Simulation Experiments in Choice Simplification: The Effects of Task and Context onForecasting PerformanceAuthor(s): Rick L. Andrews and Ajay K. ManraiSource: Journal of Marketing Research, Vol. 35, No. 2 (May, 1998), pp. 198-209Published by: American Marketing AssociationStable URL: http://www.jstor.org/stable/3151848 .

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RICK L. ANDREWS and AJAY K. MANRAI*

Experimental and protocol-based research has demonstrated convinc- ingly that consumers frequently use simplification heuristics prior to mak- ing choices. Consequently, quantitative choice models incorporating simplification strategies recently have received much research attention. Given the additional computational cost and complexity involved in esti- mating these models, the authors investigate how robustly the standard logit model accommodates choice simplification processes. The results show that when consumers screen brands using an elimination-by- aspects strategy, logit forecasts choices reasonably well, but the para- meter estimates are severely biased. However, when consumers screen through brands previously purchased, augmented by brands promoted, logit both forecasts poorly and produces biased parameter estimates, even when dedicated screening variables are included in the logit utility function. The predictive validity of logit improves somewhat when the uni- versal set size is small and when purchase event feedback is less

important.

Simulation Experiments in Choice

Simplification: The Effects of Task and

Context on Forecasting Performance

There is compelling evidence that consumers frequently use some type of simplification heuristic prior to making choices (Andrews and Srinivasan 1995; Einhom 1971; Gensch 1987; Huber and Klein 1991; Lussier and 01- shavsky 1979; Manrai 1995; Payne 1976; Siddarth, Bucklin, and Morrison 1995; Wright and Barbour 1977). The first stage of the choice process, which typically relies on effi- cient noncompensatory decision rules, simplifies the deci- sion by reducing the number of brands consumers consider. The second stage appears to employ more cognitively de- manding but more accurate compensatory strategies to eval- uate the considered brands and make the final choice.

The effects of decision task and context factors on the forecasting accuracy of the ubiquitous compensatory deci- sion model have been studied widely (e.g., Einhom 1971; Johnson and Meyer 1984; Johnson, Meyer, and Ghose 1989). In general, the compensatory logit choice model is thought to be fairly robust to variations in the choice task and context conditions, though simulation experiments have

*Rick L. Andrews is Assistant Professor of Marketing and Ajay K. Man- rai is Associate Professor of Marketing, Department of Business Adminis- tration, University of Delaware (e-mail: andrewsr@udel.edu; manraia@ udel.edu). The authors thank the editor and four JMR reviewers for their contributions to this study.

Journial of Marketing Research

Vol. XXXV (May 1998), 198-209

revealed some unfavorable environments (e.g., Johnson, Meyer, and Ghose 1989). However, if consumers screen some brands prior to choice, as is suggested strongly by the literature, then choice set misspecification is an inherent problem, because logit assumes that all consumers evaluate the full set of available brands, known as the universal set (Shocker et al. 1991). The effect of this misspecification on the forecasting accuracy of logit is not known, though bias in parameter estimates is certainly to be expected (Swait 1984; Williams and Ortuzar 1982).

Recent research has extended logit choice models to re- flect the preliminary choice simplification strategies that consumers appear to use in verbal protocols (Gensch 1987). For example, studies by Andrews and Srinivasan (1995), Gensch and Soofi (1995), Roberts and Lattin (1991), and Siddarth, Bucklin, and Morrison (1995) develop extensions of the multinomial logit choice model that explicitly model the formation of choice sets prior to choice. These studies consistently demonstrate that models accounting for choice simplification forecast consumer choices better than logit, which does not incorporate choice simplification strategies.

However, the forecasting advantages of two-stage choice simplification models relative to logit are at the expense of simplicity and ease of implementation. Estimation software is not readily available, and the time and computer resources required to implement these models often greatly exceed the

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Simulation Experiments in Choice Simplification

requirements for logit models. Given this trade-off between forecasting accuracy and complexity, it is possible that log- it will accommodate choice simplification effects reason- ably well under some circumstances.

In this study, we examine how robustly the standard log- it choice model accommodates two-stage choice processes. Two choice simplification strategies supported by the liter- ature, an elimination-by-aspects (EBA) screening (Fader and McAlister 1990; Manrai and Sinha 1989; Tversky 1972) and screening by brands previously purchased augmented by promoted brands (the promotion expansion strategy; see Siddarth, Bucklin, and Morrison 1995), are used to simulate consumer choices. In addition to logit, the promotion ex- pansion model, an EBA-based model, and a model with a compensatory screening process (all of which have been shown to predict scanner panel choices quite well; cf. An- drews and Manrai 1998) are fitted to the simulated choices in an effort to examine the predictive validity of logit when consumers' choice simplification strategies are not modeled explicitly. We also examine the effect of the misspecifica- tion on the diagnostic validity of logit.

A major difference in the EBA and promotion expansion screening strategies is the role of first-stage screening at- tributes in the second-stage compensatory evaluation of considered brands. A consumer using the EBA screening strategy to form a choice set would not use the screening at- tribute again in the second-stage evaluation of considered brands because all considered brands share the screening at- tribute, but this is not so for a consumer using a promotion expansion strategy. Thus, a follow-up simulation attempts to determine whether logit is more capable of mimicking the promotion expansion screening strategy when the screening attribute used in the first stage is not used in the second- stage compensatory evaluation. In this simulation, the dedi- cated attributes used only for screening are incorporated di- rectly into the utility specification of the logit model.

