Importance of modelling heteroscedasticity of survey index data in fishery stock assessments

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Many fish stock assessments use a survey index and assume a stochastic error in the index on which a likelihood function of associated parameters is built and optimized for the parameter estimation. The purpose of this paper is to evaluate the assumption that the standard deviation for the difference in the log-transformed index is approximately equal to the coefficient of variation of the index, and also to examine the homo- and heteroscedasticity of the errors. The traditional practice is to assume a common variance of the index errors over time for estimation convenience. However, if additional information is available about year-to-year variability in the errors, such as year-to-year coefficient of variation, then we suggest that the heteroscedasticity assumption should be considered. We examined five methods with the assumption of a multiplicative error in the survey index and two methods with that of an additive error in the index: M1, homoscedasticity in the multiplicative error model; M2, heteroscedasticity in the multiplicative error model; M3, M2 with approximate weighting and an additional parameter for scaling variance; M4-M5, pragmatic practices; M6, homoscedasticity in the additive error model; M7, heteroscedasticity in the additive error model. M1-M2 and M6-M7 are strictly based on statistical theories, whereas M3-M5 are not. Heteroscedasticity methods M2, M3, and M7 consistently outperformed the other methods. However, we select M2 as the best method. M3 requires one more parameter than M2. M7 has problems arising from the use of the raw scale as opposed to the logarithm transformation. Furthermore, the fitted survey index in M7 can be negative although its domain is positive.

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