Comparisons of elimination by aspects and suppression of aspects choice models based on choice response time

JOURNAL

OF MATHEMATICAL

PSYCHOLOGY

32, 341-349

(1988)

Theoretical

Note

Comparisons of Elimination by Aspects and Suppression of Aspects Choice Models Based on Choice Response Time JEROME R. BUSEMEYER Purdue

Uniuersitr

BARBARA Ohio

FORSYTH

State

Uninersit~

AND GEORGIE NOZAWA Purdue

Unitlersil~

A. A. J. Marley (1981, Psychomefrika, 46, 421428) proposed an extension of A. Tversky’s (1972a. Psychological Revietv, 79, 281-299) elimination by aspects (EBA) model of preferential choice that generates choice response time predictions. In this article we propose a similar extension of F. Restle’s (1961, Psychology of judgmenr rind choice, New York: Wiley) suppression of aspects (SOA) model of conflict resolution. Although both models make identical predictions regarding binary choice probability, we show that it is possible to empirically distinguish the two models with a parameter-free test based on choice response time. Previous research on preferential choice response time supports the extended SOA model over the extended EBA model for the binary choice case, but the opposite is found when more than two alternatives are presented. ‘(71 1988 Academic Press. Inc.

An important empirical finding from research on preference is that the similarity between two alternatives influences choice probability (e.g., Rumelhart & Greeno, 1971). One of the earliest models capable of explaining this effect was Restle’s (1961) suppression of aspects (SOA) model. Later, Tversky (1972a, 1972b) Send all correspondence to: Jerome R. Busemeyer, Psychological Sciences. Purdue University, West Lafayette, IN 47907. The authors thank S. E. Edgell, A. A. J. Marley, and Barbara Mellers for their comments. Requests for reprints should be sent to Jerome R. Busemeyer, Psychological Sciences, Purdue University. West Lafayette, IN 47907. This work was partially supported by NSF Grant BNS 8710103.

341 0022-2496/88 CopyrIght

‘i: 198X by Academtc

$3.00 Press. Inc.

342

THEORETICAL

NOTE

proposed a more general model of this effect called the elimination by aspects (EBA) model. Although the SOA and EBA models were derived from different hypothetical choice processes, they make identical predictions for binary choice probability. Suppose A and B are two alternatives. The measure of the unique valued aspects contained in A and not B is symbolized by a6, and the measure of the unique valued aspects contained in B and not A is symbolized by bti. According to both models, the probability of choosing A over B, ‘P[A > B], equals P[A > B] = a6/(a6 + ba).

(1)

Given that the SOA and EBA choice processes differ, it is natural to wonder whether these two models can be empirically distinguished on the basis of choice response time. Restle (1961) and Tversky (1972a, 1972b) did not attempt to derive choice response time predictions from either model. More recently, however, Marley (1981) proposed a natural extension of the EBA model that does generate response time predictions. The purpose of this article is to describe a natural extension of the SOA model that also generates response time predictions, and to show how one can empirically distinguish the latter two models with a parameter-free test based on response times obtained from a single binary choice problem. The remaining part of this note is organized as follows. First, the predictions for choice response time derived from the extended SOA model are described. Then the SOA model is compared to Marley’s (1981) extended EBA model. Finally, these and other models are evaluated on the basis of empirical results reported by Petrusic and Jamieson (1978). Although this note is primarily concerned with binary choices, extensions to multiple (two or more) choice problems are described briefly at the end. 1. SOA Choice Process According to the SOA model (cf. Chaps. 2 and 4 of Restle, 1961), each of the two competitive alternatives is described by a finite set of unique valued aspects, i.e., valued aspects that are a property of one alternative and not the other. Because each alternative provides unique valued aspects that cannot be obtained from the competing alternative, a state of conflict is induced. An attempt to resolve the conflict is made by suppressing aspects. Perhaps if some feature is ignored, then one alternative will clearly dominate the other and a choice can be made. If the first suppression does not resolve the conflict, then another aspect is suppressed, and if this second suppression also fails, then more aspects are suppressed one at a time. This process of sequential suppression continues until one alternative is depleted of all its unique aspects, and consequently, the other alternative dominates. Figure 1 illustrates the suppression of aspects choice process. The horizontal axis represents the number of unique valued aspects that remain in alternative B after suppressing a total of n unique aspects, denoted b(n). The vertical axis represents the number of unique valued aspects that remain in alternative A after suppressing a total of n unique aspects, denoted a(n). The choice process starts at the upper

THEORETICAL

343

NOTE

a@).W)

/

r’

I

a(d.b(d \ I-n ”

,

0

b5

NUMBER

FIG.

