying degrees of success. However, in recent years, thanks to the use of modern methods of treatment and diagnostics based on the latest achievements of science and technology, the chances of obtaining successful results have increased significantly. Therefore, it is important to find the exact methods [8-9] for description, research, evaluation and monitoring of the diagnosis process, which makes the task of determining the likelihood of disease based on existing data on the patient’s condition relevant.
If the study is associated with a large number of interdependent factors that exhibit significant natural variability, then for a sufficiently effective description of the complex pattern of their influence, there is only one way - using the appropriate statistical method [10]. If there is a need to determine the probability of falling into one of two classes of the disease, one of the simplest and most effective methods is the binary classifier. The quality of the classification can be evaluated for each input variable separately. If the number of factors or the number of data categories is very large, it is necessary to use the computing power of the computer [11] so that the desired results can be obtained in a fairly short time, which will reduce the likelihood of errors in the diagnosis, and will also make it as quick and efficient as possible.
Thus, the aim of the study is to determine the likelihood of a patient's disease [1213] with specified diagnostic characteristics based on Data Mining methods, which will improve the accuracy of diagnosis.
The health status of each individual is influenced by a number of factors, such as: age, gender, illness, place of residence, temperature, blood condition, etc. [14]. The objective of the study is to identify and analyze methods that allow us to assess the likelihood of illness [15] of a patient with specified diagnostic characteristics.
This task is referred to the classification tasks “with the teacher”, during which the test system is trained using the “stimulus-reaction” examples. It is required to find dependency that shows which patients belong to the "Healthy" class and which patients belong to the “Sick” class. For such a task, it is rational to use logistic regression, which is widely used to find the probabilities of an event with given characteristics [16]. 2
Estimation of logistic regression parameters based on the method of assessing chances and probabilities Consider a sample of patients based on data from the source [17]. For each patient, health information is known. The explanatory variable in this case is the result of an electrocardiogram (ECG) at rest. This variable is polytomic. Each patient can belong to the three classes “Normal”, “Hyp” and “Abnormal” according to ECG results. Two events are also considered: the patient is sick (y = 1) and healthy (y = 0).
It is necessary to evaluate the parameters of the logistic equation for this problem and determine the probability with which it will belong to the “Sick” class, i.e. evaluate his state of health. Probability that the output variable y 1 for the given value of the explanatory variable x will be P( y 1| x) x , and the probability that y 0 at a given value x will be equal to P( y 0 | x) 1 x .
The conditional average for logistic regression in this case is determined as in
formula (
The function is defined on an infinite interval and takes values in a range [0, 1]. Required to find the best estimates of parameters 0,1,2 . We will organize the information about patients based on the data [17] in the form of a Table 1. In the Table 1 in the line with the “Normal” class, the quantization variables will be equal to: С1 С2 0 . In line with class “Hyp” С1 1,С2 0 . In the line with the class “Abnormal” С1 С2 1.
The chances of being a patient with a sick heart for all categories of conditions of
the electrocardiogram are estimated by the formulas (
56 Chy1,C1 96 0.58 Chy1,C2 79
1.16 68 Chy1,C3 3 OR С2 Chy1,C2 2.01
С1 Chy1,C1 OR С3 Chy1,C3 5.17
С1 Chy1,C1
The odds ratio for the “Hyp” categories to the “Normal” category is estimated by the
formula (
Estimating the likelihood of a disease using a Bayesian classifier Consider a sample of 30 patients with input variables defined in the nominal scale (Table 2) based on data from the source [17]. For analysis, we use the following signs: age (in years), blood sugar, patient gender, ECG result. According to the Table 2, the pair correlation coefficients were calculated, the values of which are in the interval [-0.303; 0.078], which indicates a low correlation between the input variables. Thus, in this case, you can use the Bayesian classifier, the application of which for this case is considered in detail in [18].
We denote by C1 the class “Sick” for whom the state of the disease is present (the value of the resulting variable is “yes”). Through C2, we can designate the class of patients “Healthy”, which have no signs of illness (the value of the resulting variable is “no”). The use of the Bayesian classifier does not make it possible to obtain the form of a statistical dependence based on the training sample, however, it makes it possible to determine the probability that a patient with given characteristics will fall into one or another class. For example, we define that a patient aged 50 to 59 years, with blood sugar less than 120 units, a man and with the result of ECG “Hyp” will fall into the class “Sick”.
