Appendix 7
Tooth eruption pattern
We used non-linear mixed-effect regression models to fit logistic curves to each of
the traits in the three molars, by pooling all the samples into age classes that
contained at least 14 samples (mean = 31.0, s.d. = 26.0). This produced a total of 17
age classes (i.e. 0:3, 4, 5, 6, 7, 8, 9, 10, 11:16, 17:19, 20, 21, 22, 23:26, 27:30, 31:41
and 42:50 months). In each logistic model sample and technician were included as
random effects. To provide a measurement of the error of the logistic fitting we
bootstrapped our data set by removing 17 random samples at a time (one sample
per each age class) and refitted the model, this was done 10000 times for each of
the eruption traits of each molar. Using the logistic parameters of the models of each
bootstrap we predicted y (i.e. the percentage of samples with cusp eruption trait
present) within the x range [0:40] months with a step = 0.1 month. Then the quantiles
0.05, 0.50 and 0.95 of the 10000 predicted y values at each step of the [0:40] range
were calculated to represent the variability of the model for each cusp eruption trait.
These values were plotted and used to calculate thirteen new age classes (Table 4)
by maintaining or pooling the existing ones, with the condition that each new age
class was that at which x values ≥ 90% occurred in just one of the cusps but not yet
in the others. This was carried out by using the models fitted with the 0.05 quantiles,
as they provided the most conservative value of age class (i.e. a probability of 95%
that ≥90% of animals within that age classes had that eruption stage present in that
particular cusps trait). There was no attempt to partition the sample by sex, as we
considered that for this method it was a priority to have as many samples as possible
within each age class.
The models were fitted using the function nlmer of the R package lme4 (Bates &
Maechler, 2009), and the SSlogis R function that evaluates the logistic curve and its
gradient, by calculating the initial estimates of the parameters of the logistic model.
Table 4. Predicted age classes in relation to the eruption of molars in red deer using
data of the mixed effects logistic regression models in Figure 1. See Table 3 for
definitions of molars, cusps and eruption stages.
Age class
Molar and cusps
Eruption stage
≤2
M1,2,3 (C1,2,3)
0
3:5
M1 (C1)
1
6
M1 (C2)
1
7:8
M1 (C1)
2
9:15
M1 (C2)
2
16
M2 (C1)
1
17
M2 (C2)
1
18:21
M2 (C1)
2
22:26
M2 (C2)
2
27:31
M3 (C1)
1
32
M3 (C2)
1
33-38
M3 (C1)
2
>38
M3 (C2)
2
(months)
References
Bates and Maechler. lme4: Linear mixed-effects models using S4 classes. R
package version 0.999375-31. 2009.