Guía para fomentar la RSE en el sector camaronero

In the following figures we observe a series of box-plots for the zero-truncated joint model under a simulation study. Let us recall that α1 = 0.5 and α2 = 4, hence

the first value of α represents a high heterogeneity in the population, in contrast to the second value in which a smaller heterogeneity is assumed. We contrast the results predicted by the joint model always represented to the left of the figures, and the results presented by the truncated joint distribution, always shown at the right side of the figures. In order to keep the consistency with previous models under simulation, here we have 15 simulation runs for each value of the frailty term, with 1500 simulated values of pre-randomization counts and post-randomization times,

Deviance residuals for Cowling’s model, alpha1 Observed residuals −4 −2 0 2 200 400 600 800 1000 population 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (a)α1= 0.5

Deviance residuals for Cowling’s model, alpha2

Observed residuals −4 −2 0 2 4 200 400 600 800 1000 population 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (b)α2= 4

Figure 5.20: Plots of the deviance residuals for Cowling’s model, the first graph corresponds to the population withα1 = 0.5 and the second graph to the population

withα2 = 4.

under a Poisson-gamma mixture model.

The first great contrast between the findings is the observed bias for the estimates ofα, shown in Figure 5.21, is that for the joint model the estimation bias lies around zero, with particularly narrow values for α1, which is well represented

for the truncated model as well. Meanwhile, when heterogeneity is small in the population, the estimation bias for the joint model is much larger, but for the truncated joint model the estimations are concentrated a unity of distance away. This might be one of the causes of why, even when the joint truncated model fits the estimates for λ1|α2 and λ2|α2 particularly well (Figure 5.22, to the right), the

estimates for all values of ψ under α2 deviate from the true value progressively

(Figure 5.23), which also tends to happen for the values of ψunder α2 in the joint

model, but not so strongly.

In general, for the cases when the heterogeneity is high, both the joint model and the truncated joint model seem to fit well forα,λand ψ.

−12 −10 −8 −6 −4 −2 0 Bias in alpha alpha1 alpha2

Figure 5.21: Cowling vs Truncated forα

−1.5 −0.5 0.0 0.5 1.0 1.5 Bias in lambda

lam1 lam2 lam1 lam2

alpha 1 alpha2

Figure 5.22: Cowling vs Truncated forλ

−0.5

−0.3

−0.1

0.1

Bias in psi

psi1|lam1 psi1|lam2 psi1|lam1 psi1|lam2 alpha 1

alpha 2

Figure 5.23: Cowling vs Truncated forψ

Figure 5.24: Box-plots for the biases between the expected and the estimated values ofα,λand ψ.

As an attempt to understand the behavior of the truncated joint model when the heterogeneity of the population varies, we have performed a simulation study in which we only estimate the values ofα. We proceed by simulating the occurrences of 1500 individuals as before, with the same values of α1, α2, λ1, λ2, ψ1 and ψ2.

The difference lies in how we fit the truncated model. Instead of using a particular initial valueα0 ofαfor the process of optimization of the log-likelihood, we provide

a series of 9 initial values of α1 ranging from 0.1 to 1, and nine initial values for

the estimation of α2 ranging from 0.1 to 7. In Figure 5.25(a) we observe that,

when the heterogeneity is high in the population, the truncated joint model tends to fit well for α even when the initial value is understated. For the case when the initial value is too small, the model concentrates the estimate either very close to the original value, or quite far away, as observed for the case in whichα0 = 0.1. The

bias however increases noticeably as the initial value surpasses the true value of the frailty term.

For the case when the original value ofαis larger, depicted in Figure 5.25(b), we observe that the closest values of the parameter estimates are obtained, again, when the initial value is smaller than the actual value of the parameter. The general behavior of the estimates tend to be more erratic around the true value, but the same increasing trend of the bias is observed when the initial value surpasses the true value ofα.

Estimation bias for alpha=0.5 Initial value Estimation bias −0.8 −0.6 −0.4 −0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1

(a) Estimation bias for α= 0.5.

Estimation bias for alpha=4

Initial value Estimation bias −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.1 0.5 1 2 3 4 5 6 7

(b) Estimation bias for α= 4.

Figure 5.25: Estimation bias for simulated data under assumption of α = 0.5 and

α = 4 respectively, when fitting a zero-truncated model with initial values for α

shown in the horizontal axis.

We conclude by this simulation study that the joint truncated model tends to identify better the value of the frailty term when the heterogeneity in the pop- ulation is large, and that the initial value provided for the optimization is of great consequence for the estimations.