Analysis of variance of parameter estimates: F tests and t tests

The problem of comparing and pooling experimentally independent estimates of a parameter such as a Michaelis constant (K) has been treated as a simple analysis of variance of "within" and "between" set deviations from the fitted variable (v). As applied to assessing the reproducibility of multiple estimates of the same K, this is identical to the procedure of Duggleby (Anal. Biochem. 189, 84-87, 1990). However, the theory developed here shows that applying Duggleby's procedure to the comparison of two experiments (each consisting of multiple data sets) depends critically on the assumption of equal errors within and between the individual sets, i.e., Fvb vw = s2wv/s2bv is close to 1. Application of the method when this is not the case will underestimate the common error (s2rv), overestimate its associated degrees of freedom (vr = vb+vw), and may suggest apparently significant differences where there are none. The theory also shows that this situation is an instance of the Fisher-Behrens problem and shows how Welch's solution can be applied. This gives the between set error s2bv as the corrected estimate of the common error and the corrected degrees of freedom as a simple function of vb, vw, and Fvb vw. When the nine prephenate dehydratase data sets which originally showed three apparently significant differences were reanalyzed in this way, all the variations in K were found to be within the range of the experimental error.