FOX: RANDOM VARIABILITY AND PARAMETER ESTIMATION 



that the most sensitive estimate is H, followed 

 in succession by the estimates of m, K, q, and r. 

 The estimates of the management implications 

 Cmax and /opt are, for all practical purposes, the 

 same among statistical models, but less similar 

 than among levels of precision. This may be 

 offered as an argument against considering al- 

 ternative statistical models. But consider the 

 plot of the data in Figure 2; one could draw 

 an average line by eye through the data and 

 arrive at estimates of Cmax and /opt just as ac- 

 curate as those estimated by the sophisticated 

 least-squares search technique. The point is that 

 with good data most rational statistical pro- 

 cedures should provide similar estimates of Cmax- 

 and /opt. One cannot be certain that this will 

 be so with data of lesser quality or different 

 range. The values of m which determine the 

 shape of the yield curve, on the other hand, are 

 very different between Models and 3. This 

 could have a significant effect on an economic 

 analysis of the yield curve. 



In the absence of other criteria for choosing 

 a particular statistical model, the "fit" — least 

 sum of squared residuals — is often selected 

 (Glass, 1969), and is perhaps a reasonable cri- 

 terion if the goal is interpolation. The goal here 

 is to obtain the best possible parameter estimates 

 in oi'der to make, in essence, extrapolations or 

 predictions. In the latter case the best criterion 

 is not the "fit", but the degree of assumption 

 fulfillment. Statistical Model 3 provided esti- 

 mates that were least influenced by the addition 

 of error — comparing the parameters' precision 

 between A'^ equalling 1 and 5 — inferring the 

 greatest confidence in its estimates. It was also 

 seen from the simulation study that Model 3 

 best fulfilled the assumptions of a least-squares 

 procedure. Model 3, ironically, "fits" the data 

 the worst, although only by about 6 /r . 



Pienaar and Thomson (1969) have suggested 

 the utilization of an important tool for selecting 

 a statistical model which best fulfills the as- 

 sumptions of the estimating procedure — resid- 

 uals examination. Various plots of the residuals 

 suggested by Draper and Smith (1966) were 

 made for the four statistical models (Figure 3). 

 Each statistical model gives a mean residual 

 near zero fulfilling one of the least-squares as- 



sumptions (Figure 3A). Plots of residuals 

 against time (Figure 3B) indicate: 1) variation 

 increases with time in Model from 1934 

 through 1961, violating the assumption of con- 

 stant residual variance; 2) Model 1 tends to 

 over-correct as there is a propensity for var- 

 iation to decrease from 1940 through 1967; and 

 3) Models 2 and 3 are nearly identical in con- 

 trolling time-oriented variation. Runs — consec- 

 utive residuals of the same sign — are evident 

 in all four models, indicating violation of the 

 assumption of residual independence. There 

 are only ten runs in Model 3 giving a proba- 

 bility less than 0.01 that the arrangement of 

 signs is random (Figure 3B) . Draper and Smith 

 (1966) suggest, however, that unless the ratio 

 of degrees of freedom to number of observations 

 is small (here 29/34) , the effect can be ignored. 

 The dependence of consecutive residuals is un- 

 doubtedly due to vitiation of the assumption of 

 no time lags in the fish population. With changes 

 in fishing effort the age structure of the popu- 

 lation is altered as well. It might be possible 

 to average out these effects by considering a 

 time pei'iod longer than one year, say the aver- 

 age life-span of an individual. That would be 

 about 3 years for a yellowfin tuna, the approx- 

 imate mean length of the runs. However, that 

 would also reduce the number of observations 

 to eleven and the fishing effort, assumed con- 

 stant in integration of the model, would vary 

 considerably. 



An increase in residual variation with deter- 

 ministic catch is obvious for Model (Figure 

 3C), again violating the assumption of constant 

 residual variance. As in the time plot. Model 1 

 tends to over-correct for the phenomenon ex- 

 hibited by Model 0. Models 2 and 3 stabilize 

 the variance as might be expected. In the final 

 plot, residuals against fishing effort, the same 

 conclusions may be reached (Figure 3D) . 



Models 2 and 3 apparently fulfill the assump- 

 tions of the least-squares procedure while Models 

 and 1 violate the assumption of constant re- 

 sidual variance. Invoking the previously men- 

 tioned criterion for choosing between Models 2 

 and 3, the best statistical model for this fishery 

 is Model 3. 



577 



