4 
Fishery Bulletin 1 15(1) 
Table 1 
Definitions for notation used to describe parameters in estimating abundance and confidence intervals for 
the constituent groups that compose the run of steelhead (Oncorhynchus mykiss) in Snake River of the 
Pacific Northwest during spawning year 2011. 
Parameter Definition 
A Number of age groups 
B Number of bootstrap samples 
Cg Window count in stratum s; s=l,2,...S 
F Abundance of female steelhead 
G Number of genetic stocks 
k Number of categories in the compositional variable of interest 
L Lower bound of a confidence interval 
M Abundance of male steelhead 
iVg Number of wild steelhead trapped in stratum s; s=l,2...S 
^ Estimated proportion of wild steelhead in stratum s; s=l,2...S 
Estimated proportion of group i of the variable of interest (A, G,F / M) in stratum s; s=l,2...S 
Tg Number of wild steelhead subsampled in stratum s; s=l,2,...iS 
S Number of time strata 
tg Number of steelhead trapped in stratum s; s=l,2...S 
U Upper bound of a confidence interval 
Wg Estimated abundance of wild steelhead in period s; s=l,2...iS 
the wild compositional data for some strata. We could 
have pooled the compositional data over the season if 
we had assumed that we had sampled a fixed propor- 
tion of the wild fish for each stratum or if we had as- 
sumed that the composition proportions were constant. 
However, if neither condition was true, we had to fo- 
cus on obtaining sufficient samples in each stratum to 
obtain stable estimates of composition proportions by 
stratum 
We defined the proportion of wild females in a stra- 
tum (%s) as P{female\wild, stratum s). Then we had 
(ftpi, ^Mi)) •••> the conditional probabilities 
for wild females and males for strata 1, s. These pro- 
portions were estimated from data obtained from the 
subsample of the trapped fish; for example, for females 
■^Fs = ^Fs /(^Fs + ^Ms)- Given estimates of these prob- 
abilities from wild fish examined in the trap, we used 
the following equation to estimate female abundance: 
'^ = l:li4F.Wi=E?.,«F,P.C„ (4) 
and to estimate wild male abundance 
(5) 
For the pooled estimators, we dropped the summa- 
tion and subscript. Similar estimates were made for 
A (ages, BY2004-BY2008 in this study), G (genetically 
identified stocks), and AxG age groups for each stock. 
That is, the number of wild steelhead in any group is a 
weighted sum of the stratified window counts, in which 
the weights are estimates of the probabilities of being 
wild and being a member of a particular group, includ- 
ing any combinations of the compositional variables. 
To find CIs for these estimates, we had to account 
for the variability of both the trap data and the com- 
positional data. For the asymptotically normal inter- 
val, we used Goodman’s (1960) estimated variance of 
a product: 
2 „2 
„2 „2 
-S' S- 
Ps 
( 6 ) 
yielding 
where s|g = (1 - |-)iiFs G - ^Fs )/(^s - D- 
For the pooled case, we used the following equation: 
2 yu 2 ,->22 
Sp — - S. 
2 „2 
( 8 ) 
where s|^ = (l-:^)TYp(l-'Rp)/(r.-l); and (9) 
TYp = the proportion of wild females from the 
pooled sample. 
Similar formulae for M and estimates of age, stock, 
and ages by stock follow in the next paragraph. 
The bootstrap process described previously for ob- 
taining the Cl for the number of wild fish was extended 
by adding a conditional bootstrap loop based on the 
sex, age, and stock of wild fish in the trap. We defined 
pseudoreplicates parametrically, using 
(F’s*,M*) ~ binomial(rg,(7Yps,'rYMs)| Tg.wild) and (10) 
r * F' * Ml \ 
(■^Fs 
Bootstrap values for the total number of wild females, 
Fj ,...,Fg, were determined with the following equation: 
