FISHERY BULLETIN: VOL, 69, NO. 3 



Graham (1935), Schaefer (1954, 1957), and 

 others when m = 2, and the exponential model 

 discussed by Fox (1970) if the limit is taken 

 as m -> 1. 



This type of production modeling is a stock 

 assessment approach which has extreme math- 

 ematical and data requirement simplicity. 

 Therein lies its primary virtue; for example, 

 equation (2) contains only four parameters 

 whereas the simplest Beverton and Holt (1957) 

 type of model providing the same relation con- 

 tains at least nine parameters. Estimation of 

 the parameters of equation (2) requires only 

 catch and fishing effort data while at the very 

 least, the Beverton and Holt approach addition- 

 ally requires age structure information. Dis- 

 cussion of the different assumptions for imple- 

 menting each approach can be found in Schaefer 

 and Beverton (1963). The generalized produc- 

 tion model provides for a wide variety of shapes 

 for the production curve and thus coupled with 

 its mathematical simplicity represents an im- 

 portant tool for successfully managing exploi- 

 tation. 



Procedures for estimating the parameters of 

 production models can be found in Schaefer 

 (1954, 1957), Ricker (1958), Chapman, Myhre 

 and Southward (1962), Gulland (1969), and 

 Pella and Tomlinson (1969). However, it ap- 

 pears that in all cases, except Schaefer (1957), 

 random variation about the deterministic pre- 

 dictions of the production model has been largely 

 ignored in choosing a statiMical model for esti- 

 mating the parameters. Perhaps this is because 

 of the apparent formidable nature of such var- 

 iation. On the other hand, such variation may 

 often be approximated in a simple manner to 

 allow better estimates of the parameters than 

 if ignored altogether. It is conceded that the 

 generalized production model is at the very best 

 only a good approximation of the actual biologi- 

 cal dynamics, but this should not imply that 

 better parameter estimates are unwarranted, un- 

 less its prime vii'tue of mathematical simplicity 

 is compromised in the course of such action. 



Several statistical models for estimating the 

 parameters of mathematical models of biological 

 I'elationships have been discussed variously by 

 Zar (1968), Glass (1969), Hafley (1969), and 



Pienaar and Thomson (1969). While to the 

 nonstatistician these papers may bear a strong 

 resemblance to quibbling over apparent minor 

 differences of results in the face of large data 

 variability, the improper statistical model can 

 lead to misleading conclusions or to significant 

 errors, as several of the above authors demon- 

 strated. Statistical models differ on the assump- 

 tion about the manner in which variation or 

 error enters the deterministic biological model. 

 The technique employed by Pienaar and Thom- 

 son (1969) to assess fulfillment of the assump- 

 tions about variation is the graphing and ex- 

 amination of i-esiduals, the differences between 

 the observed data and those predicted by the 

 model. Extensive discussion on the examination 

 and analysis of residuals can be found in Ans- 

 combe (1961), Anscombe and Tukey (1963), 

 and Draper and Smith (1966). 



This paper presents a discussion of the nature 

 of simple random variability and its relation to 

 estimating the parameters of the generalized 

 production model. An illustration of residuals 

 examination in selecting the appropriate sta- 

 tistical model for the parameter estimating 

 technique of Pella and Tomlinson (1969) is in- 

 cluded. Data from the fishery for yellowfin tuna, 

 Thunnus albacares, in the eastern tropical Pa- 

 cific Ocean were utilized in the illustration. 



STATISTICAL MODELS 



Schaefer (1957) recognized that the produc- 

 tion model is not deterministic and represented 

 environmentally induced variation as an additive 

 term consisting of a random variable tj multi- 

 plied by population size. In terms of the gen- 

 eralized production model 



dP/dt = KPt — HPt — qfP, + rjP, . (3) 



His parameter-estimating procedure used a finite 

 difference approximation of equation (3) di- 

 vided through by Pt for the case when m = 2. 

 By summing over many time periods the effects 

 of variation are eliminated since the expected 

 value (or mean) of -q is zero. Schaefer's form- 

 ulation of the error term, while rea.sonable and 

 convenient for his estimating technique, pro- 



570 



