392 © 2024 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq PISSN: 1609-4042, EISSN: 2521-3407 IHJPAS. 2024, 37(4) A Modified Multivariate Bayesian Logistic Model with Application to Health Datasets Azza Mustafa Abd Al Kader Al Kusaem* Department of Business Administration, College of Administration and Economics, University of Mosul, Mosul, Iraq. *Corresponding Author. Received: 26 January 2023 Accepted: 16 March 2023 Published: 20 October 2024 doi.org/10.30526/37.4.3245 Abstract The application of Bayesian strategies for binary logistic estimation is demonstrated in this article. A modified method of the Bayesian logistic model using the Metropolis-Hasting algorithm is derived and applied to three simulation data sets. We compared the new model with existing classification methods: support vector machine, artificial neural network and regular logistic model. The modified model was used to classify the heart disease dataset. The data came from a database intended for UCI Data Science (https://www.kaggle.com). The clarification accuracy and the time required are checked and compared with other standard methods. It has been shown that the presented model has the best accuracy and efficiency compared to the different classification methods. All calculations were performed with the program R version 4.2.2. Keywords: Bayesian, classification, heart disease, logistic regression, posterior distribution. 1. Introduction The Bayesian algorithm, Markov chain Monte Carlo (MCMC), is a widely used technical algorithm used to estimate the parameters of the posterior distribution in the model [1, 2]. The standard estimation of the logistic regression model has some limitations that can be overcome with possible replacement methods. This paper aims to introduce a different approach using Bayesian analysis. Multinomial logistic regression can predict categorical placement based on multiple independent variables, such as the possibility of belonging to a category on a related variable. The independent variables can be either categorical or continuous. Multinomial logistic regression is often considered an appealing analysis because it does not require homoscedasticity, linearity, or normality [3]. Discriminant function analysis is an excellent substitute for multinomial logistic regression, which requires these assumptions. One of the most common classification methods for health datasets is logistic regression. The maximum likelihood or ordinary least square estimator can be used to estimate the regression coefficients [4, 5]. https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://orcid.org/0000-0003-2322-8991 mailto:aza.mustafa@uomosul.edu.iq IHJPAS. 2024, 37(4 ) 393 The dependent variable in logistic regression is either binary or dichotomous, i.e. the logistic regression contains only data coded as “1” (true, yes, healthy, success, pregnant, etc.) or “0” (false, no, sick, failure, not pregnant, etc.) [6]. Many standard classifications such as Support Vector Machine (SVM) [2, 7], Artificial Neural Networks (ANN) [8] and regular logistic regression [9] can be used for classification. However, working with a large dataset leads to inefficiency and time consumption [10, 11]. This paper presents a new Bayesian multivariate model for classifying datasets. The modified model is applied to three simulation datasets and then used to classify the heart disease dataset. This article is structured as follows: In section two, a general idea for multivariate Bayesian binary logistic linear regression is explained. The Bayesian formulation model is discussed in section three. Simulation studies and experimental results are presented in section four. The real data set is presented in section five. Section six discusses the results. 