Advances in multidimensional unfolding Busing, F.M.T.A.

Advances in multidimensional unfolding

Busing, F.M.T.A.

Citation

Busing, F. M. T. A. (2010, April 21). Advances in multidimensional

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introduction

Multidimensional unfolding is an analysis technique that creates configu- rations for two sets of objects based on the pairwise preferences between elements of these two sets. The distances between the objects correspond as closely as possible with the given preferences between them, such that high preferences correspond to small distances and low preferences to large distances. For example, in 1972, 42 respondents (21 mba students and their spouses) rank ordered 15 breakfast items according to their preference. Unfold- ing now portrays both respondents and items as points in a configuration, as illustrated in Figure 1.1, such that respondents are closest to their first ranked item and furthest from their last ranked item. Moving away in any direction from a respondent’s point thus decreases his/her preference for an item. The respondent’s point itself, the so-called ideal point, thus makes up the high- est point on the respondent’s preference surface, which has the shape of a single-peaked function.

The rank numbers for the 15 breakfast items, albeit 1 to 15 for all respon- dents, are thus retrieved by the distances in the configuration, whether the respondents actually like the breakfast items or not. Furthermore, it is not said that when a blueberry muffin is most preferred by two respondents the muffin is also equally liked by these respondents. To cope with the differ- ences in the preference scale within and between respondents, the actual rank numbers are allowed to be changed in magnitude as long as the order per respondent is maintained. The subsequent respondent-conditional mono- tone transformation of the rank numbers is optimally determined by the least squares unfolding technique, and consequently provides a metric solution from nonmetric data, creating distances from rank orders, respectively.

Notwithstanding the conceptual appeal, unfolding has not been used much in applications in the last few decades. As Heiser and Busing (2004) put it:

“Applications of multidimensional unfolding lag seriously behind, undoubt- edly due to the many technical problems that formed a serious obstacle to successful data analysis …” (Heiser & Busing, 2004, p. 27). The serious ob- stacle concerns degenerate solutions: Solutions that are perfect in terms of optimization of the least squares loss function, but useless in terms of interpre- tation of the unfolding solution. It is a problem that has become a trademark for unfolding. A mature analysis technique should operate faultlessly and this could hardly be claimed of unfolding. The freedom of the monotone transformation, the transformation that changes the nonmetric rank orders

introduction

toast pop-up

buttered toast and jelly

English muffin and margarine

corn muffin and butter

blueberry muffin and margarine

cinnamon toast hard rolls and butter

toast and marmalade

buttered toast toast and margarine

cinnamon bun Danish pastry

glazed donut

coffee cake

jelly donut

Figure 1.1 PREFSCAL unfolding solution for the breakfast data (Green and Rao, 1972) with 42 respondents (represented by dots) and 15 breakfast items.

into metric pseudo-distances, allows the different rank numbers to become (almost) identical. With the distances between the respondents and the items also (almost) identical and in addition equal to the transformed rank numbers, the solution can be achieved which is perfect in terms of fit, but completely worthless in terms of interpretative use. The (nonmetric) unfolding model is no longer identified as the freedom of transformation is such that any arbitrary data set results in a degenerate solution.

This monograph discusses the type of unfolding analysis that suffers from the degeneracy problem. It is characterized by an alternating least squares

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minimization procedure for a multidimensional unfolding model that allows for optimal transformations of the data, irrespective the conditionality of the data. As such, it deviates from other types of unfolding analysis on model specification and minimization procedure. Unfolding irt models (Andrich, 1988, 1989; Roberts & Laughlin, 1996; Roberts, Donoghue, & Laughlin, 2000), for example, exhibit single-peaked, nonmonotonic functions for unidimen- sional polytomous responses, intended for items that discriminate respondents, whereas probabilistic unfolding (Sixtl, 1973; Zinnes & Griggs, 1974; de Soete, Carroll, & DeSarbo, 1986; DeSarbo, de Soete, & Eliashberg, 1987; Ennis, 1993;

MacKay & Zinnes, 1995; Hojo, 1997; MacKay, 2001; Hinich, 2005; MacKay &

Zinnes, 2008) uses a different modeling strategy, using maximum likelihood estimation to obtain the model parameters.

The remainder of this monograph is composed as follows. Chapter 2 discusses the history of the degeneracy problem at length, specifically the sci- entific contributions that uncover, discuss, and resolve the obstinate problem.

Multidimensional unfolding can not be considered a fully fledged analysis technique with this inconvenient problem on the side. The next two chap- ters, Chapters 3 and 4, offer solutions for the degeneracy problem, the former for metric unfolding only and the latter for all possible data transformations.

With the degeneracy problem eliminated, multidimensional unfolding can be developed into a valuable analysis technique. Chapter 5 discusses one such a development: The addition of independent predictor variables to the un- folding model not only enhances the interpretation of the solutions, it also enables us to make predictions. Depending on the available information, this restricted unfolding model uses demographical information on respondents to predict respondent locations and item attributes to predict item locations, or vice versa, that is, the model uses additional locations to predict the variable values. Chapter 6 investigates the extent to which preferences can be missing while still maintaining a proper unfolding solution. This monograph finally discusses some topics for further research.

The technical appendix describes the implementation of the algorithm developed in Chapter 4. A strong extract of the program ’in development’, prefscal , belongs to the categories module of ibm spss statistics since version 14.0 (autumn 2005). The glossary provides insight in the (degen- erate) solution types used throughout the monograph. It may be said that multidimensional unfolding is a truly amazing technique, which can handle all kinds of distance-like data, uses a simple and transparent minimization method (implementation of prefscal), and produces commonly understand- able graphical results. Too bad it was not working from the beginning.

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