Other studies argue that choice simplification strategies are used more commonly when the universal set size is large (e.g., Johnson and Payne 1985; Payne 1976) and when pur- chase event feedback is a more important predictor of choices (Andrews and Srinivasan 1995; Siddarth, Bucklin, and Morrison 1995). Two simulations test whether the abil- ity of logit to accommodate choice simplification effects im- proves when the universal set size is small and when pur- chase event feedback is less important.

It is important to identify the conditions under which logit adequately describes and predicts consumer choice behavior so that (1) our ability to forecast choices is not needlessly poor and parameters are not needlessly biased and (2) a complicated two-stage model is not used when a vastly simpler logit would have only incrementally poorer predic- tive and diagnostic performance. In addition, this study be- gins to address the validity of two-stage models. For example, do two-stage models forecast choices well only because they use purchase event feedback more efficiently than logit does? Or perhaps they forecast well because they are a more effec- tive means of modeling cross-sectional heterogeneity in con- sumer decision making. If so, then perhaps two-stage models are not really modeling the formation of choice sets as they are intended to do. When modeling an unobserved (possibly unobservable) construct such as the choice set, it is important to make every effort to validate the construct being modeled.

The simulation methodology used in this study is similar to those of Johnson and Payne (1985) and Johnson, Meyer, and Ghose (1989). Johnson and Payne (1985) examine the performance of risky choice heuristics (e.g., expected value, Maximin) under different decision complexity and domi- nance conditions. Johnson, Meyer, and Ghose (1989) inves- tigate the ability of a linear compensatory model to predict the choices made by several noncompensatory rules in envi- ronments with varying interattribute correlations.

HYPOTHESES

In this section, we discuss how the ability of the logit model to accommodate two-stage choice processes robustly should be affected by the following environmental vari- ables: (1) the screening rules consumers use to simplify the choice problem, (2) whether consumers use the screening variable(s) to evaluate the brands in the choice set (which is defined as the set of brands not screened from the universal set) and make the final choice, (3) the number of alternatives in the universal set, and (4) the length of consumers' pur- chase histories.

Screening Rule Used to Form Choice Set

The ability of the logit model to accommodate choice simplification effects should depend on the actual screening rule consumers use to form choice sets. In this study, two types of noncompensatory screening strategies are used to generate the simulated choices: one in which consumers' sets of previously purchased brands are augmented by brands currently on promotion (e.g., Siddarth, Bucklin, and Morrison 1995) and one based on an EBA strategy in which consumers consider only promoted brands (e.g., Fader and McAlister 1990) or previously purchased brands. In addi- tion to the two noncompensatory screening strategies, a third condition contains choices that involve no choice sim- plification strategies. We do not generate any choices formed by a compensatory screening mechanism because there is little empirical evidence in support of such a strat- egy. Intuitively, two-stage choice processes make more sense if they enable the consumer to avoid processing attribute information for each brand. A compensatory screening model would require the consumer first to evalu- ate the utility of each brand to determine whether it should be included in the choice set. Having already evaluated the utility of each brand, the consumer could make a choice directly, without forming a choice set, by choosing the brand with the highest overall utility.

If none of the sample consumers uses simplification strategies prior to choice, it is possible that the logit model could have better forecasting accuracy on a holdout sample than the more sophisticated two-stage models, because the latter models could find spurious choice simplification ef- fects in the estimation data. However, this possibility is small because the multinomial logit model is nested in all the two-stage models used as benchmarks in this study. The more likely outcome is that the logit model should produce the same forecasting accuracy as the two-stage models when choices are generated with no preliminary screening.

We expect that the logit model would be better able to ac- commodate choice simplification effects when the noncom- pensatory strategy is of the EBA form than when the strate- gy involves promotion-augmented sets. In our data-genera-

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JOURNAL OF MARKETING RESEARCH, MAY 1998

tion procedures, consumers forming choice sets according to EBA-like strategies will consider the set of previously purchased brands, the set of promoted brands, or even the universal set. The logit model might accommodate these screening strategies reasonably well by assigning a large co- efficient to the brand loyalty variable (in the case of con- sumers considering only previously purchased brands) or the promotional variable (in the case of consumers consid- ering only promoted brands). One complication is that some consumers will screen according to past purchases, some will screen using promotions, and some will not screen at all, but one set of logit model parameters has to accommo- date all these possible behaviors.1 It is possible that this het- erogeneity in consumer screening strategies could hinder the forecasting performance of the logit model.

The promotion expansion screening strategy involves consumers considering the set of previously purchased brands, the set of previously purchased brands augmented by promoted brands, or the full set of available brands. In addition to the potential difficulties with heterogeneity de- scribed previously, it might be difficult for the logit model to reflect the consumer's consideration of this augmented set because it reflects the joint application of two screening strategies.

Hi: The logit model is more capable of accommodating choice simplification effects when consumers form choice sets us- ing an EBA-like screening strategy than when they form promotion-augmented choice sets containing previously purchased brands.

