## 5 Real Data Analysis
### Model Selection  
In this section, we select the most appropriate model for the airline data, which consists of 371 months and 179,528,198 flight records. As described in Section 3.2, we first perform the link function selection using \(\overbrace{AIC}\) and five-fold cross-validation based on RASMR with the delta ratio split (Algorithm 2). In this step, all variables listed in Table S8 are included, and the candidate link functions include logit, cloglog, probit, and cauchit. According to the link function selection results listed in Table 8, we select the logit link for our model (see Figure S14 in the Supplementary Materials for the fitted model based on \"glm2\" function in [58] and the RASMR representative data). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Simulation** & \multicolumn{3}{c}{**\(K\)-Means (\(K\) = 1000), True Link Function = Logit**} \\ \hline
**Approach** & \multicolumn{3}{c}{**RASMR with the Delta Ratio Split, Threshold = 0.1**} \\ \hline
**Setup** & **Logit** & **Cloglog** & **Probit** & **Cauchit** \\ \hline mzNormal & \(4.0\times 10^{-5}\) & \(3.2\times 10^{-2}\) & \(6.1\times 10^{-5}\) & \(8.2\times 10^{-3}\) \\  & \((9.9\times 10^{-6})\) & \((3.7\times 10^{-2})\) & \((2.4\times 10^{-5})\) & \((3.2\times 10^{-3})\) \\ nzNormal & \(1.6\times 10^{-3}\) & \(1.5\times 10^{-2}\) & \(2.7\times 10^{-4}\) & \(8.8\) \\  & \((5.6\times 10^{-4})\) & \((1.4\times 10^{-2})\) & \((8.5\times 10^{-5})\) & \((4.5)\) \\  ueNormal & \(2.0\times 10^{-4}\) & \(9.9\times 10^{-2}\) & \(1.2\times 10^{-3}\) & \(2.3\times 10^{-2}\) \\  & \(3.4\times 10^{-3}\) & \(9.6\times 10^{-1}\) & \(2.4\times 10^{-3}\) & \(3.3\times 10^{-2}\) \\ mixNormal & \(3.8\times 10^{-4}\) & \(1.9\times 10^{-1}\) & \(1.8\times 10^{-4}\) & \(7.3\times 10^{-2}\) \\  & \((1.5\times 10^{-4})\) & \((1.4\times 10^{-1})\) & \((1.8\times 10^{-4})\) & \((3.0\times 10^{-2})\) \\ \(T_{3}\) & \(6.3\times 10^{-2}\) & \(1.0\times 10^{-1}\) & \(5.2\times 10^{-2}\) & \(2.7\times 10^{-2}\) \\  & \(3.4(\times 10^{-2})\) & \((5.6\times 10^{-2})\) & \((2.5\times 10^{-2})\) & \((1.5\times 10^{-2})\) \\ EXP & \(2.3\times 10^{-6}\) & \(4.2\times 10^{-4}\) & \(5.8\times 10^{-6}\) & \(1.4\times 10^{-2}\) \\  & \((1.8\times 10^{-6})\) & \((9.0\times 10^{-5})\) & \((7.7\times 10^{-7})\) & \((9.7\times 10^{-3})\) \\ BETA & \(4.6\times 10^{-6}\) & \(7.3\times 10^{-5}\) & \(2.6\times 10^{-5}\) & \(1.0\times 10^{-2}\) \\  & \((3.9\times 10^{-7})\) & \((3.4\times 10^{-5})\) & \((3.4\times 10^{-6})\) & \((3.8\times 10^{-3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average (std) of RMSEs (\(10^{-3}\)) over 100 simulations using RASMR with the delta ratio split. As described in Section 3.3, we further perform the variable selection for the airline data by implementing the stepwise selection strategy with an initial screening based on MR and fine selection using RASMR (see Section S4 in the Supplementary Materials). The selection criterion is \(\widetilde{AIC}\) again. When using \"glm2\", we estimate the dispersion parameter by Pearson's Chi-square statistic divided by its degree of freedom (see Figure S15 in the Supplementary Materials). If \(n\), \(N\), and \(p\) represent the number of representatives, the full data size, and the number of parameters in the model, respectively, the degrees of freedom for Pearson's Chi-square test are \(n-p\) for RASMR and \(N-p\) for the full data. To estimate the dispersion parameter of full data, the RASMR dispersion parameter estimate needs to be adjusted by the degrees of freedom, i.e., \(\hat{\phi}=\hat{\phi}_{\textit{RASMR}}\cdot(n-p)/(N-p)\), where \(\hat{\phi}\) is the dispersion parameter estimated from the full data and \(\hat{\phi}_{\textit{RASMR}}\) is the estimate from the RASMR representative dataset. After such an adjustment, the estimated dispersion parameters are 3291.14 and 3341.67 for the model with all variables and the model with the selected variables, respectively, which indicates strong over-dispersion.