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The far greater amount of adult survival information in the live-recapture likeli- hood,Llive, makes it appealing to further multiply this into the joint likelihood. However, as Llive is drawn from the same data as the adult abundance likeli-

hood, La, these likelihoods are not independent. Indeed, birds contribute data to both Llive and La on every between-year encounter. To accommodate this lack of independence we initially split the data into two groups (one of 35 sites and the other of 36 sites); one is then used to derive Llive and the other inde- pendent set to form La. To ensure both split data sets provide good coverage over the duration of the study (which covered few sites in the early years), sites are initially stratified according to when they first joined the CES scheme, and within each of these strata half the sites are randomly assigned to each group. Secondly, we ignore the issue of non-independence and derive La and Llive from the full 71-site data. In both cases the likelihood for the ring-recovery data, Ldead, is also incorporated to provide more direct information onφj,t so that:

Ljoint =La×Lj×Ldead×Llive. (3.7) The juvenile abundance likelihood, Lj, which is independent ofLa and Llive, is derived from the full 71-site data in each case.

The posterior means of key parameters, At, Jt, Pt, φa,t, and φj,t, from the full and split data integrated analyses are compared to each other, and their baseline estimates, in Figure 3.6. As there are considerably more data on adult survival from the live-recaptures than the limited ring-recoveries, independent estimates of φa,t via Llive are used as the baseline.

Initially ignoring the non-independence between Llive and La, and concentrat- ing on the full 71-site integrated analysis, we note that estimates of At, Jt, Pt, and φa,t are all very similar to their baseline estimates. Conversely there have been large changes to φj,t. This is to be expected, since we have limited direct information on φj,t, and substantially more on the other demographic parame- ters; the integrated model thus alters φj,t to make the derived At correspond to the adult count data. For example, in Figure 3.4a the integrated model (with- out Llive) has explained the increase in adult abundance from 1998, 1999 and 2003 by increasing both adult and juvenile survival (Figure 3.4d, e), for which little is otherwise known. Now, because the baseline likelihood Llive is also a part of the integrated model, bringing in substantially more information about adult survival, φa,t is not adjusted (Figure 3.6d); rather the integrated model responds to account for the increase in adult abundance by just increasing φj,t in these years (Figure 3.6e).

Adult Abundance 0.4 0.8 1.2 a Juvenile Abundance 0.4 0.8 1.2 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 3.6 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 and CES data in the estimation of adult survival. The analysis in which all 71 sites

provide information toLlive and La, is represented by bold right-hand lines. The bold grey

centre line represents the split data analysis. a)At, b)Jt, c)Pt, d)φa,t, e)φj,t.

The integrated approach also tends to result in an improved precision from the sharing of information between the component models. This is most noticeable for juvenile survival (Figure 3.6e). Here the very limited direct information on φj,t from the ring-recovery data is augmented in the integrated model by the information from the population model, Equation (3.1).

The full and split data integrated analyses produce consistent results (Figure 3.6). Using the CES data as two independent groups of 36 sites and 35 sites, we find At, Pt and φa,t, resemble their baseline, though slightly less so than previously when the full data were used (Figure 3.6a, c, d). As fewer data are now used to formLlive and La this is to be expected. For φj,t both the full and split data integrated analyses estimate trends more similar to each other than either is to the imprecise baseline (Figure 3.6e). As the full data are used to deriveLj in both the integrated models Jt is virtually unchanged (Figure 3.6b).

For the integrated models, the 95% symmetric credible intervals for At, Pt, φa,t, and φj,t from the split data analysis are wider, but not dramatically so, than those from the full data analysis (Figure 3.6a, c, d, e). An increase in uncertainty is to be expected, as when the data are split the parameters are being estimated from less information.

The proportion of transients τ, a parameter of Llive, also requires estimation under the fully integrated model. The posterior mean (standard deviation) ofτ from the full 71-site data analysis is 0.371 (0.0134), and the split data analysis is 0.359 (0.0200), with the 95% symmetric credible intervals overlapping. As expected, the estimate is more precise under the full data analysis due to the greater amount of data.

Interestingly the integrated model has estimated φj,t to be greater thanφa,t in several years - 1999, 2003 and 2004 (Figure 3.6d, e). This could be the result of permanent emigration causing negative bias in the estimates of φa,t derived from the CES live-recapture data. We explore this further in Chapter 5. A nice feature of the integrated model is that it produces estimates of “true seasonal” productivity (Ps

t), the number of fledged juveniles per breeding pair per year, through the combination of k and Pt (see Equation 2.4). These es- timates are consistent with published analyses of Nest Record Scheme (NRS) data. Assuming an equal proportion of female and male birds, the integrated model, using the full 71-site data, estimates on average 3.0 juveniles per pair per year. The split data analysis estimates, on average, 2.9 juveniles per pair per year. Nest failure probabilities, the proportion of nests failing completely due to predation or desertion for example, are approximately 0.015 per day at the egg stage and 0.018 per day at the chick stage for Sedge Warblers (Baillie et al., 2009b). As the duration of laying/incubation and fledging are typically 17-19 and 13 -14 days respectively (Cramp, 1992), this gives a proportion of successful nests of about 0.60. With an average clutch size of 5 eggs (Robinson, 2005), and typically 1 brood per year (Cramp, 1992), this corresponds to 3.0 juveniles per pair per year. Though this figure ignores the (unknown) number of second broods, the losses of individual chicks up to (and shortly after) fledg- ing in otherwise successful nests and the uncertainty in the estimates, the close agreement with the integrated model is reassuring.

