ABSTRACT
The concept behind the methods employed to obtain estimates of survival rates is based on the idea that each age class of a species will have a length distribution. As the fish grow older the mean of the age specific length distribution will increase in a manner described by a growth function and the variance may also change. As fish age and approach the maximum size for the species in question, it is almost certain that age-specific distributions will overlap. By compositing the age-specific distributions, one can obtain the population (all ages) distribution of lengths. The length-frequency model employed in this study attempts to estimate the mean and variance of each age-specific length distribution by fitting the composite model to the population length-frequency histogram. After obtaining the parameters for the age-specific length distributions, it is then possible to compute survival estimates as the ratio of expected frequency at age i + 1 divided by the expected frequency at age i. The length-frequency model employed here was similar in structure to the length-frequency models used in the ELEFAN system[20]. In the Ohio River model, the length-frequency data are represented by a mixture of scaled beta density functions, whereas the ELEFAN system uses normal density functions. The classical beta density described in any mathematical statistics book is defined on a domain of {0, 1} and is defined as follows:
xa-1 (1-x)b-1 f (x; a,b) = ––––––––– 0 < x < 1, a > 0, b < 0
B (a,b)
Γ (a) Γ (b) B (a,b) = ––––––––– Γ (a + b) (3)
where Γ(b) is a complete gamma function and a and b are shape parameters for the beta density[21]. In order to use the beta density to define the probability that a randomly selected fish of a certain age falls in a particular length interval, the variable x must be rescaled from the interval {0, 1} to the interval {minimum, maximum} where minimum and maximum are defined as the minimum and maximum length for each year class of fish.