1.

A graphical

illustration

OF UNIQUE

of the suppression

ASPECTS

of aspects

IN ALT

choice

B

process

for two alternatives.

right hand corner of the figure with n =O, a(0) =a& and 6(O) =6a. The positive integers a6 and bti represent the number of unique valued aspects associated with alternative A and B, respectively, at the beginning of the choice process. The polygonal line moving from the upper right corner to the lower left corner represents a sample path of the choice process that ends after N suppressions of unique aspects, where N is such that a(N- 1) = 1, u(N) = 0, and b(N) > 0 so that alternative B is chosen. The random variable, N, represents the (minimum) number of unique aspects that must be suppressed in order to reach a decision. Common aspects belonging to both alternatives may also be suppressed in between suppressions of unique aspects. If c(0) = c is defined as the number of common aspects at the start of the, choice process, and c(N) is the number of common aspects remaining at the end of the choice process, then M= c - c(N) is the number of common aspects that are suppressed prior to making a decision. Thus, the total number of common plus unique aspects suppressed prior to making a decision equals M + N. Restle (1961, p. 33) assumed that aspects are suppressed randomly and without replacement, which implies that the probabilities of the sample paths in Fig. 1 can be described by a Markov chain with state transition probabilities P[u(n+1)=(x-1),b(n+1)=y)u(n)=x,b(n)=y]=x/(x+y),

@a)

P[a(n+l)=x,b(n+l)=(y-l)Ju(n)=x,b(n)=y]=y/(x+y).

(2b)

Equation (1) can be derived directly from (2) (see Appendix A). The conditional mean number of unique aspects that must be suppressed to

344

THEORETICAL

reach a decision favoring alternative expression (see Appendix A):

NOTE

B can be obtained

from the following

E[N(B>A]=(ab)(aB+bii)/(a6+1).

(3)

The corresponding value for alternative A, E[N / A > B], can be obtained from (3) by reversing the roles of a6 and b&. Note that the difference

is positive if a6 > bti, negative if ab < bti, and zero if ab = bG. The conditional mean number of common aspects that are suppressed prior to making a decision favoring alternative B is denoted E[Ml B>A], and E[Ml A > B] is the corresponding value when alternative A is chosen. The difference E[Ml B > A] - E[M) A > B] is greater than or equal to zero if a6 > ba, it is less than or equal to zero if a6 < bti, and it is equal to zero if a6 = bti (see Appendix A). 2. Choice Response Time Assumptions

Choice response time (denoted CRT) can be decomposed into three components. The first component, denoted NT, is the sum of the N random durations produced by the suppression of the unique aspects (NT = X( 1) + . . + X(N), where X(k), k = 1, N are positive valued random variables). The second part, denoted MT, is sum of the M random durations produced by the suppression of the common aspects (MT= Y(1) + ... + Y(M), where Y(k), k = 1, M are positive valued random variables). Finally, the third part is the random duration of the time required for extraneous activities such as stimulus encoding and motor movements (denoted ET). Thus, CRT = NT+ MT+ ET. For a given pair of choice alternatives, the following relatively weak distribution assumptions are employed. The random duration produced by suppressing each unique aspect is assumed to have a common mean pL,. Similarly, the random duration produced by suppressing each common aspect is assumed to have a common mean pLu. Finally, the mean duration of the extraneous activities is assumed to be a constant denoted pp. Note that these three parameters (p,, pv, p,) are constant across alternatives for a given choice pair, but they may vary across different choice pairs. The mean decision time conditioned on the choice of alternative B over A equals EICRTIB>A]=p,J?[N(B>A]fpL,..EIM(B>A]+p,.

(4)

By considering Eqs. (1 ), (3), and (4), one can derive the following parameter-free test of the SOA model. According to ( 1). P[A > B] > .5 implies a6 > bti; according to (3), ab>bii implies E[NJB>A]>E[NlA>B] and E[MIB>A]z E[Ml A > B]; according to (4), P[A > B] > P[B>

A] implies E[CRT(

B> A] > E[CRT(

A > B].

