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

\[  D_{rcs} = \begin{cases}  \sqrt {\sum _ d(X_{rd}-X_{cd})^2} &  \mbox{for COEF=IDENTITY:}\\ &  \text {Euclidean distance}\\[2ex] \sqrt {\sum _ d V_{sd}^2(X_{rd}-X_{cd})^2}&  \mbox{for COEF=DIAGONAL:}\\ &  \text {weighted Euclidean distance} \end{cases}  \]

Partition indexes are

\[  \begin{array}{llll} P_{rcs} &  = &  1 &  \mbox{for CONDITION=UN} \\ &  = &  s &  \mbox{for CONDITION=MATRIX} \\ &  = &  (s-1)N+r &  \mbox{for CONDITION=ROW} \end{array}  \]

The estimated transformation for each partition is

\[  \begin{array}{llll} T_ p(d) &  = &  d &  \mbox{for LEVEL=ABSOLUTE} \\ &  = &  B_ pd &  \mbox{for LEVEL=RATIO} \\ &  = &  A_ p+B_ pd &  \mbox{for LEVEL=INTERVAL} \\ &  = &  B_ pd^{C_ p} &  \mbox{for LEVEL=LOGINTERVAL} \end{array}  \]

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

\begin{eqnarray*}  Q_{rcs} & =&  O_{rcs} \\ R_{rcs} & =&  Q_{rcs}^ f - [T_{P_{rcs}}(D_{rcs})]^ f \end{eqnarray*}

For LEVEL=ORDINAL, the residuals are computed as

\begin{eqnarray*}  Q_{rcs} & =&  T_{P_{rcs}}(O_{rcs}) \\ R_{rcs} & =&  Q_{rcs}^ f - D_{rcs}^ f \end{eqnarray*}

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

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