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 leastsquares 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 badnessoffit criterion that the MDS procedure tries to minimize is
