Hedgerow analysis: trend estimation
To evaluate the effect of loss of hedgerow on British farmland bird populations, we constructed a composite time series of hedgerow length from two main types of data available since the 1950s, namely: (i) point estimates of hedgerow lengths in England or Britain (England and Wales), resulting from field censuses (Robinson & Sutherland 2002); and (ii) estimates of hedgerow rates of changes from repeated aerial photograph surveys or from field based surveys. These hedgerow length or hedgerow change estimates rely on different protocols, refer to different geographical units, and resulting estimates were reported either with or without associated errors. When both length and change in length estimates were produced, as in the Institute of Terrestrial Ecology surveys (Barr et al. 2003; Petit et al. 2003), we chose to use the change in length estimates, because these have a greater precision (see Tables 1 and 2).
Table 2. Hedgerow rate of change data Start year  End year  Change (×1000 km)  se  Source  Comments 


1990  1993  −54·6  9·6  Barr, Gillespie & Howard (1994)  England & Wales 
1984  1998  −123  13·4  CS2000 website^{1}  England & Wales 
1990  1998  0  4·8  CS2000 website^{1}  England & Wales 
1946  1963  −137  NA  Pollard, Hooper & Moore (1974, p. 61)  England & Wales 
1946  1954  −34  NA  Pollard, Hooper & Moore (1974, p. 64, table 2)  Eastern England^{2} 
1954  1962  −103  NA  Pollard, Hooper & Moore (1974, p. 64, table 2)  Eastern England^{2} 
1962  1966  −150  NA  Pollard, Hooper & Moore (1974, p. 64, table 2)  Eastern England^{2} 
1966  1970  −85  NA  Pollard, Hooper & Moore (1974, p. 64, table 2)  Eastern England^{2} 
Assuming that temporal changes had been fairly smooth at the national level, we used a Bayesian additive, statespace model to estimate yearspecific values of hedgerow availability over the period 1960–2000. This required an observation equation, to relate modelled values or differences in values to the available estimates, and a system equation, which in our case enforced smoothness on the modelled values.
Our model was informed by the likelihood associated with the two types of data (direct hedgerow estimates and rate of change estimates), as follows.
For data on changes in hedge length, we assumed the observation process as:
 (eqn 2)
where C_{jk} is the change in length between years j and k, is the associated variance of the change estimate, either taken from the source publication when provided, or given an informative prior.
For the system process of the statespace model, the modelled value, H_{i} in year i, was smoothed according to the second order difference, Diff2, of the time series of modelled values, defined as follows:
 (eqn 3)
The choice of using the second over the firstorder difference was based on the assumption that not only should hedgerow length change smoothly over time, but the pace and direction of the change should also vary smoothly over time. This is because national trends in hedgerow removal or addition is a slow process that is driven either by the gradual mutation of farming systems and farmers’ culture or by political orientations. We achieved this by the random effect distributional assumption on Diff2:
 (eqn 4)
which has been implemented computationally by specifying
 (eqn 5)
with
 (eqn 6)
The amount of smoothing was controlled by the variance parameter, sd_{D}^{2}, which was largely informed by the data, having been given a vague halfnormal prior (Gelman 2006): HalfNormal(0, 50^{2}).
Some of the data on hedgerow length or on change in hedgerow length were provided by the source publication without an associated standard error. Rather than considering these data as known without error, we chose to use an informative prior on the standard deviation of the observation process, formed as follows, for data on hedge length (eqn 7) and for changes in hedge length (eqn 8):
 (eqn 7)
 (eqn 8)
where x_{i} and x_{jk} are drawn from a beta(2, 4) distribution. se_min_{C} and se_min_{H} are the minimum standard error values we are prepared to accept for data on hedgerow change and for data on absolute length, when they are unknown. It is highly unlikely that the precision of the data without reported errors could be greater than any of the data from the Institute for Terrestrial Ecology, the latter being the most rigorous and intensive surveys conducted. The minimum value of the standard error was thus set at 14 and 30 respectively, which in each case is the largest standard error reported from standardized surveys (respectively, aerial photographs analysis and the UK’s Institute of Terrestrial Ecology’s countryside surveys in Pollard, Hooper & Moore 1974). The minimum standard error is then expanded by a factor (1 + 3*x), which varies from 1 to 4, with a mode around 1·75 and a mean around 2, to allow for larger but reasonable uncertainty about the estimates. As no information is available on the shape and upper bound of the standard error distributions, these choices have to be based on what is believed to be sensible, with an infinite number of subtle variations. Alternatively, one could also use the distribution of the stated standard errors to form a prior for the missing standard deviations. However in our case, we think that the latter data may not be representative of the former. To assess the sensitivity of the results to the specification of our informative priors, we carried out a sensitivity analysis on the prior by repeating the analysis with an expansion factor of either (1 + 1*x_{i}) or (1 + 7*x_{i}), equivalent to decreasing (expansion factor between 1 and 2, mean = 1·33) and increasing (expansion factor between 1 and 8, mean = 3·33) the value of the standard deviation of the data in the observation process. The latter case is probably too large, as a survey as imprecise as having a standard error near 100 000 km (30*3·33) would not be very useful to carry out. For comparison, we also ran a model without the data with unknown error.
Given the heterogeneous nature of the data, some approximations were required: (i) data from ‘England and Wales’ and from ‘Mainland Britain’ have been assumed to refer to the same geographical unit, as the remainder, Scotland, holds a very small proportion of mainland Britain’s hedgerows despite accounting for 36% of the landmass (HainesYoung et al. 2000); and (ii) data on changes in length from plots in England were assumed to represent the rates of change elsewhere in mainland Britain, and have been rescaled accordingly (Table 2).
The model was fitted with OpenBUGS 2.2.0 called from R’s BRugs library (Lunn et al. 2009). The code and data are given in Appendix S1 (Supporting Information), using the same variable names as in the methods description (eqns 1–8). We used 1 000 000 updates, of which one in 100 was kept to reduce memory storage needs, after a burnin set of 6000 updates. Computation time for the hedgerow model alone (without bird data) was 240 s on a 2·4 GHz, P9400 Intel Core Duo CPU.