
ALTERNATE  ALT=NONE  NO  N
ALTERNATE  ALT=MATRIX  MAT  M  SUBJECT  SUB  S
ALTERNATE  ALT=ROW  R <=n>

determines what form of alternatingleastsquares algorithm is used. The default depends on the amount of memory available.
The following ALTERNATE= options are listed in order of decreasing memory requirements:
 ALT=NONE

causes all parameters to be adjusted simultaneously on each iteration. This option is usually best for a small number of subjects
and objects.
 ALT=MATRIX

adjusts all the parameters for the first subject, then all the parameters for the second subject, and so on, and finally adjusts
all parameters that do not correspond to a subject, such as coordinates and unconditional transformations. This option usually
works best for a large number of subjects with a small number of objects.
 ALT=ROW

treats subject parameters the same way as the ALTERNATE=MATRIX option but also includes separate stages for unconditional
parameters and for subsets of the objects. The ALT=ROW option usually works best for a large number of objects. Specifying
ALT=ROW=n divides the objects into subsets of n objects each, except possibly for one subset when n does not divide the number of objects evenly. If you omit =n, the number of objects in the subsets is determined from the amount of memory available. The smaller the value of n, the less memory is required.
When you specify the LEVEL=ORDINAL option, the monotone transformation is always computed in a separate stage and is listed
as a separate iteration in the iteration history. In this case, estimation is done by iteratively reweighted least squares.
The weights are recomputed according to the FORMULA= option on each monotone iteration; hence, it is possible for the badnessoffit
criterion to increase after a monotone iteration.

COEF=IDENTITY  IDEN  I
COEF=DIAGONAL  DIAG  D

specifies the type of matrix for the dimension coefficients.
 COEF=IDENTITY

is the default, which yields Euclidean distances.
 COEF=DIAGONAL

produces weighted Euclidean distances, in which each subject can have different weights for the dimensions. The dimension coefficients that PROC MDS outputs are related to
the square roots of what are called subject weights in PROC ALSCAL; the normalization in PROC MDS also differs from that in PROC ALSCAL. The weighted Euclidean model is related to the INDSCAL
model (Carroll and Chang, 1970).

CONDITION  COND=UN  U
CONDITION  COND=MATRIX  MAT  M  SUBJECT  SUB  S
CONDITION  COND=ROW  R

specifies the conditionality of the data (Young 1987, pp. 60–63). The data are divided into disjoint subsets called partitions. Within each partition, a separate transformation is applied, as specified by the LEVEL= option. The three types of conditionality
are as follows:
 COND=UN

(unconditional) puts all the data into a single partition.
 COND=MATRIX

(matrix conditional) makes each data matrix a partition.
 COND=ROW

(row conditional) makes each row of each data matrix a partition.
The default is CONDITION=MATRIX. The CONDITION= option also determines the default value for the SHAPE= option. If you specify
the CONDITION=ROW option and omit the SHAPE= option, each data matrix is stored as a square and possibly asymmetric matrix.
If you specify the CONDITION=UN or CONDITION=MATRIX option and omit the SHAPE= option, only one triangle is stored. See the
SHAPE= option for details.

CONVERGE  CONV=p

sets both the gradient convergence criterion and the monotone convergence criterion to p, where . The default is CONVERGE=0.01; smaller values might greatly increase the number of iterations required. Values less than
0.0001 might be impossible to satisfy because of the limits of machine precision. (See the GCONVERGE= and MCONVERGE= options.)

CUTOFF=n

replaces data values less than n with missing values. The default is CUTOFF=0.

DATA=SASdataset

specifies the SAS data set containing one or more square matrices to be analyzed. In typical psychometric data, each matrix
contains judgments from one subject, so there is a onetoone correspondence between data matrices and subjects.
The data matrices contain similarity or dissimilarity measurements to be modeled and, optionally, weights for these data. The data are generally assumed to be dissimilarities unless you use
the SIMILAR option. However, if there are nonmissing diagonal values and these values are predominantly larger than the offdiagonal
values, the data are assumed to be similarities and are treated as if the SIMILAR option is specified. The diagonal elements
are not otherwise used in fitting the model.
Each matrix must have exactly the same number of observations as the number of variables specified by the VAR statement or
determined by defaults. This number is the number of objects or stimuli.
The first observation and variable are assumed to contain data for the first object, the second observation and variable are
assumed to contain data for the second object, and so on.
When there are two or more matrices, the observations in each matrix must correspond to the same objects in the same order
as in the first matrix.
The matrices can be symmetric or asymmetric, as specified by the SHAPE= option.

DECIMALS  DEC=n

specifies how many decimal places to use when displaying the parameter estimates and fit statistics. The default is DECIMALS=2,
which is generally reasonable except in conjunction with the LEVEL=ABSOLUTE option and very large or very small data.

DIMENSION  DIMENS  DIM=n <TO m < BY=i>>

specifies the number of dimensions to use in the MDS model, where number of objects. The parameter i can be either positive or negative but not zero. If you specify different values for n and m, a separate model is fitted for each requested dimension. If you specify only DIMENSION=n, then only n dimensions are fitted. The default is DIMENSION=2 if there are three or more objects; otherwise, DIMENSION=1 is the only
valid specification. The analyses for each number of dimensions are done independently. For information about choosing the
dimensionality, see Kruskal and Wish (1978 , pp. 48–60).

EPSILON  EPS=n

specifies a number n, 0 < n < 1, that determines the amount added to squared distances computed from the model to avoid numerical problems such as division
by 0. This amount is computed as equal to n times the mean squared distance in the initial configuration. The distance in the MDS model is thus computed as
where sqdist is the squared Euclidean distance or the weighted squared Euclidean distance.
The default is EPSILON=1E–12, which is small enough to have no practical effect on the estimates unless the FIT= value is
nonpositive and there are dissimilarities that are very close to 0. Hence, when the FIT= value is nonpositive, dissimilarities
less than n times 100 times the maximum dissimilarity are not permitted.

FIT=DISTANCE  DIS  D
FIT=SQUARED  SQU  S
FIT=LOG  L
FIT=n

specifies a predetermined (not estimated) transformation to apply to both sides of the MDS model before the error term is
added.
The default is FIT=DISTANCE or, equivalently, FIT=1, which fits data to distances.
The option FIT=SQUARED or FIT=2 fits squared data to squared distances. This gives greater importance to large data and distances
and lesser importance to small data and distances in fitting the model.
The FIT=LOG or FIT=0 option fits log data to log distances. This gives lesser importance to large data and distances and greater
importance to small data and distances in fitting the model.
In general, the FIT=n option fits nthpower data to nthpower distances. Values of n that are large in absolute value can cause floatingpoint overflows.
If the FIT= value is 0 or negative, the data must be strictly positive (see the EPSILON= option). Negative data might produce
strange results with any value other than FIT=1.

FORMULA  FOR=0  OLS  O
FORMULA  FOR=1  USS  U
FORMULA  FOR=2  CSS  C

determines how the badnessoffit criterion is standardized in correspondence with stress formulas 1 and 2 (Kruskal and Wish 1978, pp. 24–26). The default is FORMULA=1 unless you specify FIT=LOG, in which case the default is FORMULA=2. Data partitions are defined by
the CONDITION= option.
 FORMULA=0

fits a regression model by ordinary least squares (Null and Sarle, 1982) without standardizing the partitions; this option cannot be used with the LEVEL=ORDINAL option. The badnessoffit criterion
is the square root of the error sum of squares.
 FORMULA=1

standardizes each partition by the uncorrected sum of squares of the (possibly transformed) data; this option should not be used with the FIT=LOG option. With the FIT=DISTANCE
and LEVEL=ORDINAL options, this is equivalent to Kruskal’s stress formula 1 or an obvious generalization thereof. With the
FIT=SQUARED and LEVEL=ORDINAL options, this is equivalent to Young’s sstress formula 1 or an obvious generalization thereof.
The badnessoffit criterion is analogous to , where R is a multiple correlation about the origin.
 FORMULA=2

standardizes each partition by the corrected sum of squares of the (possibly transformed) data; this option is the recommended
method for unfolding. With the FIT=DISTANCE and LEVEL=ORDINAL options, this is equivalent to Kruskal’s stress formula 2 or
an obvious generalization thereof. With the FIT=SQUARED and LEVEL=ORDINAL options, this is equivalent to Young’s sstress
formula 2 or an obvious generalization thereof. The badnessoffit criterion is analogous to , where R is a multiple correlation computed with a denominator corrected for the mean.

GCONVERGE  GCONV=p

sets the gradient convergence criterion to p, where . The default is GCONVERGE=0.01; smaller values might greatly increase the number of iterations required. Values less than
0.0001 might be impossible to satisfy because of the limits of machine precision.
The gradient convergence measure is the multiple correlation of the Jacobian matrix with the residual vector, uncorrected
for the mean. (See the CONVERGE= and MCONVERGE= options.)

INAV=DATA  D
INAV=SSCP  S

affects the computation of initial coordinates. The default is INAV=DATA.
 INAV=DATA

computes a weighted average of the data matrices. Its value is estimated only if an element is missing from every data matrix.
The weighted average of the data matrices with missing values filled in is then converted to a scalar products matrix (or
what would be a scalar products matrix if the fit were perfect), from which the initial coordinates are computed.
 INAV=SSCP

estimates missing values in each data matrix and converts each data matrix to a scalar products matrix. The initial coordinates
are computed from the unweighted average of the scalar products matrices.

