How to cope with negative estimates of components of variance in ecological field studies

Ecologists often need to estimate components of spatial or temporal variation. The most widely used method in ecology uses the observed and expected mean squares in an analysis of variance. A more general approach, which can be used for balanced and unbalanced designs, is based on residual maximal likelihood (REML). This method is less well known by ecologists and requires specialist software. If the design is balanced, the two methods are equivalent, except for one important respect: estimates from analysis of variance can be negative whereas REML estimates cannot. The purpose of this note is to point out a simple modification to the analyses of variance which yields the same estimates as REML for many of the designs commonly used in ecological studies. This modification has been available in the mathematical literature for over 30 years, but appears not to be well known amongst ecologists. It is useful in many cases of balanced analytical designs.