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PSYCHOMETRIKA--VOL.61, NO. 3, 485--508 SEPTEMBER1996

A STOCHASTIC MULTIDIMENSIONAL UNFOLDING APPROACH REPRESENTING PHASED DECISION OUTCOMES

FOR

WAYNE S. DESARBO MARKETING DEPARTMENT PENNSYLVANIA STATE UNIVERSITY DONALD R. LEHMANN MARKETING DEPARTMENT COLUMBIA UNIVERSITY GREGORY CARPENTER MARKETING DEPARTMENT NORTHWESTERN UNIVERSITY INDRAJIT SINHA MARKETING DEPARTMENT TEMPLE UNIVERSITY This paper presents a stochastic multidimensional unfolding (MDU) procedure to spatially represent individual differences in phased or sequential decision processes, The specific application or scenario to be discussed involves the area of consumer psychology where consumers form judgments sequentially in their awareness, consideration, and choice set compositions in a phased or sequential manner as more information about the alternative brands in a designated product/ service class are collected. A brief review of the consumer psychology literature on these nested cognitive sets as stages in phased decision making is provided. The technical details of the proposed model, maximum likelihood estimation framework, and algorithm are then discussed. A small scale Monte Carlo analysis is presented to demonstrate estimation proficiency and the appropriateness of the proposed model selection heuristic. An application of the methodology to capture awareness, consideration, and choice sets in graduate school applicants is presented. Finally, directions for future research and other potential applications are given. Key words: consumer psychology, multidimensional scaling, maximum likelihood, consideration sets, multidimensional unfolding, successive categories analysis.

1.

Introduction

M o s t psychological t h e o r i e s o f decision m a k i n g / c h o i c e c h a r a c t e r i z e t h e p r o c e s s as sequential, a d a p t i v e , a n d / o r dynamic, involving v a r i o u s heuristics in s c r e e n i n g t h r o u g h alternatives. Janis (1968) p o s i t e d a five stage s e q u e n t i a l d e c i s i o n p r o c e s s w h e r e "as he surveys t h e a l t e r n a t i v e s d u r i n g the early stages, t h e d e c i s i o n m a k e r dismisses o r e l i m i n a t e s f r o m f u r t h e r c o n s i d e r a t i o n any a l t e r n a t i v e that a p p e a r s to b e t o o ineffectual o r t o o costly a m e a n s o f d e a l i n g with t h e c h a l l e n g e " (Janis & M a n n , 1977, p. 174). T h e d e c i s i o n m a k e r n a r r o w s d o w n his list o f a l t e r n a t i v e s to those t h a t a p p e a r to b e successful o p t i o n s w i t h o u t entailing i n t o l e r a b l e costs o r risks. I n Janis's (1968) l a t e r stages, m o r e t h o r o u g h search a n d

Requests for reprints should be sent to Wayne S. DeSarbo, Department of Marketing, Smeal School of Business, Pennsylvania State University, University Park PA 16802. 0033-3123/96/0900-94213500.75/0 © 1996 The Psychometric Society

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evaluation is performed on the "surviving" alternatives, where focus is placed on pros and cons in an effort to select the best course of action. According to Payne (1976, 1982), the exact process of this narrowing down of alternatives appears to depend upon the difficulty of the decision task. Payne, Bettman, and Johnson (1990) state that decision problems consist of three basic components: the alternatives available to the decision maker, events or contingencies that relate actions to outcomes as well as their associated probabilities, and the values associated with the outcomes. Decision tasks become more onerous with more alternatives, multiple contingencies, and multiple conflicting dimensions of value. And, depending upon the difficulty of the decision task, there is much research to support a variety of information processing strategies (Einhorn 1970a, 1970b, 1970c) being utilized to narrow the set of alternatives down at different stages of the decision process (see Abelson & Levi, 1985; Bettman, Johnson, & Payne, 1993; Johnson & Meyer, 1984; Lussier & Olshavsky, 1979). Here, as with Janis's (1968) theory, more time consuming compensatory processes appear to be utilized with smaller numbers of alternatives as one approaches a final decision. Further elaboration of this decision process in the consumer psychology literature involving brand choice is provided in the next section. 2.

