Hunter et al,: Fecundity, spawning, and maturity of Microstomus pacificus 



113 



these four comparisons; F values ranged from 0.01 to 

 0.14, with the degrees of freedom being 1 and 45 for 

 each comparison. Thus the advanced yolked oocytes in 

 Dover sole are randomly distributed within the ovary, 

 and tissue samples can be taken from any location or 

 lobe without bias. 



Optimal number of tissue samples 



To develop a procedure for estimating the number of 

 tissue samples needed for estimating total fecundity, 

 we first considered the general fecundity model. The 

 true total fecundity (Yp) is the condition where all the 

 advanced yolked oocytes in the ovary are counted, and 

 the relation between female weight (W) and fecundity 

 is defined as 



Yp = f(W) + A 



(Eq. 2) 



where f(W) = a + bW, and A is the error term. The 

 variance of A, o^a. measures the deviation of the data 

 set (Yp.W) to the model f(W). As it was impractical 

 to count all advanced yolked oocytes in the ovary, Yp- 

 is estimated from counts of oocytes in weighed tissue 

 samples, expressed as oocytes per gram of tissue or 

 oocyte density. The precision of a fecundity estimate 

 can be increased by increasing the number of tissue 

 samples taken per female. On the other hand, if the 

 amount of labor for fecundity work is fixed, then in- 

 creasing the number of tissue samples per fish would 

 reduce the number of fish that can be sampled. Thus 

 we needed to know the minimum number of tissue 

 samples necessary to guarantee a goodness-of-fit of the 

 model to the data set. 



We determined the optimum number of tissue 

 samples by minimizing the variance of sample variance 

 of A (o2(s2a))- This procedure led to using the ratio of 

 the variance of oocyte counts between tissue samples 

 within fish (o^e) to the variance around the regression 

 line (o^a), i.e., Q = a'^Ja'^t^. The smaller the 0, the 

 fewer tissue samples are needed. 



Let's denote for the ith fish, i = 1, . . ,n, 

 Wj = fish weight, 

 Ypi = total number of advanced yolked oocytes in the 



ovary, 

 yij = advanced yolked oocyte count in the jth tissue 



sample, j = l,. .,m, 

 Zjj = weight of the jth tissue sample, 

 Zj = formalin wet weight of ovary, 

 m = number of tissue samples from an ovary, 

 Mj = maximum number of tissue samples in an 



ovary, 

 Ypij = estimate of total number of advanced yolked 



oocytes in the ovary from the jth tissue sample 



■^ Z„ 



Ypi = estimated total number of advanced yolked 

 oocytes in the ovary and is used for all analyses 

 in fish fecundity in later sections 





, and 



m 



Ypi = estimate of total fecundity from the regression 

 model. 



We write Ypij as 



Ypij = Ypi + (Ypij - Ypi) 



= f(Wi) + Ai + ei3 (Eq. 3) 



where e^ = Ypjj - Ypi . The estimated total number of 

 advanced yolked oocytes in the ovary is 



m 



I Ypij 

 Ypi = — = f(W) + Ai + e, 



m 



and 



o^ = o2a + 



= f(W) + I 

 ( M-m \ „ 

 m 



(Eq. 4) 



(Eq. 5) 



Thus the variance around the regression line o-^ based 

 upon the data set (Ypi,Wi) is composed of two vari- 

 ance components: one is o^a ^"d the other is o^g- The 

 sample counterparts for o-^ and o^g ^^re s^i and s-g: 



s2, = 



[YF-f(W)]2 

 n-q 



(Eq. 6) 



is the mean square error from a regression analysis on 

 (Ypi, Wi) where q is the number of regression coeffi- 

 cients and n is the number of fish, and 



1 l(YFij-Yp,)2 

 n(m-l) 



(Eq. 7) 



is the within-sample variance (Hunter et al. 1985). The 

 estimate (s^a) of the variance around f(W) when Yp 

 is known (o^a) can be estimated by subtraction: 



