Statistical Properties of the Relative Weight (W_r) Index and an Alternative Procedure for Testing W_r Differences between Groups

Abstract

Fishery managers commonly test for differences in relative weight (W _r ) between or within populations using standard parametric or nonparametric statistical tests. However, the statistical properties of the W _r index have not been studied; consequently, we cannot be certain that W _r data satisfy the theoretical assumptions upon which many standard tests are based. In particular, we do not know if W _r data are independent and identically distributed (i.i.d.), which is an assumption made by most standard statistical tests and which can increase the type I error rate of a statistical test when it is violated. We derived approximations to the expectation, variance, and covariance of the W _r index using the Delta method and estimated these parameters with sample data for muskellunge Esox masquinongy, largemouth bass Micropterus salmoides, and black crappie Pomoxis nigromaculatus. Additionally, we devised a likelihood ratio test (R-test) that tests for differences in W _r between groups and that does not assume i.i.d. data. We applied the R-test to the sample data and compared the results with those obtained from traditional statistical tests. The statistical properties of the W _r index were conditionally dependent on fish length. This conditionality prevents the covariance from equaling zero across an entire fish sample, so that W _r data will be inherently correlated. For the sample data, parameter estimates increased with fish length and typically were above realistic W _r values. This bias was attributed to variability in sample length-weight data. Traditional statistical tests were more liberal than the R-test in declaring statistical significance when testing for differences in W _r between groups. Based on our derivations, we conclude that W _r data will violate an i.i.d. assumption to some extent. To avoid falsely declaring statistical significance when testing group differences, we recommend that the R-test be used with W _r data.