Appendix B. Simulation study.

The coverage properties of the Wald intervals from DC were assessed via simulations from the Beverton-Holt-Poisson state space model. 1000 time series with one missing observation, as in Gauses’ data set of *Paramecium aurelia* were generated according to the Stochastic Beverton-Holt model with Poisson sampling error. The parameters used for the simulations were the ML estimates for the first time series of *P. aurelia* from Gauses’ data set, namely:. For each simulated data set, the ML estimates from DC along with the 95% Wald confidence intervals were calculated with clones each time. The percentage of times the 95% Wald confidence intervals contained the true parameter values were 88.1%, 88.3% and 84.7% for respectively. This is well within what would be expected from a small time series data set. The coverage of the PL confidence intervals is known to be better than the Wald intervals’ coverage (Pawitan 2001) and this characteristic applies also to hierarchical models (Lee, Nelder and Pawitan 2006). The PL confidence intervals coverage properties were not investigated here due to the amount of computer time involved in such calculations. Furthermore, the results obtained would give us a limited vision of the particular model under study, as each parameter in each hierarchical model is different. Testing properly the reliability of the DC-method means testing the reliability of the ML estimation method, and various books about the qualities of ML estimates exist (see for instance, Pawitan 2001 cited in the main text). Of particular interest for this study’s context would be testing the performance of ML estimation for hierarchical models. However, a plethora of such models is available in the ecology literature. Therefore, testing the performance of ML estimation for hierarchical models would imply carrying a literature review and performing a vast array of simulations for many ecological models and types of data available. Finally, we show here the sampling distribution of the parametric bootstrap ML estimates from DC is also shown in Fig. B1. Besides, the bootstrap, Wald and PL confidence intervals for the ML estimateare:

P. aurelia, 1st replicate |
(2.5% LCL, 97.5% UCL) |

Profile Likelihood |
(1.893,2.453) |

Parametric Bootstrap |
(1.982,2.433) |

Wald asymptotic variance |
(1.905,2.355) |

FIG. B1. Sampling distribution of the ML estimates obtained through Parametric Bootstrap of the Stochastic Beverton-Holt State space model for the first replicate of Gauses’ P. aurelia data set. The ML model parameter estimates from which the 1000 bootstrap replicates were drawn are shown as vertical red lines. Each time, the ML estimates were found using DC with 40 clones. |

LITERATURE CITED

Lee, Y., Nelder, J. and Pawitan, Y. 2006. Generalized linear models with random effects. Chapman and Hall, New York, New York, USA.