Dominance approach vs. ideal-point approach in item selection

Dominance approach (Coombs, 1964; Likert, 1932)

• It is about measuring people's ability
• It uses items of high internal consistency.
• Therefore, if a person scores low on one item, he/she should be score low on the total scores as well. Likewise, if I score higher on the item than you do, my ability would be dominant over your ability.
• In IRT terminology, DIF (Differential Item Functioning) refers to "a difference in the probability of endorsing an item for members of a reference group (e.g., US workers) and a focal group (e.g., Chinese workers) having the same standing on the latent attribute measured by a test." It is related to dominance approach.

Ideal-point approach (Thurstone, 1928)

• It is about measuring people's attitude
• Individuals will endorse an item to the degree that it reflects
• More neutral items should be included

Metric MDS and software

Metric MDS include the followings (Borg & Groenen, 2005, p. 203):

• ratio MDS:
• (disparities) = b * (proximities in terms of dissimilarities; short for 'prox' below)


• interval MDS:
• (disparities) = a + b * (prox)


• logarithmic MDS:
• (disparities) = log(prox)
  
• (disparities) = b * log(prox)
  
• (disparities) = a + b * log(prox)


• exponential MDS
• (disparities) = exp(prox)
• (disparities) = b * exp(prox)
  
• (disparities) = a + b * exp(prox)


• power MDS (which includes square root with q = 0.5):
• (disparities) = (prox)^q
• (disparities) = b * (prox)^q
  
• (disparities) = a + b * (prox)^q


• polynomial MDS (i.e., spline MDS without interior knots)
• (disparities) = a + b * (prox) + c * (prox)^2
• (disparities) = a + b * (prox) + c * (prox)^2 + d * (prox)^3

However, softwares are not always clear about the kinds of metric MDS they are performing. Based on my own testing as of 04/21/09, here is a table of comparison:

Software Package	Program, version, date	Metric MDS supported
MATLAB 7.8.0.347 (R2009a)	mdscale() 1.1.6.9, 12/01/08 Criterion = 'metricstress'	Ratio only
smacof in R 0.9-0 (05/24/08)	smacofSym(), metric = TRUE	Ratio only
SPSS 17.0.0 (08/23/08)	Proxscal version 1.0	Ratio, Interval, Spline
SYSTAT 12.02.00	Multidimensional Scaling Shape = Square (similarities model)	Interval (Linear), Log, Power

To date, no program in any of these software packages provide combinations of two or more than two transformations, but these could be very helpful. For example, log + polynomial may be of interest, because log may be used to normalize residuals, while polynomial may be able to pick up the trend of the data. That is,

• (disparities) = a + b * log(prox) + c * log(prox)^2
• (disparities) = a + b * log(prox) + c * log(prox)^2 + d * log(prox)^3

Eigendecomposition and Singular Value Decomposition

Eigenvalue and eigenvector are those satisfying the following eigenequation:

matrix(transformation) * eigenvector = eigenvalue * eigenvector

Thus, if we can find such a eigenvector and therefore a eigenvalue, their interpretations are: after being linearly transformed by the matrix, eigenvector still has the same direction. Eigenvalue can thus be considered some essential part of the matrix, or the characteristic value of the matrix. Eigenvector can be considered a tool to extract such essential part of the matrix.

A nice explanation can be founded here; see also Borg and Groenen (2005) Chapter 7.

Eigendecomposition: matrix A = QΛQ'
Thus, AQ = QΛQ'Q = QΛ, where Λ is a diagonal matrix of eigenvalues

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Singular Value Decomposition: matrix A = PΦQ'

P is a matrix of left singular vectors, Φ is a diagonal matrix with singular values, Q is a matrix of right singular vectors. The naming choice of "singular" probably is similar to that of "eigen", because the expressions of the two decompositions are very similar and probably referring to the essential and unique quality of the matrix.

Unfolding Models

On the top of a folded handkerchief is the ideal point, representing the highest degree of preference for a particular individual, i.e., the optimal choice within a given set of items. The closer the item is to the ideal point, the higher the preference is of this individual; thus, the individual prefers choice 1 to choice 2. While different individuals have different ideal points on the handkerchief, unfolding the handkerchief will give us a 2D diagram showing all ideal points and all the items on a common space.

Some applications of unfolding models (adapted from this):

Applicaton 1: In American Idol, a set of judges rate a set of contestants. Unfolding would display the ideal point of each judge as a point, and each contestant as a point. Three pieces of information will be revealed: (a) Judges with similar ideal points would cluster; (b) Contestants rated similarly would cluster; (c) The closeness between the ideal point of a judge and a contestant indicates how high the judge would rate the contestant.

Application 2: A set of TV brands (e.g., Panasonic, Sony, ...) were rated on a set of attributes (e.g., price, quality, style, ...). In the matrix, the rows are the brands and the columns are the attributes. Unfolding would display (the ideal point of) each brand as a point and each attribute as a point. Three pieces of information: (a) Similar brands (in terms of ideal points) would cluster; (b) Similar attributes would cluster; (c) Brands rated highly on a particular attribute would appear close to that attribute.

Application 3: Unfolding can also be used to display relationships that may not be symmetric, such as desire between people, trade-flows between nations, and journal citation frequency. Each journal would appear as both a row and a column. The matrix would contain the citation frequency of the row-journal by the column-journal. Self-citing is excluded. Unfolding would produce a diagram in which each journal would appear as two points: citing others and being cited by others. Clusters would have the obvious interpretation, and the distance between a journal’s two points would reflect the imbalances in its citation.