Sensitivity to the Split

To investigate the sensitivity to the split of the data used to create separate data sets for Llive and La, the split data integrated analysis is repeated with the data sets used to form Llive, or La, switched. Posterior means of Jt, φa,t, φj,t and the derived At and Pt parameters from these two split data integrated analyses are compared in Figure 3.7.

Adult Abundance 0.4 0.8 1.2 a Juvenile Abundance 0.4 0.8 1.2 b Productivity 0.4 0.8 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 3.7 Posterior means, and the 95% symmetric credible intervals, from the split data

integrated analyses where the data is split into two groups; one is then used to deriveLlive

and the other independent set to formLa, (grey left-hand lines) and then the data sets are

switched (black right-hand lines), for a)At, b)Jt, c)Pt, d)φa,t, e)φj,t.

The estimated indices, At,Jt, and Pt, are similar between both split data inte- grated analyses (Figure 3.7a, b, c) as are the estimated survival probabilities, φa,t and φj,t (Figure 3.7d, e). Further, switching the data sets also produces similar estimates of τ (posterior mean = 0.383, standard deviation = 0.0173) and seasonal productivity (on average 3.2 young per pair per year).

The main discrepancy between the two split data integrated analyses occurs in 2002; under the first split analysis an increase in φj,t and a decrease in φa,t

is estimated, whereas the converse is the case when the data sets are switched (Figure 3.7d, e). However, the majority of the differences are minor, and the integrated analysis seems relatively insensitive to the split.

Interestingly the 95% symmetric credible intervals of At and Pt from the first split analysis are substantially narrower than those when the data sets are switched (Figure 3.7a, c). This is because the indices are relative to a reference year (in this case 1987) and site. The two split analyses were required to have a different reference site, and this change affects the posterior variation ofβa _and consequently the precision of Atand Pt. Note that posterior variance ofβawas lower in the first split analysis (0.0100) than in the second (0.0783) resulting in the observed increase in precision.

To further verify the insensitivity of the split, the analysis was repeated with multiple other random splits, albeit all of 36 and 35 sites, respectively. Analo- gous results were obtained. In Figure 3.8 the results from two of these further random splits, both with the data sets used to form Llive, or La, also switched, are presented. There is general agreement in the estimated annual trends, but precision is highly affected by the split. This is particularly true for At and Pt (Figure 3.8a, c) as a result of changes in the reference site used. A random split (prior to analysis) will not give biased or misleading results, but as clearly demonstrated in Figure 3.8, it can be sub-optimal in terms of precision.

The Importance of Ring-Recovery Data

Despite ring-recovery data being limited, their presence is crucial to the in- tegrated analysis. In the absence of ring-recovery data, or informative prior information, φj,t would be estimated entirely indirectly via the joint likelihood and its role in the deterministic population model, Equation (3.1), as is the other “freely” estimated parameter, the productivity constant k. In such cir- cumstances φj,t and k are confounded, and we obtain a parameter redundant model. Although their product, kφj,t ∈[0,∞], is estimable integration becomes less meaningful - the baseline estimates forAt,Jt,Ptandφa,tare simply returned with kφj,t taking the values required to match the population model, Equation (3.1), to the observed data. This is readily verified by removing Ldead from the integrated analysis, that is settingLjoint =La×Lj×Llive. The estimable quan- tity kφj,t, however, does provide an index for juvenile survival which can still be used for detecting whether juvenile survival rates are decreasing for example, a useful quantity in the absence of any direct information pertaining to φj,t.

Adult Abundance 0.5 1.0 1.5 a Juvenile Abundance 0.4 0.8 1.2 b Productivity 0.4 0.8 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 3.8 Posterior means, and the 95% symmetric credible intervals, from two further split data integrated analyses where the data is split into two groups; one is then used to

derive Llive and the other independent set to form La. On the left, (grey lines) are the

results from the first additional analysis, giving the initial split and the reversal of the split respectively. On the right, (black lines) are the results from the second additional analysis.

The solid line denotes the posterior means from the baseline model. Forφa,tthe CES data

is adopted as the baseline. a)At, b) Jt, c)Pt, d) φa,t, e)φj,t.

Estimates ofkφj,tfrom the integrated analysis with, and without,Ldeadincluded are not exactly the same (Figure 3.9) demonstrating that even the sparse ring- recoveries have some limited influence on the population model (Equation (3.1)). In addition, an increase in precision is generally realised by including the ring- recovery data, albeit sparse.

In an analogous situation to the absence of direct information on φj,t, if k were allowed to be time-dependent in the population model, given by Equation (3.1), the baseline estimates ofAt,Jt, Pt, φa,t and φj,t would be returned, with kt taking the values required to match the population model to the observed data. However, time-invariance in k can be assumed under the constant effort protocol that underpins the CES scheme and the estimation of abundance and productivity indices from the resulting count data.

1990 1995 2000 2005 k φj 0.5 1.0 1.5

Figure 3.9 Posterior means, and the 95% symmetric credible intervals ofkφj,t, from the in-

tegrated analysis with (bold right-hand lines), and without (thin left-hand lines) ring-recovery

data. In both cases all 71-sites contribute to Llive andLa.