2. Multivariate Bayesian binary logistic regression model Binary logistic regression is a special form of regression in which the binary response variable is linked to a discrete or continuous set of explanatory variables. The key point here is that the predicted values of the response variable are modelled based on the mixture of values provided by the predictors in linear regression [12]. Let 𝑋 = (𝑋1, 𝑋2, … , 𝑋𝑝) be a set of 𝑝 explanatory variables and 𝑌 be a binary response variable; then the binary Logistic regression model can be written as follows, 𝑙𝑜𝑔𝑖𝑡(𝑝) = log ( 𝑝 1−𝑝 ) = 𝛽0 + 𝛽𝑥𝑖 + ⋯ + 𝛽0 + 𝛽𝑋𝑘′ (1) Which models the log odds of the probability of “present” as a function of explanatory variables. Assume that 𝑌𝑖 ∼ 𝐵𝑖𝑛(𝑛𝑖, 𝑝𝑖), i.e., 𝑌𝑖 is a binary logistic model, then 𝜋𝑖 = 𝑃𝑟𝑜(𝑌𝑖 = 1|𝑋𝑖 = 𝑥𝑖) = 𝑒𝛽0+𝛽1𝑥𝑖 1+ 𝑒𝛽0+𝛽1𝑥𝑖 (2) The maximum likelihood estimator (MLE) for the parameters, {𝛽0, 𝛽1}, is obtained by finding (𝛽0̂, 𝛽1̂) that maximizes: 𝐿(𝛽0, 𝛽1) = ∏ 𝑝𝑖 𝑦𝑖(1 − 𝑝𝑖)𝑛𝑖−𝑦𝑖 =𝑛 𝑖=1 ∏ 𝑒{𝑦𝑖(𝛽0+𝛽1𝑥𝑖)} 1+𝑒𝛽0+𝛽1𝑥𝑖 𝑛 𝑖=1 (3) 3. Bayesian Formulation The Bayesian model can be treated like a classification problem. Researchers can infer the individual from the model parameters and the data. From a set of likely divergent opinions about a particular condition, one of two outcomes can be inferred: Does individual j actually have a particular disease state (y = 0 if no and y = 1 if yes) [13, 14]. The Bayesian hierarchical logistic regression model addresses this problem to account for the variability of outcomes stemming from both informants and informative priorities. Bayes' theorem and a generative model can be applied when we have data to calculate the posterior probability distribution of the model parameters using the prior distribution as a function of the predictors (X) and the response (Y). In general, the three critical components associated with parameter estimation in the Bayesian system are the prior distribution, the likelihood function, and the posterior distribution. Bayes' theorem formally combines these three elements: 𝑃𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟 = 𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 × 𝑃𝑟𝑖𝑜𝑟 IHJPAS. 2024, 37(4 ) 394 Basically, the above expression means that the knowledge in the sample (reflected in the likelihood function) is combined with data from other sources (summarised in the prior distribution) to obtain the posterior distribution. All available information about the parameters of the model is incorporated into the posterior distribution. [15],[16] deals in detail with the principle of Bayesian analysis. The likelihood contribution from the 𝑖𝑡ℎ individual is binomial, 𝐿𝑖 = ( 𝑒𝛽0+𝛽1𝑥𝑖 1+ 𝑒𝛽0+𝛽1𝑥𝑖 ) 𝑦𝑖 × (1 − 𝑒𝛽0+𝛽1𝑥𝑖 1+ 𝑒𝛽0+𝛽1𝑥𝑖 ) (1−𝑦𝑖) (4) Since individual subjects are presumed to be separate from one another, the probability function for a data set of 𝑛 subjects is then the probability distribution. 𝐿 = ∏ ( 𝑒𝛽0+𝛽1𝑥𝑖 1+ 𝑒𝛽0+𝛽1𝑥𝑖 ) 𝑦𝑖 × (1 − 𝑒𝛽0+𝛽1𝑥𝑖 1+ 𝑒𝛽0+𝛽1𝑥𝑖 ) (1−𝑦𝑖) 𝑛 𝑖=1 (5) By assuming the multivariate normal prior on 𝛽; i.e. 𝛽𝑖 ∼ 𝑁(𝜇𝑖 , 𝜎𝑖 2), we get 𝑓(𝛽) = 1 √2𝜋𝜎2 𝑒 {− 1 2 ( 𝛽𝑖−𝜇𝑖 𝜎𝑖 ) 2 } (6) As a result, the posterior distribution is calculated by multiplying the probability function by the prior: Posterior = ∏ ( eβ0+β1xi 1+ eβ0+β1xi ) yi × (1 − eβ0+β1xi 1+ eβ0+β1xi ) (1−yi) n i=1 × 1 √2πσ2 e {− 1 2 ( βi−μi σi ) 2 } (7) We can use the Metropolis-Hasting procedure to sample from the above posterior distribution. The following section will present the algorithm in detail. 3.1 Markov chain Monte Carlo If a sequence of numbers follows the below graphical model, it is a Markov chain is 𝑃(𝑋5|𝑋4, 𝑋3, 𝑋2, 𝑋1) = 𝑃(𝑋5|𝑋4). As a result, the probability of reaching a specific state is solely determined by the chain's previous state [10]. Using the full joint density function, the Metropolis-Hasting (MH) algorithm can be used. The MH technique is an iterative method that generates a Markov chain sequence to estimate the posterior distribution's parameters [17,18]. The following steps summarize the Metropolis-Hasting algorithm [19, 20]: Metropolis-Hasting algorithm 1. Initialize 𝑥(0) 2. For i = 0 to K−1  calculate 𝑢 ∼ uniform distribution[0,1]  calculate 𝑋⋆ ∼ 𝑞(𝑋⋆|𝑋(𝑖))  If 𝑢 < 𝑅(𝑋(𝑖) , 𝑋⋆) = min{1, 𝐹(𝑋⋆)𝑞(𝑧(𝑖)|𝑍⋆) 𝐹(𝑧(𝑖))𝑞(𝑋⋆|𝑍(𝑖)) }  𝑋(𝑖+1) = 𝑋⋆  𝑒𝑙𝑠𝑒  𝑋(𝑖+1) = 𝑋(𝑖) We can carefully choose the proposal distribution 𝑞. The MH algorithm assumes a symmetric random walk for the proposal distribution, i.e., 𝑞(𝑥|𝑦) = 𝑞(𝑦|𝑥). 𝑝(𝑥) does not even have to be IHJPAS. 