The surgery-first-orthognathic approach is characterized by minimal presurgical orthodontic treatment and orthognathic surgery followed by postsurgical orthodontic treatment. This application has recently been emphasizing its advantages, which include increased patient cooperation, effective compensation, and a shortened treatment period. However, because presurgical orthodontic treatments such as dental decompensation and arch coordination are rarely performed in the surgery-first-orthognathic approach, postsurgical occlusal instability clinically leads to more severe forward mandibular postoperative movement than in conventional surgical orthodontic treatment for skeletal class III deformity.

Our study was triggered by these questions: What variables (CBCT-generated cephalometric measurements) account for the final mandibular setback movements? How does postsurgical occlusal vertical dimension (VD) affect it?

To answer these questions, we applied PLS-PM analysis to the assessment of the treatment outcome for class III SFA cases. From this modeling, we confirmed that the PLS-PM as derived in our study signified the effect of occlusal VD on the extent of final mandibular setback and that an increase in VD leads to a decrease in the absolute amount of the final setback and, ultimately, more severe postsurgical skeletal changes (postsurgical relapse) (Fig. 2).

Tu and co-workers introduced a novel approach to high correlations among the variables using PLS and path-modeling analysis in periodontal research. PLS analysis was used for providing a simple relationship among multiple measures with two or more variables [14, 15]. It could be used for successfully summarizing the inter-correlations among many variables obtained from the analysis, such as the cephalometric analysis data. Lowe and co-workers used PLS analysis to assess the interrelations between obstructive sleep apnea (OSA) outcome variables and computer tomographic, cephalometric, and demographic predictor variables [8].

In the present study, \( {\mathrm{B}}_{10}^{\mathrm{s}},\kern0.5em {\mathrm{Me}}_{10}^{\mathrm{s}},\kern0.5em {\mathrm{A}}_1^{\mathrm{s}} \) and \( {\mathrm{Cp}}_1^{\mathrm{f}} \), along with VD_{10,} consisted of MVs, which are observable values by definition in the PLS-PM framework (Table 3, Fig. 2), as well as the LVs LV_{1}, LV_{10}, LV_{overall} and LV_{setback} and their presumed interrelationships (Tables 3 and 4, Fig. 2). Lee et al. built the multiple regression models for the estimation of the final mandibular extent in class III SFA cases. From their study, we re-analyzed these data for this research. From the previous study, the predictors \( \left({\mathrm{B}}_{10}^{\mathrm{s}},\kern0.5em {\mathrm{Me}}_{10}^{\mathrm{s}},\kern0.5em {\mathrm{A}}_1^{\mathrm{s}},\kern0.5em {\mathrm{Cp}}_1^{\mathrm{f}},\kern0.5em \mathrm{V}{\mathrm{D}}_{10}\right) \) identified in the general multiple regression model for all 40 patients consisted of the measurement components or, equivalently, the manifest variables (MVs) [9]. The other variables were not significant for predictors in the general model, as Lee and co-workers [9] had already reported. The LVs LV_{1}, LV_{10}, LV_{overall}, and LV_{setback} were not arbitrary but conceptualized the various postsurgical physiologic responses to surgery and orthodontic treatment that ultimately influence the final setback amount of the mandible [11]. These variables were used to evaluate the associations between these variables and the final mandibular setback extent, to determine which variables account for the final mandibular setback extent.