INITIAL  IN=SASdataset

specifies a SAS data set containing initial values for some or all of the parameters of the MDS model. If the INITIAL= option is omitted, the initial values are computed from the data.

LEVEL=ABSOLUTE  ABS  A
LEVEL=RATIO  RAT  R
LEVEL=INTERVAL  INT  I
LEVEL=LOGINTERVAL  LOG  L
LEVEL=ORDINAL  ORD  O

specifies the measurement level of the data and hence the type of estimated (optimal) transformations applied to the data or distances (Young 1987, pp. 57–60; Krantz et al. 1971, pp. 9–12) within each partition as specified by the CONDITION= option. LEVEL=ORDINAL specifies a nonmetric analysis, while
all other LEVEL= options specify metric analyses. The default is LEVEL=ORDINAL.
 LEVEL=ABSOLUTE

permits no optimal transformations. Hence, the distinction between regression and measurement models is irrelevant.
 LEVEL=RATIO

fits a regression model in which the distances are multiplied by a slope parameter in each partition (a linear transformation).
In this case, the regression model is equivalent to the measurement model with the slope parameter reciprocated.
 LEVEL=INTERVAL

fits a regression model in which the distances are multiplied by a slope parameter and added to an intercept parameter in
each partition (an affine transformation). In this case, the regression and measurement models differ if there is more than
one partition.
 LEVEL=LOGINTERVAL

fits a regression model in which the distances are raised to a power and multiplied by a slope parameter in each partition
(a power transformation).
 LEVEL=ORDINAL

fits a measurement model in which a leastsquares monotone increasing transformation is applied to the data in each partition.
At the ordinal measurement level, the regression and measurement models differ.

MAXITER  ITER=n

specifies the maximum number of iterations, where . The default is MAXITER=100.

MCONVERGE  MCONV=p

sets the monotone convergence criterion to p, where , for use with the LEVEL=ORDINAL option. The default is MCONVERGE=0.01; if you want greater precision, MCONVERGE=0.001 is
usually reasonable, but smaller values might greatly increase the number of iterations required.
The monotone convergence criterion is the Euclidean norm of the change in the optimally scaled data divided by the Euclidean
norm of the optimally scaled data, averaged across partitions defined by the CONDITION= option. (See the CONVERGE= and GCONVERGE= options.)

MINCRIT  CRITMIN=n

causes iteration to terminate when the badnessoffit criterion is less than or equal to n, where . The default is MINCRIT=1E–6.

NEGATIVE

permits slopes or powers to be negative with the LEVEL=RATIO, INTERVAL, or LOGINTERVAL option.

NONORM

suppresses normalization of the initial and final estimates.

NOPHIST  NOPRINT  NOP

suppresses the output of the iteration history.

NOULB

causes missing data to be estimated during initialization by the average nonmissing value, where the average is computed according
to the FIT= option. Otherwise, missing data are estimated by interpolating between the Rabinowitz (1976) upper and lower bounds.

OCOEF

writes the dimension coefficients to the OUT= data set. See the OUT= option for interactions with other options.

OCONFIG

writes the coordinates of the objects to the OUT= data set. See the OUT= option for interactions with other options.

OCRIT

writes the badnessoffit criterion to the OUT= data set. See the OUT= option for interactions with other options.

OITER  OUTITER

writes current values to the output data sets after initialization and on every iteration. Otherwise, only the final values
are written to any output data sets. (See the OUT=, OUTFIT=, and OUTRES= options.)

OTRANS

writes the transformation parameter estimates to the OUT= data set if any such estimates are computed. There are no transformation
parameters with the LEVEL=ORDINAL option. See the OUT= option for interactions with other options.

OUT=SASdataset

creates a SAS data set containing, by default, the estimates of all the parameters of the PROC MDS model and the value of
the badnessoffit criterion. However, if you specify one or more of the OCONFIG, OCOEF, OTRANS, and OCRIT options, only the
information requested by the specified options appears in the OUT= data set. (See also the OITER option.)

OUTFIT=SASdataset

creates a SAS data set containing goodnessoffit and badnessoffit measures for each partition as well as for the entire
data set. (See also the OITER option.)

OUTRES=SASdataset

creates a SAS data set containing one observation for each nonmissing data value from the DATA= data set. Each observation
contains the original data value, the estimated distance computed from the MDS model, transformed data and distances, and
the residual. (See also the OITER option.)

OVER=n

specifies the maximum overrelaxation factor, where . Values between 1 and 2 are generally reasonable. The default is OVER=2 with the LEVEL=ORDINAL, ALTERNATE=MATRIX, or ALTERNATE=ROW
option; otherwise, the default is OVER=1. Use this option only if you have convergence problems.

PCOEF

produces the estimated dimension coefficients.

