502 



Fishery Bulletin 104(4) 



for the coordinates. A fast Fourier transform (FFT) was 

 calculated as a Cartesian FFT. The Cartesian FFT uses 

 the 128 x-y coordinates as complex numbers (a + ib), 

 where a is the real component and Ib the imaginary 

 component, representing the amplitudes of the cosine 

 and sine waves, respectively. The resultant 128 set 

 of complex numbers or descriptors were subsequently 

 normalized for differences in otolith position by set- 

 ting the 0"'^ descriptor to 0. and for size and rotation 

 of the otolith by dividing all the descriptors with the 

 first descriptor. The normalized descriptors (a' + ;b') 

 were used to calculate the absolute value (harmonic) 

 of each descriptor according to the following equation 

 (Christopher and Waters, 1974): 



Harmonic = J a'^f + {ib') . 



The harmonics were then used in combination with 

 the other morphological variables and shape indices to 

 compare otoliths between cohorts and among regions 

 and reefs within regions. 



The higher the number of equidistant points and 

 subsequent complex numbers included in the model, 

 the closer is the fit to the original shape. The main 

 features of the otolith shape, however, are generally 

 captured by the first 10-20 harmonics (e.g., Campana 

 and Casselman, 1993; Friedland and Reddin, 1994). 

 The minimum number of Fourier descriptors required 

 to explain at least 90% of the recorded shape of the 

 otoliths in our study was calculated similarly to the 

 range-finding procedure of Smith et al. (2002). A total 

 of 128 descriptors were collected from two randomly 

 selected otoliths from each reef and cohort (24 in to- 

 tal) and normalized for position, size and rotation as 

 described above. The shape of each otolith was recon- 

 structed (by computing the inverse FFT) by using all 

 the descriptors and then reconstructed by using only 

 the first and last descriptors. The Euclidian distance 

 between the inverse FFT using all the descriptors and 

 the inverse FFT using only the first and last descrip- 

 tors was defined as the maximum percent error of re- 

 construction, i.e. 100% reconstruction error (Smith 

 et al., 2002). Because the Cartesian descriptors are 

 asymmetrical around the middle frequency, both ends 

 of the array are required in the reconstruction. Oto- 

 lith shape was reconstructed, therefore, by using the 

 first two and last two descriptors, the first three and 

 last three descriptors, and so on until the first and 

 last 22 descriptors were used. This range-finding test 

 allowed us to estimate the decrease in mean percent 

 reconstruction error by using more and more descrip- 

 tors and it was estimated that 14 of the first and last 

 descriptors were required for the reconstruction error 

 to be less than 10%. See Figure 2 for a comparison of 

 the otolith shape reconstructed from the first and last 

 14 descriptors and all 128 descriptors. These descrip- 

 tors, therefore, were used in the statistical analyses to 

 compare the spatial and temporal patterns in otolith 

 shape of P. leopardus. 



Statistical methods 



The assumption of normality and homogeneity of vari- 

 ance for each morphological variable was examined by 

 using Shapiro-Wilk's and Levenes tests, respectively, 

 and homogeneity of the group covariance matrix by Box's 

 M test. Variables of circularity, breadth and area were 

 log||, -transformed and Fourier harmonics 2, 4, 9, 11-14, 

 120-121, 114-118, and 123-127 square-root-transformed 

 to conform to the assumption of normality and homoge- 

 neity of variances. 



A relationship between otolith shape and otolith 

 growth rate (assumed to be correlated to fish length) 

 may confound spatial or temporal differences in oto- 

 lith shape (Campana and Casselman, 1993). We mini- 

 mized the potential for such effects by 1) including only 

 fish with a fork length (FL) between 280 and 514 mm 

 (overall FL for four-year-olds sampled during the ELF 

 experiment in 1995 and 1999 ranged from 250 to 551 

 mm), and 2) standardizing morphological variables by 

 fish FL where a significant relationship existed between 

 the variable and FL before further analyses. The effect 

 of FL on each morphological variable was examined by 

 analysis of covariance (ANCOVA; Winer et al., 1991). 

 Our primary interests in these analyses were 1) to test 

 whether morphological variables differed with FL for 

 any group of samples; and 2) if so, to test whether the 

 slopes of regressions of morphological variable on FL 

 were homogeneous among groups. If a significant re- 

 gression was detected and homogeneous among groups, 

 the effect of FL was removed from each measurement 

 by using the relationship 



0,. 



Kb 



0+b.iFL-MFL. 



where O 



otA = otolith morphological measurement of 

 fish / adjusted to mean fork length of 

 group j; 

 O,^ = original otolith morphological measure- 

 ment for fish i from group /; 

 b. = slope of the relationship 0,^:FL common 

 to all groups; 

 FL^^ = fork length offish / in group j; and 

 MFL.^ = average fork length within group j. 



If significant slopes of the relationship differed among 

 groups, the correction for FL was made separately for 

 each group by using the equation above, but by replac- 

 ing the common slope (6.) with the group-specific slope 

 (6^1. These corrections had the effect of scaling all mor- 

 phological variables from all otoliths to their predicted 

 group mean FL. 



Multivariate analysis of variance (MANOVA; Tabach- 

 nick and Fidell. 1983) was used to investigate the ef- 

 fects of sex (females, males, and individuals in the 

 process of changing sex; i.e., transitional fish) on otolith 

 shape (MANOVA). A total of 302 of the 351 individuals 

 had been examined for sex, of which 166 were females, 

 123 males and 13 transitional fish. Separate MANOVAs 

 were computed for the one- and two-dimensional shape 



