396 
Fishery Bulletin 1 10(4) 
Variance of the estimate of D,: 
Recall that was estimated as 
/ 
ItiDi 
D,=^— 
We expressed the L i in terms of p and made substitu- 
tions to obtain 
Covariance of ID,, pi: 
Recall 
/ i 
i=i 
P=Jj>r 
1=1 
tf.D, ±p,Di i _ 
D t =—, — — =2>,». 
2>, p “ 
where = p ( /p. 
Here, we did not substitute z { for pjp because the set of 
p t was common to both functions. Recall that the covari- 
ance of 2 functions of random variables was 
C ov(g(x),h(x)) = 2^C ov{x it Xj 
i>j 
dg_dh_ 
dx t <)x ] 
This expression rescaled the values of p L so that they 
summed to one. In this case, we observed a given number 
of “samples” of trackline from the patches, and z i was the 
proportion of all the patch trackline that was in patch i. 
This calculation was still a binomial distribution, except, 
in this case, we ignored the background category and 
were concerned only with the patches. 
We applied the delta method to this sum of products 
of random variables: 
V 
2>,'j 
=l(' , h] £ [°-T + r[a>hf) 
+2%E[D l ]E[z J ]Cov(D i ,z J ). 
i*j 
The variance of z i could be obtained with the same 
method as that for the variance of p, with an adjusted 
n L . We substituted sample statistics for expected values 
to obtain the estimated variance of Dp 
I p 2 A -df 
n-I+l 
(■ n-I-l)p 2 
+ d 2 
P 2 >h, 
\ 
y 
In theory, the term for Co w(D i , z) should be a nonzero 
value. For example, consider a case with 2 patches. If 
the proportion in one patch is large, the proportion in 
the other patch is small, and CPUE and patch length 
(the proportion) are correlated, then the CPUE would be 
small in the patch with the small proportion. However, 
as the number of patches becomes much greater than 2, 
the covariance between patches and density decreases 
as 2 --> 0. We assumed this covariance was negligible: 
V 
=l( v h] £ [ D .T+r[a]sp,] 2 ). 
i ' ' 
In our application, g(x) = D x and h(x) = p. 
We applied the delta method: 
Coi >(A» p)=ZZ 
i= 1 ;=1 
f _ I _ \ 
D,p-^p k D 
Cov(Di,p i )^ + Cov(p i ,p j ) 
We used the argument above that Cov (D i , p .) , where i 
* j, can be ignored. This argument leaves only the Cov 
(D l , p •), which, in the sampling design of the TAPAS, 
was expected to be a nonzero value (i.e., the length of a 
given patch is correlated with the CPUE of that patch): 
Cov (D 1 , p)=^Cov [D, , p t 
/ I 
+11 
«=i y=i 
C ov[p i ,p j 
D,p-^p k D k 
) M 
(Pf 
We substituted sample statistics to obtain the covari- 
ance of LD 1; p] : 
C6u[^D 1 ,pJ = 
Ypdv(d t ,p^ 
II 
1 = 1 7=1 
Cov[p ,, 
d,P~Y,P k d k 
I 4=1 
(Pf 