(5)

THEORETICAL

NOTE

In other words, the SOA model predicts that choice probabilities response times are inversely ordered for a pair.

345 and mean choice

3. EBA Choice Process For binary choices, the EBA process terminates as soon as the first unique aspect is sampled (cf. Tversky, 1972a). The process begins by selecting a valued aspect. If a unique aspect from alternative A is selected, then alternative B is eliminated (and consequently A is chosen since there are only two alternatives). If a unique aspect of alternative B is selected, then alternative A is eliminated (and consequently B is chosen). If the selected aspect is a property common to both alternatives, then neither alternative is eliminated and another aspect is selected. This process continues until a unique aspect of A or B is eventually selected resulting in the choice of A or B, respectively. Marley (1981) extended the EBA model to account for choice response time by assuming that the distribution of latencies to select a unique aspect from alternative A is represented by a Poisson process with a rate parameter, ~6. Similarly, the distribution of latencies to select a unique aspect from alternative B is represented by a Poisson process with a rate parameter, bti. The distribution of latencies to select a common aspect is represented by another Poisson process, but this event would not eliminate an alternative. The three Poisson processes are assumed to be independent and parallel. Marley’s model makes the following predictions for mean decision time conditioned on the alternative selected: E[CRTl

A > B] = E[CRT(

B > A] = l/(a6 + ba).

(6)

(see page 276 of Townsend & Ashby [1983] and set KO = K, = 1). In other words, mean choice response time is independent of the alternative chosen.’ 4. Alternative Models and Empirical Results There are very few articles on preferential choice that include measures of mean choice response time conditioned on the alternative chosen. However, an experiment by Petrusic and Jamieson (1978) was designed specifically for this purpose. Petrusic and Jamieson (1978) were interested in testing the empirical implications of several psychophysical models of choice response time. Most relevant are the stochastic counter models which include the simple random walk (or difference) model, the simple accumulator (or recruitment) model, and the runs model (see ‘The present authors have explored another extension of the EBA model. Using exactly the same Markov chain representation of the EBA model described by Tversky (1972b, Table l), it follows that the probability of sampling n common aspects prior to sampling a unique aspect, given that alternative B is chosen over A, equals c” (1 - c), where c is the probability of sampling a common aspect. Note that this conditional probability is independent of the alternative chosen. If decision time is a monotonically increasing function of the number of common aspects selected prior to making a final choice, then relation (6) follows.

346

THEORETICAL NOTE

Audley & Pike, 1965, for details). As Petrusic and Jamieson (1978) point out, the simple accumulator model predicts an inverse relation between choice probability and choice time similar to (5). The (unbiased) simple random walk model predicts that mean choice time is independent of the response chosen similar to (6). The predictions for the runs model are less clear-cut--either relation (5) or (6) can occur depending on the run length parameter of the model (cf. Vickers, 1979, p. 59). Petrusic and Jamieson (1978) investigated three different types of choice tasks-preference between gambles, acceptability of opinion statements, and aesthetic appeal of geometric forms. The gambles were of the form “win $x with probability p or lose !$y with probability (1 - p).” In general, the stimuli were carefully constructed so that it was reasonable to assume that the mean of the extraneous time was constant across alternatives within a pair. The results consistently favored the prediction derived from the extended SOA model (5), and consistently violated the qualitative prediction derived from the extended EBA model (6), for all three types of stimuli (see Fig. 1 of Petrusic & Jamieson).2 Although Marley’s (1981) version of the EBA model provides an inadequate description of choice response times for binary choices, he has pointed out that the Poisson model can be modified by assuming that more than one unique aspect favoring an alternative must be accumulated before a decision is reached, similar to the counter model of Townsend and Ashby (1983, Chap. 9, with K0 = K, > 1). However, this modified model no longer generates the same choice probabilities as the EBA model. 5. Multiple (Two or More) Choice Problems

