Applications in adaptive cluster sampling of Gulf of Alaska rockfish

Fishery Bulletin, July, 2003 by Dana H. Hanselman, Terrance J. Quinn, II, Chris Lunsford, Jonathan Heifetz, David Clausen

[x.sub.k] = the number of units in the network.

The HT estimator is based on the probability of sampling a network given the initial tows sampled and involves the number of distinct networks sampled (in contrast to the HH estimator which is based only on the initial tows). The HT estimator often outperforms other estimators as seen in simulation studies (Su and Quinn, 2003). Both estimators use the network samples and initial random samples, but not the edge units. This sample size is referred to as v' (convention established by Thompson (1990) and used in Thompson and Seber (1996)). To include edge units into the estimates Thompson and Seber (1996) and Salehi (1999) used the Rao-Blackwell theorem, which is a complex method that could theoretically result in more precise estimates. However, it had little effect for the 1998 survey data (<1% improvement, Hanselman, 2000); therefore these calculations were not used in our study.

When a stopping rule is used, the theoretical basis for the adaptive sampling design changes. It may result in incomplete networks that overlap and are not fixed in relation to a specified criterion---changing with the pattern of the population. In contrast, the nonstopping-rule scheme has disjoint networks that form a unique partition of the population for a specified criterion. This partitioning is the theoretical basis for the unbiasedness of [[micro].sub.HH] and [[micro].sub.HT]. Thus with a stopping rule, some bias may be introduced.

Recent simulation studies (Su and Quinn, 2003) have estimated the bias induced by using a stopping rule on each estimator with order statistics, but not with a fixed criterion. Because the use of a fixed criterion is design unbiased, its estimate should be less biased by the stopping rule than a sample with order statistics. Therefore, we can use the Su-Quinn simulation results to approximate the maximum bias induced by the stopping rule. With a stopping rule of three and the HH estimator, the maximum positive bias is 17% for a highly aggregated simulated population. With a stopping rule of three and the HT estimator, the maximum bias is approximately 12%. Considering our design, we accepted the tradeoff of relatively small bias for gains in precision and logistical efficiency.

Additionally, nonparametric bootstrap methods were adapted from Christman and Pontius (2000) and we used the HH version of the estimates to examine bias from our survey. Five thousand resamples were performed by using n for the SRS bootstrap, and the sample size from the original criterion value of 220 kg/km (v') was used for the ACS bootstrap. Bootstrap distributions of the data were examined for SRS and ACS designs to examine the capability of each design to clearly demonstrate a central tendency.

We evaluated two hypotheses: 1) Adaptive sampling would be more effective in providing precise estimates of POP biomass than would a simple random survey design; and 2)Assessment of POP abundance would benefit more from an adaptive sampling design than would SR-RE because POP are believed to be more clustered in their distribution than SR-RE. SRS estimates were obtained from the initial random tows, and variance estimates were calculated for the initial sample size (n) and for the equivalent sample size that included the adaptive tows but not the edge units (v'). This procedure makes the theoretical comparison fair because each estimate is based on the same number of samples. Total sample size including edge units (v) was not used in the theoretical precision comparison but was considered when efficiency issues were examined later. These hypotheses were assessed by comparing the standard errors (SEs) of ACS to those of SRS. Substantial reductions in SE with ACS for POP would support the first hypothesis, whereas no reductions of SE using ACS for SR-RE would support the second hypothesis. This comparison is qualitative because relevant significance tests are unavailable and the two methods are different in terms of efficiency.