# MDS Preference Analysis

• Cody S. Ding
Chapter

## Abstract

Discuss the fundamental concepts of MDS preference analysis. An example of real data is provided to illustrate interpretation of the results. Single-ideal point MDS analysis is also explained.

## Keywords

Preference model Vector representation Ideal-point Single-ideal point

Using MDS model for analysis of individual preference is probably one of the most interesting aspects of MDS, but it is also a confusing part in that the concept of individual preference can be defined in various ways. For example, in individual differences MDS analysis discussed in Chap. , individual weight can be thought as preferences, although the dimension weight of individuals typically indicates the dimensional saliency. Sometimes even the mean score computed from a set of items can be thought of as an indicator of an individual’s preference, with a higher score indicating higher preference. However, the preference MDS analysis has its theoretical roots back in 1950s when Coombs (1950, 1964) first introduced the unfolding MDS with J scale and I scale and proposed a distance model for preference data, often known as the unfolding model. It is the only method that has explicit methodologically theoretical foundation for assessment of individual preferences. In this chapter, we discuss the basic ideas of preference MDS analysis and its potential applications in educational and psychological research.

## 8.1 Basic Ideas of Preference MDS

The ideas of MDS preference analysis can be described in terms of vector model or distance model (i.e., ideal point model ). A vector preference model is the one in which an individual’s preference is represented by a vector (i.e., a straight line) in a latent configuration of variables, where the angular separation of the vectors corresponds to the data dissimilarities. On the other hand, in distance model both individual preference and behaviors are represented as points in the configuration, where the distance between points in a space represents the data dissimilarities. Figure 8.1 shows a result of MDS preference analysis using both types of models based on a hypothetical data that are preference ratings by 30 individuals with regard to five behaviors (reading, writing, drinking, TV watching, and playing sport) on a scale of 1–5, with 5 being most preferred. As can be seen in Fig. 8.1, some individuals prefer playing sports (vectors that point toward 5 and 6 o’clock), one prefers reading (7 o’clock vectors), some prefer TV watching and drinking (vectors point toward 10 o’clock). Six people prefer anything but playing sports and reading (vectors toward between 11 and 1 o’clock), while seven people prefer anything but writing and drinking (vectors toward 8 o’clock). The same preference is represented by distance model on the right of Fig. 8.1.

In vector model, the direction of the vectors indicates the individual’s preference; that is, the direction of the vector indicates the direction that is most preferred by the individuals, with preference increasing as the vector moves from the origin. On the other hand, the relative lengths of the vectors indicate fit of individuals, with the squared lengths being proportion of variance in preference that can be accounted for by the model. As can be seen in Fig. 8.1, some vectors (e.g., individuals 10, 17, and 22) have a shorter length, indicating that the model does not account well for these individuals’ preferences. For example, only 18% of preference for individual 10 is accounted for by the model. Thus, to interpret the biplot (plot of row and column) of preference, look for directions through the plot that show a continuous change in some attribute of the behaviors, or look for regions in the plot that contain clusters of behavior points and determine what attributes the behaviors have in common. Behavior points that are tightly clustered in a region of the plot represent behaviors that have the same preference patterns across the individuals. Vectors that point in roughly the same direction represent individuals who have similar preference patterns.

In distance model, the solution consists of a configuration of n variables or items points that assess behaviors and i individual points where each individual is represented as being at a ‘maximal’ or ‘ideal’ point, located in such a way that the distances from this point to the behavior points are in maximum agreement with the individual’s preference ratings or rankings. The position of the ‘ideal point’ is interpreted as the one point in the space where the individual’s preferences are at a maximum, and her preference decreases in every direction. This is often termed a ‘single peaked preference function’ since it assumes that there is only one point of maximum preference and that preference decreases from this point.

Normally the behavior points corresponding to the most popular or consensual rankings will lie at the center of the space, and the least popular ones at the periphery. That is, highly popular behaviors will tend to be projected into the center of the individual points so that behaviors can be close to most individual’s ideal points and highly unpopular behaviors will be located at the outside of a configuration. Thus, if a behavior is sufficiently unpopular it can be located virtually anywhere on the periphery as long as it is at a maximum distance from the ideal points. But each behavior must be preferred by at least one individual.

An important distinction between the vector and distance models is that distance model can accommodate more I-scales, as long as the number of behavior points is large compared to the number of dimensions, the size of the isotonic regions is small, especially towards the center of the configuration, and they become increasingly well-represented by a point. For this reason, behavior or item points in the central part of a configuration are normally the most stable, while those at the periphery can usually be moved around fairly freely without affecting the goodness of fit . As seen in Fig. 8.1, playing sport and writing are the most preferred behaviors since they located close to the center of the configuration.