A related issue is how robust choice simplification mod- els are to misspecification of the process that underlies choice set formation. It has been suggested that compen- satory processes are quite capable of representing noncom- pensatory processes (Dawes and Corrigan 1974; Einhom 1970; Einhorn and Hogarth 1975; Johnson and Meyer 1984). However, Johnson, Meyer, and Ghose (1989) pro- vide convincing evidence that compensatory models fit compensatory processes better than they fit noncompen- satory processes. Thus, we do not expect that a compen- satory choice simplification model would fit as well as a noncompensatory choice simplification model, because the choices are generated using noncompensatory screening strategies. It is not known how robust the noncompensatory models are to variations in the noncompensatory screening strategy used to generate the choices.

Overlap of Choice Simplification Attributes and Choice Attributes

Another argument as to why logit should accommodate choices made with an EBA screening better than it accom- modates choices made with promotion expansion screening is that consumers are using the same attribute data in two capacities with the promotion expansion screening but not with the EBA screening. Logit might be less able to accom- modate choice simplification effects when attributes have more than one function. First, consider a promotion expan- sion screening in which the promotional variable a con-

lThis argument suggests that latent segment models (e.g., Kamakura and Russell 1989) and random effects models (e.g., Chintagunta, Jain, and Vil- cassim 1991) might be capable of accommodating choice simplification effects.

sumer uses to screen alternatives from the choice set also is used in the second stage to evaluate the brands remaining in the choice set. The consumer is able to use the same promo- tional information in making a choice, because some brands in the choice set might be promoted, whereas others might not be promoted. In this scenario, two-stage models make more efficient use of the promotional variable than logit does because they model two separate, conceptually distinct functions of the promotional variable. Therefore, two-stage models should have a significant forecasting advantage over logit when a screening attribute is used again to make a choice from the choice set (the overlap condition).

Second, consider a scenario in which a consumer does not use screening attributes to evaluate the brands remaining in the choice set. For example, if a consumer only considers promoted brands (Fader and McAlister 1990), then all brands in the choice set will share that attribute, and it no longer will be useful in evaluating the considered brands. In this scenario, the forecasting advantage of the two-stage models should diminish because attribute information is used in only one capacity. Logit then should be better able to mimic effectively the effects of the screening variable. The misspecification in the logit formulation should be rel- atively small because two-stage models are not making more efficient use of the screening attributes.

HI argues that logit should be capable of mimicking EBA-like screening strategies but not promotion expansion screening strategies. The implication of H2, given subse- quently, is that, when consumers use the promotion expan- sion screening strategy, logit mimics the screening better when separate attributes are used for the screening and eval- uation stages.

H-: When consumers make choices using the promotion expan- sion screening strategy, the logit model is more capable of accommodating the screening when an attribute used for choice simplification is not used again to evaluate brands in the choice set.

Number of Alternatives in the Universal Set

Choice behavior appears to be sensitive to the number of brands in the universal set (Johnson and Payne 1985; Olshavsky 1979; Payne 1976; Payne and Braunstein 1978). There is evidence that choice heuristics decrease in accuracy as the number of alternatives and outcomes for risky choices increase (Johnson and Payne 1985; Thomgate 1980). It is well known that consumers are more likely to use choice simplification strategies when the universal set is large (e.g., Payne 1976).

We expect that the logit model is more capable of accom- modating two-stage choice processes when the universal set is small than when it is large. When the universal set is small, say 5 brands, and a consumer's true choice set con- sists of, say, 3 brands, the misspecification in the logit for- mulation involves only the 2 brands that are available but not considered. These 2 brands should have zero choice proba- bilities if they are not considered, but the logit model likely will assign nonzero (though perhaps small) choice probabil- ities to them. However, when the universal set is large, say 20 brands, and the consumer's choice set contains only the 3 brands, then the misspecification involves 17 brands that should have zero choice probabilities. There is evidence that choice set sizes do not vary drastically across product cate-

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Simulation Experiments in Choice Simplification

gories (Hauser and Wernerfelt 1990), though universal set sizes do vary significantly across product categories.

Hi argues that logit should be quite capable of mimicking EBA-like screening strategies but not promotion expansion screening strategies. The implication of H3, given subse- quently, is that, when consumers use the static heuristic screening strategy (which assumes that a consumer's choice set size and composition do not change over time), logit mimics the screening better when there are fewer brands in the universal set.

H3: The logit model is more capable of accommodating static heuristic screening when the universal set is small than when it is large.

Number of Observations Per Consumer2

A plausible theory of choice set formation is that con- sumers form choice sets that consist of previously purchased brands, and recent two-stage models for scanner panel choices (e.g., Andrews and Srinivasan 1995; Siddarth, Bucklin, and Morrison 1995) have recognized the impor- tance of prior purchases. However, one issue that has been addressed less in choice literature (for an exception, see Roberts and Lattin 1991) is the question of how consumers simplify the choice if there is not a temporal dimension to choice behavior. Logit might be more capable of accommo- dating choice simplification strategies with purely cross- sectional data than with panel data because there is no temporal dimension to cross-sectional data.

