Formulas

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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(·): 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

$\begin{tabular} {p{.25in}p{.1in}p{1.5in}p{3in}} D_{rcs}\space &=& \sqrt{\displ... ...DIAGONAL:} \linebreak \phantom{for }weighted Euclidean distance} \\end{tabular}$

Partition indexes are

$\begin{tabular} {p{.3in}p{.1in}p{1.1in}p{1.8in}} P_{rcs}\space &=& 1\space & {\... ...or CONDITION=MATRIX} \ &=& (s-1)N+r\space & {\rm for CONDITION=ROW}\end{tabular}$

The estimated transformation for each partition is

$\begin{tabular} {p{.3in}p{.1in}p{1.1in}p{1.8in}} T_p(d)\space &=& d\space & {\r... ...EVEL=INTERVAL} \ &=& B_pd^{C_p}\space & {\rm for LEVEL=LOGINTERVAL}\end{tabular}$

For LEVEL=ORDINAL, T_p(·) is computed as a least-squares monotone transformation.

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

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

For LEVEL=ORDINAL, the residuals are computed as

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

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

$S_p &=& 1 & {\rm for FORMULA=0} \ &=& U_p \displaystyle{\sum_{r,c,s | P_{rcs}=p}... ..._{r,c,s | P_{rcs}=p} (Q_{rcs}-\overline{Q}_p)^2F_{rcs} } & {\rm for FORMULA=2}$

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

$\sqrt{\displaystyle{\sum_{r,c,s} \frac{R_{rcs}^2 F_{rcs} }{S_{P_{rcs}}} } }$

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