Year: 2015
Type: erratum
Abstract: Methods in Ecology and EvolutionVolume 6, Issue 9 p. 1067-1068 CorrigendumFree Access Corrigendum This article corrects the following: An examination of index-calibration experiments: counting tigers at macroecological scales Arjun M. Gopalaswamy, Mohan Delampady, K. Ullas Karanth, N. Samba Kumar, David W. Macdonald, Nigel Yoccoz, Volume 6Issue 9Methods in Ecology and Evolution pages: 1055-1066 First Published online: February 23, 2015 First published: 23 June 2015 https://doi.org/10.1111/2041-210X.12400Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Our recent paper (Gopalaswamy et al. 2015) concluded that, largely due to uncertainties in detection probabilities of elusive wildlife under field conditions, there are greater risks of inaccuracy in -based index-calibration methods than has previously been realised. We have discovered two errors in the detail of the algebra we presented, and we report here how their correction further strengthens our general conclusion regarding the risks of inaccuracy inherent in this method. We are grateful to the scientific colleagues who encouraged us to explore the problematic derivation and thereby led us to these improvements. To be specific, we refer to the derivation of the lower bound of for Model 2 (when is a variable; when k→∞) mentioned in the Appendix 1–5. The term V1/V2 that appears in the denominator in statement 5, page 3 of Appendix 1–5, is incorrect. This should instead be V2/V1. And, in the application of the Taylor's ‘first-order’ approximation method on page 4 of Appendix 1–5, the fourth statement has an incorrect a in the denominator. This should have been b. Consequences of these corrections Theoretical The lower and upper bounds for for Model 2, in the case where is a variable, will now read as below. Prior to the correction of this error, we incorrectly implied that the lower bound formula for (when is a variable) was a piecewise function. This is not the case, and we show instead that it is a monotonically increasing function up to a limit defined by the lower bound formula (e.g. when a = b→∞). We additionally caution readers about the computation of the lower bound of using the above formula when a = b < 10. The derivation of the lower bound is based on an application of the Taylor's ‘first-order’ approximation method, and hence, the Jensen's inequality (Bolker 2008) for a strictly convex function becomes relevant. The function in our case, g(x) = −ln(1−x), 0 < x < 1, is a strictly convex function because , so . Since p∼Beta(a, b), then E(p) = a/(a + b). Jensen's inequality then says . If , and we know that , we have The inequality will then suggest that the approximate lower bound will lie below the actual lower bound. However, Jensen's inequality is not useful for us to assess approximations of variance, . So the net approximation, relative to the true lower bound, is unclear to us. However, the monotonic nature of the revised figure (Fig. 1) suggests that deviations from the true lower bound may not be large enough to mislead inference. We also remind readers that we have, in the first place, already made an approximation that . Figure 1Open in figure viewerPowerPoint Revised Figure 4. in Gopalaswamy et al. (2015) Plot of lower bound values when, a and b, V, Q and μ are varied using Model 2: = variable. Note, here μ is represented as three categories - μ<33 is represented by an empty square, 33<μ<66 is represented by a crossed-out square, and 66<μ<100 is represented by a dark square. Application Following these corrections, the lower bound estimate of predicted for IC-Jhala (in Table 3) should be 0·182 (0.0124). And the lower bound estimate of predicted for IC-Karanth (in Table 3) should now read as 0·057 (0·004). This result means that the estimated from IC-Karanth is now anomalously low when compared to the lower bound estimate of the predicted for IC-Karanth (Z score = −14·15, ). The results, following these corrections, strengthen our view that the assumed coefficient of variation of 1 was not sufficient enough to describe heterogeneity in detection rates that occurred in the IC-Karanth experiment. This also suggests that our conservative approach of mapping the site-specific detection probability p to the individual-specific detection probability may have led to an overestimation of . This could have resulted in overestimation of predicted values in Table 3. The large deviations of the estimates in the two IC experiments from the theoretical estimates is also indicative of a high degree of sampling variation (Thompson 2002). There are two additional sources of sampling variation that may be relevant here. The first relates to the chance variation that occurs during the selection of sites for conducting IC experiments. One selection of a set of sites for the conduct of an IC experiment could yield an statistic that is different from another selection when detection probability varies across sites. The second relates to the sampling variation that can occur at individual sites. For example, each trail segment within a site can yield a different sign count based on trail-segment level detection probabilities. A detailed exploration of both these sources of sampling variation in the two tiger IC experiments will perhaps provide further insights regarding variances about the component parts. Finally, and as a minor point, the specific citation ‘Karanth and Kumar (2005)’, should be replaced by the more inclusive reference ‘IC-Karanth’ in the figure legend of Fig 5 in the original article. We regret not having numerically tested our formula ( is a variable case), at very large values of k initially. Such an exercise would have provided a trend showing a convergence to some bound that would have been quite different to the faulty lower bound previously arrived at. Our original omission highlights the advantage of rigorous numerical testing at every step along the way when ecologists undertake mathematically intensive work. However, the new results presented in this corrigendum do not detract from the general conclusions reached, and principles illustrated, in our original paper. On the contrary, the cleaner results strengthen our view that serious risks are inherent in -based index-calibration experiments. Acknowledgements We thank Murray Efford for his blog on the MEE website, which encouraged us to take another closer look at the derivations of the lower bound graph for the is a variable model. References Bolker, B.M. (2008) Ecological Models and Data in R. Princeton University Press, Princeton, NJ, USA. CrossrefGoogle Scholar Gopalaswamy, A.M., Delampady, M., Karanth, K.U., Kumar, N.S. & Macdonald, D.W. (2015) An examination of index-calibration experiments: counting tigers at macroecological scales. Methods in Ecology and Evolution, DOI: 10.1111/2041-210X.12351. Wiley Online LibraryWeb of Science®Google Scholar Thompson, S. K. (2002) Sampling. Wiley, New York, NY, USA. Google Scholar Citing Literature Volume6, Issue9September 2015Pages 1067-1068 FiguresReferencesRelatedInformation
Source: Methods in Ecology and Evolution
Language: English
Subfield: Ecology
Field: Environmental Science
Domain: Physical Sciences
SDG: Life on land
Open Access status: bronze
OpenAlex ID: W4252218868