Restle (1961, pp. 69-73) proposed a sequential binary elimination process for choice problems with more than two alternatives. An initial pair of alternatives is selected for comparison and the loser is eliminated. The winner is retained and is compared with another alternative. This process continues until each alternative is compared with the previous winner, and the final winner that survives all the 2 The results shown in Fig. 1 of Petrusic and Jamieson (1978) can be used to test the relations in (6) and (7) as follows. Each pair was presented on five trials. If one alternative in a pair was chosen on x trials, then the other member was chosen on (5 -x) trials. Thus the mean choice response time for alternatives chosen on x trials can be compared with the mean choice response time for alternatives chosen on (5 - x) trials. The results shown in Figs. 2 and 3 of Petrusic and Jamieson (1978) do not provide a direct test of relations (5) or (6) because unlike Fig. 1, each member of each pair of alternatives was included in the computation of different means on different replications. Thus it is not possible to isolate the means that need to be compared. Another problem is that the results shown in Figs. 2 and 3 of Petrusic and Jamieson (1978) are difficult to interpret because they are confounded with a very large decrease in latency due to practice as indicated in their Table 1. For example, they found that the mean choice response time for an alternative chosen only once on the fifth trial was faster than the mean choice response time for an alternative chosen twice-once on the Iifth trial and once on an earlier trial. But this difference may have resulted from the fact that responses on the fifth trial were generally much faster than responses on earlier trials.

THEORETICAL

NOTE

347

comparisons is finally chosen. The mean response time for a given sequence of comparisons and selections equals the sum of the individual mean binary choice response times produced by the sequence. Marley (1981) developed a stochastic process model compatible with Tversky’s (1972a, 1972b) multiple choice EBA model. Each aspect is associated with a separate Poisson process producing a family of Poisson processes that race in parallel. Marley (1987) provides closed form solutions for choice response time probabilities for a class of models closely related to the EBA model. Perhaps the simplest way to empirically test these two multiple choice models is to investigate the effects of manipulating the number of alternatives on the mean choice response time. According to a sequential processing model, increasing the number of equally attractive alternatives should increase mean choice response time. According to a parallel processing model, increasing the number of equally attractive alternatives may decrease mean choice response times3 Keisler (1966) investigated the effects of number of alternatives on mean choice response time using a preferential choice task. One group of children was asked to choose one of four equally attractive well-known candy bars, and a second group was asked to choose one of two candy bars (where the two candy bars presented to the second group were randomly sampled from the original set of four presented to the first group). The results indicated that mean choice response time for four alternatives was less than or equal to the mean choice response time for two alternatives (despite the fact that the mean time to make an identification response was longer for the four alternative problem). While this result is consistent with a parallel processing model such as Marley’s (1981) version of the EBA model, it is inconsistent with a sequential processing model such as the one proposed for the SOA model. In conclusion, the SOA model provides a better description for the amount of time it takes to make a binary decision than known extensions of the EBA model, but the opposite seems to be true when four alternatives are presented. However, these results were obtained from a restricted set of choice stimuli, and further research is needed before a general conclusion can be supported. Nevertheless, this analysis of past research provides further evidence for changes in choice strategies as a function of the size of the choice set (cf. Payne, 1982). Furthermore, these results demonstrate the utility of using choice response time to distinguish among competing models of preferential choice that are indistinguishable on the basis of choice probability.

3 For example, if the ensemble of aspects associated with all n alternatives can be partitioned into n sets of equally valued unique aspects (a unique set for each alternative) and one set of aspects common across all alternatives, then the parallel Poisson model predicts that the mean time to make a decision equals l/(n u), where u equals the rate parameter associated with each of the equally attractive set of unique aspects associated with each of the n alternatives (see Marley, 1987, Example 1).

480,‘32/3-IO

THEORETICALNOTE

348

APPENDIX

A

Derivation of Eqs. ( 1) and (3). Equations ( 1) and (3) can be derived from simple combinatorial principles. For convenience, let C(n, r) = n!/[(n - r)!r!] be the number of ways to choose r objects from n objects. Define the event Bj as the set of all sample paths having the property N = (a6 + j), and a(N) = 0, and b(N) = (bti - j). All paths in Bj must pass through the state a(N - 1) = 1, b(N - 1) = (ba - j). In other words, event Bj occurs when all aspects of alternative A are suppressed, j aspects of alternative B are suppressed, and alternative B is chosen. In a similar manner define Aj as the event that occurs when all aspects of alternative B are suppressed, j aspects of alternative A are suppressed, and alternative A is chosen. The number of sample paths in Bj equals C(a6 + j- 1, a6 - 1). The transition probabilities shown in (2) imply that each path in Bj has the same probability equal to ljC(a6 + bci, ah). Finally, the probability of the event Bj equals P[B,]=C(a6+j-l,a6-1)/C(ati+bti,a@

