Performs individual‑differences multidimensional scaling for metric data using alternating least squares.

Routine MSIDV performs multidimensional scaling analysis according to an alternating optimization algorithm. Input to MSIDV consists of symmetric dissimilarity matrices measuring distances between the row and column objects. Optionally, similarities can be input, and these can be converted to dissimilarities by use of the ICNVT option. In MSIDV, the row and column objects (stimuli) must be identical. Dissimilarities in multidimensional scaling are used to position the objects within a d = NDIM dimensional space, where d is specified by the user. Optionally, in the individual differences scaling model (MODEL ≠ 0), the weight assigned to each dimension for each subject may be changed.

The data input in X must be in a special symmetric storage form. For this storage mode, the input array X contains only the upper triangular part of each dissimilarity matrix and does not contain the diagonal elements (which should all be zero anyway). Storage of symmetric data in X is as follows: X(1) corresponds to the (1, 2) element in the first matrix (which is a measure of the distance between objects 1 and 2), X(2) corresponds to the (1, 3) element, X(3) corresponds to the (2, 3) element, etc., until all t = q(q ‑ 1)/2 off‑diagonal elements in the first matrix are stored, where q = NSTIM. The t + 1 element in X contains the (1, 2) element in the second matrix, and so on.

Missing values are indicated in either of two ways: 1) The standard missing value indicator NaN (not a number), specified via routine AMACH(6) (Reference Material) may be used to indicate missing values, or 2) Negative elements of X may be used to indicate missing observations. In either case, missing values are estimated as the mean dissimilarity for the subject and used as such when computing initial estimates, and they are omitted from the criterion function when optimal estimates are computed.

Routine MSIDV assumes a metric scaling model. When no transformation is specified (ITRANS = 1), then each datum (after transforming to dissimilarities) is a measure of distance plus a constant, αm. In this case, the constant (which is always called the “intercept”) is assumed to vary with subject and must first be added to the observed dissimilarities in order to obtain a metric. When a transformation is specified (ITRANS ≠ 1), the meaning of αm changes (with respect to metrics). Thus, when ITRANS = 1, the data is assumed to be interval (see the chapter introduction) while when ITRANS ≠ 1 ratio data is assumed. A scaling factor, the “slope”, is also always estimated for each subject.

When ISTRS = 1 or 2, the criterion function in MSIDV is given as

where δijm denotes the predicted distance between objects i and j on subject m,

denotes the corresponding dissimilarity (the observed distance), νm is the subject weight, f is one of the transformations f(x) = x2, f(x) = x, or f(x) = ln(x) specified by parameter ITRANS, αm is the intercept added to the transformed observation within each subject, and βm is the slope for the subject. For ISTRS = 0, the criterion function is given as

where nm is the number of nonmissing observations on the m‑th subject. Assuming fixed weights, the first derivatives of the criterion for ISTRS = 0 are identical to the first derivatives of the criterion when ISTRS = 1 or 2, but with weights

ISTRS can, thus, be thought of as changing the weighting to be used in the criterion function.

The transformation f(x) specified by parameter ITRANS is used to obtain constant within‑subject variance of the subject dissimilarities. If the variance of the log of the observed dissimilarities (about the predicted dissimilarities) is constant within subject, then the log transformation should be used. In this case, the variance of a dissimilarity should be proportional to its magnitude. Alternatively, the within‑subject variance may be constant when distances (or squared distances) are used.

The distance models for δijm available in MSIDV are given by:

where Λ denotes the configuration (CFL) so that λik is the location of the i‑th stimulus in the k‑th dimension, where d is the number of dimensions, and where wmk is the weight assigned by the m‑th subject to the k‑th dimension (W).

Weights that are inversely proportional to the estimated variance of the dissimilarities (about their predicted values) within each subject may be preferred because such weights lead to normal distribution theory maximum likelihood estimates (when it is assumed that the dissimilarities are independently normally distributed with constant residual variance). The estimated (conditional) variance used as the inverse of the weight νm for the m‑th subject in MSIDV (when ISTRS = 0) is computed as

where the sum is over the observations for the subject, and where nm is the number of observed nonmissing dissimilarities for the subject. These weights are used in the first derivatives of the criterion function.

When ISTRS = 1, the within‑subject average sum of squared dissimilarities are used for the weights. They are computed as

Finally, when ISTRS = 2, the within‑subject variance of the dissimilarities is used for the weights. These are computed as follows

where

denotes the average of the transformed dissimilarities in the stratum.

Initial estimates of all parameters are obtained through methods discussed in routine MSINI. After obtaining initial estimates, a modified Gauss‑Newton algorithm is used to obtain estimates for the parameters that optimize the criterion function. The parameters are optimized sequentially as follows:

An iteration is defined to be all of the Steps 1, 2, and 3. Convergence is assumed when the maximum absolute change in any parameter during an iteration is less than 10−4 or if there is no change in the criterion function during an iteration.

A modified Gauss‑Newton algorithm is used in the estimation of all parameters. This algorithm, which is discussed in detail by Merle and Spath (1974), uses iteratively reweighted least squares on a Taylor series linearization of the parameters in δijm. During each iteration, the subject weights, which may depend upon the parameters in the model, are assumed to be fixed.

All models available are overparameterized so the resulting parameter estimates are not uniquely defined. For example, in the Euclidean model, the columns of X can be translated or “rotated” (multiplied by an orthonormal matrix), and the resulting stress will not be changed. To eliminate lack of uniqueness due to translation, model estimates for the configuration are centered in all models. No attempt at eliminating the rotation problem is made, but note that rotation invariance is not a problem in many of the models given. With more general models than the Euclidean model, other kinds of overparameterization occur. Further restrictions on the parameters to eliminate this overparameterization are given below by the model transformation type specified by ITRANS. In the following, wlk ∈ W, where W is the matrix of subject weights. The restrictions to be applied by model transformation type are

The following example concerns some intercity distance rankings. The data are described by Young and Lewyckyj (1979, page 83). The driving mileages between various cities in the United States are ranked, yielding a symmetric ordinal dissimilarity matrix. These rankings are used as input to MSIDV. A Euclidean model is fit. The resulting two‑dimensional scaling yields results closely resembling the locations of the major cites in the U.S. Note that MSIDV assumes continuous, not ranked, data.

The original rankings are given as:

The second example involves three subjects’ assessment of the dissimilarity between rectangles that vary in height and width. An analysis is performed in k = 2 dimensions using the individual‑differences scaling model. The estimated subject weights, wmk, indicate how each subject weight the dimensions. The raw data are given as follows: