184 Measurement and Analysis of Biodiversity number of observed species, Sobs As discussed previously, Sobs is a biased estimator of s To apply the jackknife technique to Sobs we need three assumptions: The capture probabilities may vary across species, The capture probabilities not change during the census, The bias in Sobs decreases at least as fast as 1/t Under these assumptions, Burnham and Overton (1978) propose a generalized jackknife estimator for species richness, simplified by Smith and Van Belle (1984): ! tÀj iÀj k k X k 1X SJk ẳ Sobs Fj 1ịi t iịk 13ị t k! jẳ1 i i¼j i where k gives the order of the estimator SJk does not require that we actually the resampling To better understand the properties of SJk, we consider how the first-order jackknife estimate of Sobs, written SJ1, would be produced Suppose we compute a statistic, D(X1,y, Xt) ¼ Dt from t observations of X Now, we want to consider the effect of each observation in turn by removing it from the data set However, we not want the order in which we the removal to matter Within a given sampling occasion, there may be time of day or other local differences For example, during bird banding operations we may consistently catch flycatchers later in the day than warblers because it may take time for flying prey to become active Then, resampling within a trapping occasion may introduce heterogeneity Thus, we shall conduct jackknife resampling by adding or removing entire quadrats Let us remove the ith quadrat and compute D without this quadrat Call this value D(i) Do this for each i The mean of these D(i), Pt Diị 14ị Dt1 ẳ iẳ1 t gives rise to the first-order jackknife estimate DJ1 ẳ tDt t 1ịDt1 ð15Þ Now, DJ1 will be less biased than Dt if EẵDt ẳ s ỵ N X bi iẳ1 ti ð16Þ where the bi not depend on t If the bias of Dt decreases like 1/tFthat is, b1a0Fthen the bias in DJ1 decreases like 1/t2 Suppose D ¼ Sobs If assumption holds, then we can easily get SJ1 ẳ Sobs ỵ t1 F1 t 17ị The higher order estimates are considerably more complicated but depend on the assumptions in the same way As we increase the order of the jackknife, we get greater bias reduction Unfortunately, we can expect that removing data to reduce bias might increase estimator variance We see from Eq (13) that SJk is a linear combination of the Fi with constant coefficients, ai, k given t and k Burnham and Overton (1978) give the unconditional variance as VARSJk ị ẳ t X iẳ1 a2i;k EẵFi EẵSJk s 18ị Note that as we increase the order from k to k ỵ the coefficient of Fk ỵ becomes nonzero and therefore increases the variance Furthermore, a quick check of the lower order coefficients reveals that they also increase with k Therefore, as we expected, the variance increases with k Thus, we tradeoff bias reduction against increased noise Two methods have been proposed for choosing the order of the jackknife The first method tests the hypothesis that E[SJk SJ,k ỵ 1] ẳ using the test statistic SJ;kỵ1 SJk Ti ẳ q VARSJ;kỵ1 SJk 9Dị 19ị If we can reject this hypothesis for k ¼ then we proceed to test successive values of k until we fail to reject The difficulty with this test is that quadrat information comes in discontinuous jumps so that the order used in estimation may jump around during the course of a census Burnham and Overton (1978) suggest an interpolation technique that smoothes this behavior The power of the jackknife technique is that it requires very little knowledge about the distribution of the data It is thus nonparametric An argument like the one presented previously on species observed tells us that assumption is reasonable for finite collections of species The first two assumptions, however, require biological information For example, suppose we sample a site containing migratory animals Then, when we sample determines, to a large extent the distribution of number of individuals observed If we sample across a productivity gradient then we would not expect the distribution of a given species’ abundance, (X1,1,y, X1,t), to be symmetric That is, the quadrats would not be exchangeable This changes the importance of each quadrat on removal, making some have a deterministically larger effect Consequently, the effective rate of bias reduction may become much smaller than 1/t2 Nonetheless, with attention to census design, such issues can be minimized If in addition we assume that the pi are iid random variables, then Burnham and Overton (1978) show that F is sufficient for the distribution of the capture histories This result does not depend on the resampling protocol This assertion will apply to some of the other estimators we examine Bootstrap Estimator (SB) Bootstrap estimation begins by constructing a surrogate data set by sampling the original data set with replacement We shall bootstrap to remove bias in Sobs, so we shall once again focus on incidence-based data Smith and van Belle (1984) analyzed the bootstrapping process to derive Sobs X Yi t SB ¼ Sobs ỵ 20ị t iẳ1 Just as in the jackknife estimator, we analyze the resampling process to get the estimate that would result if we actually did the resampling We would like to think of resampling as a surrogate for doing multiple replicate