Formulas |

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*c*th 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*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 *f*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