# A Multivariate Approach to Determine the Dimensionality of Human Facial Asymmetry

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{2}), all with self-reported European-derived ancestry and no history of any significant facial trauma or facial surgery, or any medical condition that might alter facial structure. The facial scans originated from a mixture of several studies at the University of Pittsburgh, Pennsylvania State University and Indiana University–Purdue University, Indianapolis. The scans were made using two stereophotogrammetry systems: the VECTRA H1 camera (Canfield Scientific, Parsippany, NJ, USA); and the 3dMDface system (3dMD, Atlanta, GA, USA).

## 3. Results

#### 3.1. Facial Directional Asymmetry in Men and Women

_{622,622}= 0.40, p < 0.001).

#### 3.2. Estimating Fluctuating Asymmetry

#### 3.3. Correlations between C-FA and F-DA Scores

## 4. Discussions

^{2}= 0.58). Therefore, about 40% of the average asymmetry is not shared between men and women (Figure 4). In addition, the amount of between-individual variation in this morphological direction is larger in men (Figure 5). Clearly, if one wants to generate FA estimates from human faces, corrections for DA should be done separately for men and women, and both the average directional asymmetry, as well as the between-individual variation in DA (which we termed F-DA), should be taken into account.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Ethics Statement

## References

- van Dongen, S. Fluctuating asymmetry and developmental instability in evolutionary biology: Past, present and future. J. Evol. Biol.
**2006**, 19, 1727–1743. [Google Scholar] [CrossRef] [PubMed] - van Valen, L. A Study of Fluctuating Asymmetry. Evolution (N.Y.)
**1962**, 16, 125. [Google Scholar] [CrossRef] - Hill, A.K.; Cardenas, R.; Wheatley, J.R.; Welling, L.L.M.; Burriss, R.P.; Claes, P.; Apicella, C.L.; Mcdaniel, M.A.; Little, A.C.; Shriver, M.D.; et al. Are there vocal cues to human developmental stability? Relationships between facial fluctuating asymmetry and voice attractiveness. Evol. Hum. Behav.
**2017**, 38, 249–258. [Google Scholar] [CrossRef] [Green Version] - Grammer, K.; Thornhill, R. Human (Homo sapiens) facial attractiveness and sexual selection: The role of symmetry and averageness. J. Comp. Psychol.
**1994**, 108, 233–242. [Google Scholar] [CrossRef] [PubMed] - Simmons, L.W.; Rhodes, G.; Peters, M.; Koehler, N. Are human preferences for facial symmetry focused on signals of developmental instability? Behav. Ecol.
**2004**, 15, 864–871. [Google Scholar] [CrossRef] - Palmer, A.; Strobeck, C. Fluctuating Asymmetry Analyses Revisited. In Developmental Instability: Causes and Consequences, 1st ed.; Polak, M., Ed.; Oxford University Press: Oxford, UK, 2003; pp. 279–319. [Google Scholar]
- Leary, R.F.; Allendorf, F.W. Fluctuating asymmetry as an indicator of stress: Implications for conservation biology. Trends Ecol. Evol.
**1989**, 4, 214–217. [Google Scholar] [CrossRef] - Watson, P.J.; Thornhill, R. Fluctuating asymmetry and sexual selection. Trends Ecol. Evol.
**1994**, 9, 21–25. [Google Scholar] [CrossRef] - Klingenberg, C.P. A developmental perspective on developmental instability: Theory, models and mechanisms. In Developmental Instability: Causes and Consequences, 1st ed.; Polak, M., Ed.; University Press: Oxford, UK, 2003; pp. 14–34. [Google Scholar]
- van Dongen, S.; Gails, F.; Ten Broek, C.; Heikinheimo, K.; Wijnaendts, L.C.D.; Delen, S.; Bots, J. When right differs from left: Human limb directional asymmetry emerges during very early development. Laterality Asymmetries Body Brain Cogn.
**2014**, 19, 591–601. [Google Scholar] [CrossRef] - Palmer, A.R. Fluctuating asymmetry analyses: A primer. In Developmental Instability: Its Origins and Evolutionary Implications; Springer: Dordrecht, The Netherlands, 1994; Volume 93, pp. 335–364. [Google Scholar]
- Palmer, A.R.; Strobeck, C. Fluctuating Asymmetry: Measurement, Analysis, Patterns. Annu. Rev. Ecol. Syst.
