3.5.1 Independent Variables (Medication Characteristics)

We analyzed and selected relevant columns for the study exploratory data analysis (EDA), correlations, and feature importance with random forest.

3.5.2 Dependent Variable: Price Change

The dependent variable “price change” captures variations in ARV medication prices. This binary variable, encoded as “1” for price increase and “0” for decrease, allows us to analyze trends in procurement prices over time.

3.5.3 Data Analysis Methods and Machine Learning Modeling

3.5.3.1 Logistic regression

In this phase of the study, logistic regression was employed as one of the primary machine learning techniques for analyzing medication prices. Logistic regression remains a cornerstone in predictive modeling, particularly in domains where the outcome of interest is binary or categorical. Its flexibility, interpretability, and ability to handle complex relationships make it indispensable in the analysis of healthcare data, including pharmaceutical pricing dynamics ^{Hastie, Tibshirani, and Friedman (2021a)} and ^{Hastie, Tibshirani, James, and Witten (2021b)}.

According to the logistic regression approach described by ^{Hastie et al. (2021a)} in their book The Elements of Statistical Learning, the procedure for analyzing medication prices using logistic regression is as follows:

Through the logistic function, we examine the relationships between these variables and their impact on the dependent variable.

We formulated a logistic regression model in Python with the aim of anticipating whether medication prices exhibit an increase or a decrease. To accomplish this, a function is employed to model p(X), producing values that are exclusively binary, zero or one for all X values. In logistic regression, the logistic function is introduced within this context.

The logistic regression formula is expressed as:

Where p(X) is the probability that Y = 1 given X = x, β₀ is the intercept term, and β₁, …, β_p are the coefficients of the predictor variables X₁, …, X_p, also known as the regression coefficients.

The logistic regression formula is:

This function converts the weighted sum of the input variables and coefficients into a probability value, which falls within the range of 0 and 1. If this probability surpasses the threshold of 0.5, the model indicates a prediction for the positive category; otherwise, it predicts the negative category.

In the analysis of medication prices, logistic regression offers a robust framework for modeling the likelihood of price changes based on various predictor variables. Its application in forecasting price fluctuations in pharmaceutical markets has yielded valuable insights for policymakers and healthcare administrators (^{Hosmer Jr et al., 2013}).

With the increasing availability of healthcare data, logistic regression serves as a powerful tool for analyzing trends in medication pricing and procurement. Its ability to handle large datasets and incorporate diverse predictor variables makes it indispensable for understanding the drivers of price variations in the pharmaceutical sector (^{Agresti, 2015}).

3.5.3.2 Random forest

The implementation of random forest allowed for a more sophisticated and accurate analysis. This technique combines multiple decision trees to identify subtle patterns and nonlinear relationships in the data.

Aligned with ^{Hastie et al.’s (2021b)} description of the random forest methodology, this approach involves the creation of decision trees using a set of training data. However, a distinct feature of this technique is that during the development of each tree, a random subset of predictors is chosen from the complete set of available predictors when evaluating a potential split. Among these selected predictors, only one is used for the actual split. Moreover, with each split, a new subset of predictors is randomly sampled, typically with the number of predictors being close to the square root of the total number of predictors (m ≈ √p).

Random forest is recognized for its versatility and capability to handle complex datasets in various domains (^{Fernández-Delgado et al., 2014}). In healthcare analytics, it has been extensively utilized for predicting clinical outcomes, disease diagnoses, and medication responses.

Opting for a smaller m value when constructing random forest proves advantageous, particularly in scenarios where there is substantial correlation among numerous predictors. This strategy aids in predicting the decrease in the price of ARV medications by effectively handling the complexities and interrelationships among various factors involved.

According to ^{Marsland (2014)}), the random forest algorithm leverages ensemble learning by aggregating multiple decision trees, each trained on a bootstrap sample and a subset of features, thereby mitigating overfitting and enhancing generalization performance.

