Assessment of Sampling Stability in Ecological Applications of Discriminant Analysis

A simulation study was undertaken to assess the sampling stability of the variable loadings in linear discriminant function analysis. A factorial design was used for the factors of multivariate dimensionality, dispersion structure, configuration of group means, and sample size. A total of 32 400 discriminant analyses were conducted, based on data from simulated populations with appropriate underlying statistical distributions. Results from the simulations suggest that minimum sample sizes must exceed multivariate dimensionality by at least a factor of three to achieve reasonable levels of stability in discriminant function loadings. However, the requisite sample size would vary with respect to each of the design factors and, especially, with the overall amount of system variation. A review of 60 published studies and 142 individual analyses indicated that sample sizes in ecological studies often have met that requirement. However, individual group sample sizes frequently were very unequal, and checks of assumptions usually were not reported. We recommend that ecologists obtain group sample sizes that are at least three times as large as the number of variables measured.