Somerton and Kobayashi: Robustness of Wetherall length-based method to population disequilibna 3j_ 



3 4 5 6 7 8 9 10 11 12 13 14 15 16 

 YEARS SINCE INITIATION OF FISHING 



1 



2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 

 YEARS SINCE RECRUITMENT PULSE 



2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 

 YEARS SINCE INITIATION OF FISHING 



2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 

 YEARS SINCE RECRUITMENT FAILURE 



Figure 2 



Temporal patterns in the statistical power of a chi-square test to detect population disequilibrium at each of four levels of sampling 

 (N = 100, 500, 1000, and 5000). For the fishing-up experiment, power is shown for (a) F = 0.3 and (b) F = 0.6, starting in the third 

 year after the initiation of the fishery and continuing to equilibrium. For the recruitment perturbation experiment at F = 0.6, power 

 is shown for (c) the 1-year doubling of recruitment and (d) the 1-year absence of recruitment starting in the year in which the perturba- 

 tion occurred and continuing to equilibrium. 



13%. For F = 0.6 and a sample size of 5000 fish, the 

 interval of high power is almost identical to that at 

 F = 0.3. At lower sample sizes, however, the chi-square 

 test is considerably more powerful when F = 0.6 than 

 when F = 0.3. 



In the recruitment perturbation experiment, statis- 

 tical power is high over a broader time-interval than 

 in the fishing-up experiment. For the case of a 1-year 

 doubling of recruitment at a sample size of 5000 fish, 

 power is high between years 2 and 8, an interval that 

 includes the entire period in which disequilibrium bias 

 is >5%. For the case of a 1-year absence of recruit- 

 ment, power is high between years 2 and 8, an inter- 

 val that again includes the entire period in which 

 disequilibrium bias is >5%. Unlike the situation in the 

 fishing-up experiments, power tends to remain relative- 

 ly high with reductions in sample size. These experi- 

 ments indicate that, in terms of its ability to detect the 



likelihood of disequilibrium bias, the chi-square test per- 

 forms best on a 1-year doubling of recruitment, second 

 best on a 1-year absence of recruitment, and worst on 

 the fishing-up disequilibrium. The fishing-up case is 

 worst because the test cannot be applied to the first 

 2 years of the time-series when bias is high and larger 

 sample sizes are needed to detect disequilibrium. 



In addition to the type of disequilibrium perturba- 

 tion and size of the length-frequency sample used, 

 the statistical power of the chi-square test also varies 

 with the number of length-distributions included. 

 Thus, power is calculated for tests including two, 

 three, and four length-distributions. Since the gain in 

 power is substantial when the number of length- 

 distributions is increased from two to three but only 

 minor when the number is increased from three to four, 

 all further power simulations are based on three 

 length-distributions. 



