## 3 Research Method  
Regression analysis is a fundamental tool in healthcare research, offering a structured approach to understanding the relationships between predictor variables and outcomes of interest. By employing linear, log-linear, or nonlinear regression models, researchers can quantify the association between a response variable \(Y\) and explanatory covariates xi (where \(i=1\), \(\ldots\), \(n\), with \(n\) denoting the number of observations). This framework enables analysts to estimate the parameters of the model from observed data, predict values of \(Y\), and evaluate model performance, thereby facilitating comprehensive what-if scenarios. Regression analysis not only models observed data but also helps establish causal relationships between variables, providing a robust foundation for accurate prediction. To investigate the relationship between predictor variables and the severity of COPD, GLMs were utilized. GLMs were chosen for their interpretability and suitability for the data structure, allowing for a clear understanding of how each covariate influences the response variable. Although more complex machine learning techniques such as Random Forests and XGBoost may offer improved predictive accuracy, they were not employed in this study to maintain model transparency and enhance clinical interpretability. Standard linear regression models assume homoscedasticity that the variance of errors is constant across all levels of covariates and independence of observations. The linear regression model can be expressed as follows:  
\[Y_{i}=\beta_{0}+\hat{\beta}x_{i}+\varepsilon_{i} \tag{1}\]  
where \(Y_{i}\) is the dependent variable (a parameter that has been estimated) for observation \(i\), while \(x_{i}\) is the corresponding independent variable. \(\beta_{0}\) is the intercept, \(\hat{\beta}\) represents the slope, and \(\varepsilon_{i}\) is the residual error, representing the difference between observed and predicted values. Model parameters are estimated by minimizing the sum of squared residuals, \(\sum\varepsilon_{i}^{2}\), to achieve the best fit. Slope \(\hat{\beta}\) is calculated as follows:  
\[\hat{\beta}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\overline{x})\left(Y_{i}- \overline{Y}\right)}{\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}} \tag{2}\]  
To ensure that the predicted mean values remain non-negative, it is common to apply a link function \(g(\cdot)\) such that  
\[g(\mu_{i})=\beta_{0}+\beta x_{i} \tag{3}\]  
The link function \(g(\cdot)\) transforms the expected value \(\mu_{i}\) to a scale appropriate for the model. For example, the logarithmic link \(g(x)=log(x)\) is used to guarantee non-negativity, while the logit link function is applied to scale the mean to a probability range between 0 and 1. \[(x)=log\ (x/(1-x).\]  
For multiple predictors, the linear regression model extends to  
\[Y_{i}=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots\beta_{p-1}x_{ip-1}+ \epsilon_{i} \tag{4}\]This model can be represented compactly in matrix form as  
\[Y=X\beta+\epsilon \tag{5}\]  
where  
\[Y= \begin{bmatrix}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{n}\end{bmatrix}  \tag{6}\]  
\[\beta= \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{p-1}\end{bmatrix}  \tag{7}\]  
\[\epsilon= \begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \vdots\\ \epsilon_{n}\end{bmatrix}  \tag{8}\]  
and the design matrix is  
\[X= \begin{bmatrix}1&x_{11}&x_{12}&\ldots&x_{1p-1}\\ 1&x_{21}&x_{22}&\ldots&x_{2p-1}\\ \vdots&\vdots&\vdots&\ldots&\vdots\\ 1&x_{n1}&x_{n2}&\ldots&x_{np-1}\end{bmatrix}  \tag{9}\]  
When the outcome is binary, such as disease status (presence or absence of COPD), the response variable is modeled using a binomial generalized linear model with the canonical logit link function. The logit model is expressed as follows:  
\[logit(\theta_{i})=ln(\frac{\theta_{i}}{1-\theta_{i}})=\beta_{0}+\beta_{1}x_{i 1}+\beta_{2}x_{i2}+\ldots+\beta_{n}x_{in} \tag{10}\]  
where \(x_{i1},~{}x_{i2},~{}\ldots,~{}x_{in}\) are the observed covariate values for the ith observation, and \(\beta_{0}\) represents the intercept.