Awareness, Consideration, and Choice Sets in Consumer Psychology

For many years, the standard approach for representing decision making in consumer psychology has been the traditional compensatory model. Beginning with modifications of linear attitude models (e.g., Fishbein, 1967; Rosenberg, 1956; Wilkie & Pessemier, 1973) and continuing with share-focused models (e.g., McFadden, 1978), a tremendous amount of research has focused on choice among a known set of alternatives. While such models have proved extremely useful, there are many situations where the number of alternatives is so great that consumers are clearly unaware of many of the alternatives, not to mention their attribute values. In such situations, consumers seem likely to follow some type of phased decision process (Bettman, 1979), and a number of models have been developed which formally incorporate distinct choice stages (Gensch, 1987; Hauser & Gaskin, 1984; Hauser & Shugan, 1983; Silk & Urban, 1978; Urban, Hulland, & Weinberg, 1993). Phased decision processes are, in their most simplified form, processes that, in stages, reduce the set of possible choices to smaller and smaller sets until a choice is made. The concept of choice being limited to a subset of available brands can be traced back to the work of Howard and Sheth (1969). Howard and Sheth define an evoked set as "those brands the buyer considers when he/she contemplates purchasing a unit of the product class" (p. 416). A number of studies have demonstrated the existence of evoked sets (Belonax, 1979; Belonax & Mittelstaedt, 1978; Campbell, 1969; Crowley & Williams, 1991; Narayana & Markin, 1975; Spiggle & Sewall, 1987). Limiting consideration to a set of brands is related both to memory capacity (Biehal & Chakravarti, 1986; Kardes, Kalyanaram, Chandrashekaren, & Dornoff, 1993; Nedungadi, 1990a) and to the costs versus benefits of considering a brand (Hauser & Wernerfelt, 1990; Roberts & Lattin, 1991). The notion that consumers seek several simplifying strategies to make their decisionmaking task easier, given their complex choice environment, is not a new one (see Wright, 1975). From this perspective, the concept of a phased decision process (Bettman 1979; Bettman & Park, 1980; Wright & Barbour, 1977) is particularly relevant in representing the actual choice behavior. However, while the idea of phased choice is fairly intuitive and is widely accepted, the exact process by which it occurs is not fully understood. Issues pertaining both to the identity of the stages and the intervening process by which one proceeds from stage to stage have not yet been fully addressed. One way to conceptualize

