698 



Fishery Bulletin 98(4) 



nik and Fidell, 1996). This technique is similar to multiple 

 regression analysis in that one or more independent vari- 

 ables (the three set stages in our study) are used to pre- 

 dict a single dependent, categorical variable (the stocks). 

 Linear probability models accommodate all types of inde- 

 pendent variables (numerical and categorical) and they 

 do not have to be normally distributed, linearly related, 

 or of equal variance within each group. The assumptions 

 of multivariate normality and equal variance-covariance 

 matrices across groups do not have to be met, either. 

 Furthermore, logistic regression might be preferable to 

 multiple discriminant analysis because it is similar to 

 regression with its straightfoi-ward statistical tests, ability 

 to incorporate nonlinear effects, and wide range of diag- 

 nostics (Tabachnik and Fidell, 1996). 



The model produced by logistic regi-ession is nonlinear, 

 and the outcome variable is the probability of having one 

 outcome or another (in our study: one stock or the other) 

 based on a nonlinear function of the best linear combi- 

 nation of predictors, with two outcomes (Tabachnik and 

 Fidell, 1996): 



the probability of being in the other group. The procedure 

 for estimating coefficients is maximum likelihood, and the 

 goal is to find the best linear combination of predictors 

 to maximize the likelihood of obtaining the observed out- 

 come frequencies. Logistic regression can be used to fit 

 and compare models. The researcher uses goodness-of-fit 

 tests to choose the model that does the best job of predic- 

 tions with the fewest predictors. (Tabachnik and Fidell, 

 1996). 



Therefore, logistic regression analysis was applied to 

 test the predictability of stock membership (the dependent 

 or grouping variable) by gi'ouping codes (response vari- 

 ables ) during three set stages ( independent variables ). The 

 simplest model (constant-only model) was compared with 

 the full model (with the three independent variables) by 

 computing their log-likelihoods and by using x^- Degrees 

 of freedom were the difference between degi'ees of freedom 

 for the full and the constant-only models. The constant- 

 only model has 1 df (for the constant) and the full model 

 for our study had 3 df (1 df for each individual effect and 

 one for the constant); therefore x^ was evaluated with 3 df 

 If x^ was significant, the full model would be reliable 

 (Tabachnik and Fidell, 1996). 



l + e" 



where Y = the estimated probability that the (th case is 

 in one of the categories and ii is the usual 

 linear regression equation: 



u=A + B,X, + B,X. + ... + B,,X, , 



with constant A, coefficients B^. and predictors, X^ (inde- 

 pendent variables, the set stages in this study) for k pre- 

 dictors (J=l, 2, ..., k). 



This linear regression equation creates the log of the 

 odds, that is, the linear regression equation is the (natu- 

 ral log of the) probability of being in one group divided by 



Results 



Spatial patterns in evasive behavior of northeastern 

 offshore spotted dolphins 



The evasion index by set of northeastern offshore spotted 

 dolphins was averaged in 2 x 2 quadrants and the result- 

 ing contour map is shown in Figure 1. The highest evasion 

 index by set contour was 60'7f and extended approximately 

 from south of the Baja California peninsula, across the 

 Gulf of California mouth, and to the Mexican mainland 

 (approx. 20 northern latitude). The 5.5'7f and SCTr evasion 



Table 1 



Repeated-measures design to test for differences in dispersion behavior' between northeastern offshore spotted and eastern spin- 

 ner dolphins. The data (response variable) are grouping codes for case /. during set stage./, for species k (=.y,,,, ). (Mexican fleet, 

 1992-95. PNAAPD data. ) 



Set stages 



Case 



(herd in each set) 



Species 



( before chase ) (during chase) (during encirclement i 



1 to 308 

 309 to 544 



1 northeastern offshore spotted dolphin 



2 eastern spinner dolphin 



^3, 



A'-, 



Xjj 



^44 2 2 



^1 = 



' Herd dispersion: to establish if a herd dispersed during a set, the grouping codes apphed by the ob.ser\-ers in paragi'aph 3 of the 

 RMMO.SD were used. The codes are the following: 1, herd is in one group; 2, herd has divided into two or three groups; 3, herd 

 consists of more than three groups. These codes were recorded during the three set stages: before chase, during chase, and during 

 encirclement. If observers documented an ascending order in the codes (1, 2, 3), this feature was interpreted as herd dispersion 

 during the fishing operation. 



