Alternative definitions of partial ancillarity, in the presence of nuisance parameters, will be examined with regard to loss of information involved when inference is based on the conditional distribution given the values of such partial ancillary statistics. Similarly, definitions of partial sufficiency will be examined with respect to loss of information when inference is based on the marginal distribution of such partial sufficiency statistics. Finally, a generalized likelihood principle will be formulated for inference concerning parameters of interest in the presence of the nuisance parameters. This is foundational research in statistics for data structures confounded with variables other than those of research interest. Statistical methods based on partial information have become quite popular in practice and this foundational work is needed and timely.