The PLS-PM as derived in our study signified the effect of occlusal vertical dimension on the extent of final mandibular setback (stability of the mandible). As depicted in Fig. 2, an increase in VD_{10} leads to a decrease in the value of LV_{overall}, which in turn reduces the value of LV_{setback}. The result figure (Fig. 2) explains a decrease in the absolute amount of the final setback and, ultimately, more severe postsurgical skeletal changes (postsurgical relapse). In the model derivation of PLS-PM, the LVs are presumed to represent patterns of physiologic responses after the surgery until the removal of orthodontic appliances, which may or may not be in favor of the expected SFA outcome. Those predictors were manually categorized, based on their temporal dimensions according to either the postsurgical position (*T*
_{1}) or the amount of surgical movement (∆*T*
_{1} − *T*
_{0}), as LV_{1} or LV_{10}, respectively. The underlying physiologic response represented as LV_{1} is likely to involve factors dependent on the postsurgical skeletal positions of the maxilla and proximal segment, apart from the movement of the distal segment, while the physiological response represented as LV_{10} is solely dependent on the amount of horizontal setback of the distal segment of the mandible. Among those presumably related postsurgical factors such as the adaptation or healing process of the masticatory muscles or the change of occlusion due to surgery and orthodontic treatment, the question as to what constitutes LV_{1} and LV_{10} remains and should be studied further in subsequent research.

As for the outer model, the MVs of \( {\mathrm{B}}_{10}^{\mathrm{s}} \) and \( {\mathrm{Me}}_{10}^{\mathrm{s}} \) were presumed to cause LV_{10}, while those of \( {\mathrm{A}}_1^{\mathrm{s}} \) and \( {\mathrm{Cp}}_1^{\mathrm{f}} \) were presumed to cause LV_{1}, and those of \( {\mathrm{B}}_{10}^{\mathrm{s}},\ {\mathrm{Me}}_{10}^{\mathrm{s}},\ {\mathrm{A}}_1^{\mathrm{s}},\kern0.5em {\mathrm{Cp}}_1^{\mathrm{f}} \), and VD_{10} to cause LV_{overall}, in the formative sense; meanwhile, the extent of final mandibular setback \( \left({\mathrm{B}}_{20}^{\mathrm{s}}\right) \) was presumed to be caused by LV_{setback} in the reflective sense (Fig. 2) [11]. For the selection of the inner model, where the interactive relationships between LVs should be specified, various possible interactive mapping patterns were tested, and the finalized pathway was chosen based on the *R*-squares of \( {R}_{{\mathrm{LV}}_{10\kern0.5em \to \kern0.5em 1}}^2=\kern0.5em 0.9818 \) (LV_{10} effect on LV_{1}) and \( {R}_{{\mathrm{LV}}_{1,\kern0.5em 10,\kern0.5em \mathrm{all}\kern0.5em \to \kern0.5em \mathrm{setback}}}^2 = 0.8731 \) (LV_{1}, LV_{10}, and LV_{all} effects on LV_{setback}), with the acceptable GoF value of 0.7236 (Tables 4 and 5). In the finalized PLS-PM, an increase in VD_{10} was found to decrease the absolute value of the final setback amount of the mandible, in the LV_{all} pathway, as reflective of the postsurgical physiological responses to both surgery and orthodontic treatment, which in turn can be interpreted as an increase in postoperative mandibular changes.

The application of the surgery-first approach (SFA) has recently been reported, with its advantages (increased patient cooperation, effective compensation, and a shortened treatment period) [13,14,15,16,17]. However, orthodontists and oral surgeons often encounter postsurgical occlusal instability, induced from the premature contacts in SFA cases [16,17,18,19]. From the current literature and clinical experience, we hypothesized that these postsurgical changes in occlusal vertical dimension might be related to postsurgical skeletal changes (postsurgical relapse). To support the clinical observations and to detect collinearity between measurements, we adopted the partial-least-square path-modeling approach for this study. In clinical dental research, the data might be related to each other. However, we have reviewed the statistical problems such as collinearity. The deductive inference from the clinical research without considering this could sometimes be different from clinical observations, or an unclear conclusion, could be drawn. We attempted to apply the PLS-PM method to support the clinical observations with their interactions. This inference could provide clinicians with clearer cause-and-effect information. Unfortunately, this result was from one of many models with our defined latent variables. As for the shortcomings of our study, first, it was based on a 2-dimensional analysis. Therefore, for improved path modeling, a 3-dimensional analysis of postsurgical skeletal stability should follow this study. At this point, further research is required for better explanations and manifesting variables.