Knowledge Bank

One of the methods in common use for analyzing large data sets is a two-step procedure, in which subsets of the full data set are first least-squares fitted to preliminary sets of parameteres, and the latter are subsequently merged to yield the final parameters. The second step of this procedure is properly a correlated least-squares fit and requires the variance-convariance matrices from the first step to construct the weight matrix for the merge. There is, however, an ambiguity concerning the manner in which the first-step variance-covariance matrices are assessed, which leads to different statistical properties for the quantities determined in the merge. The issue is one of a priori vs a posteriori assessment of weights, which is an application of what was originally called internal vs external consistency by $Birgea$ and $Demingb$. In the present work the simplest case of a merge fit - that of an average as obtained from a global fir vs a two-step fit of partitioned data - is used to illustrate that only in the case of a priori weighting do the results have the usually expected and desired statistical properties; normal distributions for residuals, $t$-distribution for parameters assessed a posteriori, and chi-square distributions for variances.