The MDS Procedure

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

$T_{p}({\cdot })$

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

\[  \begin{tabular}{p{.25in}p{.1in}p{1.5in}p{3in}} $D_{rcs}$   &  =   &  $\sqrt {\displaystyle {\sum _ d(X_{rd}-X_{cd})^2}}$   &  {\mbox{for COEF=IDENTITY:} \linebreak Euclidean distance}   \\ &  =   &  $\sqrt {\displaystyle {\sum _ d V_{sd}^2(X_{rd}-X_{cd})^2}}$  &  {\mbox{for COEF=DIAGONAL:} \linebreak weighted Euclidean distance}   \\ \end{tabular}  \]

Partition indexes are

\[  \begin{tabular}{p{.3in}p{.1in}p{1.1in}p{1.8in}} $P_{rcs}$   &  =   &  1   &  \mbox{for CONDITION=UN}   \\ &  =   &  \Mathtext{s}   &  \mbox{for CONDITION=MATRIX}   \\ &  =   &  $(s-1)N+r$   &  \mbox{for CONDITION=ROW}   \end{tabular}  \]

The estimated transformation for each partition is

\[  \begin{tabular}{p{.3in}p{.1in}p{1.1in}p{1.8in}} $T_ p(d)$   &  =   &  \Mathtext{d}   &  \mbox{for LEVEL=ABSOLUTE}   \\ &  =   &  $B_ pd$   &  \mbox{for LEVEL=RATIO}   \\ &  =   &  $A_ p+B_ pd$   &  \mbox{for LEVEL=INTERVAL}   \\ &  =   &  $B_ pd^{C_ p}$   &  \mbox{for LEVEL=LOGINTERVAL}   \end{tabular}  \]

For LEVEL=ORDINAL, $T_{p}({\cdot })$ is computed as a least-squares monotone transformation.

For LEVEL=ABSOLUTE, RATIO, or INTERVAL, the residuals are computed as

$\displaystyle  Q_{rcs}  $
$\displaystyle = $
$\displaystyle  O_{rcs}  $
$\displaystyle R_{rcs}  $
$\displaystyle = $
$\displaystyle  Q_{rcs}^ f - [T_{P_{rcs}}(D_{rcs})]^ f  $

For LEVEL=ORDINAL, the residuals are computed as

$\displaystyle  Q_{rcs}  $
$\displaystyle = $
$\displaystyle  T_{P_{rcs}}(O_{rcs})  $
$\displaystyle R_{rcs}  $
$\displaystyle = $
$\displaystyle  Q_{rcs}^ f - D_{rcs}^ f  $

If f is 0, then natural logarithms are used in place of the fth powers.

For each partition, let

\[  U_ p = \frac{\displaystyle {\sum _{r,c,s}F_{rcs}}}{\displaystyle {\sum _{r,c,s | P_{rcs}=p}F_{rcs}}}  \]

and

\[  \overline{Q}_ p = \frac{\displaystyle {\sum _{r,c,s | P_{rcs}=p}Q_{rcs}F_{rcs}}}{\displaystyle {\sum _{r,c,s | P_{rcs}=p}F_{rcs}}}  \]

Then the standardization factor for each partition is

\[  \begin{array}{llll} S_ p & =&  1 &  \mbox{for FORMULA=0} \\ & =&  U_ p \displaystyle {\sum _{r,c,s | P_{rcs}=p} Q_{rcs}^2F_{rcs} } &  \mbox{for FORMULA=1} \\ & =&  U_ p \displaystyle {\sum _{r,c,s | P_{rcs}=p} (Q_{rcs}-\overline{Q}_ p)^2F_{rcs} } &  \mbox{for FORMULA=2} \end{array}  \]

The badness-of-fit criterion that the MDS procedure tries to minimize is

\[  \sqrt {\displaystyle {\sum _{r,c,s} \frac{R_{rcs}^2 F_{rcs} }{S_{P_{rcs}}} } }  \]