190 Measurement and Analysis of Biodiversity E½F1 EF1 that result are Sobs;k ¼ k X ^ k ¼ À P R1 C k i¼1 Similarly, we have ^gk ¼ max so that 68ị Ri iẳ1 69ị iRi Sobs;k R1 ^g ỵ ^k ^k k C C 71ị EẵSobs EẵF1 ỵ gpị2 EẵC EẵC ( ^0 ^g2 ¼ max N E½Sobs ¼ s À E½C ¼ À s X iÀ1 ð1 À pi Þt Ps pi ị iẳ1 Ps iẳ1 pi Sobs F1 ỵ ^g ^ ^ C C 79ị ^IC ẳ Sobs ỵ F1 ~g S ^ ^ C C ð80Þ ^ increases, the bias must decrease Therefore, we can exAs C pect that the assumptions of fixed pi will lead to SIC underestimating s in most cases The variance of SAC is approximately a linear combination of COV(Fi, Fj) It can be shown that E½Fi VAR Fi ị ẳ EẵFi s 81ị EẵFi E Fj COV Fi ; Fj ẳ s ð82Þ À Á The variance is thus bounded and decreases with increasing richness because Firs Now, as we increase t each of the species will be detected a different number of times with probability (even if they have the same pi) Therefore, there will be at most s nonzero Fi among (Fl,y, Ft) Thus, E[Fi]-0 as t-N As with SAC, the effects of a few dominant pi can be adjusted for using the methods of Chao et al (1993) If we truncate the data set to species with k or fewer representatives The new estimates for E[C] and E[D] that result are t Sobs;k ¼ E[D] and E[C] are functions of s and the pi Therefore, we would like to eliminate the dependence on the pi in order to estimate s We can use Taylor’s theorem to expand about the point (p1,y, ps) to get E½Sobs E½F1 ỵ gpị2 sE EẵC EẵC 78ị If ^g2 is large, the bias can be reduced by using SIC in place ^ to compute ~g Then, using C ~ for C, ^ we have of Sobs/C We define pi as the probability of detection of species i For SIC we assume that Thus, the pi can be regarded as a sequence of random variables with mean p, variance sp, and coefficient of variation g(p) Then, we find that ) ii 1ịFi 1; P tiẳ1 iFi Þ2 i¼1 ^ when ^g2 r 0:5, we get ^ ẳ Sobs/C where N 72ị Detection probabilities, pi, may vary between species The pi remain fixed during sampling 77ị Pt SIC ẳ Chaos Incidence-Based Coverage Estimator (SIC) 76ị iFi 2F2 ^ ẳ t 1ịFP C t 1ị tiẳ1 iFi and 70ị The incidence-based coverage estimator, SIC, applies to data sets that have the quadrat structure outlined for SIM Briefly, when we have the presenceabsence of data then incidencebased estimators apply Lee and Chao (1994) found ways to relax many of the assumptions present in SAM, SIM, SJK, and SAC They follow Pollock (1976) by partitioning variability into between-species, within-species, and temporal components The results involve more complex notation and more complicated proofs but use the same conceptual framework as SAC For brevity, we present only the case of unequal detection probabilities among species The estimator for incidencebased coverage is SIC ¼ ^ ¼ À P F1 C t i¼1 ( ) Pk Sobs;k iẳ1 ii 1ịRi 1; P ^k C kiẳ1 iR1 ị2 SACk ẳ Sobs Sobs;k ỵ 75ị k X 83ị Fi iẳ1 ^ k ẳ P F1 C t iẳ1 73ị Similarly, we have 84ị iFi Using the approximations EẵSobs ESobs 74ị ^gk ( Sobs;k ẳ max ^k C Pk iẳ1 ii 1ịFi Pk iẳ1 iFi ị2 ) À 1; ð85Þ