Appendix S2: Simulated data and mean-variance relationship
of the estimate
Simulated data
In order to investigate the applicability of the method for small sample sizes, we estimated the
number of reproductive attempts for simulated data sets with sample size N varying from 400
to as few as 25 nest records.
We drew laying dates from two distributions with fixed parameters: A) a skew-Normal
distribution simulating first reproductive attempts with location= -35, scale= 20 and skewness=
2; B) a Normal distribution simulating all subsequent reproductive attempts with location= 35
and scale= 20. The target proportion of first reproductive attempts was set to p1= 0.6,
corresponding to an average of 1.67 attempts per breeding pair.
Each data item was drawn from distribution A with probability p1, and from B otherwise (Figure
S2.1).
0.000 0.005 0.010 0.015
Density
Figure S2.1: Histogram of a simulated mixture of A) a skew-Normal distribution with location= 35, scale= 20 and skewness= 2 (green line); B) a Normal distribution with location= 35 and
scale= 20 (blue line). The target proportion of component A was set to 0.6 (or 1.67 nesting
attempts)
-50
0
50
exsamp
100
Mean-variance relationship of the estimate
The following simulation illustrates the effect of the number of reproductive attempts on the
standard deviation of the estimate.
We drew N= 200 laying dates from two distributions with fixed parameters: A) a skew-Normal
distribution simulating first reproductive attempts with location= -35, scale= 20 and skewness=
2; B) a Normal distribution simulating all subsequent reproductive attempts with location= 35
and scale= 20. The target proportion of first reproductive attempts (p1) was given a range of
values between 1 and 0.2, corresponding to an average of 1 to 5 attempts per breeding pair,
respectively. The simulation method followed the procedure described in simulation I. The
simulation results reported in Figure S2.2 indicate that the precision of the estimate decreases
(increase in standard deviation) from 1 to 2 reproductive attempts. For a number of
reproductive attempts equal to 1 (p1= 1), the large errors stand out of this pattern, as a
consequence of the mixture model being unstable when fitted to purely unimodal data.
0.25
0.20
0.15
0.10
0.05
sd(Estimated proportion of first clutches)
Figure S2.2: Relation between the standard deviation of p1 and the number of reproductive
attempts. The median and 95% confidence interval for the standard deviation of the proportion
of first clutches(p1) are given by the thick and the dotted lines respectively.
1
2
3
4
Number of reproductive attempts
5