**1986**, 17, 391–421. [Google Scholar] [CrossRef] - Özener, B. Fluctuating and directional asymmetry in young human males: Effect of heavy working condition and socioeconomic status. Am. J. Phys. Anthropol.
**2010**, 143, 112–120. [Google Scholar] [CrossRef] - Graham, J.; Özener, B. Fluctuating Asymmetry of Human Populations: A Review. Symmetry (Basel)
**2016**, 8, 154. [Google Scholar] [CrossRef] [Green Version] - Lens, L.; van Dongen, S. Fluctuating and directional asymmetry in natural bird populations exposed to different levels of habitat disturbance, as revealed by mixture analysis. Ecol. Lett.
**2008**, 3, 516–522. [Google Scholar] [CrossRef] - van Dongen, S. Human bodily asymmetry relates to behavioral lateralization and may not reliably reflect developmental instability. Symmetry (Basel)
**2018**, 10, 117. [Google Scholar] [CrossRef] [Green Version] - Stige, L.C.; David, B.; Alibert, P. On hidden heterogeneity in directional asymmetry—Can systematic bias be avoided? J. Evol. Biol.
**2006**, 19, 492–499. [Google Scholar] [CrossRef] [PubMed] - Ekrami, O.; Claes, P.; White, J.D.; Zaidi, A.A.; Shriver, M.D.; van Dongen, S. Measuring asymmetry from high-density 3D surface scans: An application to human faces. PLoS ONE
**2018**, 13, e0207895. [Google Scholar] [CrossRef] - White, J.D.; Ortega-Castrillón, A.; Matthews, H.; Zaidi, A.A.; Ekrami, O.; Snyders, J.; Fan, Y.; Penington, T.; Van Dongen, S.; Shriver, M.D.; et al. MeshMonk: Open-source large-scale intensive 3D phenotyping. Sci. Rep.
**2019**, 9. [Google Scholar] [CrossRef] [Green Version] - Klingenberg, C.P.; Barluenga, M.; Meyer, A. Shape analysis of symmetric structures: Quantifying variation among individuals and asymmetry. Evolution (N.Y.)
**2002**, 56, 1909–1920. [Google Scholar] [CrossRef] [Green Version] - Claes, P.; Walters, M.; Vandermeulen, D.; Clement, J.G. Spatially-dense 3D facial asymmetry assessment in both typical and disordered growth. J. Anat.
**2011**, 219, 444–455. [Google Scholar] [CrossRef] [Green Version] - Burnaby, T.P. Growth-Invariant Discriminant Functions and Generalized Distances. Biometrics
**1966**, 22, 96. [Google Scholar] [CrossRef] - Enlow, D.H. The Human Face: An. Account of the Postnatal Growth and Development of the Craniofacial Skeleton; Hoeber Medical Division, Harper & Row: New York, NY, USA, 1968. [Google Scholar]
- Ferrario, V.F.; Sforza, C.; Poggio, C.E.; Tartaglia, G. Distance from symmetry: A three-dimensional evaluation of facial asymmetry. J. Oral Maxillofac. Surg.
**1994**, 52, 1126–1132. [Google Scholar] [CrossRef] - Ferrario, V.F.; Sforza, C.; Ciusa, V.; Dellavia, C.; Tartaglia, G.M. The effect of sex and age on facial asymmetry in healthy subjects: A cross-sectional study from adolescence to mid-adulthood. J. Oral Maxillofac. Surg.
**2001**, 59, 382–388. [Google Scholar] [CrossRef] [PubMed] - Markow, T.A. Developmental Instability: Its Origins and Evolutionary Implications; Springer: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Clarke, G.M. The genetic basis of developmental stability. IV. Individual and population asymmetry parameters. Heredity (Edinb)
**1998**, 80, 553–561. [Google Scholar] [CrossRef] - Polak, M. Developmental Instability: Causes and Consequences; Oxford University: Oxford, UK, 2003. [Google Scholar]

**Figure 1.**Representation of linear measurements commonly used for asymmetry analysis. (

**a**) There is no variation between the individuals in the DA direction, therefore FA and DA vectors are not correlated. In this scenario, FA can be adequately calculated by only removing the effect of DA; (

**b**) in case the FA vector is correlated with the DA, its component in the direction of DA (F-DA) and the orthogonal component (C-FA) cannot be effectively separated.

**Figure 2.**A 2D representation of the asymmetry space. The plot on the left (

**a**) represents a case where the calculated FA vectors are orthogonal to the DA vector. The plot on the right (

**b**) shows a scenario where the calculated FA vector has a component parallel to the DA vector.

**Figure 3.**Flow chart of the algorithm used to obtain FA vectors. The template face is mapped onto each face, and the reflection is obtained. Each face and its mirror are superimposed and subtracted, and the subtraction matrix is vectorized to obtain the matrix of asymmetry vectors (TA). The average of these vectors is considered as DA, and by removing the DA from each row of the matrix, the FA vectors are calculated. Different dimensions of FA can then be explored by applying a PCA on the vectors.

**Figure 4.**(

**a**) Female and (

**b**) male directional asymmetry, amplified 10 times and visualized onto the average face; and (

**c**) heat-map of the difference between DA in both sexes. The values are scaled (amplified) equally in all three figures for visual purposes.

**Figure 5.**Histogram of the magnitudes of F-DA vectors for females (

**above**) and males (

**below**). The kurtosis measurement validates the normality of the distributions. This also suggests a lack of AS in the faces. Based on the standard deviations, we can see that the male subgroup shows a higher effect of DA on individual faces.

**Figure 6.**Heat map of DA and the first 5 PCs of C-FA for both males and females. The differences and similarities between the two groups are clear in this figure. The variations are scaled (amplified) for visual purposes. The blue color on the heat maps shows low variations, and the red regions correspond to high variations. The same scale has been used for both groups.

Abbreviation | Full Name | Description |
---|---|---|

DA | Directional Asymmetry | The average difference between two sides in a population sample |

FA | Fluctuating Asymmetry | Directionally random asymmetry resulting from random perturbations during development |

DI | Developmental Instability | The inability of an organism to buffer its development against random perturbations |

TA | Total Asymmetry | Signed difference between the two sides (left–right) |

F-DA | Fluctuating Directional Asymmetry | Individual variation of asymmetry in the dimension of DA |

C-FA | Corrected Fluctuating Asymmetry | Fluctuating asymmetry after correcting for DA and F-DA |

**Table 2.**Pearson correlation coefficients (r) between the DA vector and the first 5 PCs of the FA vectors, without F-DA correction. The first 5 PCs containing most of the variation in FA show correlation with DA. The statistically significant results are shown in bold (p < 0.001).

PC1 | PC2 | PC3 | PC4 | PC5 | ||
---|---|---|---|---|---|---|

Females | % variance explained | 21.7 | 14.7 | 11.0 | 7.0 | 5.0 |

r | 0.08 | 0.36 | −0.22 | 0.36 | −0.20 | |

Males | % variance explained | 22.1 | 16.2 | 12.0 | 6.8 | 4.8 |

r | 0.40 | 0.57 | 0.00 | −0.22 | 0.11 |

**Table 3.**Pearson correlation coefficients (r) between the absolute value of F-DA scores with the absolute value of PC scores of C-FA. Although the DA and C-FA vectors are orthogonal (i.e., reflect different dimensions of variation), there is some correlation between their scores for individual faces. Statistically significant results are shown in bold (p < 0.001).

|PC1| | |PC2| | |PC3| | |PC4| | |PC5| | ||
---|---|---|---|---|---|---|

|F-DA| | Females | 0.14 | 0.19 | 0.17 | 0.22 | 0.15 |

Males | 0.06 | 0.08 | 0.51 | 0.16 | 0.03 |

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## Share and Cite

**MDPI and ACS Style**

Ekrami, O.; Claes, P.; White, J.D.; Weinberg, S.M.; Marazita, M.L.; Walsh, S.; Shriver, M.D.; Van Dongen, S.
A Multivariate Approach to Determine the Dimensionality of Human Facial Asymmetry. *Symmetry* **2020**, *12*, 348.
https://doi.org/10.3390/sym12030348

**AMA Style**

Ekrami O, Claes P, White JD, Weinberg SM, Marazita ML, Walsh S, Shriver MD, Van Dongen S.
A Multivariate Approach to Determine the Dimensionality of Human Facial Asymmetry. *Symmetry*. 2020; 12(3):348.
https://doi.org/10.3390/sym12030348

**Chicago/Turabian Style**

Ekrami, Omid, Peter Claes, Julie D. White, Seth M. Weinberg, Mary L. Marazita, Susan Walsh, Mark D. Shriver, and Stefan Van Dongen.
2020. "A Multivariate Approach to Determine the Dimensionality of Human Facial Asymmetry" *Symmetry* 12, no. 3: 348.
https://doi.org/10.3390/sym12030348