# Quantifying Perceived Facial Asymmetry to Enhance Physician-Patient Communications
## 3 Results  
The study recruited 128 laypeople as the respondents and 113 (88.3%) of them replied with valid questionnaires, in which 43 (38%) are male and 70 (62%) are female. The invalid questionnaires include incomplete or apparently inconsistent answers, e.g., the respondent may consider a face as symmetrical but paradoxically call for a surgery. The questionnaire results present a Cronbach's Alpha value of 0.944 [37]. The category of a face, PN, PAN, or PA, is determined by which category receives the most votes from the respondents. Among the 64 stimulus faces, 9 faces are classified as PN (14%), 15 faces are classified as PAN (23%), and 40 faces are classified as PA (63%). The distribution of face categories is aligned with the process of generating stimulus faces, which are deformed from a normal face and present varied degrees of facial asymmetry.  
The asymmetry classifier models the respondents' perceptions to predict the perceived asymmetry of a face. The input layer takes the asymmetry index vector (or 14 asymmetry indices) of the corresponding 20 facial landmarks. The 14 asymmetry indices (\(\mathrm{AI}_{i},\ i=1\ldots 14\)) of a stimulus face are computed using Equation (2). The classifier is trained with 75%, validated with 15%, and tested with 15% of the 3D stimulus face images and yields an overall accuracy >99% (root mean squared error 0.000059225). The resulted inter-layer weight matrix between the input and hidden layers (denoted as \(IW^{1,1}\)) is detailed in Table 3. The inter-layer weight matrix between the hidden and output layers (denoted as \(LW^{2,1}\)) is detailed in Table 4. The bias vectors added to the hidden and output layers are listed in Table 5.  
The relative importance RIs (\(\mathrm{RI}_{i},\ i=1\ldots 14\)) are calculated using Equation (3) and are employed as the weights of facial features, respectively. As shown in Table 6, the RIs are rank-ordered, and it shows that alar curvature (\(AI\)) and subnasale (\(Sn\)) are relatively important when classifying perceived asymmetry: both having RIs greater than 0.10, which are significantly greater than those of the remaining facial features. It is also aligned withthe proposed deformation of the normal face with the emphasis on the nose and chin regions, as suggested in [1; 7; 33].  
The oAI of a stimulus face is a linear combination (SOP; sum of product) of the corresponding AI, and RI, as shown in Equation (4). Table 7 presents the overall asymmetry index (oAI), classification of perceived asymmetry (PN\(|\)PAN\(|\)PA), and the confidence value of the classification (\(\hat{\text{C}}\): percentage of the voted classification) for the 64 stimulus faces. Please note that for brevity of demonstration, only partial asymmetry quadruple \(\langle\)oAI, \(\hat{\text{AI}}\), PN\(|\)PAN\(|\)PA, \(\hat{\text{C}}\rangle\) is shown: the detailed asymmetry index vector (\(\hat{\text{AI}}\)) and detailed confidence vector (\(\hat{\text{C}}\)) are excluded. For example, \(n_{03}n_{15}\) is a stimulus face with a \(1.5^{\circ}\) nose roll and a \(8.42^{\circ}\) chin roll (as depicted in Table 2) from the normal face. Its entry in Table 7 is shown as (6.5208,PA,0.92), i.e., its oAI is 6.5208 and it is perceived as abnormal (PA) with a confidence vector of 0.92.  
The range of oAI is from 0.8182 (\(n_{01}c_{01}\)) to 8.6792 (\(n_{15}c_{15}\)). The greater the oAI of a stimulus face, the greater the possibility of it being classified as PAN or PA.