# Comparative Analysis of Local Differential Privacy Schemes in Healthcare Datasets
## 2 Preliminaries
### Correlation between Variables in Healthcare Datasets  
In the realm of healthcare datasets, the question of whether there is a correlation between variables becomes pivotal, especially considering the sensitivity of health-related information. Here, there are some examples of some kinds of datasets that can have correlations between their attributes:  
* Age and Disease Risk:
* The risk of developing many diseases increases with age.
* Certain diseases are more likely to occur in specific age groups.
* Body Mass Index (BMI) and Diabetes:
* People with a higher BMI are more likely to develop type 2 diabetes.
* Excess body weight can lead to insulin resistance, increasing the risk of diabetes.
* Blood Pressure and Cardiovascular Diseases:
* High blood pressure is a major risk factor for cardiovascular diseases.
* The risk of developing cardiovascular diseases increases with blood pressure.
* Cholesterol Levels and Atherosclerosis:
* High levels of LDL (bad) cholesterol can lead to atherosclerosis.
* The risk of developing atherosclerosis increases with LDL cholesterol levels.
* Exercise Habits and Overall Health:
* Regular exercise has many health benefits, including a reduced risk of various conditions.
* Exercise also helps to improve mental health and overall mood.
* Smoking is a major risk factor for respiratory diseases.  
R-Squared, \(R^{2}\)  
Despite the ubiquity of regression analysis in scientific disciplines, a unified standard for its evaluation remains elusive. [PERSON] et al. [19] perform a critical review, examining several commonly employed metrics and ultimately advocate for R-squared as the most informative and transparent measure in many cases. While metrics such as mean absolute percentage error (MAPE), mean absolute error (MAE), mean square error (MSE), and its rooted variant (RMSE) are frequently used, they often obscure crucial details about a regression model's performance. For example, a seemingly acceptable MAPE value can conceal significant outliers or systematic biases. In contrast, [PERSON] et al. [19] show that \(R\)-squared offers valuable clarity: high values denote accurate predictions across diverse data points, while negative values unambiguously signal a poorly fitting model. This inherent informativeness makes \(R\)-squared a compelling choice for evaluating regression analyses in various scientific contexts. \(R\)-squared is computed as  
\[R^{2}=1-\frac{MSE}{MST},\]  
where \(MSE\) is \(\sum_{i=1}^{z}(X_{i}-C_{i})^{2}\) and the mean total sum of squares (\(MST\)) is \(\sum_{i=1}^{z}(Y_{i}-\overline{Y})^{2}\). In the previous formulas, \(X_{i}\) is the predicted \(i\)th value, the \(C_{i}\) element is the actual \(i\)th value, and \(z\) is the length of the vector. In our experiments, \(X_{i}\) and \(C_{i}\) are the joint probability distributions for real and computed data on \(k\)-way. \(R\)-squared can take values in the range \((-\infty,1]\) according to the mutual relation between the ground truth and the prediction model. Hereafter, we show a brief overview of the principal cases.  
* \(R^{2}\geq 0\): With linear regression with no constraints, \(R^{2}\) is non-negative.
* \(R^{2}=0\): The fitted line (or hyperplane) is horizontal. With two numerical variables, this is the case if the variables are independent, that is, are uncorrelated.
* \(R^{2}\leq 0\): This case is only possible with linear regression when either the intercept or the slope are constrained so that the \"best-fit\" line (given the constraint) fits worse than a horizontal line, for instance, if the regression line (hyperplane) does not follow the data.