The following notation is used:

intercept for partition p

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

residual 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

Partition indexes are

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 is 0, then natural logarithms are used in place of the th 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