## 5 Regression Analysis  
The original equations of ASCE 41-17 do not include the influence of a height-to-depth ratio; however, it was revealed in the previous section that there was a considerable amount of error in the modeling parameter predictive equation proposed by ASCE 41-17 when the height-to-depth ratio was lower than 2.0. For this reason, a calibration factor was proposed to minimize error. Linear regression was used to introduce a calibration factor for the modeling parameter predictive equation of ASCE 41-17 to calculate more accurately the values of parameters \(a\) and \(b\). The measured modeling parameters extracted from the first-cycle envelope curve is shown in Table 2.  
To propose an empirical calibration factor, various candidate predictor variables frequently used in the past research [52; 53; 54; 14; 55; 15] were considered. Considered candidate predictor variables include: longitudinal reinforcement ratio (\(\rho_{l}\)), transverse reinforcement ratio (\(\rho_{t}\)), axial load ratio (\(v=P/\left(f_{c}^{\prime}A_{g}\right)\)), measured concrete compressive strength (\(f_{c}\)), column shear span (\(a_{v}\)), height-to-depth ratio (\(a_{v}\)/\(h\)), ratio of a transverse bar spacing to a column depth (\(s\)/\(h\)), ratio of a strength contribution of longitudinal reinforcements to that of concrete (\(f_{yl}A_{sl}\)/\(f_{c}^{\prime}A_{g}\)), ratio of a strength contribution of concrete axial strength to that of transverse reinforcements (\(f_{c}^{\prime}A_{g}\)/\(f_{yl}A_{st}\)). Figure 3: Absolute relative error (\(ARE\)) between the measured and ASCE modeling parameters. Among the candidate variables listed above, only the statistically significant variables used to predict the deformation parameters were selected. To determine the statistically significant variables, the difference between the calculated and calculated deformation parameters is calculated. The results are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**No. ** & **ID** & \(a_{n}\)**/_h_ & \(b\)**(mm)** & \(h\)**(mm)** & \(s\)**(mm)** & \(p_{l}\)**(\%)** & \(p_{t}\)**(MPa)** & \(f_{e}\)**(MPa)** & \(f_{gt}\)**(\%)** & \(a_{meas}\)**(\%)** & \(b_{meas}\)**(\%)** & **Reference** \\ \hline  
a linear stepwise regression analysis [56] was used. A candidate variable is considered to be statistically significant, when the \(p\)-value is less than 5%. The \(p\)-value is used for measuring the plausibility of a null hypothesis. Typically, in regression analysis, a null hypothesis is rejected if the \(p\)-value is less than 0.05. As a result of the regression analysis, the obtained equation for parameter \(a\) was written as:  
\[a_{fit}=a_{ASCE}+\left(-\frac{1}{47.78}-\frac{1}{57.49}\frac{1}{a_{\text{p}}/ h}+\frac{f_{yl}}{1988.07c_{unit}}\right), \tag{7}\]  
where \(c_{unit}\) is 6.89 when using the unit of MPa and 1.00 when using the unit of ksi. As discussed in the previous section, height-to-depth ratio \(a_{\text{p}}/h\) was found to be a significant predictor variable. The equation obtained for parameter \(b\) was described as:  
\[b_{fit}=b_{ASCE}+\left(\frac{1}{12.56}-\frac{1}{703.39\rho_{l}}\right)\geq a_{ fit}. \tag{8}\]  
In the proposed equations (Equations (7) and (8)), three variables were considered (\(a_{\text{p}}/h\), \(\rho_{l}\), and \(f_{yl}\)). It was reported that RC member behavior is affected significantly by \(a_{\text{p}}/h\)[1; 2; 3; 4; 5; 6; 7]. RC columns with lower \(a_{\text{p}}/h\) are more likely to fail in shear. Past studies [7; 8; 9] also reported that the dowel action of longitudinal reinforcements significantly affects the load-deformation responses of stocky structural components; thus, \(\rho_{l}\) and \(f_{yl}\) selected from the stepwise regression analyses can be physically meaningful variables. The significant predictor variable for both modeling parameters \(a\) and \(b\) was related to the longitudinal reinforcement, that is, the measured yield strength of a longitudinal reinforcement \(f_{yl}\) and the longitudinal reinforcement ratio \(\rho_{l}\).