In addition to vector and distance MDS preference models, the MDS preference analysis can also be discussed in terms of internal vs. external analysis. Internal preference analysis simultaneously provides estimates of the coordinates for behaviors or items, individual preference or ideal points, and a fit measure. In external preference analysis, coordinates of behaviors or item is assumed to be known, either from theory or previous analysis, and the analysis provides the estimates of individual ideal points based on the known configuration of behaviors. This type of analysis can be used as confirmatory MDS preference analysis. In the current literature, there are very few studies using this kind of analysis.

## 8.2 Preference Analysis Using PREFSCAL

In the previous sections, we mainly focus on the discussion of basic ideas of MDS preference modeling, using SAS proc prinqual (vector model) and proc transreg (ideal point model ) to analyze the hypothetical data for illustration. Another program that can perform MDS preference analysis (traditional ideal point model) is PREFSCAL in SPSS (Busing et al. 2005b). Readers interested in technical aspects of the program can consult Busing et al. (2005a). In this chapter, we focus on an application of the PREFSCAL program in studying adolescents’ general outlook. The goal is to provide readers a sense of MDS preference analysis in real world situation.

The data used here (n = 15) were a sub-sample of 486 students in grade 7. A battery of various measures that assessed psychosocial adjustments was administered to the participants in the regular classroom setting. For the purpose of this example, we used a 12-item instrument of the Life Orientation Test (LOT) (Scheier et al. 1994) that was developed to assess generalized optimism versus pessimism. The responses were coded along a 5-point Likert-type scale, ranging from “strongly disagree” to “strongly agree.” The items were scored so that high values indicate optimism (i.e., a large distance from pessimism). Examples of items include "In uncertain times, I usually expect the best." "If something can go wrong for me, it will." or "I’m always optimistic about my future." In a sense, these items assessed adolescents’ attitudinal preferences towards life.

One of the questions can be asked is: What kinds of life orientation preferences do these 15 adolescents in grade 7 show as measured by these 12-items? Given this question, an MDS preference model is a better choice. In addition, it is reasonable to assume that a rating of ‘2’ on a 5-point Likert-type scale by one individual may not be compared with the same rating by another individual since they may have a different reference point. Thus, the ‘2’s as rated by different individuals only indicate that a participant provides a rating of ‘2′ on a particular item, and the same ‘2’s do not indicate the same degree of similarity or dissimilarity. Based on these two considerations, the MDS preference model with row-conditional data type may be a better analytical technique.

A two-dimensional MDS model was specified and the PREFSCAL procedure using SPSS version 25 yielded the following fit indices:

Iterations final function value

130.716

Normalized stress

0.088

Kruskal’s stress-I

0.296

Kruskal’s stress-II

0.817

Young’s S-stress-I

0.390

Young’s S-stress-II

0.643

Goodness of fit

Dispersion accounted for

0.912

Variance accounted for

0.622

Recovered preference orders

0.927

Spearman’s rho

0.741

Kendall’s tau-b

0.618

Variation coefficients

Variation proximities

0.453

Variation transformed proximities

0.568

Variation distances

0.467

Degeneracy indices

Sum-of-squares of DeSarbo’s Intermixedness indices

0.058

Shepard’s rough nondegeneracy index

0.729

The results of the MDS preference analysis are shown in Fig. 8.2. First, the algorithm converges to a solution after 130 iterations, with a penalized stress (marked final function value) of 0.72. The variation proximities are close to coefficient of variation for the transformed proximities, indicating the solution provides discrimination between 12 optimism items. The sum-of-squares of DeSarbo’s intermixedness indices (DeSarbo et al. 1997) are a measure of how well the points of the different set are intermixed. The closer to 0, the more intermixed the solution. In here the intermixedness is 0.059, indicating that the solution is well intermixed. Shepard’s rough nondegeneracy index (Shepard 1974), which assesses the percentage of distinct distances, is 0.729, indicating 73% of distinct distances. Taken together, the results indicate the solution was not degenerate; that is, the points along the dimensions were distinctly separated.

Second, for the Goodness-of-fit indices (how well the model-based distance fit the observed distances), it is advisable to consider several measures together. Kruskal’s Stress-II is scale independent; variance accounted for (VAF) is equal to the square of correlation coefficient (i.e., r2) and is calculated over all values regardless of the conditionality of the analysis. In this example, Kruskal’s Stress-II and VAF, and recovered preference orders (RFO) are acceptable.

