Pennington et a\ Assessing the precision of frequency distributions from trawl survey samples 



75 



The data for the Naniihian deepwater hake iMt'lurciiis 

 paradoxus) were collected during bottom trawl sui-\-cys off 

 Namibia conducted by the Ministry of Fisheries and Ma- 

 rine Resources of Namibia in conjunctioTi with the Noi-we- 

 gian Agency for Foreign Aid (NORAD). For these sui-v-eys, 

 tows of 30-minute duration were made at stations along 

 transects perpendicular to the coast. ' 



The data for the deepwater hake for South Africa, were 

 collected from during bottom trawl surveys off the west 

 coast of South Africa. The sui-veys were conducted by the 

 Marine and Coastal Management Centre, South Africa, by 

 using a stratified random design. Tows of 30-minute dura- 

 tion were made at each station (see Payne et al., 1985). 



Assessing the precision of length-frequency 

 estimates 



The sample offish of a particular species measured during 

 a survey is not a random sample of individual fish from 

 the entire population but a sample of?! clusters, one cluster 

 from each station. Because fish caught together are usually 

 more similar than those in the general population, a total 

 of M fish collected in 7i clusters will contain less informa- 

 tion about the population length distribution than M fish 

 sampled randomly. One way to measure the information 

 contained in a sample of length measurements is to esti- 

 mate the number of fish that one would need to sample at 

 random (the effective sample size) to obtain the same infor- 

 mation on length contained in the cluster samples. 



The effective sample size for cluster sampling can be 

 defined and calculated as follows (Pennington and Vols- 

 tad, 1994: Folmer and Pennington. 2000). First estimate 

 the population mean fish length and its variance based 

 on the clusters of fish caught at n stations. Because both 

 the lengths and the number of fish at a station are ran- 

 dom variables, a ratio estimator is appropriate (Cochran, 

 1977). The ratio estimator, R. of the mean length is given 



by 



R = ^, 



(1) 



where M, = the number of fish caught (either actual or 

 estimated) at station /; and 

 fi, = an estimate of the average length of fish at 

 station i. 



var( 



«-I 



(Af, Mrit-t, -/?) 

 nUi-1) 



(2) 



^■here M = 



M. In. 



Next estimate the variance, a'~, of the population length 

 distribution. If/?;, fish are randomly selected at each sta- 

 tion (or if all fish are measured), then 



y V<A/, in,nx, I -R) 



(3) 



M-1 



is an estimator of cr;, 



where M - Z A/, is the total number offish caught during 

 the survey; and 

 -v, ^ = the length of the/'' fish at station /. 



For other sampling schemes at a station, first estimate 

 the number offish caught during the survey in each of L 

 length bins, then (e.g. Bhattacharyya and Johnson, 1977) 



Y,f,Sy',-R)' 



(4) 



M-1 



is an estimator of a'~. 



where ff, = the frequency offish in the k''' length bin; and 

 y^ = the bin's midpoint. 



Now if it were possible to sample m fish at random 

 from the population, then the variance of the sample mean 

 would be equal to opm. The effective sample, m,-.-, is de- 

 fined as the number of fish that would need to be sam- 

 pled at random so that the sample mean would have the 

 same precision as an estimate based on a sample of n clus- 

 ters. An estimate of the effective sample size for a particu- 

 lar cluster sample can be derived by substituting the esti- 

 mates from Equation 2 and either Equation 3 or 4 into the 

 equation 



For example, if the catch at a station is divided into strata 

 and a random sample of fish are chosen in each stratum, 

 then for that station /}, would be the stratified estimate 

 of mean length. The estimated variance of R is approxi- 

 mately given by 



0-, 



5 Anonymous. 1999. Survey of fish re.sources of Namibia. 

 Preliminary cruise report 1/99. NORAD-FAOA.INDP Project 

 GLO92/013, 52. Unpubl. manuscr. NatMIRC, PO Box 912, 

 Swakopmund, Namibia. 



= var{R). 



(5) 



The effective sample size is related to Kish's design ef- 

 fect (deff), which for estimating the mean from cluster 

 sampling is defined by (Cochran, 1977) 



deff 



var(j?) 

 al I m 



