Optimal scaling of a data vector, relative to a set of targets, is obtained through a least-squares transformation subject to appropriate measurement constraints. The targets are usually predicted values from a statistical model. If the data are nominal level, then the transformation must be identity-preserving. If the data are ordinal level, then the transformation must be monotonic. If the data are discrete, then tied data values must remain tied in the optimal transformation. If the data are continuous, then tied data values can be untied in the optimal transformation.