It is necessary to maximize the product of probabilities P( X | Ck )P Ck for k 2 ,
because there are only two classes in this problem. The prior probability of the
appearance of class C1 is calculated by the formula (
Estimation of logistic regression parameters based on the maximum likelihood estimation Consider a sample of 303 patients with input characteristics shown in Table 4 based on data from the source [17]. The resulting trait is measured in a dichotomous scale, and factor traits in metric and other types of scales. It is necessary to determine the likelihood of a patient's disease with this many characteristics.
Since the maximum likelihood estimation (MLE) is quite resource-intensive, we
will use the software from IBM – SPSS Statistics for the demonstration. This software
allows us not only to find the parameters of logistic regression, but also to evaluate the
parameters of the model and probability, and also analyze the quality of the model.
(
cholesterol fasting blood sugar <120 resting ecg maximum heart rate angina peak slope #colored vessels thal class* The most significant results are visible in the tables below. In the Table 5 presents the quality factors of the model. Criterion -2 Log probability corresponds to the correspondence between the models and the source data. The smaller this indicator, the more adequate the model.
R-squared Cox & Snell and R-squared of the Nagelkerke are stably statistically consistent, which are used in the logit. The value of an equal object is achievable. In the second sign, this drawback is eliminated. These criteria shows the share of all factor characteristics. More detailed information can be taken from the source [17]. In the table 6 presents the values of the Chi-square test.
1
147 23
18 115
93.1 91.3 92.2
A – sex(
E1
@#coloredvessels(
E2
@#coloredvessels(
E3
@#coloredvessels(
F - thal
F1 - thal(
Based on this table, you can determine the most significant factors by which you can get the smallest errors with a high probability. The general form of the regression equation for the patient will have the form similar to formula (24): g x 1, 464 A 2, 286 B1 0,971 B2 0,170 B3 0,763 C 1,724 D1 2,018 D2 1,763 E1 0, 495 E2 1, 498 E3 1, 492 F1 1, 452 F2 Then, for a male patient with a second type of chest pain that did not have a sore throat, with a bias of the first type, with vessels of the third type, as well as a thal of the first type, it will be true (25):
g x 1, 464 0,971 1, 724 1, 498 1, 492 1, 237 Then the probability that such a patient will be healthy is calculated by the formula (26): x eg x / 1 eg x 0, 77 (26) Moreover, as can be seen from the Table 10, according to Wald's statistics, the most significant are the following factors: chestpaintype (value 30.864), slope (value 24.588), coloredvessels (value 36.481).
Wald test – a statistical test used to check the restrictions on the parameters of statistical models estimated on the basis of sample data. It is the most appropriate of the three basic constraint checking tests such as the likelihood ratio test and the Lagrange multiplier test. The test is asymptotic, that is, a sufficiently large sample size is required for the reliability of the conclusions. The confidence interval (CI) of the test is also a closed form. The higher the statistics, the better.
The significance of the factors is confirmed using the appropriate level of significance. It is defined as the p-level, which is calculated during the test. The lower this level, the better.
Based on the data in Table 4, the probabilities of getting into the Healthy group were calculated for all data.
In the “Expected” and “Group” columns, you can see the probabilities of getting into the “Healthy” or “Sick” group.
The simulation results (Table 8) show high accuracy of the classification results in comparison with the classes previously known for the experimental sample (Table 4).
Based on the results, we can say that the model adequately describes this population. 5
Conclusions In this work we identified, analyzed and implemented methods that allow us to assess the likelihood of a patient's disease with specified diagnostic characteristics.
It is shown in which cases it is advisable to use certain methods to determine the probability and estimate the parameters of models. These models are not static. Calculation of parameters can be carried out every time when the amount of data about patients changes, and the use of SPSS software tools will allow calculations to be made quite quickly. The data obtained will allow a more accurate assessment of the state of health in the face of constantly changing diagnostic parameters.