Statistical Methods

Unit 9 - Correlation and regression

9.8 Length-weight relationship in Fishes
In fishes the relationship between body weight (W) and body length (L) has been empirically observed to be of the form
W=aL^b ................................... (1)
This equation is in nonlinear form. The parameters/constants ‘a’ and ‘b’ are almost universally estimated by researchers by transforming the above equation to logarithmic form and applying the least squares technique. Thus the equation actually used is
Log W =log a + b log L
The above method assumes the following multiplicative error model:
W = a L^b’ ..................................... (2)
Where a and b are constants and e’ is a random error factor.
Taking logarithm to the base ‘e’ on both sides of (2) gives rise to
In W = In a + b I n L + In e’
I.e. Y= A + BX + E .................................... (3)
where Y = lnW; X = ln L; A= ln a, b=B, E= In e’
Expression (3) is in the linear form. If it is assumed that E is distributed normally with mean zero and variance o², then the estimates of A and B can be obtained by the method of least squares.


In the above expression Y = In W and X =In L,and denote arithmetic means of Y and X values respectively. The B value given and estimate of b, whereas conventionally ‘a’ is estimates as exp (A). This method however gives biased estimate of ‘a’. To compensate for the bias the ‘a’ value obtained is multiplies by the correction factor exp (S²/2), where S² is an estimate of variance of deviations from regression. Hence, corrected a = exp (A+S²/2).
Note: If logarithm to the base 10 (common logarithm) are used then, a = antilog (A+S²/2)