## 3 Method
### Research Procedures
#### 3.3.2 Model Construction and Validation  
Micro-features, aggregated features derived from averaged combinations of micro-features, and a combination of those were used to investigate the strengths of characterizing writing quality. With qualified linguistic features, both linear and non-linear models were developed using the following six machine learning algorithms, namely Bayesian Ridge Regression [111, 112], Linear Regression [112, 113], Stochastic Gradient Descent Regression [114], Random Forest Regression [115], Decision Tree Regression [116], and Support Vector Regression [117]. The first three algorithms were used for linear AES model construction, whereas the last three algorithms were used for non-linear AES model construction. * **Bayesian Ridge Regression:** integrates ridge regression and Bayesian inference, providing a probabilistic perspective on linear regression, excelling in data-scarce environments or large feature domains by proficiently approximating a weight distribution. * **Linear Regression**: establishes a linear correlation between the input linguistic features and the predicted scores. Its straightforwardness and interpretability render it a foundational benchmark for numerous regression endeavors. * **Stochastic Gradient Descent Regression**: iteratively adjusts parameters to reduce the loss function, promoting efficiency and the capacity to scale. * **Random Forest Regression**: An ensemble method that consolidates outputs from various decision trees, renowned for its precision, robustness to over-fitting, and adeptness in handling non-linear correlations. * **Decision Tree Regression**: builds a tree-like model that bifurcates the dataset into branches predicated on attribute values, offering an intuitive method capable of deciphering intricate patterns. * **Support Vector Regression**: applies support vector machine concepts to regression, focusing on maintaining errors within a defined threshold, excelling in high-dimensional contexts and situations where error tolerance is crucial. Four evaluation metrics concerning \(R^{2}\), Quadratic Weighted Kappa (QWK), Exact Agreement, and Exact-Plus-Adjacent Agreement were used to assess the discrepancy between the predicted scores generated by the model and the scores assigned by human raters. \(R^{2}\) serves to quantify the percentage of the variance in human raters' scores that can be accounted for by the linguistic features utilized in a model. If the \(R^{2}\) of the proposed model was 0.60, 60% of the variability observed in the scores assigned by human raters could be explained by the inputs to the model. \(R^{2}\) usually varied from 0% (the model did not explain any of the variations) to 100% (the model explained all the variations). Its formula is given by the following:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}- \hat{y})^{2}},\]  
where \(y_{i}\) represents the actual scores assigned by human raters, serving as a benchmark for the AES model's accuracy. The model aims to closely match its predictions \(\hat{y}_{i}\) to these actual scores, effectively emulating human scoring behavior. \(\hat{y}\) denotes the average of all actual scores, providing a reference point for assessing variations in both actual and predicted scores. QWK was employed to evaluate the level of agreement between two raters. QWK values range from 0, indicating no match between raters, to 1, which signifies a perfect match. A great majority of studies have used QWK as a gold standard in evaluating the accuracy of AES models [3, 6, 83, 118]. Therefore, the current study adopts QWK as the metric to gauge the accuracy between human raters' scores and the scores predicted by the model. \[QWK=1-\frac{2\times\sum_{i,j}w_{ij}o_{ij}}{\sum_{i,j}w_{ij}e_{ij}},\]  
where \(w_{ij}\) is the weight assigned to the disagreement between raters for items in categories \(i\) and \(j\), \(o_{ij}\) is the observed frequency of agreement, and \(e_{ij}\) is the expected frequency of agreement under chance.