Appendix B. A discussion of the methodological aspects.
De Valpine and Hastings (2002) concluded that a state-space modeling approach (as used in our study) is advantageous over other methods when fitting population models to simulated data with added observation error. However, provided that the process model is appropriate, the ability of the state-space model to account for sampling errors depends on the observation process being adequately modeled. The reliability of our results is supported by the fact that we found no strong relationship between the precision of the data and the strength of the estimated density dependence – which is not true for the first-order autoregressive approach (where sampling errors are not accounted for) where the strength of the estimated density dependence was inversely related to the precision of the data. Moreover, simulations we have performed suggest that there is only a slight bias in the estimated density dependence and that the bias is clearly reduced compared to the case when sampling errors are ignored (see Appendix D). Taking into account the uncertainty of the estimates, we calculated the proportion of times the true density dependence were likely to be rejected. Compared to the autoregressive approach we are less likely to conclude that the density dependence is stronger than it truly is (see Tables D1 and D2): Generally, the wrong rejection rates are less than 0.1 for the state-space method and higher than 0.1 for the autoregressive approach. Using the state-space method, the sampling error is accounted for and on average the process stochasticity variance is well estimated (Table D3).
Standardizing time series by subtracting the mean is a common procedure in time-series analysis. In a first-order autoregressive approach, the time-series mean is supposed to correspond to the long-term equilibrium value defining the “carrying capacity” (Royama 1992). This is, however, likely to be a problem whenever time series are short and, especially, if there are trends. In the state-space approach we may properly assume a vague prior to the equilibrium value, meaning that uncertainty in the equilibrium value is introduced (i.e., the shorter the time series or the stronger the trends in the data, the lower the precision will be found in the posterior estimate of the equilibrium value). Estimates of density dependence became considerable stronger if, in another version of the state-space model, the “true” log-abundances are scaled by their short-term mean (see Appendix A: Table A2).
de Valpine, P., and A. Hastings. 2002. Fitting population models incorporating process noise and observation error. Ecological Monographs 72:57–76.
Royama, T. 1992. Analytical population dynamics, Chapman and Hall, London, UK.