## 3 Methodology
### Regression Analysis  
Logistic regression (LG) model is an efficient and commonly used classification model for binary outcomes, first proposed by [PERSON] (2005). Logistic regression is widely used, e.g., in analyzing the effect of patient-physician relationship ([PERSON] and [PERSON], 2006), the effect of web-based exercise as an effective complementary treatment for patients ([PERSON] and [PERSON], 2006), etc. Furthermore, a regression model typically can serve very well in predicting a dependent variable, when a good number of independent variables are avaiable. In this research, logistic regression models are developed to predict the probability of a certain health insurance claim becoming denied. The independent variables used to perform the logistic regression analysis include the topics derived from the LDA model and the medical information, such as treatments, medicines, and prices. A list of binary variables is defined based on the results of the LDA model. For each binary variable, the value one represents that a case with a certain topic. To determine if a topic is covered in the document, the number of high-frequency words of the topic utilized in the document is measured. The common thresholds are three or four, which varied across different departments. Like LDA, the logistic regression model is built for each department. Due to the large number of predictive variables, a variable selection algorithm is used to enhance the regression model. [PERSON] and [PERSON] proposed a regularization and variable selection method for the general linear regression called elastic net [PERSON] and [PERSON] (1980). This algorithm can be used to solve several common issues such as overfitting and collinearity. To realize the regression model, this research builds a generalized linear model with the elastic-net penalty by using R package glmnet [PERSON] and [PERSON] (1980). The function of glmnet is to solve the following problem:  
\[\text{in}\frac{1}{N}\sum_{i=1}^{N}l\left(y_{i},\ \beta_{0}+\beta^{T}x_{i} \right)+\lambda\left[(1-\alpha)||\beta||_{2}^{2}+\alpha||\beta||_{1}\right] \tag{3}\]  
in this study, where:  
\(y_{i}\) is the outcome of the case \(i\),  
\(x_{i}\) is the vector of predictor variables for case \(i\), such as topic, disease and drug, and \(l\) is the negative log-likelihood function. The hyper-parameter \(\lambda\) in Equation (1) controls the overall strength of the penalty. This research performed a cross validation to find the best \(\lambda\) which gives a minimum mean misclassification error. The graph in Figure 4 with the department of family medicine as example shows the binomial deviance which varies as the \(\lambda\) changes. The elastic-net penalty controlled by \(\alpha\) in the equation bridges the gap between lasso (\(\alpha=1\)) and ridge (\(\alpha=0\)). Based on the knowledge that ridge penalty shrinks the coefficients of correlated predictors towards each other while the lasso tends to pick one of them and discard the others, this research tried many different values to find the best \(\alpha\) that can minimize the overfitting problem. The predictive model was built on training dataset and validated by testing dataset. The performance of the model on the testing dataset including precision, recall and area under curve (AUC). \"Precision\" represents the proportion of actual denied GC under the predicted denials, which was calculated as precision = TP/(TP + FP), where TP is the true positive count, and FP is the false positive count. Furthermore, \"recall\" represents the proportion of predicted denied GC under the actual denials, which was calculated as recall = TP/(TP + FN), where FN is the false negative count. A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied and the sensitivity and specificity are used as the coordinates to construct the curve.