Third, some relationships among indices with different names should be noted. Dispersion Accounted For (DAF) is also referred to as the sum-of-squares accounted for (SSAF), which is equal to Tucker’s congruence coefficient. The square of Kruskal’s Stress-I is equal to normalized raw Stress. As Busing et al. (2005a) indicated, the function values of normalized raw Stress, SSAF or DAF, and Kruskal’s Stress-I are insensitive to differences in scale and sample size, and these values are suitable for comparing models with different dimensional solutions.

## 8.3 MDS Single-Ideal Point Model

In addition to the traditional vector and ideal point (i.e., distance) model, MacKay (2001) proposed a probabilistic MDS single-ideal point model. The idea of MDS single-ideal point model is the same as the other MDS preference models, which can be traced to Thurstone (Thurstone 1928) and Coombs (1950). Essentially, a probabilistic MDS single-ideal point model requires a single-ideal point solution across all individuals to be estimated rather than a number of ideal points, one for each person. A single-ideal solution represents both individuals and behaviors as a point in Euclidean space. The distance relation between individuals and behaviors as indicated by the behaviors or items provides information about the preference structure of the individuals in such a way that individuals are closer to the behaviors they prefer. The model is initially used to represent a rectangular matrix of preferences by i individuals for v objects or variables as distances between i ideal points and v actual objects or variables by estimating the coordinates of individuals and behaviors in the same latent space. Thus, a MDS single-ideal point model is a spatial model in which i individuals and v items or behaviors are represented as points in multidimensional space. The coordinates xi of an individual or a group of individuals is generally referred to as his or her (or that group’s) ideal point and, hence, it is called the single ideal-point model.

The preference of an individual or a group of individuals for a behavior or object is an inverse function of the distance between the point that represents the actual objects and the ideal point that represents the individuals. A large distance between an object and an ideal point indicate that the object has high disutility (i.e., less liked or preferred). In other words, an individual responds negatively to an actual object (a variable or item) when the attitude or behavior represented by the object or item does not closely reflect the attitude or behavior of the individual. In the MDS single-ideal point model, such disagreement occurs when the individual is located too far away from the object. On the other hand, individuals respond positively to actual items or objects that have locations similar to their own.

In the probabilistic MDS single-ideal point model, the ideal point and actual items are represented not by points but by distributions. Based on the probabilistic MDS single-ideal point model of Hefner (1958) as proposed by Mackay and his associates (MacKay 2007; MacKay and Zinnes 1986; Zinnes and MacKay 1983), the coordinates x ik of a behavior or object have the Euclidean properties:
$${d}_{\mathrm{ij}}=\sqrt{\sum \limits_{\mathrm{k}-1}{\left({\mathrm{x}}_{\mathrm{ik}}-{\mathrm{x}}_{\mathrm{jk}}\right)}^2}$$
(8.1)
where d ij is the distance random variable between item i and j, and x ik or x jk are coordinates that are assumed to be normally and independently distributed with mean μ ik and variance σ ik 2.The variance σ ik 2can be assumed equal (i.e., isotropic) or unequal (i.e., anisotropic) for each item i on each dimension k in a Euclidean space. The goal of the analysis is to estimate the mean location μ ik of coordinates x ik or x jk and variance σ ik 2, including location and variance of the ideal point. In order to obtain the parameter estimates μ ik and σ ik 2, one needs to specify the probability function of the distance random variable d ij , which depends on the variance structure and sampling properties. The detailed discussion of how a probability MDS single-ideal point model is derived may be found in MacKay and his associates (MacKay 1989, 2007; MacKay and Zinnes 1986; Zinnes and MacKay 1983, 1992).

MacKay et al. (1995) indicated one primary reason why probabilistic MDS models are of particular interest in modeling preferences characterized by a single-ideal point. The probabilistic MDS is able to estimate mean (i.e., centroid) location and variance of preferred behaviors. When variability in preferred behavior exists, or when there are measurement errors inherited in single-items of an instrument, it is desirable to take such variability or measurement errors into consideration. Technically, for each ideal point or actual behavior or item, i, there is a corresponding k-dimensional random vector Xj that has an x variate normal distribution with mean vector uj and covariance matrix Σj. Individuals’ choices are assumed to be based on values sampled from the Xj distributions. If an individual has a consistently preferred behavior, be it actual or ideal, then we expect the diagonal elements of the covariance matrix Σj to be small. However, if the individual does not have a consistently preferred behavior or there are more measurement errors, the diagonal elements of the Σj are expected to be large.