PCONFIG

produces the estimated coordinates of the objects in the configuration.

PDATA

displays each data matrix.

PFINAL

displays final estimates.

PFIT

displays the badnessoffit criterion and various types of correlations between the data and fitted values for each data matrix,
as well as for the entire sample.

PFITROW

displays the badnessoffit criterion and various types of correlations between the data and fitted values for each row, as
well as for each data matrix and for the entire sample. This option works only with the CONDITION=ROW option.

PINAVDATA

displays the sum of the weights and the weighted average of the data matrices computed during initialization with the INAV=DATA
option.

PINEIGVAL

displays the eigenvalues computed during initialization.

PINEIGVEC

displays the eigenvectors computed during initialization.

PININ

displays values read from the INITIAL= data set. Since these values might be incomplete, the PFIT and PFITROW options do not
apply.

PINIT

displays initial values.

PITER

displays estimates on each iteration.

PLOTS<(globalplotoption)> <= plotrequest <(options)>>
PLOTS<(globalplotoptions)> <= (plotrequest <(options)> <... plotrequest <(options)>>)>

specifies options that control the details of the plots. When you specify only one plot request, you can omit the parentheses
around the plot request.
ODS Graphics must be enabled before plots can be requested. For example:
ods graphics on;
proc mds plots(flip);
run;
ods graphics off;
For more information about enabling and disabling ODS Graphics, see the section Enabling and Disabling ODS Graphics in Chapter 21: Statistical Graphics Using ODS.
The global plot option is as follows:

FLIP

flips or interchanges the Xaxis and Yaxis dimensions for configuration and coefficient plots.
The plot requests include the following:

COEFFICIENTS(ONE)

combines the INDSCAL coefficients panel of plots into a single plot. By default, the display consists of a panel with two
plots. The vectors are displayed in the left plot, and the labels are displayed in the right plot. The right plot provides
a magnification of the region of the vector endpoints. In contrast, the single display, requested by COEFFICIENTS(ONE), is
more compact, but there is less room for vector labels. It is often easier to identify the vectors in the default display.

NONE

suppresses all plots.
By default, a fit plot is produced. When more then one dimension is requested, plots of the configuration are also plotted.
For individual differences models with more than one dimension, the subject weights or coefficients are plotted. When more
than one value is specified for the DIMENSION= option, the badnessoffit plot is produced.

PTRANS

displays the estimated transformation parameters if any are computed. There are no transformation parameters with the LEVEL=ORDINAL
option.

RANDOM<=seed>

causes initial coordinate values to be pseudorandom numbers. In one dimension, the pseudorandom numbers are uniformly distributed on an interval. In two or more dimensions, the pseudorandom
numbers are uniformly distributed on the circumference of a circle or the surface of a (hyper)sphere.

RIDGE=n

specifies the initial ridge value, where . The default is RIDGE=1E–4.
If you get a floatingpoint overflow in the first few iterations, specify a larger value such as RIDGE=0.01, RIDGE=1, or RIDGE=100.
If you know that the initial estimates are very good, using RIDGE=0 might speed convergence.

SHAPE=TRIANGULAR  TRIANGLE  TRI  T
SHAPE=SQUARE  SQU  S

determines whether the entire data matrix for each subject or only one triangle of the matrix is stored and analyzed. If you
specify the CONDITION=ROW option, the default is SHAPE=SQUARE. Otherwise, the default is SHAPE=TRIANGLE.
 SHAPE=SQUARE

causes the entire matrix to be stored and analyzed. The matrix can be asymmetric.
 SHAPE=TRIANGLE

causes only one triangle to be stored. However, PROC MDS reads both upper and lower triangles to look for nonmissing values
and to symmetrize the data if needed. If corresponding elements in the upper and lower triangles both contain nonmissing values,
only the average of the two values is stored and analyzed (Kruskal and Wish 1978 , p. 74). Also, if an OUTRES= data set is requested, only the average of the two corresponding elements is output.

SIMILAR  SIM<=max>

causes the data to be treated as similarity measurements rather than dissimilarities. If =max is not specified, each data value is converted to a dissimilarity by subtracting it from the maximum value in the data set or BY group. If =max is specified, each data value is subtracted from the maximum of max and the data. The diagonal data are included in computing these maxima.
By default, the data are assumed to be dissimilarities unless there are nonmissing diagonal values and these values are predominantly
larger than the offdiagonal values. In this case, the data are assumed to be similarities and are treated as if the SIMILAR
option is specified.

SINGULAR=p

specifies the singularity criterion p, . The default is SINGULAR=1E–8.

UNTIE

permits tied data to be assigned different optimally scaled values with the LEVEL=ORDINAL option. Otherwise, tied data are
assigned equal optimally scaled values. The UNTIE option has no effect with values of the LEVEL= option other than LEVEL=ORDINAL.