Log-normal variation belts for growth curves.

Prediction (confidence) or tolerance belts compound the uncertainty of sample estimates with the estimated extent of individual variation. The latter is therefore better described by variation belts, in which sample estimates are simply substituted for population parameters. Variation belts can provide valuable graphical indications concerning the goodness of fit of postulated error models. While multiplicative least-squares (MLS) methods appear appropriate in principle for biological growth, they are unsatisfactory in practice when logarithmically transformed data are heteroscedastic. Heteroscedastic multiplicative error models can be fitted by iteratively reweighted multiplicative least squares (IRMLS), but unacceptable negative or infinite residual variance estimates and unreasonably wide variation belts are occasionally obtained. These difficulties can be prevented by constrained iteratively reweighted multiplicative least squares (CIRMLS). Examples are presented concerning the metabolic allometry of white rats, the somatic growth of male elephant seals, and the growth of an experimental population of Paramecium caudatum.