Fifty-year trends in UK hunting bags of birds and mammals, and calibrated estimation of national bag size, using GWCT’s National Gamebag Census

Abstract

The Game and Wildlife Conservation Trust’s National Gamebag Census (NGC) has been collecting voluntary bag returns from shoots across the UK since 1961. Methods similar to the ones used for bird census data are applied to NGC data to derive annual bag indices for the UK, assess temporal trends and evaluate changes in bags over 50, 25 and 12 years for 30 bird species and 15 mammal species, as well as for numbers released of four bird species. Total UK bags and numbers released in the 2004 and 2012 seasons are obtained by splitting up aggregate bags from two independent surveys (PACEC 2006, 2014) in relation to their NGC species composition. These are used to calibrate NGC species-specific bag indices and obtain estimates of total UK bags and numbers released for the 2016 season. Over 50 years, large changes have taken place in the bags and numbers released of some species whereas bags of others have remained approximately constant. This work contributes to pan-European efforts seeking a rational assessment of hunting effects within a policy combining conservation and the sustainable use of wildlife, in line with national and international legislation.

Appendix

Calibrating NGC bag indices against total UK bags—statistical derivation

Maximum likelihood estimation (Cox and Hinkley 1974) is used to derive a formula for a calibration coefficient, assumed constant, that scales up an NGC index value to an estimate of total bag. Given that NGC index values are derived from a linear predictor on the logarithmic scale, the same scale is used for the statistical calculations.

Let n be the number of calibration points for which there exist independent estimates ${i}_j\sim N\left({\mu}_j,{\sigma}_j^2\right)$ of the NGC bag index and ${b}_j\sim N\left({\beta}_j,{\tau}_j^2\right)$ of the total bag, both on the logarithmic scale, with 1 ≤ j ≤ n. The problem is to determine a constant k such that μ_j + k = β_j, or equivalently μ_j + k − β_j = 0, 1 ≤ j ≤ n, by maximising the likelihood $\mathrm{L}\left(k\right|{i}_j,{b}_j\Big)={\prod}_{j=1}^nP\left({i}_j+k-{b}_j=0\right)$.

Using the expectation and variance of i_j + k − b_j, the likelihood and log-likelihood follow:

$$ \mathrm{E}\left({i}_j+k-{b}_j\right)=\mathrm{E}\left({i}_j\right)+\mathrm{E}(k)-\mathrm{E}\left({b}_j\right)={\mu}_j+k-{\beta}_j=0 $$

$$ \mathrm{Var}\left({i}_j+k-{b}_j\right)=\mathrm{Var}\left({i}_j\right)+\mathrm{Var}(k)+\mathrm{Var}\left({b}_j\right)={\sigma}_j^2+0+{\tau}_j^2={\delta}_j^2,\mathrm{where}\ {\delta}_j^2={\sigma}_j^2+{\tau}_j^2. $$

$$ \mathrm{L}\left(k\right|\left\{\left({i}_j,{b}_j\right)\right\},j=1\dots n\Big)=\prod \limits_{j=1}^n\left[\frac{1}{\sqrt{2\pi {\delta}_j^2}}{e}^{-\frac{{\left({i}_j+k-{b}_j\right)}^2}{2{\delta}_j^2}}\right] $$

$$ \ln\ \mathrm{L}(k)=-\frac{1}{2}\sum \limits_{j=1}^n\ln \left(2\pi {\delta}_j^2\right)-\sum \limits_{j=1}^n\frac{{\left({i}_j+k-{b}_j\right)}^2}{2{\delta}_j^2} $$

The log-likelihood lnL is maximised when its first derivative is 0:

$$ \frac{\partial \ln \mathrm{L}}{\partial k}=-\sum \limits_{j=1}^n\frac{i_j+k-{b}_j}{\delta_j^2}=-\sum \limits_{j=1}^n{w}_j\left({i}_j+k-{b}_j\right)=0,\mathrm{where}\ {w}_j=\frac{1}{\delta_j^2} $$

$$ \hat{k}=\frac{1}{\sum_{j=1}^n{w}_j}\sum \limits_{j=1}^n{w}_j\left({b}_j-{i}_j\right) $$

$$ \mathrm{Var}\left(\hat{k}\right)={\left[\frac{\partial^2\ln \mathrm{L}}{\partial {k}^2}\right]}^{-1}=\frac{1}{\sum_{j=1}^n{w}_j} $$

Hence the estimate $\hat{k}$ of k may be seen as a weighted average of the differences between the total bag and the NGC bag index at the n calibration points, with weights equal to the inverse of the variances of the differences. The variance of $\hat{k}$ is the inverse of the sum of the weights and $\mathrm{SE}\left(\hat{k}\right)=\sqrt{\mathrm{Var}\left(\hat{k}\right)}$.

A goodness-of-fit test is given by the deviance D, approximately distributed as χ² with n − 1 degrees of freedom, calculated as twice the difference between the log-likelihood of the saturated model and ln L($\hat{k}$):

$$ D=\sum \limits_{j=1}^n\frac{{\left(\Big({i}_j+\hat{k}-{b}_j\right)}^2}{\delta_j^2} $$

Given an NGC bag index value i₀ (on the logarithmic scale) that is not one of the calibration points, the total bag b₀ (also on the logarithmic scale) is estimated as ${\hat{b}}_0={i}_0+\hat{k}$, and $\mathrm{SE}\left({\hat{b}}_0\right)=\sqrt{\mathrm{SE}{\left({i}_0\right)}^2+\mathrm{SE}{\left(\hat{k}\right)}^2}$.