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



309 



ing recruitment, the initial population length-frequency 

 distribution is set at the equilibrium distribution with 

 a fishery producing F = 0.6. Two types of recruitment 

 perturbations are tested: (1) a year with twice the nor- 

 mal recruitment, and (2) a year with a complete absence 

 of recruitment. After each recruitment perturbation 

 is introduced, the simulation is run until the popula- 

 tion is again in equilibrium. Since disequilibrium bias 

 may be only one component of the total bias of a 

 parameter, the disequilibrium bias is isolated by ex- 

 pressing it as a percentage difference relative to the 

 equilibrium value rather than the known value of a 

 parameter. 



Population equilibrium is statistically tested by com- 

 paring a catch length-frequency distribution in any one 

 year to those from the previous two years by using a 

 chi-square test of independence. Rationale for this type 

 of test is that a population in equilibrium should, ex- 

 cept for sampling variability, produce catch length- 

 frequencies that are identically distributed over time. 

 Significance of the chi-square test therefore indicates 

 that the population is not in equilibrium. Performance 

 of the chi-square test first requires the construction of 

 3-by-N frequency tables, where N is the number of size 

 categories that are jointly defined over the 3-year se- 

 quence. To do this, the largest and smallest size cate- 

 gories in the catch containing at least five individuals 

 in each of the three years are determined. Then for 

 each year, all size categories less than the lower bound 

 of the joint interval are pooled, as are all size categories 

 greater than the upper bound. Once the table is con- 

 structed, the chi-square test can be performed accord- 

 ing to the procedures described in Conover (1971). 



The statistical power of the test is examined in a 

 Monte Carlo experiment in which the test is applied 

 to catch length-frequencies generated by the popula- 

 tion simulation model. For each disequilibrium experi- 

 ment, annual catch length-frequencies are subsampled, 

 with replacement, with subsample sizes of 100, 500, 

 1000, and 5000 fish. At each level of subsampling, a 

 chi-square test of independence is then performed on 

 each 3-year sequence of subsampled length-frequencies. 

 Subsampling and testing are replicated 100 times, and 

 the power is estimated as the proportion of the 100 

 tests that is significant at the 5% level. 



To determine whether disequilibrium bias could 

 be reduced by averaging a series of catch length- 

 frequencies over time, the Wetherall method is also ap- 

 plied to simulated catch length-frequencies after they 

 have been time-averaged. First, the catch length- 

 frequencies for each year are converted to proportions 

 by length. Second, the time-series of catch proportions 

 for each size interval is smoothed by using a 3-year 

 centered running average. Third, the smoothed catch 

 proportions at length for each year are then multiplied 



by the actual catch (in numbers) for that year to recover 

 the true sample size. 



In addition to disequilibrium bias, also examined are 

 two types of bias that can influence Wetherall esti- 

 mates of Z/K and L^, even when the population is in 

 equilibrium. The focus of this examination is on the 

 variation in these biases as a function of the chosen 

 value of L c . To do this, the model first is run until 

 equilibrium conditions are reached. Then for each 

 chosen value of L c , the Wetherall method is applied to 

 the catch length-frequency distribution, which ex- 

 periences size selection by the fishery, and to the 

 population length-frequency distribution. Type I bias 

 in the estimates of Z/K is expressed as the percentage 

 difference between the estimates based on catch 

 length-frequencies and those based on population 

 length-frequencies. Type II bias in the estimates of Z/K 

 is expressed as the percentage difference between the 

 estimates based on population length-frequencies and 

 the known values of Z/K. 



Results and discussion 



As Wetherall et al. (1987) have cautioned, population 

 disequilibrium indeed leads to biased estimates of Z/K 

 and L^ , but the temporal pattern and magnitude of 

 this bias vary greatly with the type of equilibrium 

 perturbation. In the fishing-up experiment, disequilib- 

 rium bias in the estimates of Z/K is initially large and 

 negative, but later becomes positive before decaying 

 to zero (Fig. la). Increasing the fishing mortality rate 

 increases the magnitude of the positive bias and 

 shortens the time in which the maximum positive bias 

 is reached. The magnitude of fishing mortality, 

 however, appears to have little effect on the magnitude 

 of the initial negative bias. Disequilibrium bias in the 

 estimates of L m changes with time and with fishing 

 mortality rate in a manner similar to the bias in Z/K, 

 but the bias in L^ is almost always positive and con- 

 siderably smaller than the bias in Z/K (Fig. lb). In the 

 recruitment perturbation experiment, when recruit- 

 ment is doubled for 1 year, the disequilibrium bias in 

 the estimates of Z/K is oscillatory in time and has a pro- 

 nounced positive peak and a less-pronounced negative 

 peak (Fig. lc). When recruitment is eliminated for 1 

 year, the disequilibrium bias in Z/K is again oscillatory, 

 but the positive and negative peaks occur 2 years later 

 and the magnitude of the bias, especially that of the 

 negative bias, is smaller. Disequilibrium bias in L^ 

 varies with time in a manner similar to the bias in Z/K, 

 but again the bias is smaller in magnitude (Fig. Id). 

 Thus, the following two points emerge from these 

 simulations: (1) Biases in both parameters can be large 

 but vary tremendously as the size structure of the 



