# Optimized Weighted Nearest Neighbours Matching Algorithm for Control Group Selection
## 5 Results  
### Results of the Scenario I  
As was described before, the data-generating process in this scenario was identical to the one used in [34] to illustrate the efficiency of the proposed method on a widely used benchmark dataset. The results of the algorithms were evaluated from different aspects. The individual balance values for the observed covariates are presented in Figure 2. When comparing balances, a method which could achieve the best results on all variables can not be selected. However, comparing the WNNEM and the WNNSA methods, it can be seen that the variable-wise balance is a little more diverse in the case of the WNNSA method than in the case of the WNNEM method. This result is not surprising since the WNNSA method does not always select the nearest neighbour from the candidates but balances the entire matching. To assess the balance of the groups by variables, we also calculated the SMD values for all covariates and all matching methods. The SMD values for all matching and covariates were less than 0.1, which confirms that according to the general evaluation principles, all results on all covariates can be seen as well-balanced. The advantage of using the WNNSA method can be seen when looking at the similarity of control and case groups at the group level. To illustrate this, group-level quality indicators were also calculated. The results are presented in Table 1. Table 1 contains the minimal, average and maximal quality values of the control group selections performed on the generated 100 datasets. As DDI, NNI, and GDI metrics are distance measures, lower values mean more similar selected control groups. In contrast, in the case of the Hansen and Bowers test (HB), a higher value means higher similarity. The maximum possible value of the HB test is 1. It can be seen in Table 1 that the SM method did not perform well on this dataset, and it could not select a full-sized control group. This fact can be inferred from the high values of the dissimilarity indices. Although the PSM methods performed well in distribution-based evaluations (HB, DDI), they under-performed by one order of magnitude in pairwise similarities (NNI, GDI) compared to the NNM, MMM, WNNEM and WNNSA methods. The NNM, MMM, WNNEM and WNNSA methods all performed well in the pairwise evaluations, but in the distribution-based evaluations, the WNNEM and WNNSA methods slightly outperformed the NNM and MMM methods. Comparing the WNNEM and WNNSA methods, we can see that the WNNEM method preferred selecting the nearest neighbours, while the WNNSA method aimed to achieve a global optimum. Across the four metrics, the WNNSA method scored best on three metrics and only on the NNI index (emphasizing nearest neighbour selection) scored slightly worse than algorithms that prefer  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**NNM**} & \multicolumn{3}{c}{**MMM**} & \multicolumn{3}{c}{**SM**} \\ \hline  & \(min\) & \(avg\) & \(max\) & \(min\) & \(avg\) & \(max\) & \(min\) & \(avg\) & \(max\) & & \\ \hline HB(\(p\)) & 0.583 & 0.976 & 1.000 & 0.347 & 0.960 & 1.000 & 0.512 & 0.873 & 1.000 & & \\ DDI(\(d\)) & 0.006 & 0.015 & 0.035 & 0.004 & 0.015 & 0.030 & 0.504 & 0.574 & 0.631 & & \\ \hline NNI(\(d\)) & 0.054 & 0.065 & 0.075 & 0.057 & 0.066 & 0.078 & 0.504 & 0.574 & 0.631 & & \\ GDI(\(d\)) & 0.062 & 0.075 & 0.095 & 0.059 & 0.076 & 0.097 & 0.504 & 0.574 & 0.631 & & \\ \hline  & \multicolumn{3}{c}{**PSM\_02**} & \multicolumn{3}{c}{**PSM\_DYN**} & \multicolumn{3}{c}{**WNNEM**} & \multicolumn{3}{c}{**WNNSA**} \\ \hline  & \(min\) & \(avg\) & \(max\) & \(min\) & \(avg\) & \(max\) & \(min\) & \(avg\) & \(max\) & \(min\) & \(avg\) & \(max\) \\ \hline HB(\(p\)) & 0.813 & 0.978 & 1.  
nearest neighbour selection. However, this is understandable as WNNSA aims to achieve a global optimum, not a local optimum. Nevertheless, the difference is marginal.