Testing Component Contributions in Finite Discrete Mixtures: Detecting Specific Populations in Mixed Stock Fisheries

Synopsis Mixed stock analysis (MSA) is used to estimate the relative contributions of distinct populations in a mixture of organisms, generally via conditional maximum likelihood estimation using a baseline of learning samples from all potentially contributing populations. MSA is increasingly used to judge the presence or absence of specific populations in specific mixture samples. This is commonly done by inspecting the marginal bootstrap confidence interval of the contribution of interest. This method suffers from a number of major statistical deficiencies, including zero power to detect even a perfectly identifiable population at the low contribution levels of interest. In contrast, the likelihood ratio test has 100% power to detect any positive contribution from this ideal population. Both methods are compared in a power analysis using a 17-population baseline of sockeye salmon (Oncorhynchus nerka) from the Kenai River, Alaska, watershed. Power to detect a contribution varies with the population(s) relative identifiability, contribution level, mixture sample size, and analysis method. The power analysis shows the likelihood ratio is more powerful than the bootstrap method, with equality only at 100% power. Power declines for both methods as contribution declines, but the bootstrap method declines faster and goes to zero. Power quickly declines for both methods as population identifiability declines, though the likelihood ratio test is able to capitalize on the presence of ‘perfect identification’ characteristics, such as private alleles in genetic markers. Given the baseline-specific nature of detection power, MSA researchers are encouraged to conduct a priori power analyses. Introduction Mixed stock analysis (MSA) is used to estimate the relative contributions of distinct populations in a mixture of organisms. This is an important tool in wildlife management and research, with genotypes commonly used as natural markers to distinguish major populations or stocks (e.g., genetic stock identification) (Begg et al. 1999, Pearce et al. 2000). Increasingly, MSA is used to judge the presence or absence of specific stocks in specific mixture samples. For example, management of an interception fishery may be heavily influenced by the presence of a specific threatened, weakened, or politically high profile stock (e.g., salmon of Canadian origin harvested by Alaskan fisheries). MSA can overestimate stocks contributing little or nothing to a mixture (Pella & Milner 1987). Managers and researchers require a method for testing whether a specific nonzero stock contribution is really a biased estimate of zero. In practice, one checks the limit of the contribution’s 95% bootstrap lower confidence interval: a stock contribution with zero lower interval limit is deemed ‘statistically indistinguishable from zero’ (Seeb & Crane 1999). This method is statistically flawed. Using the interval as a test assumes that the contribution estimate is a pivotal statistic, which it is not. Further, such marginal tests implicitly employ an inappropriate measure of distance between compositions (Aitchison 1992). Of greatest practical importance, the method is shown, below, to have very low or even zero power exactly in the settings of interest. Consider an ideal marker and an ideal population: a gene for which Population A is fixed for an allele that is unique among the other populations in the baseline – a private allele. Population A is perfectly identifiable, so a mixture sample of size n containing πA times n Population A individuals will produce a nonzero contribution