One explanation as to why choice simplification models seem to forecast more accurately than logit models is that they take advantage of purchase event feedback in a manner that the logit model cannot. Logit models for scanner data incorporate purchase event feedback only through loyalty variables, which frequently are defined as exponentially smoothed purchase histories. Two-stage models also define consumers' choice sets on the basis of past purchases, in ad- dition to incorporating the loyalty variable in their utility functions. Perhaps this more efficient usage of purchase event information is partly responsible for the superior fore- casting accuracy of two-stage models.

The logit model should be more capable of accommodat- ing choice simplification effects when less purchase history information is available for each consumer. When a short purchase history is available for each consumer (say, two purchases), consumers' complete sets of considered brands might not be revealed to the researcher, and choice sets in two-stage models will be misspecified. If the misspecifica- tion is severe enough, the estimation process will result in a two-stage model very similar to a logit model (for which all brands are in the choice set) to prevent a consumer from pur- chasing a brand believed to be outside the choice set (the log-likelihood becomes an infinitely large negative number in this case).

When there is ample opportunity to observe a consumer's purchasing patterns and choice set prior to the estimation period (say, ten purchases are observed), there is more pur- chase history information that the two-stage models can use to their advantage. The logit model therefore should be at a disadvantage because the two-stage models are using a large

2We thank a JMR reviewer for suggesting this line of inquiry.

amount of purchase information more efficiently than the logit model is.

HI argues that logit should be capable of mimicking EBA-like screening strategies but not promotion expansion screening strategies. The implication of H4, given subse- quently, is that, when consumers use the promotion expan- sion screening strategy, logit mimics the screening better when less purchase history information is observed.

H4:When consumers make choices using the promotion expan- sion screening strategy, the logit model is more capable of accommodating the screening when less purchase history in- formation is used to initialize loyalty variables and choice sets.

SIMULATION

Experimental Desiglls

Data. We run four simulation experiments to test the hy- potheses. Each experiment varies one factor (e.g., the screening rule consumers use to simplify the choice prob- lem), and 20 data sets are generated for each level of the ex- perimental factor. Each data set contains 1000 purchase oc- casions that are used for parameter estimation and another 1000 purchases that are used as a holdout sample. The 1000 observations in each estimation data set are simulated so that they represent 200 consumers with 5 purchases each, and these same 200 consumers make another 5 purchases in the holdout period. To initialize choice sets and loyalty vari- ables, 2 initialization purchases prior to the estimation sam- ple are generated randomly for each of the 200 consumers.

To make the simulation as comparable as possible to scanner data applications, two binary variables intended to represent promotional activities and two normally distrib- uted continuous variables representing variables such as price are generated to represent the attributes consumers use to make choices. It is not uncommon for choice models to use a comparable number of attributes, apart from loyalty variables and brand-specific constants (which are also included in the models estimated here). The attribute data are generated randomly for each purchase occasion, reflect- ing that the purchase environment facing the consumer con- stantly is changing. The two normally distributed continuous variables are generated such that the correlations of the attributes across brands average a moderate .30, which reflects some degree of similarity among brands. The binary variables are generated such that they take values of one about 16% of the time, which is comparable to the pro- motion frequency observed in scanner data.

Given the attribute data, the choices are generated. To be as consistent as possible with the literature on choice sim- plification, our assumption is that consumers' choice sim- plification strategies are heterogeneous. Therefore, for the promotion expansion strategy, we allow some consumers to consider only previously purchased brands and some con- sumers to consider the set of previously purchased brands augmented by currently promoted brands. (Because there are two binary promotional variables, there are actually two possible strategies involving augmented sets.) In addition, the simulation allows some consumers to perform no screening at all prior to choice, so there are four possible screening strategies consumers can use under the promotion expansion heuristic.

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To determine which of the four screening strategies a consumer uses, we generate a set of four uniformly distrib- uted random variables for each consumer, normalize them so that they sum to one, find their cumulative sum, and then generate a random draw from a uniform distribution to pick which strategy the consumer uses. For example, if we ran- domly generate the numbers .4325, .8425, .7028, and .8124 from a uniform distribution, normalize them (producing .1550, .3020, .2519, and .2912 after rounding), and find their cumulative sum, we have the probability vector .1550, .4570, .7088, and 1.0000. If our random draw from a uni- form distribution is .0821, then the consumer uses the first screening strategy, because the random number falls in the range [.0000-. 1550), which is the first interval of the prob- ability vector. Although a consumer is assumed to use the same screening strategy across purchase occasions, the composition of the choice set can change across purchase occasions as the consumer tries new brands or as different brands are promoted.

Turning now to the procedure for modeling the con- sumer's choice from the choice set, consumer c's utility for brand i at purchase occasion t is

(1) . t j + t Uc.i c,i c,i'

where the deterministic component of the brand's utility is

(2) v = PX, + Pc,X + P3X, + PX4,

the Xkci is the randomly generated value of attribute k for consumer c and brand i at occasion t, and the P3 are impor- tance weights for the attributes. For each simulation, the 3k parameters are drawn from a standard normal distribution, but for each consumer, a small amount of noise is added to reflect heterogeneity in preferences:

(3) = + c oc - N(0,.25), k = 1, ...4.

The random components of utility in Equation 1, ?e ,, are independently distributed random variables with a double exponential distribution. These errors reflect errors in pro- cessing, variations in preferences across purchase occasions, in-store distractions, omitted variables, and so forth. By def- inition, these errors have a mean of .575 and a variance of 1.622 (Johnson and Kotz 1970).

The true probability that consumer c will choose brand i at occasion t is then

(4) 8t.i exp (U,)

.m exp( m) C.m nil

where 5, is 1 if brand i is included in consumer c's choice set on purchase occasion t, and 0 if not. Given the choice probabilities in Equation 4, a random draw from a uniform distribution is used to pick the chosen brand, similar to the procedure described previously for choosing which screen- ing strategy the consumer uses.

Turning now to the experimental design features specific to each simulation, Simulation 1 varies the screening rule used to form choice sets across three conditions. In the first condition, no consumers use choice simplification strate- gies. In the promotion expansion condition, consumers can

use one of four possible screening strategies, which produce (1) the set of previously purchased brands, (2) the set of pre- viously purchased brands augmented by brands having a value of one on a binary promotional variable, (3) the set of previously purchased brands augmented by brands having a value of one on a second binary promotional variable, and (4) the universal set of brands. Similarly, in the EBA screen- ing condition, consumers can use one of four possible screening strategies. However, set 2 is replaced with only those brands having a value of one on the first binary pro- motional variable, and set 3 is replaced with only those brands having a value of one on the second binary promo- tional variable. In accordance with the hypotheses, Simula- tions 2 and 4 are based exclusively on the promotion expan- sion strategy, with Simulation 3 based on Siddarth, Bucklin, and Morrison's (1995) static heuristic screening strategy (which we discuss subsequently).

To facilitate the recovery of the true parameter values in the investigation of diagnostic validity, Simulation I has two unique features. First, no consumer-specific noise is added to the P values (Equation 3). Second, the brand with the largest choice probability (Equation 4) is assumed to be the "chosen" brand, rather than using a random draw in con- junction with the probability vector to pick the chosen brand. Although these sources of noise are most likely pre- sent in real purchase situations, reducing the amount of noise or random error in Simulation 1 produces a clearer picture of the models' abilities to recover the true parameter values.

Simulation 2 studies the effects of the same attribute be- ing used to screen brands from consideration and to make a choice from the choice set. For the condition in which the screening attribute is not used again to make a choice, two new binary variables are created so that the binary variables used in the utility component are not the same as the bina- ry variables used to define choice sets. For the condition in which the screening attribute is used again in choosing from the choice set, the binary variables used in defining choice sets are the same ones that appear in the utility com- ponent. All other simulations assume that the screening at- tributes also are used in making a choice from the choice set, because this is more likely to be the case in actual scan- ner applications.

For Simulation 3, the size of the universal set varies over two levels, 5 and 20. The simulation is designed so that choice set size remains constant under the two experimental condi- tions. To accomplish this, we allow only two possible simpli- fication strategies for this simulation, one in which the con- sumer does not perform any screening and one in which the consumer's choice set consists of three randomly determined brands. Thus, the size and composition of the choice set do not change over purchase occasions as a result of promotions or purchases of previously untried brands. This data- generation process is similar to the static heuristic screening process described by Siddarth, Bucklin, and Morrison (1995). All other simulations fix the number of alternatives in the uni- versal set at 12 and allow four possible screening strategies.

Simulation 4 varies the number of initialization observa- tions per consumer across two levels: 2 and 10. All other simulations use 2 observations per consumer, except Simu- lation 3, which randomly generates static choice sets of three brands for each consumer.

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Simulation Experiments in Choice Simplification

Models fit to data. To test the hypotheses, four models were fit to each data set. The basic utility specification across all models is

(5) v j = 2X + P3Xlci + 4X'Xc, + + ci + B

i= 1,2,... 12,

where the &; are brand-specific constants, and the brand loyalty variable is

0 if brand i was not purchased att - 1

(6) BLtKc = k BL' + (I -)< 1 if brand i was purchased att- 1.

The brand loyalty variable is included in the utility specifi- cation for two reasons: (1) This is standard practice in choice modeling literature and (2) It is quite possible that the logit model will use this purchase event feedback to accommodate choice simplification effects.

The model with no choice simplification effects is the multinomial logit model, which estimates choice probabili- ties as

(7) P(i)t =

E e Lexp( tm) m

The compensatory screening model is based on the com- peting destinations model developed by Fotheringham (1988):

(8) exp(vcI) p(i E C)

P(i)l = /

Eexp.n ') p(m e C)t nl

The probability that alternative i is included in consumer c's choice set C, p(i E C)t, is modeled as

(9) p(i C)t = )(y + , +Y X + yX + y3BLt),

where the X variables are the two randomly generated binary promotional variables, and ys are parameters to be estimated. The normal distribution function (P(.) is used to transform the linear combination in Equation 9 so that it falls in the range [0,11 (see Andrews and Srinivasan 1995).

Each of the two noncompensatory models is based on

(10) P(i)t =0 P(i IC )t +0 P(i IC )t c I Ic 2 2c

+0 P(i IC3t +04P(i I C)t,

which reflects the possibility that the consumer could use any of four screening strategies to form choice sets (e.g., consider only previously purchased brands), as discussed in the previous section. The 0 weights, which are probabilities, are constrained to sum to one and are estimated empirically. Because they are constrained, only 3 of the 4 weights must

be estimated.3 It is necessary to model the consumer's choice simplification strategy probabilistically because the analyst cannot observe the actual strategy the consumer uses but must instead infer it from the choice data. The condi- tional probabilities P(i I C,)t are the probabilities that the consumer chooses brand i given the choice set Cj and are computed as

(II) at .expf vt .

P(ilC )t = I x ': X6t exp( v ) c.m c,m

ni

where bt is I if brand i is included in consumer c's choice set Cj on purchase occasion t, and 0 if not. For all models, the brand with the highest choice probability is assumed to be the chosen brand.

Assuming 12 alternatives (which is the case in all experi- ments except Simulation 3), the logit model requires I I brand-specific constants, 4 attribute weights, I loyalty coef- ficient, and I loyalty smoothing parameter, for a total of 17 parameters. The compensatory screening model requires an additional 4 parameters (Equation 9) for a total of 21, whereas the two noncompensatory models require an addi- tional 3 parameters (3 of the 4 0 weights), for a total of 20.

Several criteria were used to determine which models would be fit to the data. First, the models must be amenable to estimation with numerical optimization procedures that require only a single run. Because of heavy computational requirements, we could not estimate models that required multistage estimation procedures and grid searches (e.g., the Bayes models suggested by Siddarth, Bucklin, and Morrison 1995). Second, the number of parameters required for the first-stage component should not depend on the number of alternatives in the universal set (e.g., the Dogit model by Gaudry and Dagenais 1979). Third, the model should not become computationally infeasible with large universal sets (e.g., the parameterized independent availability model by Andrews and Srinivasan 1995). Fourth, the model should not frequently become unstable and/or fail to converge dur- ing the estimation process (e.g., the parameterized logit cap- tivity model by Swait and Ben-Akiva 1987). We chose mod- els, subject to these four requirements, that have performed best on scanner data sets in our previous experiences (An- drews and Manrai 1998).

Hypothesis Test Results

For each experiment, we used an analysis of variance (ANOVA) to assess the effects of the experimental factors. The dependent variable used is the percentage of correct predictions made by a model on the holdout observations. We chose this variable as the accuracy measure, because it is not necessary to adjust the percentage correct predictions for the number of parameters used to estimate the model. An overparameterized model frequently will make fewer cor- rect predictions on the holdout sample, even without any adjustment for the number of parameters. In addition, the percentage correct predictions measure enables us to deter-

3Our approach is to use a logit transformation to estimate these proba- bilities, which requires the estimation of three utilities with the fourth util- ity set to zero, much as we would estimate a logit model with only brand-specific constants.

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mine whether there are meaningful differences in fit among models; for example, model A produced 5% more correct predictions than did model B.

For a typical experiment that varies a factor over two lev- els, the ANOVA would have 160 observations on the per- centage correct predictions (20 data sets per factor level x 2 levels x 4 models estimated per data set). In most of the ex- periments, the design is a two-factor experiment with re- peated measures on one factor (model type is the within fac- tor). Each ANOVA will explain variation from the factor of interest (e.g., the size of the universal set), model type, the interaction between these factors, and data sets (analogous to subjects).

Table I contains the cell means for each experiment. The ANOVA results themselves provide little insight into rela- tive model performance and therefore are not reported, but the mean square errors from the ANOVAs are used to de- termine significant differences in pairwise comparisons in the paragraphs that follow.

Screening rule used to form a choice set. According to the cell means for Simulation 1 in Table 1, the forecasting ac- curacy of the four models does not differ when consumers use no choice simplification strategies prior to choice. Giv- en that model type mean differences (within levels of screening rule) of .0074 are required for statistical signifi- cance at the .05 level in paired comparisons,4 there are no significant differences among the models when consumers

4The Bonferroni adjustment for multiple comparisons is used.

use no choice simplification strategies. Therefore, the choice simplification models do not identify spurious choice simplification effects, and there is no harm in fitting choice simplification models when no such effects generated the data, at least from a forecasting standpoint.

HI argues that logit should be more capable of accommo- dating choice simplification effects when consumers use EBA-type screening strategies than when they use promotion expansion screening strategies. The results of Simulation I support this hypothesis. When the choices are generated using an EBA strategy, the logit model makes only 1.11 % fewer correct predictions than the correctly specified EBA model (which is greater than the difference of .74% required for statistical significance at p = .05), with the logit model making .73% fewer correct predictions than the promotion expansion model. Although the logit model makes .41 % more correct predictions than the compensatory screening model, this is not a significant difference at p = .05.

However, when the promotion expansion strategy is used to generate choices, the logit model produces 4.92% fewer correct predictions than the correctly specified promotion expansion model and 5.96% fewer correct predictions than the EBA model. Thus, as is expected, logit is not as capable of accommodating choice simplification strategies based on promotion-augmented sets. It is interesting that the EBA model forecasts choices made with promotion-augmented sets better (at p = .05) than the promotion expansion model does, even though the true screening rule is based on the promotion expansion strategy. This finding suggests that the

Table 1 HYPOTHESIS TESTING RESULTS: PREDICTIVE VALIDITY

Simulation 1: Effect of Screening Rule Used to Form Choice Set Means (% Correct Predictions on Holdout Observations)

Model

Data Logit CompensatorY Promotion Expansiont' EBA"' A verage

No Simplification .4374 .4372 .4387 .4384 .4379 Promotion Expansion .4640 .4773 .5132 .5236 .4945 EBA .4218 .4177 .4291 .4329 .4254 Average .4411 .4441 .4604 .4650 .4526

Simulation 2: Effect of Overlap of Choice Simplification Attributes and Choice Attributes Means (% Correct Predictions on Holdout Observations)

Attribute Overlap .4789 .5201 .5575 .5571 .5284 No Overlap .4689 .5078 .5509 .5510 .5 197

Average .4739 .5139 .5542 .5541 .5240

Simulation 3: Effect of Numnber of Alternatives in the Universal Set Means (% Correct Predictions on Holdout Observations)

5 Alternatives .3891 .4078 .4152 .4155 .4069 20 Alternatives .2561 .2579 .2933 .2933 .2752 Average .3226 .3328 .3543 .3544 .3410

Simulation 4: Effect of Number of Observations per Consumer Means (% Correct Predictions on Holdout Observations)

2 Observations .2113 .2269 .2266 .2226 .2219 10 Observations .2252 .2454 .2566 .2557 .2457 Average .2183 .2361 .2416 .2391 .2338

aBased on the promotion expansion model by Siddarth, Bucklin, and Morrison (1995). bBased on EBA screening strategy.

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Simulation Experiments in Choice Simplification

EBA model is quite robust to misspecification of the choice simplification strategy. It is important that two-stage models be able to accommodate robustly different simplification strategies, given a significant degree of heterogeneity in consumers' decision strategies (e.g., Payne 1976).

To investigate further the consequences of fitting models that do not describe correctly the data-generation process, we compute the average coefficient estimates across the 20 data sets in each cell for each of the four attributes in the utility functions (Equation 5), as well as the average brand loyalty coefficient, by model and data type, in Table 2. For example, the average coefficient estimated by the logit mod- el for continuous attribute I when there are no simplification effects in the data is -.3196, with the true value of that co- efficicnt being -.3263. The number in bold for each data- generation process represents the average coefficient esti- mate that results from fitting the correct model to the data.

Several implications emerge from this analysis. Even though the logit model did not give up much predictive ac- curacy to the noncompensatory models when the choices were generated using an EBA process, the misspecification produced severe bias in the coefficients. The logit model tends to underestimate the coefficients of the continuous at- tributes that were not used to simplify the choices. For ex- ample, the average coefficient estimate for continuous at- tribute 2 for the logit model fit to EBA data is -.7383, whereas the correct value is -1.5650, a 53% error; there is also a 53% error in the estimation of continuous attribute 1.

Analysis of diagnostic validity for the other experiments5 shows that the logit model's estimates for the continuous at- tributes not used in screening were smaller in absolute val- ue than the true coefficients, without exception. One expla- nation for this finding is that the logit model produces an av- erage coefficient for two segments of consumers, one seg- ment that does not even consider a brand and therefore has attribute sensitivities of zero, and another segment that may be highly attribute sensitive.

However, the logit model significantly overestimates the coefficient of discrete attribute 1 in an effort to accommo- date the use of this variable in the EBA screening process (1.3935 compared with the true value .2055). The coeffi- cient of discrete attribute 2 is not overestimated, most likely because the true value of the coefficient happens to be large. The simulation experiments show that the size of the esti- mated logit coefficient for an EBA screening attribute de- pends much more on how frequently the attribute is used for screening than on the importance of the attribute in the util- ity computation. Thus, the estimated coefficient for discrete attribute 2 would be nearly as large as the current value even if the true utility function coefficient were much smaller.

Notice that the brand loyalty coefficients are very large for the logit model when there are choice simplification ef- fects in the data (4.8978 and 3.6121). These coefficients are

5Only the results for Simulation I are shown in detail because the results of the other simulations are quite consistent, except as noted in the text.

Table 2 DIAGNOSTIC VALIDITY FOR SIMULATION 1: AVERAGE P COEFFICIENT ESTIMATES ACROSS DATA SETS

BY ATTRIBUTE, MODEL, AND DATA TYPE

Continuous attribute I (true value = -.3263)

Model

Data Logit Compensatory Promotion Expansion EBA

No Simplification -.3196a -.3201 -.3232 -.3239 Promotion Expansion -.2237 -.2250 -.3125 -.3254 EBA -.1537 -.1524 -.2180 -.2597

Cohtinuous attribute 2 (true value = -1.5650)

No Simplification -1.5572 -1.5589 -1.5736 -1.5790 Promotion Expansion - 1.1487 -1.1508 -1.5915 -1.6324 EBA -.7383 -.7357 -.9794 -1.1783

l)icrete attribute I, used for screening when there are choice simplification effects (truie itvalue = .2055)

No Simplification .2283 -.2471 .2159 .2156 Promotion Expansion .7993 .1096 .3062 .3596 EBA 1.3935 -5.1195 1.5111 .2813

Discrete attribute 2, used for screening when there are choice silnplification e.fects (true value = 2.3368)

No Simplification 2.3117 .5366 2.3116 2.2497 Promotion Expansion 2.2423 1.6905 2.4436 2.2319 EBA 2.1035 .8744 2.5419 2.1362

Brand loyalty coefficient (true value = 0)

No Simplification .0394 1.2195 -.1879 -.0803 Promotion Expansion 4.8978 2.2309 .0213 -.0819 EBA 3.6121 1.1690 -1.7420 -.5810

aBold number in each row is the average coefficient estimate that results from fitting the "correct" model to the data.

