from a given length; this is not a biologically reason- 

 able model. Traditionally, length has been viewed as 

 the independent variable that is measured with no 

 eiTor and weight as the random dependent variable 

 that is measured with error. The validity of these 

 assumptions is beyond the scope of this paper and 

 will not be discussed. We have concerned ourselves 

 with the appropriateness of two statistical models, 

 the log-linear and nonlinear models. The log-linear 

 model, with log-additive error, was written as 



be compared because they are minimal estimates in 

 their respective sample spaces. We chose to present 

 the "R-" and "F" statistics for the log-linear model 

 as an indication of precision, but did not use the 

 statistics in deciding best fit, since they cannot be 

 compared to those obtained for the nonlinear 

 model. 



Our criteria for best fit of the models were based 

 on measures of appropriateness, namely, whether 

 the error terms have the following properties: 



In Wi = Inb^ + «2 1"^/ + '"^2/' ^-^ 



The arithmetic equivalent of this model can be writ- 

 ten as 



but this equation should not be construed to be the 

 model. The nonlinear model, with additive error, 

 was written as 



'E[^^^ 



Var( e;.) 



= or £[63,] = 

 o\ orVarCCj,) = 



(4) 



W = h L^^ + (. ■■ 



(3) 



The evaluation of the goodness of fit of regres- 

 sion lines can be divided into distinct tests of preci- 

 sion (or significance) of the regression and of the 

 appropriateness of the model. The appropriateness 

 of a model (Equation 2 or 3) was tentatively ac- 

 cepted, and the model was fitted to the data. The 

 precision of this fit can then be measured by the 

 "F" test and the "/" test, both of which test Hjs|: a 

 = and Ha^: a ^ , or the "/?-" statistic^ the 

 "proportion of total variation about the mean Y [W] 

 explained by regression" (Draper and Smith, 1966). 

 All of these tests are equivalent and basically mea- 

 sure the usefulness of the regression as a predictor. 

 To be able to perform any of these tests, the ran- 

 dom error term must be normally distributed. The 

 distribution of e ', . = Ine.,^. was tested for the log- 

 linear model by calculating R.A. Fisher's statistics 

 for skewness (Gl) and kurtosis (G2, measuring the 

 amount of peakness or bimodality). A model can 

 fail in the significance tests because the model is 

 incoiTect or because the sample size is small rela- 

 tive to the amount of variability in the data. In addi- 

 tion, if a model is nonlinear in its parameters, it is 

 not possible to test for significance because the var- 

 iance estimates are biased, making it superfluous to 

 test the distribution of the error term,e,j.. 

 Moreover, the residual sums of squares for linear 

 and nonlinear least squares fitting routines cannot 



that is, the error terms have a mean equal to zero 

 and a constant variance. The error terms for the 

 log-linear model must have a mean equal to zero, 

 since an intercept term was included in the model 

 (Draper and Smith, 1966, p. 87). For the nonlinear 

 model, it is not readily apparent that the error term 

 must be equal to zero: hence, the mean was calcu- 

 lated. The residuals were plotted against the depen- 

 dent and independent variables to check for con- 

 stant variance. If variance is constant, the residuals 

 appear as a horizontal band along the variable axes 

 (Draper and Smith, 1966. p. 86). 



The final regression coefficients, or coefficients 

 of allometry, were tested using the hypothesis 

 scheme H^: « = 3.0, Ha: " / 3.0 (Steel and Tor- 

 rie, 1960, p. 171). 



In reporting the results of the various statistical 

 tests, the following convention was used: "NS" in- 

 dicates not significant at the 0.05 level, "*, **, ***" 

 indicate significance at the 0.05, 0.01, 0.001 levels, 

 respectively; and "d.f." stands for degrees of 

 freedom. 



RESULTS 

 Growth Stanzas 



The weight-length data for each species were first 

 plotted with logarithms of weight versus logarithms 

 of fork length in order to subjectively check for more 

 than one growth stanza (Tesch, 1968). Blue marlin 



^From this statement, the estimated value of the log-error 

 term, «;,. may be taken as zero which in turn indicates that€,, 

 in the anthmetic equivalent to the log-linear model (Equation 2) 

 may be taken as equal to one. If the arithmetic equivalent to the 

 log-linear model were designated as a separate model, it does not 

 follow that £ [e,,J = I or that Var(e,,) =02^ 



129 



