FISHERY BULLETIN: VOL. 79, NO. 1 



considering a much more general and widely 

 applicable problem than that discussed by Allen 

 (1976) and by Gallucci and Quinn (1979). For the 

 technique presented here, the only assumption 

 that is made is that the forms of the population 

 growth curves can be specified. No parameters 

 are assumed to be known or equal, and no distri- 

 butional assumptions are made. 



The solution, described in the following sec- 

 tions, of our growth curve comparison problem is 

 not obtained via a classical statistical hypothesis 

 testing approach. That is, we do not formulate an 

 appropriate null hypothesis and derive a crite- 

 rion which dictates when and only when it should 

 be rejected. Instead, in the spirit of Geisser and 

 Eddy (1979), we formulate various possible mod- 

 els and give a data analytic approach to selecting 

 the one model preferred by the data. 



A difficulty, which plagues classical hypothesis 

 testing, does not exist for the approach described 

 here. It is the necessity of specifying a signifi- 

 cance level. Historically, significance levels such 

 as 0.10, 0.05, and 0.01 have routinely been used 

 for tests without objective justification. Yet the 

 choice of a significance level affects the conclu- 

 sions arrived at. For example, it is quite possible 

 to find that a hypothesis can be rejected at the 

 0.05 level, but not at the 0.01 level. Further, the 

 choice of a significance level affects the probabil- 

 ity of rejecting a false hypothesis. Lowering the 

 significance level usually lowers the probability 

 of rejecting a false hypothesis. Requiring one to 

 specify a significance level presumes that one has 

 a sound basis for controlling the probability of 

 rejecting one of two possible hypotheses when it 

 is true, that one can objectively assign a signifi- 

 cance level which controls this probability, and 

 that controlling this probability is more crucial 

 than controlling the probability of not rejecting 

 the hypothesis when it is false. The contention 

 here is that for many, if not most, scientific 

 investigations, the consequences of rejecting one 

 hypothesis when it is true are no more serious 

 than rejecting the other when it is true. That is, 

 often an investigator has no reason to favor 

 either hypothesis, but merely wants to know 

 which one is more reasonable, given the data that 

 has been collected. 



THE TWO POPULATIONS CASE 



Suppose that two populations of fish are being 

 studied. For example, the first population might 



96 



consist of all fish of a given species inhabiting one 

 area while the second might consist of all fish of 

 this species inhabiting a different area. Suppose 

 we are interested in comparing the growth curves 

 associated with the two populations. 



We consider two possible models, say Mi and 

 M2. The model Mi specifies that the growth 

 curves are the same, while under M2 the two 

 growth curves differ. 



Let X and y represent, respectively, the age and 

 length of a fish. Then we rewrite Mi and M2 as 



Mi: y=fix;6) + e (1) 



no matter which of the two popula- 

 tions fish belongs to. 



M2: y = fiix;di) + e (2) 



if the fish belongs to the first 

 population. 



y = f2ix;d2) + e (3) 



if the fish belongs to the second 

 population. 



Here fix; 6), fiix; di), and fiix; 62) are each 

 functions of x. 6, di, and 62 are each vectors of 

 unknown parameters and f, fi , and /2 are speci- 

 fied except for the values of elements of 6, 61 , and 

 02. Essentially, f, fi, and [2 represent three dif- 

 ferent growth curves which are specified except 

 for the values of unknowm parameters present in 

 each. The function f represents the expected 

 length of a fish whose age is x, assuming equal 

 growth curves for the two populations, while /"i 

 and f2 are, respectively, the expected lengths for 

 fish of age x from the first and second populations, 

 assuming the growth curves differ. As usual, e 

 represents the unknown, random error term. 



We now give a data analytic approach for 

 selecting one of the two possible models Mi or 

 M2 . The data used to make the selection are pairs 

 of age-length measurements for samples of fish 

 from each of the two populations. 



Let {xii,yu), ixi2,yi2),---,ixin,yin) represent a 

 sample of n pairs of observations of x and y from 

 the first population and (^21, ^21), (^22, y22),--, 

 ix2m, y^m) represent a sample of m pairs of obser- 

 vations of X and y from the second population. 

 These n + m pairs of observations are the data 

 gathered by the investigator and we want to use 

 these data to select either Mi or M2 . 



Assume, for the moment, that M2 is correct. 

 Forj - 1, 2,...,n, let ^k^) represent the vector of 



