Variation in Dental Tissues: Using Bayesian Multilevel Modelling to Explore Intra- and Inter-Individual Dental Variation

Objectives Dental variation within populations and, even more so, within individuals is far less well understood than variation between populations. This is problematic as a single tooth type is often used as a representative of the whole dentition, despite a lack of understanding of intra-tooth type relationships. This research investigates the variation of dental tissues and proportions within and between individuals. Materials and Methods Upper and lower first incisor to second premolar tooth rows were obtained from 30 individuals (n=300), from 3 archaeological samples. The teeth were micro-CT scanned and surface area and volumetric measurements were obtained from the surface meshes extracted. Dental variation of these measurements on a tooth and individual level was studied using Bayesian Multilevel Modelling. Results The individual and tooth level variation differed by dental measurement, ranging between 9.5%-47.5% and 52.6-90.5% respectively. Enamel volume had the highest degree of individual-level variation in contrast to coronal dentine volume that had the lowest of individual-level variation. Tooth type, isomere, and position in field all showed a significant effect on the dental measurements examined in this study. Discussion Tooth selection and sampling strategies should consider individual and tooth-level variation, with at least one tooth from each type and isomere included in analyses. This will ensure that any population-level differences are not masked by variability between teeth. The low level of coronal dentine volume individual variation indicates that it is particularly useful in studies with small sample sizes.

Dental variation within populations and, even more so, within individuals is far less well understood 1 5 than variation between populations. This is problematic as a single tooth type is often used as a 1 6 representative of the whole dentition, despite a lack of understanding of intra-tooth type relationships.

7
This research investigates the variation of dental tissues and proportions within and between 1 8 individuals. , with a value of 1,000 sufficient for stable estimates (Bürkner, 2017). The Bulk ESS is a 1 4 3 Table 2 Definition of dental measurements measure for the overall sampling efficiency in the bulk of the distribution, whereas the tail ESS 1 4 4 corresponds to the 5% and 95% quantiles. Model assumptions were checked using a QQ-plot.

4 5
A stepwise approach was used to construct each model, adding a single parameter at a time.

4 6
The models were compared using an approximate leave-one-out (LOO) cross-validation, using the loo

5 4
For each dental measurement a two-level model was constructed: tooth (level 1) and  Figure 3).

5 9
On a tooth level, variation between teeth has been associated with tooth type (Lombardi, 1975;Townsend and Brown, 1979;Harris and Bailit, 1988;Dempsey et al., 1995;Harris, 2003), and 1 6 1 between upper and lower dentitions, which are the result of two different developmental programmes 1 6 2 (Ferguson et al., 2000(Ferguson et al., , 2001Cobourne and Mitsiadis, 2006;Sperber, 2006). Position in field was 1 6 3 included as a predictor as morphogenetic field theory models suggest that there should be distinct 1 6 4 patterns of heritability within each tooth class (Butler 1939;2001;Dahlberg 1945). According to 1 6 5 these models, there are gradients of variation in tooth size and shape from the mesial, or pole tooth, to 1 6 6 distal members of each tooth class. The distal later developing tooth, apart from the lateral incisor, 1 6 7 shows greater variation (Line, 2001;Townsend et al., 2009a).

6 8
On the individual level, variation in dental size has been found by both sex and age. Tooth 1 6 9 size has been used in archaeological and modern dentitions to sex individuals using discriminant 1 7 0 function analysis (Fernée et al., 2020;Viciano et al., 2011;Tardivo et al., 2015). Tooth size variation 1 7 1 by age is largely the product of degenerative processes, with undergoing a range of macroscopic and 1 7 2 microscopic degenerative processes, including wear, changes in root chamber morphology, root

7 5
A normal likelihood was fitted to each dental measurement model with an identity link 1 7 6 function. Normal distributions were used for each of the random effects (levels) with a weakly 1 7 7 informative prior distribution of normal (0, 10), and weakly informative prior of student t (3,0,74.2) 1 7 8 was used for the fixed effects (for mathematical structure see supporting information 2).

7 9
The data was fitted into 8 models: 1) a null two-level random-intercept model (M n ), 2) a two-      Variation in the ESA MLM was 58.6% at the tooth level and 41.4% at the individual level (Table 4).

4
For the fixed effects, tooth type (I< C<PM), isomere (L<U), position in field (D<P) and degree of  (Table 4). Variation in the EVol MLM was 52.5% at the tooth level and 47.5% at the individual level (Table 4).

5 7
Tooth type (I<C<PM), isomere (L<U), position in field (D<P) and degree of wear (1>4) significantly Variation in the RSA MLM was 70.0% at the tooth level and 30.0% at the individual level (Table 4).

6 3
Tooth type (I<PM<C), isomere (L<U) and position in field (D<P) significantly affected RSA ( Figure   2 6 4 5). The remaining predictors did not significantly affect RSA ( Figure 5).  Variation in the RVol MLM was 73.8% at the tooth level and 26.2% at the individual level (Table 4).

7 4
Tooth type (I<PM<C), isomere (L<U) and position in field (D<P) significantly affected RVol ( Figure   2 7 5 5). The remaining predictors did not significantly affect RVol (Figure). Variation in the CDVol MLM was 90.5% at the tooth level and 9.5% at the individual level (Table 4).

0 7
The degree of variability at the tooth level, although greater than at the individual level in 3 0 8 most measurements, is lower than expected. These findings suggest that teeth within the same 3 0 9 individual exhibit a considerable degree of similarity, providing justification for the use of dimensions 3 1 0 from a single tooth or tooth class in inter-population studies. However, tooth type was found to 3 1 1 contribute the most variation to the models for most dental measurements, followed by tooth isomere.

1 2
Consequently, it is recommended that when possible, at least one tooth from each tooth type and 3 1 3 isomere should be included in analyses. This will ensure that the full range of variation within the 3 1 4 dentition is captured, and that any population-level differences are not masked by the inherent analyses of dental variation.

1 8
Enamel volume was found to have the highest degree of individual-level variation, 3 1 9 accounting for 47.5% of residual variation in the final model. This high degree of variation at the 3 2 0 individual level suggests that detecting true between-sample differences in enamel volume may be 3 2 1 more difficult than for other dental measurements examined in the study. Consequently, studies that 3 2 2 rely solely on enamel volume as a metric may be limited in their ability to accurately characterise 3 2 3 population-level variation. It is important for researchers to carefully consider the sources of 3 2 4 individual variation in enamel volume, such as age, sex, and diet, and to account for these factors 3 2 5 when conducting comparative studies.

2 6
In contrast to the other dental measurements examined in this study, coronal dentine volume 3 2 7 displayed the least variation between individuals, with only 9.5% of residual variation attributed to the and environmental factors in dental development and evolution.

4 5
The study did not find any significant effect of sex or side on the dental measurement models 3 4 6 under investigation. It is possible that the lack of significant effect of sex on dental measurements is 3 4 7 due to sexual dimorphism being more pronounced in certain types of teeth, such as canines, as

5 1
The small number of right-sided teeth included in the sample may have contributed to this result, as 3 5 2 the left side was preferentially selected and the right was only included if the left side was unavailable 3 5 3 or did not meet the inclusion criteria.

2 2
The role of effectors of the activin signalling pathway, activin receptors IIA and IIB, and Smad2, in       18.6%