for j = 0 to (bti - 1) and zero otherwise. The probability obtained by reversing the roles of a6 and ba. The probability of choosing alternative B is obtained index j to yield

of the event Ai can be by summing

across the

P[B>A]=xP[B,]=bti/(bti+a@,

and (1) is obtained from 1 - P[B > A]. The probability of suppressing n = (a6 + j) unique aspects given that alternative B is chosen equals P[N=a6+j\B>A]=P[Bj]/P[B>A] =C(a6+j-1,a6-l)/C(a6+bii-l,a6),

for j = 0 to (bti - l), and P[N = n 1B > A] = 0 otherwise. The probability of suppressing n unique aspects given that alternative A is chosen, denoted P[N = n 1A > B], is obtained by reversing the roles of a6 and bci. The conditional mean number of unique aspects that must be suppressed to reach a decision favoring B is obtained by summing across n to yield E[NlB>A]=~n.P[N=n(B>A]=a6.(ab+bt?)/(a6+1),

which is Eq. (3). The conditional mean for alternative obtained by reversing the roles of a6 and ba. The marginal mean equals E[N]=P[A>B].E[N(A>B]+P[B>A].E[NlB>A] =a6.ba/(bii+

l)+ba.a6/(a6+

1).

A, E[N( A > B], can be

THEORETICAL

349

NOTE

It may be worth noting that P[A > B]/P[B

> A] = (E[NI

B > A] - E[N])/(E[N]

- E[Nl A > B]).

Finally, we need to show that if a6 > bti then E[Ml B> A] zE[M(A > B], where M is the number of common aspects that are suppressed prior to making a choice. This conclusion follows from two properties of the SOA model. The first property is that for a fixed value of IV, the probability distribution of M is independent of the alternative chosen. This follows from the assumption that common aspects are randomly sampled from the total number of aspects remaining at each stage of the suppression process. The mean of M conditioned on N is denoted m(N) = E[Ml N]. Note that m(N) is an increasing function of N. The second property is called likelihood ratio ordering (cf. Ross, 1983). Denote L(n) as the likelihood ratio L(n)=P[N=nIB>A]/P[N=n)A>B].

Note that L(n) is a nondecreasing function of n when a6 > bij. The likelihood ratio ordering property together with the fact that m(N) is an increasing function implies that E[m(N) 1B> A] 2 E[m(N) (A > B] (see Ross, 1983, Chap. 8; also see Townsend & Ashby, 1983, Chap. 9). Finally, note that E[MIB>A]=E(E[MIN]IB>A)=E[m(N)IB>A] and similarly E[MIA>B] = E[m(N)

1A > B]. REFERENCES

R. J., & PIKE, A. R. (1965). Some alternative stochastic models of choice. Brirish

AUDLEY,

Mathematical

and Sfatistical

Psychology,

Journal

of

21, 183-192.

C. A. (1966). Conflict and number of choice alternatives. Psychological Reports, 18, 603-610. A. A. J. (1981). Multivariate stochastic processes compatible with “aspect” models of similarity and choice. Psychometrika, 46, 421428. MARLEY, A. A. J. (1987). An overview of random utility models with Weibull marginals and closedform choice probabilities and choice reaction times. Unpublished manuscript. PAYNE, J. W. (1982). Contingent decision making. Psychological Bulletin, 92, 382-402. PETRUSIC, W. F., & JAMIESON, D. G. (1978). Relation between probability of preferential choice and time to choose changes with practice. Journal of Experimental Psychology: Human Percepfion and PerforKEISLER, MARLEY,

mance,

4, 471482.

(1961). Psychology ofjudgment and choice. New York: Wiley. Ross, S. M. (1983). Stochastic processes. New York: Wiley. RUMELHART, D. L., & GREENO, J. (1971). Similarity between stimuli: An experimental test of the Lute and Restle choice models. Journal of Mathematical Psychology, 8, 370-381. TOWNSEND, J. T., & ASHBY, F. G. (1983). Stochastic modeling of elementary psychological processes. London/New York: Cambridge Univ. Press. TVERSKY, A. (1972a). Elimination by aspects: A theory of choice. Psychological Review, 79, 281-299. TVERSKY, A. (1972b). Choice by elimination. Journal of Mathematical Psychology, 9, 341-367. VICKERS, D. (1979). Decision processes in perception. New York: Academic Press. RESTLE,

F.

RECEIVED:

September 3, 1987