Moreover, in financial domains, random forest has proven effective for tasks such as credit risk assessment, stock market prediction, and fraud detection (^{Liaw & Wiener, 2002}). Its ability to handle large volumes of data, nonlinear relationships, and highdimensional feature spaces makes it a valuable tool for analyzing pharmaceutical pricing dynamics and forecasting changes in medication costs.

3.5.3.3 K-Nearest Neighbors Method

The K-nearest neighbors (KNN) method, as described by ^{Hastie et al. (2021b)}, plays a crucial role in our predictive approach. This method relies on the idea that a new data point’s category can be predicted by examining the categories of nearby data points. It assumes that data points with similar characteristics tend to belong to the same category, providing useful insights for classification and prediction tasks and improving our understanding of price changes.

Implementation of the KNN technique entails the following procedural steps:

By following these steps, the KNN method facilitates informed predictions regarding price changes, thereby enhancing decision-making processes within our predictive framework.

KNN demonstrates its effectiveness particularly in situations where there is no clear functional relationship between the features and categories, as well as when the data exhibits nonlinear structures. However, it is important to consider that KNN may present significant computational consumption, especially in extensive datasets, due to the distance calculation required for all points in the training set.

In the context of this research, the KNN method was employed in conjunction with the two predictive approaches mentioned earlier: logistic regression and random forest, in order to explore its predictive potential and complement the comprehensive analysis of ARV medication prices, following the methodology proposed by ^{Hastie, Tibshirani, and Friedman (2021a)}, and ^{Hastie, Tibshirani, James, and Witten (2021b)} and ^{Badal and Sungkur (2023)}.

We separated data into training and testing sets with 80% of the data for training and 20% for testing. Prior to this separation, the data underwent preprocessing that normalized and standardized the variables, ensuring the consistency and accuracy of the results.

Given the significance of KNN in pharmaceutical price prediction, it is noteworthy to mention the seminal work by ^{Cover and Hart (1967)} which underscore KNN’s versatility, efficacy, and simplicity in pharmaceutical price forecasting by leveraging similarities among medication attributes and purchase patterns.

5.3.1 Analysis with Logistic Regression

During training, we adjusted the model coefficients using the gradient descent algorithm. This involved iteratively updating the coefficients to minimize the logarithmic loss function, which measures the difference between the predicted probabilities and the actual outcomes. Feature selection techniques were employed to identify the most influential variables affecting price changes in ARV medications. Additionally, cross-validation was performed to assess the model’s generalization ability and to mitigate overfitting.

We thoroughly evaluated the logistic regression model using data collected on ARV medications and their price changes. In addition to traditional evaluation metrics like precision, sensitivity, and specificity, supplementary techniques such as cross-validation and bootstrapping were utilized to validate model robustness. Calibration plots were generated to scrutinize the calibration of predicted probabilities across diverse thresholds.

We generated a confusion matrix, shown in Table 9, in order to analyze the correct and incorrect classification of the model’s predictions. The confusion matrix offers a comprehensive overview of the outcomes, encompassing true positives, false positives, true negatives and false negatives (see Table 9).

We calculated precision, sensitivity, and specificity metrics to assess the overall performance of the model in classification. We obtained the following metrics:

Although the logistic regression model showed reasonable ability to predict price changes in ARV medications, it revealed certain limitations in its predictive capacity in some instances. Despite an overall accuracy of 89.29%, we identified areas for improvement in sensitivity and specificity.

To enhance the performance of the logistic regression model, we will consider incorporating additional data to enrich the model.

5.3.2 Analysis with Random Forest

Throughout the training phase, we made adjustments to fine-tune the model’s settings using the Grid Search method. Parameters such as the number of trees in the forest, maximum tree depth, and minimum number of samples required to split a node were fine-tuned to enhance predictive accuracy. Feature importance analysis was conducted to identify significant predictors of ARV medication price changes.

