Measures of dispersion as constraints 

 for length-frequency analysis 



Karim Erzini 

 Margarida Castro 



Unidade de Gencias e Tecnologias dos Recursos Aquaticos 

 Universidade do Algarve, 8000 Faro. Portugal 



Length-frequency analysis (LFA) 

 methods are widely used in popu- 

 lation dynamics studies, particu- 

 larly for tropical fish species that 

 may be difficult or impossible to age 

 by the traditional methods of read- 

 ing growth rings on hard parts. LFA 

 is characteristically subjective, and 

 numerous authors have warned 

 against its indiscriminate use, 

 pointing out that estimated para- 

 meters may be questionable or even 

 meaningless if the biology of the 

 species is not taken into consider- 

 ation or if the sampling was inad- 

 equate (e.g. Castro and Erzini, 

 1987; Macdonald, 1987; Morgan, 

 1987; Basson et al., 1988; Erzini, 

 1990). Biological information can be 

 incorporated into these studies to 

 obtain better results by using aged 

 subsamples, time series of length- 

 frequency distributions, or by con- 

 straining parameters to be esti- 

 mated (Macdonald, 1987; Morgan, 

 1987). Constraints are based on 

 assumptions concerning mortality, 

 the relative abundance of the com- 

 ponent age classes, the type of 

 growth pattern or growth curve, the 

 shape of the length-at-age distribu- 

 tions, the magnitude of the variabil- 

 ity in length at age, and the pat- 

 tern of this variability with age or 

 size. 



Our objective was to develop 

 simple models, relating variability 

 in length at age to life history and 

 environmental parameters that 

 could be used to select appropriate 

 starting values and constraints for 

 length-frequency analysis. We as- 



sumed that both the magnitude of 

 variability in length-at-age and the 

 size-and-age-dependent trends are 

 related to species-specific life his- 

 tory and environmental character- 

 istics. We demonstrate that mea- 

 sures of dispersion for particular 

 lengths can be estimated on the ba- 

 sis of easily estimated parameters). 



Methods 



The data set used in this study con- 

 sisted of 468 records representing 

 168 species and 50 families (Erzini, 

 1991). The following measures of 

 variability in length at age were 

 calculated: standard deviation of 

 mean length at age (SD), variance 

 of mean length at age (V), and co- 

 efficient of variation of mean length 

 at age (CV). The following life his- 

 tory and environmental param- 

 eters were also compiled: von 

 Bertalanffy K and L m , the Gallucci 

 and Quinn (1979) growth param- 

 eter co( intrinsic rate of growth), the 

 growth performance index (j)' (Long- 

 hurst and Pauly, 1987), maximum 

 observed age, age at 0.95 L_, spawn- 

 ing pattern, spawning duration 

 (months), geographic location, and 

 environmental regime (tropical, 

 temperate, and boreal). Spawning 

 patterns were described as continu- 

 ous, continuous with one major 

 peak, continuous with two peaks, 

 discrete with one peak, and discrete 

 with two peaks. Only data sets that 

 were not based on LFA, composite 

 samples, or back-calculated lengths 



at age were included in the analy- 

 sis. 



Stepwise multiple regression 

 with selection of variables by maxi- 

 mum R 2 improvement (SAS Insti- 

 tute Inc., 1985) was used to evalu- 

 ate the relative effectiveness of life 

 history and environmental para- 

 meters in predicting three mea- 

 sures of dispersion (SD, V, and CV). 

 Qualitative variables such as envi- 

 ronmental regime and spawning 

 pattern were represented by indi- 

 cator variables with values of and 

 1 (Neter et al., 1983). For each 

 qualitative variable consisting of m 

 classes, m— 1 indicator variables 

 were formed. Preliminary plots and 

 simple and quadratic regressions 

 were used to guide the transforma- 

 tion and creation of new variables, 

 such as mean length at age squared 

 for the stepwise regression, result- 

 ing in a total of 19 independent 

 variables. Only data where the 

 sample size corresponding to the 

 measures of dispersion was at least 

 10 were used. 



After multiple linear regression 

 was used to identify the most im- 

 portant explanatory variables, 

 simple linear regression was used 

 to examine the trends in variabil- 

 ity in length at age for data grouped 

 into discrete classes of these vari- 

 ables. Three-dimensional smoothed 

 plots of measures of dispersion as 

 functions of the independent vari- 

 ables and the classification para- 

 meters were also used to investigate 

 trends in variability in length at age. 



Results 



The multiple regression models 

 show that the SD models have the 

 highest R 2 values whereas the CV 

 models have the lowest (Tables 1- 

 3). The SD and the V are strongly 

 influenced by size and certain 



Manuscript accepted 18 March 1994. 

 Fishery Bulletin 92:865-871. 



865 



