Integrated Model with Ring-Recovery Data

A wide range of analyses are possible in the integrated context. Initially we form a model by multiplying the likelihoods La and Lj for the adult and ju- venile yearly-site counts from the CES data with the information from the ring-recoveries via Ldead. These two data sets, and the three component likeli- hoods, are independent. Posterior means ofJt,φa,t,φj,t and the derived At and Pt parameters are shown in Figure 3.4 along with their baseline estimates. For φa,t the appropriate baseline is from the ring-recovery data.

Adult Abundance 0.6 0.8 1.0 a Juvenile Abundance 0.4 0.7 1.0 b Productivity 0.6 0.9 1.2 c Adult Survival 0.0 0.5 1.0 d 1990 1995 2000 2005 Juvenile Survival 0.0 0.5 1.0 e

Figure 3.4 Posterior means, and the 95% symmetric credible intervals, from the “baseline” models, denoted by thin left-hand lines, and the integrated model which uses ring-recovery

data only in the estimation of adult survival, denoted by bold right-hand lines, for a)At, b)

Jt, c)Pt, d)φa,t, e)φj,tusing Sedge Warbler data. Forφa,tthe ring-recovery baseline is used.

Comparing the baseline and integrated models reveals that although the de- rived adult abundance indices from the integrated model are very similar to their baseline (Figure 3.4a) there are some differences in the estimates of adult survival (Figure 3.4d) and juvenile survival (Figure 3.4e). This would suggest that the adult abundance data, through the recursive population model (Equa- tion (3.1)), is driving the estimation of adult and juvenile survival, which is to be expected as the ring-recovery data are relatively sparse. Conversely, as the derived adult abundance indices and their baseline are almost exactly the same, it would seem that the limited ring-recovery data has practically no influence on the recursive population model.

Figure 3.4 illustrates changes in the estimates of adult and/or juvenile survival in the integrated model, not reflected in their baseline models, to produce derived adult abundance indices that correspond to the observed adult count data. Note that productivity, and consequently juvenile abundance, are much reduced

after 1995, and that since then only three years (1998, 1999 and 2003) have been followed by an increase in adult abundance (Figure 3.4a). These years (along with the imprecise terminal years) are also those in which adult and juvenile survival is greatly increased under the integrated analysis (Figure 3.4d, e). Estimates of Jt and Pt arise from considerably more data, the CES count data, than those of adult and juvenile survival from the ring-recovery data, thus the former are more resistant to change once the component models are integrated. Estimates of Jt and/or Pt in these three years are not sufficient so the integrated model increases the estimates of φa,t and φj,t to account for the greater number of adult birds caught. Therefore, it is the paucity of ring- recovery data that explains the discrepancies between the baseline values and the estimates from the integrated analysis in Figures 3.4d and 3.4e.

Integration also noticeably improves the precision in the estimates of adult and juvenile survival (Figure 3.4d, e). In the baseline φa,t and φj,t are freely estimated from the sparse ring-recovery data, unrestricted by any assumptions relating them to adult abundance. In the integrated case, however, the limited information on φa,t and φj,t from the ring-recovery data is augmented by the population model, given by Equation (3.1), which relates the number of adults in consecutive years to productivity and survival. This extra information improves the precision in the estimates of φa,t and φj,t.

Simulation of a Rich Ring-Recovery Data Set

To make the observed CES adult count data match the underlying population model, given by Equation (3.1), the integrated analysis alters estimates of φa,t and φj,t for which there is limited direct information available. By simulating a large ring-recovery data set, with parameters consistent with the observed NRR data but that provides precise inference on φa,t and φj,t, we illustrate how the other demographic parameters (At,Jt, andPt) are now likewise affected. Cohort recovery totals in the observed Sedge Warbler NRR data range from 0 -37. Selecting recovery totals to range between 1000 -2000, and setting the survival parameters equal to the posterior means from the independent analysis of the Sedge Warbler NRR data, a large ring-recovery data set is simulated. Posterior means of the key demographic parameters (At,Jt, Pt, φa,t,φj,t) from the integrated model which adopts this large simulated ring-recovery data set are shown in Figure 3.5.

Adult Abundance 0.6 1.2 1.8 a Juvenile Abundance 0.4 1.0 1.6 b Productivity 0.5 0.9 1.3 c Adult Survival 0.0 0.5 1.0 d 1990 1995 2000 2005 Juvenile Survival 0.0 0.5 1.0 e

Figure 3.5 Integrated analysis with a large, simulated, ring-recovery data set. Posterior means, and the 95% symmetric credible intervals, from the “baseline” models, denoted by

thin left-hand lines, and the integrated model denoted by bold right-hand lines, for a) At,

b)Jt, c)Pt, d)φa,t, e) φj,t using Sedge Warbler data. For both φa,t and φj,t the simulated

ring-recovery data are used as the baseline.

The baseline estimates of φa,t and φj,t, from the simulated ring-recovery data, are now extremely precise (Figure 3.5d, e). The integrated model now responds to discrepancies between the baseline estimates of the demographic parameters and the underlying population model, Equation (3.1), by adjusting the estimates ofAt,Jt, andPtfor which uncertainty, in comparison, is greater (Figure 3.5a, b, c). For example, the integrated model estimates a higher At in 1988 and 1989, compared to the baseline (Figure 3.5a), in response to the high, but precisely estimated, φa,t and φj,t values in the proceeding years (Figure 3.5d, e).