We thoroughly evaluated the random forest model using standard evaluation metrics to understand its performance and predictive capability. In addition to traditional evaluation metrics, such as precision, sensitivity, and specificity, techniques such as out-of-bag error estimation and permutation feature importance were employed to evaluate the model’s robustness and generalization ability. Moreover, visualization techniques, such as partial dependence plots and feature interaction analysis, were utilized to gain insights into the complex relationships between predictors and price changes.

The confusion matrix offers a comprehensive breakdown of true positives, correct positives, correct negatives, false positives, and false negatives, providing valuable insights into the model’s performance, as shown in Table 10 (see Table 10).

We assessed the random forest model’s discriminative ability between positive and negative instances using the ROC curve. We quantified the quality of the model by calculating the AUC.

The precision of the random forest model, 98.53%, suggests that the model has a high level of confidence in identifying instances where prices are likely to increase. With a sensitivity that identifies 95.74% of actual price increases in ARV medications, this implies that the model effectively captures the majority of instances where prices are indeed observed to increase. The specificity, 98.53% of the time, the model accurately predicts instances where prices remain stable or decrease.

The overall accuracy of the random forest model indicates that the model’s predictions, both positive and negative, are correct 97.32% of the time. It demonstrates the model’s effectiveness in providing accurate forecasts of price changes in ARV medications.

To address any potential overfitting of the model, we will consider further optimization of hyperparameters and inclusion of more training data to improve the model’s generalization. Additionally, feature selection techniques may be explored to identify the most relevant attributes contributing to the model’s predictions.

5.3.3 K-Nearest Neighbors (KNN) Method

The KNN method, leveraging similarity between nearby data points, was employed to predict price changes of ARV medications based on drug characteristics. Prior to model training, dataset preprocessing was performed to ensure compatibility with the KNN algorithm. Techniques such as partitioning, normalization, and standardization of independent variables were employed.

For analysis purposes, we partitioned the dataset into training sets comprising 80% of the data and test sets comprising the remaining 20%.

For each data point in the test set, we calculated the Euclidean distance between that point and all points in the training set. The Euclidean distance is calculated using the formula:

Where n is the number of features, x_i and y_i are the values of feature i in the test point and training point respectively.

In our analysis, we chose a K value which signifies the number of nearest neighbors used to predict the label of a data point within the test set. We set the K value to 5. We determined the most common label among the K nearest neighbors and assigned that label to the test point. If the majority of neighbors experienced a price increase, the test point was labeled “1” (price increase), and if the majority experienced a decrease, it was labeled “0” (price decrease).

Once we assigned labels to all test points, we constructed the confusion matrix to evaluate the KNN model’s performance on the test set. The confusion matrix compares the model’s predictions against the actual outcomes, providing insight into the accuracy of the classifications.

The KNN model demonstrated a precision of 97.67%, a sensitivity of 94.33%, and a specificity of 97.67% in predicting price changes in ARV medications. The confusion matrix associated with these results is shown in Table 11 (see Table 11).

Below is a comparative analysis of the logistic regression, random forest, and K-nearest neighbors (KNN) models in terms of precision, sensitivity, and specificity for predicting price changes in ARV.

Although the KNN model provided reasonable capability to predict price changes in ARV medications, it also revealed certain limitations in its predictive capacity in some instances. The precision results for each method are detailed in Table 12 (see Table 12).

The results show that the random forest demonstrated 98.53% precision in predicting price changes, followed by KNN and logistic regression. This suggests that the random forest is the most suitable method for analyzing and predicting price fluctuations in ARV medications.

Additionally, we observed that the models yielded consistent results with conventional data analysis. We identified influential factors in procurement prices, such as medication type, quantity procured, and duration of consolidated purchasing strategy. We detected complex interactions among these variables, highlighting the importance of considering multiple dimensions when analyzing ARV drug prices.

Specifically, the logistic regression model indicated that the duration of consolidated purchasing strategy had a significant effect on reducing procurement prices. On the other hand, the random forest model underscored the importance of medication type in price determination. The KNN model identified clustering patterns among different ARV medications, suggesting the existence of complex relationships between them in terms of prices and characteristics.