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phased choice is as a gradual narrowing down of attribute levels considered potentially acceptable until, ultimately, only one combination of levels (and hence brand/alternative) remains. This is the basic spirit of Tversky's Elimination by Aspects model (Tversky, 1972) and derivatives such as Elimination by Tree and Hierarchical Elimination methods (Tversky & Sattath, 1979). The general approach of narrowing down the choice alternatives based on attribute levels has been adopted in two-stage choice models (Gensch, 1987) as well as concept learning systems (Currim, Meyer, & Lee, 1988). This narrowing occurs due to a desire to simplify the decision (see Simon, 1957; Shugan, 1980) and tends to eliminate extreme alternatives (note that the aversion to extreme alternatives has been well documented, see Simonson & Tversky, 1992). As more and more "extreme" alternatives are eliminated, the feasible region shrinks toward the ultimate choice. Notice that this reducing tolerance for extreme alternatives can be viewed as a basic learning phenomenon related to aging/ maturity (e.g., children become more "discriminating") and expertise (experts are more sensitive to smaller differences on important attributes, Alba & Hutchinson, 1987). Shocker, Ben-Akiva, Boccara, and Nedungadi (1991) present an exposition of a nested or hierarchical choice set framework involving sequential elimination in their review paper on the role of consideration sets in consumer decision-making (see also Boccara, 1989; Brisoux & Cheron, 1990; Brisoux & Laroche, 1980, 1981; Brown & Wildt, 1992; Klenosky & Rethans, 1989; Negungadi, 1987). In their scheme, at any given purchase situation, a consumer is capable of retrieving a finite set of alternatives, either from memory or the external environment, which they call the awareness or knowledge set. Within the awareness set, the members which are congruent and diagnostic to the given choice problem are then actively processed by the consumer for further consideration; hence the name consideration set. Though basically unobservable, consideration sets can be deduced from protocol statements or information search records. A number of direct questioning methods have also been used. These include asking about brands that are seriously considered (Narayana & Markin, 1975; Nedungadi, 1990b) and/or brands that are acceptable (Brisoux & Laroche, 1981). Perhaps the most prevalent view of consideration sets is that they are the result of various cost/benefit tradeoffs (Hauser & Wernerfelt, 1990; Roberts & Lattin, 1991). Consideration sets are often viewed as resulting from various constraints (Ratneshwar & Shocker, 1991; Swait, 1984). They also are influenced by market factors such as the order of entry (Lehmann & Pan, in press). It is possible that with ongoing information search about the alternatives, several alternatives will be discarded (or a few others added) as the choice process unfolds. Consequently, immediately prior to the choice stage, the consumer may only be deliberating on a reduced set of alternatives, called the choice set. As Shocker et al. (1991) note, this conceptualization assumes a dynamic nature of consideration set evolution as the consumers actively add and delete alternatives before the final decision. Figure 1 displays the relationship of these three nested cognitive sets: the awareness, consideration, and choice sets. As noted earlier, the notion of a nested alternative set framework, despite having intuitive appeal, has not explicitly been addressed within an empirical modeling framework. But given the current topicality of and burgeoning interest in consideration set research (see Shocker et al., 1991), it is clear that there is a definite need to provide an analytical approach to modeling this important consumer choice phenomenon. This paper provides a stochastic multidimensional unfolding approach for paramorphically representing the awareness, consideration, and choice sets of actual consumers in a spatial manner (we are not positing a model of decision making). Thus, the paper extends the work of Brisoux and Laroche (1981) and Roberts and Lattin (1991) who geometrically describe consideration sets. The advantages of using an MDS framework in this context are transparent: it provides a natural approach toward representing the alternative set evolution in

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AWARENESS SET

CONSIDERATION SET

CHOICE SET FIGURE 1. Nested cognitive sets.

a multidimensional spatial configuration. Further, it is possible to interpret the dimensions as the underlying attributes for the preferred vs. discarded alternatives. Finally, distance between brands and consumers in the derived space are interpretable (see DeSarbo & Rao 1984, 1986) with threshold parameters estimated to define the boundaries of these nested cognitive sets. In subsequent sections, we describe the model and the maximum likelihood estimation framework. A small scale Monte Carlo analysis is provided to investigate estimation

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proficiency as well as model selection heuristics. Then, we discuss an application of the model to the selection of a graduate business school for students in two universities, and analyze the spatial representations of the resulting nested alternative sets. Finally, we discuss the implications of this procedure and outline future research possibilities. 3. A.

Mapping Awareness, Consideration, and Choice Sets

The M o d e l

Our objective is to represent the three distinct phases of choice, the awareness, consideration, and choice sets, as nested concentric spheres in a continuous dimensional space. Let: i j t Y~)

= = = =

1,..., 1,..., 1, . . . , {~

I consumers; J brands in a specified product class; T replications (e.g., time periods); if consumer i is aware of brand~ in time t, else;

y(2) {~ qt =

if consumer i considers purchasing brandj in time t, else;

Y~3t) = {~

else; if b r a n d j is in c o n s u m e r i's choice set in time t,

k = 1 , . . . , K dimensions; aig = k-th coordinate of consumer i; bjk = k-th coordinate of brandj.

