Formulas
The following notation is used:
 A_{p}

intercept for partition p
 B_{p}
 slope for partition p
 C_{p}
 power for partition p
 D_{rcs}
 distance computed from the model between objects r and c for
subject s
 F_{rcs}

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
 O_{rcs}
 observed dissimilarity between objects r and c for subject s
 P_{rcs}
 partition index for objects r and c for subject s
 Q_{rcs}
 dissimilarity after applying any applicable estimated transformation for
objects r and c for subject s
 R_{rcs}

residual for objects r and c for subject s
 S_{p}
 standardization factor for partition p
 T_{p}(·)
 estimated transformation for partition p
 V_{sd}
 coefficient for subject s on dimension d
 X_{nd}
 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, T_{p}(·) 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
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.