Formulas
The following notation is used:
- Ap
-
intercept for partition p
- Bp
- slope for partition p
- Cp
- power for partition p
- Drcs
- distance computed from the model between objects r and c for
subject s
- Frcs
-
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
- Orcs
- observed dissimilarity between objects r and c for subject s
- Prcs
- partition index for objects r and c for subject s
- Qrcs
- dissimilarity after applying any applicable estimated transformation for
objects r and c for subject s
- Rrcs
-
residual for objects r and c for subject s
- Sp
- standardization factor for partition p
- Tp(·)
- estimated transformation for partition p
- Vsd
- coefficient for subject s on dimension d
- Xnd
- coordinate for object n on dimension d
Summations are taken over nonmissing values.
Distances are computed from the model as
![\begin{tabular}
{p{.25in}p{.1in}p{1.5in}p{3in}}
D_{rcs}\space &=&
\sqrt{\displ...
...DIAGONAL:} \linebreak \phantom{for }weighted Euclidean
distance} \\end{tabular}](images/mdseq8.gif)
Partition indexes are
![\begin{tabular}
{p{.3in}p{.1in}p{1.1in}p{1.8in}}
P_{rcs}\space &=& 1\space & {\...
...or CONDITION=MATRIX} \ &=& (s-1)N+r\space & {\rm for CONDITION=ROW}\end{tabular}](images/mdseq9.gif)
The estimated transformation for each partition is
![\begin{tabular}
{p{.3in}p{.1in}p{1.1in}p{1.8in}}
T_p(d)\space &=& d\space & {\r...
...EVEL=INTERVAL} \ &=& B_pd^{C_p}\space & {\rm for LEVEL=LOGINTERVAL}\end{tabular}](images/mdseq10.gif)
For LEVEL=ORDINAL, Tp(·) is computed as a
least-squares monotone transformation.
For LEVEL=ABSOLUTE, RATIO, or INTERVAL, the residuals are computed as
![Q_{rcs} &=& O_{rcs} \R_{rcs} &=& Q_{rcs}^f - [T_{P_{rcs}}(D_{rcs})]^f](images/mdseq11.gif)
For LEVEL=ORDINAL, the residuals are computed as
![Q_{rcs} &=& T_{P_{rcs}}(O_{rcs}) \R_{rcs} &=& Q_{rcs}^f - D_{rcs}^f](images/mdseq12.gif)
If f is 0, then natural logarithms are used in place of
the fth
powers.
For each partition, let
![U_p = \frac{\displaystyle{\sum_{r,c,s}F_{rcs}}}
{\displaystyle{\sum_{r,c,s | P_{rcs}=p}F_{rcs}}}](images/mdseq13.gif)
and
![\overline{Q}_p = \frac{\displaystyle{\sum_{r,c,s | P_{rcs}=p}Q_{rcs}F_{rcs}}}
{\displaystyle{\sum_{r,c,s | P_{rcs}=p}F_{rcs}}}](images/mdseq14.gif)
Then the standardization factor for each partition is
![S_p &=& 1 & {\rm for FORMULA=0} \ &=& U_p \displaystyle{\sum_{r,c,s | P_{rcs}=p}...
..._{r,c,s | P_{rcs}=p}
(Q_{rcs}-\overline{Q}_p)^2F_{rcs} }
& {\rm for FORMULA=2}](images/mdseq15.gif)
The badness-of-fit criterion that the MDS procedure tries to
minimize is
![\sqrt{\displaystyle{\sum_{r,c,s} \frac{R_{rcs}^2 F_{rcs} }{S_{P_{rcs}}} } }](images/mdseq16.gif)
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.