This nested or phased decision process is described as a sequential phenomena shown in Figure 2. The tree implies that ify~lt ) = 0, then y~) = y~) = 0 by definition (Node (1) not aware). If y~jlt) = 1 and y~) = 0, then y~) = 0 by definition (Node (2) = aware, but not considered). Node (3) (aware, considered, but not in choice set~ occurs whenv!~ ) = v!~) = 3 , -'ljr -" lJf 1, but y!!,) = 0. Node, (4) (aware, considered, in choice set] describes the case where -'tJt v!~) = v!~ ) (3) q~t ~ -'t.lt = Ybi = j" Inus, umess a consumer is aware of a brand, the brand cannot be in the consumer's consideration or choice sets. Similarly, unless the consumer is aware of and considers a brand, the brand cannot be in the consumer's choice set. Here, the terminal nodes define branches which represent the probabilities /

P ( N o d e 1) = p(,,(1) = 0),

(1)

P ( N o d e 2) = P(yij, -. (1) = 1, y~) = 0),

(2)

P ( N o d e 3) = l'(yiy -- (1) . (3) , = 1, y}~)= 1, Yij, P(Node

4) =

Ptyij,"(1) =

0);

(3)

1,- Y~i," (2) = 1, y})3)= 1).

(4)

=

As such, it is trivial to show that P ( N o d e 1) + P ( N o d e 2) + P ( N o d e 3) + P ( N o d e 4) = 1

(5)

by definition of joint probabilities. We now aefine a latent, unobservable distance construct in a K dimensional space which represents the squared Euclidean distance (perturbed by error) between the spatial location of the consumers (aik'S) and the brands (bjk's):

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PSYCHOMETRIKA

Aware:

No

Consider:

Awareness Set

Yes

Consideration Set

No

Yes

Choice:

Choice Set

No

Yes

/

/

Terminal Node: (1)

/

(2)

\

(3)

(4)

FIGURE 2.

A nested, sequential process: four terminal nodes defined. K

Zq, = E (aik -- byk) 2 + eqt,

(6)

k=l

where the error component (eijt) is assumed iid N(0, o'~1). This latent distance construct is defined such that:

Y(lt)

=

v(2) :i# = y}~/=

1

if Zij t ~ C} 1),

0

else;

'

(7)

0

if Zij, ,(') - 4,,~(')

Y~12(1-y~2))J * 1 ( ' ) _ w(1)~(2){1

ahi

(A-3)

k

,,,, Oc~

)

4,,(')

Yijt,(l)(l--YiflJ , l ( . 7 ~ ~ j 2 ( . ) ]

(~b2(') )] * 2 ( " ) ~ ~)3(" ) '

~ -2h,], J

(A-5)

where: *n(') = standard normal c d f for n-th stage evaluated at (.), n = 1, 2, 3. 4)n(') = standard normal distribution for n-th stage evaluated at (-). For external analyses, the appropriate set of partial derivatives are set equal to zero. With these partial derivatives specified, the conjugate gradient procedure with automatic restarts can be described as follows: (i) Start with initial parameter estimates ~//I), and set the iteration counter I C = 1. rk here denotes a vector, stacking all the ~arameters to be estimated. (ii) Set the first search direction S O) = - V (In L) 0), where V (In L) (1) denotes the gradient vector of the log-likelihood function evaluated at ~0). (iii) Find ik(2) using the relationship ik(2) = ik(l) + h(1)S('),

(A-6)

where A0) is the optimal step-size in the gradient direction S (0. A quadratic interpolation method is used for estimating the optimal step-size. Set IC = 2. (iv) Calculate V (In L ) (m) and set the (new) search direction:

WAYNE S. DESARBO ET AL,

S ~1c) = - V (In L ) ~1cl + 9 S Ilc- 1),

505

(A-7)

if I C = 2 or if a restart is needed. In this algorithm, restarts are made every M iterations (M is the number of parameters to be estimated) or when the search direction is not "sufficiently downhill". If this is the case, then set R = I C and go to Step (vi). R reflects the number of the iteration where a restart is made. Otherwise, set: S ~1c) = - V (In L ) - IC*). Phase 111." Normalization and Output

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Manuscript received 6/6/94 Final version received 2/20/95