Dressel and Norcross: Using poststrafication to improve abundance estimates from multispecies surveys 



473 



of the six sampling years. Unsuitable habitat (nonhabitat 

 area) was denned for each species as ranges of depth 

 and percent sand in which the species was never caught. 

 Within the habitat area, the area of high fish density for 

 each year was defined as ranges of depth and percent 

 sand associated with CPUEs in the 75 th -100 th percentile 

 of nonzero catches in the five other years. The area of low 

 fish density was defined as the remaining habitat area 

 not incorporated in the HFD area. 



In order for the poststratification method to estimate 

 abundance accurately (high precision and low bias), the 

 size of each stratum must be known or closely approxi- 

 mated (Cochran, 1977; Scheaffer et al., 1996). When 

 using habitat variables to determine stratum sizes, the 

 accuracy of stratum sizes defined by the boundaries is 

 heavily dependent upon the number and distribution 

 of habitat variable measurements. For our study, 243 

 depth and percent sand measurements collected over 

 the six years at trawl locations were used to determine 

 stratum boundaries. The ranges of depth and percent 

 sand that defined the four areas for each species were 

 contoured over the study area by using a minimum 

 curvature algorithm (Surfer, 1995). The size of each 

 stratum in relation to the size of the entire study area 

 was then visually estimated to the nearest square ki- 

 lometer. Although not used in our study, a digital rep- 

 resentation of the size of each stratum and the size of 

 the study area is recommended to produce more precise 

 estimates. 



To assess the advantages and disadvantages of using 

 poststratification to estimate abundance, three esti- 

 mates of total abundance were calculated and compared 

 for each species in each year. An unstratified estimate 

 of total abundance was calculated from samples across 

 the entire survey area, with no differentiation with 

 regard to habitat. The unstratified estimate of total 

 abundance was calculated with the standard simple 

 random sampling equation 



The estimate poststratified by habitat was calculated 

 as 



i sl =lN,y r 



where f s , = the estimated population total; 



L = the number of strata (here L=2, habitat and 



nonhabitat); 

 N : = the total number of possible samples in stra- 

 tum ; (samples were standardized to 1000 m 2 , 

 therefore iV, x 1000 m 2 =stratum size); and 

 y i = the mean CPUE in stratum i. 



A third estimate, poststratified by habitat and fish 

 density, was calculated with the same poststratification 

 estimator with L = 3. This poststratification estimator 

 used the HFD area of that year as one stratum, the 

 LFD area of that year as the second stratum, and the 

 nonhabitat area as the third. An approximate variance 

 estimator (Scheaffer et al., 1996), 



V P G M 



, N(N-n)^N, 2 



£IB>- 



was used to estimate the variance of each poststratifica- 

 tion estimator, 



where V = the estimated poststratified variance of i st , 

 the estimated population total; 

 N = the total number of possible samples in the 



survey area; 

 n = the total number of samples taken; 

 N t = the total number of possible samples in 



stratum i; and 

 s ; 2 = the sample variance in stratum i. 



i = Ny, 



where i = the estimated population total; 



N = the total number of possible samples in the 



survey area; and 

 y = the mean CPUE of all sites sampled in a 



year. 



The estimated variance for the unstratified estimator 

 was calculated as 



The first term of the variance equation is the variance of 

 a stratified sample mean under proportional allocation. 

 The second term shows the amount of increase in vari- 

 ance expected from post- rather than prestratification 

 (Scheaffer et al., 1996). 



Relative efficiency statistics were calculated for pair- 

 wise comparisons of the precision of the unstratified 

 and the two poststratified estimates. Pairwise com- 

 parisons of the estimates were made for each species in 

 each year. Relative efficiency was calculated as 



V(i) = N 



2 s 



2\ 



m 



R.E. 



Va 



where V(f ) = the estimated variance of the population 

 total estimate; 

 N = the total number of possible samples in the 



survey area; 

 n = the total number of samples taken; and 

 s 2 = the sample variance. 



where V^ represents the variance of an unstratified 

 estimate or a stratified sample with fewer strata than 

 the estimate of variance represented by V B . 



The variance of an estimate is directly affected by the 

 sample size (Zar, 1996). In our study, three total abun- 

 dance estimates and their respective variances were 