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Other variants of unfolding models:

1. External unfolding models. Besides the preference data, we also have a pre-existing coordinate matrix of the choice objects.
2. Vector model of unfolding. Representing individuals by preference vectors instead of ideal points. Because it is the direction of the vector that matters, the preference vectors are usually scaled to have equal length.
   
3. Weighted unfolding.
   

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Some terms and programs:

1. In marketing, unfolding model is known as perceptual mapping.
2. In marketing, MDPREF ("MultiDimensional PREFerence") performs internal unfolding analysis, whereas PREFMAP ("PREFerence MAPping") performs external unfolding analysis.
   

Procrustes analysis

The purpose of Procrustes analysis is to fit one MDS solution (configuration, map), B, to another one, A, and eliminate superficial differences between B and A, by means of rotating, mirror-reflecting, dilating/magnifying, shrinking, or shifting/moving B, without changing either's shape.

Application 1. A is the physical location map, whereas B is the travel-time map produced by MDS. In Procrustes analysis, we fit B to A, which allows us to display B on the top of A and to spot differences.

Application 2. Y is easy to interpret, whereas the initial X is not. In Procrustes analysis, we fit X to Y in order to interpret X.

Application 3. F is the result from the female participants, whereas M is that from the male participants. In Procrustes analysis, we fit M to F (or F to M) so that we can compare the results from males and females on the same page (provided that the fitting is satisfactory).

Application 4. CH is is the result from Chinese participants, whereas AM is that from American participants. In Procrustes analysis, we fit CH to AM (or AM to CH) so that we can compare the cross-cultural results on the same page (provided that the fitting is satisfactory).

MDS and social psychology

Searching JPSP by scholar. The 12 results found are categorized as the following:

A. Structure of Emotion

1. Russell (1980) A circumplex model of affect: 28 emotion-denoting adjectives are reduced to a 2D space: pleasure-displeasure and arousal-sleepiness.

• In the same year, Russell and Pratt (1980) also talked about the two dimensions on the meaning that persons attribute to environments.
  
• Russell and Bullock (1985) followed up on Russell (1980) to show that the two dimensions reveal a basic property of the human conception of emotions, rather than represent an artifact that is due to semantic relations learned along with the emotion lexicon.
• Russell, Weiss, and Mendelsohn (1989) followed up to develop a single-item scale, the Affect Grid, to quickly assess affect along the dimensions of pleasure-displeasure and arousal-sleepiness.
  
• Feldman (1995) interpreted the 2D as valence-focus and arousal-focus and suggested their relation to Positive Affect and Negative Affect.
• Barrett (2004) followed up on Feldman (1995) to talk about how valence-focus and arousal-focus are related to cognitive structure of emotion language vs. phenomenological experience.
• Extending Russell's model, Larsen, McGraw, and Cacioppo (2001) argued that people can feel happy and sad at the same time; they do not have to experience positive-negative emotions in a bipolar way.
  

B. Structure of Self-Other Relationship:

2. Falbo (1977) Multidimensional scaling of power strategies: 16 strategies of "How I Get My Way." reduced to a 2D space: (a) rational/nonrational and (b) direct/indirect.

3. Bartholomew and Horowitz (1991) examined a model of individual differences in adult attachment in which two underlying dimensions, the person's internal model of the self (positive or negative) and the person's internal model of others (positive or negative), were used to define four attachment patterns. (as seen in General Discussion)

4. Wiggins, Phillips, and Trapnell (1989) interpersonal circumplex: dominant/submissive and agreeable/cold-hearted.

• Gurtman (1992) applied this to plot individuals' profiles of high/low trust and high/low Machiavellianism.

5. Walker and Hennig (2004) studied the underlying 2D for the three exemplars of morality: just, brave, and caring, and found different 2D for each of them.

6. Abele and Wojciszke (2007) found that a large number of trait names can be organized into the 2D space of agency and communion.

7. Grouzet et al. (2005) found that 11 types of goals can be organized into a 2D space of intrinsic (e.g., self-acceptance, affiliation) versus extrinsic (e.g., financial success, image), and self-transcendent (e.g., spirituality) versus physical (e.g., hedonism). This results has cross-cultural validity.

(Incomplete) list of MDS researchers

1. Warren S. Torgerson:


• former professor at John Hopkins
• developed MDS while he was a PhD student
• known for the classical scaling (aka., Torgerson scaling) in MDS
• Solution from Torgerson scaling can be used as initial configuration; however, it is a rational configuration and is prone to local minima


2. Louis E. Guttman:


• former president of the Psychometric Society
• developed Guttman loss function in SYSTAT


3. Roger N. Shepard:


• former president of the Psychometric Society
  
• professor of cognitive psychology at Stanford University (Emeritus)
• known for Shepard diagram
  


4. Joseph B. Kruskal:


• former president of the Psychometric Society
• former president of the Classification Society of North America
• developed stress formula 1 and formula 2
  
• developed the program of KYST (Kruskal, Young, & Seery, 1973)


5. Forrest W. Young:


• former president of the Psychometric Society
• professor of quantitative psychology at the University of North Carolina at Chapel Hill (Emeritus)
• developer of ALSCAL (alternating least squares scaling) (available in SPSS)
  


6. J. Douglas Carroll:


• former president of the Psychometric Society
• professor of management and psychology at Rutgers University
• developer of INDSCAL (individual differences scaling)


7. Jan de Leeuw:


• former president of the Psychometric Society
• developer of smacof package in R


8. Lawrence J. Hubert:


• former president of the Psychometric Society
• developer of combinatorial analysis
• developer of dynamic programming
• developer of city-block MDS


9. Ingwer Borg and Patrick J. F. Groenen:


• authors of the Bible book of MDS: Borg, I., & Groenen, P.J.F. (2005). Modern multidimensional scaling. 2nd edition. New York: Springer.