446 



Fishery Bulletin 100(3) 



year there is no annual variance component. The variance 

 of the diet composition for a prey group for a single season 

 is the sum of three components corresponding to the three 

 stages: estimation of biomass proportion <cy^-), sampling 

 of occasions (a,^), and sampling of years (oj-^). Following 

 Cochran ( 1977), the estimators for each component are 



where the elements in Z and P for the /''' prey group are 

 partitioned into ii ^ and 



, We also used the delta- 

 method to obtain approximate variances and covariances 

 in I of the biomass estimates for a single occasion 



y. T., 



J^J^VariU,,^,) 



• 2 _ .v=l (=1 



i'-. 



Y, T„ 



or = 



' I v=l 1 = 1 



v=l _v=l 



i-yE"^.v-'^.'' 



7,(7,-1) 



We used the delta-method (Taylor series approximation) 

 (Seber 1973) to estimate the variance of the estimated 

 proportion for occasion / which expressed in matrix nota- 

 tion is 



yar^<^^,j,,) = Varf, 



■p-^p. 



where /3 is a vector of partial-derivatives of ;r,,^,, with 

 respect to 6^,^, for r=l, w and 1 is the w x w variance-cova- 

 riance matrix of 6^„, for ;'=l,(o. Likewise, the variance of p 

 was estimated by using 



Va7-(p„ ) 



+ o;. 



y< T,. 



SX^°'-.'p.v,) 



where 



■2 v=l (=1 



and 



Vaii{p,j^.,) = Var^ 



I^. 



t^L 



^prp. 



The first term in the variance is the uncertainty re- 

 sulting from the variability in the estimate of the mean 

 weight. For salmonid species, the second term measures 

 the uncertainty resulting from apportioning the number of 

 unknown salmonids into prey groups (i.e. estimating '!,,,,,). 

 For nonsalmonid species, all hard parts were classified 

 into less specific prey groups; therefore a„ = 0. The group- 

 specific estimates of biomass within season are uncorre- 

 lated except for the estimates for salmonid species (iefO 

 which are correlated because of the apportionment of the 

 items which were identified as salmonid but could not be 

 identified to species nor to the juvenile and adult sizes. For 

 salmonid groups r and vff , the estimated covariance is 



Coi'(6,„,,6.,,,,, ) = Coi'(;v,,,,,n,j,„ )uv,u'>,- 



The total amount of biomass represented by the scats is 

 the sum across all prey groups: 



^v<=X^'^-VM 



and its variance was estimated as 



Varib 



jyt ' 



uarib,,^., ) +2 



•jyi ' 



rsw v>rvEii/ 



COi'ffeavMV- 



The variance of n , was estimated assuming a binomial 

 distribution: 



Varin,^ 



,2 r„(i-x„^ 



and the covariance between salmonid groups was esti- 

 mated as . . 



"(Vf 



O,, 



C0l'( «„„,/"!,, 



We assumed that the weights of the prey consumed were 

 independent samples from a group- and season-specific 

 distribution with mean ^,,. and variance a,-'. The param- 

 eters ^,, and a^j were estimated with the predicted weights 

 from the sample of o* measurable hard parts: 



^^.J = '^'y 



-^ and a': = 



°;-i 



To incorporate uncertainty in regression parameters and 

 prediction error for the measured hard parts and variabil- 

 ity in mass for the unmeasured prey (n - o* ), the vari- 

 ance of ^ should be estimated by 



