2

Option manual mixture of 2betaregfunction: The levels of the observed values of d_{j,k,e} were dichotomized. Then, we fit 2 Beta distribution by applying the betareg function (or the equivalent betamixfunction with 1 as the number of components) on each level of dichotomisation. All levels of dichotomisation were compared, from that separating the two left values from the others, to that separating the two right values from the others. The two distributions were then mixed by applying the weights w_{1,j} and w_{2,j}=1−w_{1,j} to each distribution. The weights w_{1,j}∈(0,0.05,0.10,0.15,…,0.95,1) were tested. The models obtained with the different levels of dichotomisation and with the different weights were compared using the criteria for goodness of fit described in Section 1 of the Additional file 7. The fit with the lowest criteria was retained for the comparison with the other 2 methods.

3

Option manual mixture of abetamixfunction and abetaregfunction: A mixture of betamix function and manual mixture: We mixed: (i) a first Beta distribution obtained on the left level of dichotomisation (the one obtained with method 2), (ii) a mixture of a second and a third distribution, obtained by applying to the right level of dichotomisation the betamix function with 2 as the number of components. The weights given to those distributions were: (i) for the first distribution the w_{1,j} was obtained through method 2, (ii) for the second and third distribution, the weights w_{2,j} and w_{3,j} were obtained through the ’betamix’ procedure, multiplied by (1−w_{1,j}).
