The following notation is used:
slope for partition p
power for partition p
distance computed from the model between objects r and c for subject s
data weight for objects r and c for subject s obtained from the cth WEIGHT variable, or 1 if there is no WEIGHT statement
value of the FIT= option
number of objects
observed dissimilarity between objects r and c for subject s
partition index for objects r and c for subject s
dissimilarity after applying any applicable estimated transformation for objects r and c for subject s
standardization factor for partition p
estimated transformation for partition p
coefficient for subject s on dimension d
coordinate for object n on dimension d
Summations are taken over nonmissing values.
Distances are computed from the model as
The estimated transformation for each partition is
For LEVEL=ORDINAL, is computed as a least-squares monotone transformation.
For LEVEL=ABSOLUTE, RATIO, or INTERVAL, the residuals are computed as
For LEVEL=ORDINAL, the residuals are computed as
If f is 0, then natural logarithms are used in place of the fth powers.
For each partition, let
and
Then the standardization factor for each partition is
The badness-of-fit criterion that the MDS procedure tries to minimize is