The model fit can be tested using information criterion statistics, such as CAIC (Bozdogan 1987), BIC (Schwarz 1978), or log-likelihood ratio tests. Thus, we can test various models with respect to kinds of variances assumed, latent groups in the data, or the number of ideal-points that may need to reflect individuals’ typical behavior. The ability to test hypotheses about the structure of the variance can also be just as interesting to researchers as the ability to test hypotheses about the location of actual objects and ideal points. For example, a psychologist might have an interest in knowing if the variability in a client’s anxiety behaviors about a positive event and a negative event were the same as a result of interventions or a part of developmental processes.

The hypothetical data used in previous example is used again to illustrate what MDS single-ideal point model can inform us. Figure 8.3 shows the results of the analysis using PROSCAL (MacKay and Zinnes 2014). The circle indicates the variance around the points, which is 1.76, assuming equal variance across all behaviors. It seems that playing sport and writing are fairly close to the ideal point. Table 8.1 shows the distance between the ideal point and each of the five behaviors. As shown in Table 8.1, the distance between ideal point and playing sport and writing is the smallest, which corresponds to the I scale. The result from MDS single-ideal point analysis is fairly consistent with that from traditional distance model as shown in Fig. 8.1 in which playing sport and writing are close to the center of the configuration.
Table 8.1

Distance between ideal point and each of the five behaviors

Ideal point

I scale

2.06

3.03

Writing

1.91

2.93

TV watching

2.00

2.97

Drinking

2.45

3.27

Sport

1.27

2.70

If we assume that there will two groups of individuals (e.g., males vs. females), we can also use MDS single-ideal point model to conduct group-based single-ideal point analysis, with each group having its own single-ideal point. If no group information is known a priori, mixture MDS single-ideal point analysis can be conducted, where each individual is assigned to be a group based on the estimated probability of group membership. In type of analysis has not been performed in psychological and educational research, but it has potentially interesting applications.

To show how MDS single-ideal point model can be used to test multiple vs. single ideal points in the preference data, the data of the same 15 adolescents in the preference modeling via PREFSCAL shown in previous section are used for a such purpose. Based on Fig. 8.2, it seemed that there may be two groups of adolescents with different life orientation preferences. In other words, a model of two-ideal points seemed to underlie the data. However, it is also possible that a single-ideal point may be adequate to account for the differences in these adolescents’ preference. Thus, we used maximum likelihood MDS single ideal-point analysis to test a single-ideal point vs. a two-ideal point preference model. Of course, there were other possible models such as a two-dimensional vs. a one-dimensional model or a different combination of dimensionality and ideal points can also be tested.

The single-ideal vs. the two-ideal two-dimensional solutions estimated by PROSCAL are shown in Fig. 8.4. The hypotheses were tested using information criterion statistics, such as CAIC (Bozdogan 1987) or BIC (Schwarz 1978). The CAIC value for the single-ideal solution was 1882.67, whereas the two-ideal solution was 1880.66. The CAIC difference between the two models was less than 10, indicating that the single-ideal model was adequate to account for individual differences in life orientation preference (Burnham and Anderson 2002). This finding was consistent with what was found in traditional ideal point analysis conducted in PREFSCAL, in which a group of nine adolescents was not responsive to the items, and six adolescents indicated life orientation preference.

## 8.4 Conclusion

In this chapter, we discussed the fundamentals of MDS preference analysis using both simulated and real data. We used data that are more likely to be seen in educational and psychological setting rather than data that are less commonly seen such as car rating. Our discussion is more of practitioner-oriented; that is, only focused on the practical aspects of MDS preference analysis, omitting many technical components of the models. Also, we do not discuss external preference analysis in this chapter because we want to discuss it in a later chapter where issues related to measurement and hypothesis testing are presented. We want to make external preference analysis be more useful in today’s research situations by relating it to what we are commonly seen in current research.

MDS preference analysis can have many applications such as in longitudinal data analysis. In here we only attempt to pique researchers and practitioners interests to use this method. One research problem can be addressed from a different angle using various methods, which can either shed more lights or provide some unique information, at least possibly providing a validity triangulation on some research questions. MDS preference analysis will not replace more popular methods of preference analysis but it can be another method for addressing the same research issues from a different perspective.

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## Authors and Affiliations

• Cody S. Ding
• 1
• 2
1. 1.Department of Education Science and Professional ProgramUniversity of Missouri-St. LouisSt. LouisUSA
2. 2.Center for NeurodynamicsUniversity of Missouri-St. LouisSt. LouisUSA