2024, 37(4 ) 395 the full Bayesian probability but simply is required to be proportionate to it. This is clear since the Bayes denominators will cancel out. 4. Simulation Studies The posterior distributions for the parameters are applied in this section (Eq. 1 and Eq. 2). Datasets of size 100, 200 and 400 are simulated. First, a set of five variables are simulated uniformly from a uniform distribution U(0 1), and then 𝛽0 = 2.41, 𝛽1 = −1.11 𝛽2 = 4.72, 𝛽3 = 0.31, and 𝛽4 = 4.10 are used. By applying the MH algorithm for 10000 repetitions, the parameters were obtained by calculating the posterior sample mean [21, 22]. In Figure 1, the posterior samples are plotted as histograms, and a red line remarks the true values for these parameters. Obviously, the histograms are approximately normal, and the true values are close to the sample's mean. In addition, the model is used to forecast the response for specific different values. The responses are plotted as histograms, Figure 1, for all the prediction samples, and the exact value are notified with the blue lines. Figure 1. Posterior mean for the parameters. Table 1. compares the exact parameter values with the classical logistic estimation, and posterior sample mean. Also, 95% credible intervals (𝐶. 𝐼.)are created for all the parameters. The exact values lie inside the 95% 𝐶. 𝐼. which implies that the approximation is acceptable [23]. Table 1. Estimation of parameters using classical logistic and posterior distribution. True value Classical Logistic Posterior mean 𝟗𝟓% 𝑪. 𝑰. 𝜷𝟎 2.41 2.12 2.45 (2.23 , 2.53) 𝜷𝟏 -1.11 -1.43 -1.18 (-1.37 , -1.04) 𝜷𝟐 4.72 4.83 4.72 (4.62 , 4.82) 𝜷𝟑 0.31 0.21 0.25 (0.12 , 0.42) 𝜷𝟒 4.10 4.17 4.12 (4.02 , 4.17) 𝝈𝟐 0.18 0.16 0.17 (0.15 , 0.19) b0 2.2 2.3 2.4 2.5 2.6 2.7 b1 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 −0.9 b2 4.4 4.5 4.6 4.7 4.8 4.9 5.0 b3 0.0 0.1 0.2 0.3 0.4 0.5 b4 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 s 2 0.14 0.16 0.18 0.20 0.22 0.24 IHJPAS. 2024, 37(4 ) 396 5. Real Dataset Application The proposed method has been applied to the real health dataset. We used the heart disease dataset as an application to our work. The dataset has been downloaded from the UCI data science website (https://www.kaggle.com). It contains patients’ health history and many medical indicators for each patient. In the following section, a brief description of the heart disease dataset is given. Heart Disease Dataset Analysis The description of the variables is given as shown in Table 2. The dataset consists of 303 observations and 12 variables: age, sex, cp, trestbps, chol, fbs, restecg, thalach, exang, slop, ca, and thal. The aim is to forecast whether an individual has heart disease depending on the features or not [24]. Table 2. Descriptive table for the variables No. Predicters Descriptive 1 age The age in years for patients 2 sex Male = 1; Female = 0 3 cp the chest pressure that was felt: Value 1 denotes normal angina, Value 2 denotes atypical angina, Value 3 denotes non-anginal discomfort, and Value 4 denotes asymptomatic 4 trestbps On admission to the facility, the patient's resting blood pressure was measured in millimeters of mercury (mm Hg). 5 chol cholesterol levels in milligrams per deciliter 6 fbs fasting blood sugar is over 120 mg/dl: true = 1; false = 0 7 restecg Resting electrocardiographic measurement: regular = 0, st-t wave abnormality = 1, potential or definite left ventricular hypertrophy according to Estes' criteria = 3. 8 thalach Attained optimum heart rate 9 exang Angina caused by workout: yes = 1; no = 0 10 slop the slope of the most difficult workout segment: Value 1 indicates an upslope, value 2 indicates a smooth surface, and value 3 indicates a downslope. 11 ca total of major vessels colored by flourosopy (0 - 3) 12 thal thalassemia is a form of blood disorder: Standard=3, fixed=6, and reversible=7. The correlation matrix for predictors in the heart disease dataset are shown in Figure 2. All the predictors do not have significant correlations. However, the 𝑠𝑙𝑜𝑝𝑒 has a positive correlation with 𝑡ℎ𝑎𝑙𝑎𝑐ℎ, 𝑎𝑔𝑒 has negative correlation with 𝑡ℎ𝑎𝑙𝑎𝑐ℎ, and 𝑡ℎ𝑎𝑙𝑎𝑐ℎ has positive and negative correlation with both 𝑠𝑙𝑜𝑝 and 𝑒𝑥𝑎𝑛𝑔; respectively. The data set is visualized in Figure 3. where it is plotted as a scatter plot. The x-axis and y-axis are chosen to be chol and trestbps, respectively. The chest pain experienced (cp) is used to note the observations for all the individuals. https://medium.com/@halima23121998/heart-disease-uci-logistic-regression-in-r-b95b821088e6?sk=def7a489c8ce9b249048e903d80f8591 IHJPAS. 2024, 37(4 ) 397 Figure 2. Correlation matrix for predictors in heart disease dataset. Figure 3. Scatter plot ( trestbps vs. chol). 6. Results and discussion The dataset was split up into training (212 observations) and testing (91 observations) parts. SVM, ANN, classical logistic, and Bayesian logistic classification methods are applied for testing and training datasets. The methods were compared according to their accuracy and consuming time. Table 3. compares four methods by calculating the accuracy for both training and testing datasets and the total time-consuming. The best accuracy exists in the Bayesian Logistic method, which is 91.38% in the training dataset and 90.92% in the testing dataset. This indicates the Bayesian Logistic method is the best. Also, Bayesian Logistic consumes less time than other methods. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 re s te c g c p th a la c h s lo p e e x a n g s e x th a l fb s c a c h o l a g e tr e s tb p s restecg cp thalach slope exang sex thal fbs ca chol age trestbps 80 120 160 200 100 200 300 400 chol tr e st bp s 0.00 0.25 0.50 0.75 1.00 target cp 0 1 2 3 Scatterplot (trestbps vs. chol) IHJPAS. 2024, 37(4 ) 398 Table 3. Comparison among SVM, ANN, Classical Logistic, and Bayesian Logistic methods. Methods Accuracy Total time Training dataset Testing dataset SVM 88.60% 84.42% 7 seconds ANN 86.80% 80.55% 9 seconds Classical Logistic 90.10% 86.39% 6 seconds Bayesian Logistic 91.38% 90.92% 4 seconds The accuracy curve for both classical and Bayesian logistic is shown in figure 4. We can see the value of AUC for Bayesian logistics is higher than classical logistic. In general, the best model exists with a higher AUC. For example, the accuracy in the testing dataset in Bayesian Logistic is 90.92 %. The model can discriminate between individuals (patients) with heart disease and no heart disease with an excellent prospect. In general, the receiver operating characteristic (ROC) curve with a level of 0.7 appears to be very good, and hence the true positives are maximized. The highest quantity of patients with disease is not recognized as well. The higher the AUC, the more the model distinguishes between people who have the disease and others who do not. Figure 4. shows a comparison between Bayesian and classical logistic classifiers. Overall, the Bayesian logistic classifier performs better than classical logistic. By allowing only a few samples (less than 100), the steep slopes of the recall curves for the point estimate classifiers (top row, red lines) suggest that they better identify patients than the classical logistic classifier. A smaller slope indicates less progress in performance as more patients are included. These trends are also reflected in the overall measurement of balanced accuracy. Figure 4. Accuracy curve for both classical and Bayesian Logistic. Classical Logistic vs Bayesian Logistic False positive rate T ru e p o s it iv e r a te 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Bayesian Logistic Classical Logistic IHJPAS. 2024, 37(4 ) 399 7. Conclusion The Bayesian Markov Chain Monte Carlo (MCMC) technique is presented in this article as an alternative method for estimating the logistic regression model. We derived a Bayesian logistic model by using a particular type of MCMC, the Metropolis-Hastings (MH) algorithm. The MH algorithm was introduced and used to obtain the estimated parameters in the new method. The Bayesian logistic model was used to overcome some limitations of the classical logistic model. The actual data of the heart dataset is used in this paper to classify healthy and non-healthy people. The modified method is compared with the classical logistic, SVM and ANN. It is shown that our method performs better than other methods and takes less time. Acknowledgment The authors are very grateful to the executive manager and editorial board members of the College of Administration and Economics Conflict of Interest The authors declare that they have no conflicts of interest. 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Bayesian Formulation 3.1 Markov chain Monte Carlo 4. Simulation Studies 5. Real Dataset Application Heart Disease Dataset Analysis 6. Results and discussion 7. Conclusion References