The MDS Procedure

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

f

value of the FIT= option

N

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 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