Statistical properties of the relative weight (Wr) index and an alternative procedure for testing Wr differences between groups

Fishery managers commonly test for differences in relative weight (W,) 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.