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JOURNAL OF MARKETING RESEARCH, MAY 1998

attempting to pick up the formation of choice sets consisting of previously purchased brands. Purchase event feedback did not play a role in utility computations when the data were generated (Equation 2), so the brand loyalty coeffi- cients should be zero. Analysis of Simulations 3 and 4 showed that the brand loyalty coefficients estimated by the logit model were even larger when the universal set was larger and when more purchase history information was available for each consumer.

When there are noncompensatory screening effects, the compensatory screening model estimates the coefficients of continuous variables not used in the screening process about as well as logit does. However, the compensatory screening model does an extremely poor job of estimating the coeffi- cients of discrete variables used to form choice sets. Note the average coefficient for discrete attribute 1 when the EBA strategy was used to generate choices (-5.1195). Over- all, the compensatory screening model does not do an ac- ceptable job of accommodating noncompensatory screening processes.

The promotion expansion model coefficient estimates closely match the true values when there are no simplifica- tion effects (indicating no spurious simplification effects). Also, the promotion expansion model does a good job of finding the correct coefficient values when the true screen- ing strategy is promotion expansion. But the model does not estimate the coefficients well when the data are generated using an EBA strategy. Similar to logit, the promotion ex- pansion model tends to underestimate the coefficients of continuous variables not used for screening and overesti- mate the coefficients of discrete variables used for screen- ing. For example, for continuous attribute 2, the promotion expansion model produces an average coefficient of -0.9794 compared with the correct value of-1.5650, but for discrete attribute 1, the promotion expansion model pro- duces an average coefficient of 1.5111 compared with the correct value of .2055.

The EBA-based model estimates closely match the cor- rect values when there are no simplification effects (again indicating no spurious simplification effects), but the model also estimates the coefficients quite well when the promo- tion expansion strategy generates the choices. Although the EBA model does only a fair job of recovering the parame- ters when the true screening process is EBA, the coefficients are much closer to the true values than those produced by the other three models. It is evident from the relatively low forecasting accuracy for EBA data (Table 1) and the diffi- culties in parameter recovery (Table 2) that modeling and forecasting choices produced by an EBA process are diffi- cult. Overall, however, the results of this simulation suggest that the EBA-based model is clearly the robust model of the group, with little bias in the coefficients when there are no simplification effects or when the simplification effects are generated by a promotion expansion process.

Overlap of choice simplification attributes and choice at- tributes. The findings from Simulation 1 showed that logit does a better job of forecasting choices made by an EBA process than by a promotion expansion process. H? argues that, when the choices are generated by a promotion expan- sion process, the logit model should be more capable of ac- commodating the screening effects when the attribute used to screen brands is not used again to make a choice from the

choice set. However, the ANOVA results for Simulation 2 show that neither the overlap main effect nor the "model type x overlap" interaction is statistically significant. An im- portant finding is that the logit model is not capable of using the dedicated screening attribute (the attribute used exclu- sively for screening in the "no overlap" condition) to ac- commodate promotion expansion effects. The logit models in the "no overlap" condition contain two more parameters (those for the dedicated screening attributes) than those in the "attribute overlap" condition, yet the predictive accura- cy is no better.

The two-stage models average 4.12%-7.86% more cor- rect predictions than logit in the "attribute overlap" condi- tion and 3.89%-8.21% more correct predictions in the "no overlap" condition. H2 argues that the multinomial logit model should be better able to mimic the effects of the pro- motion expansion screening in the "no overlap" condition. This is not the case.

We observed in Simulation 1 that the logit model could not match the predictive ability of the noncompensatory models when the choices were generated using the promo- tion expansion strategy. In that simulation, the attribute vari- able used for screening also was used in the utility compu- tation. Simulation 2 extends this finding to show that adding dedicated screening attribute data to the logit model does not improve its ability to predict choices generated using the promotion expansion strategy. So the predictive advantage of the noncompensatory models over logit is due not to their ability to extract more information from the same attribute but to their ability to use attribute data in a complex, non- compensatory manner.

Number of alternatives in the universal set. H3 argues that the logit model should be more capable of accommodating static heuristic screening effects when the universal set is small, because the misspecification in the logit formulation then would involve fewer brands. The pattern of cell means reported in Table I supports this hypothesis.

Differences between models within levels of universal set size are significant in multiple paired comparisons when the difference is .0063 or more. When the universal set has 5 brands, all three two-stage models predict better than logit, with the prediction accuracy advantage ranging from 1.87% to 2.64%. When the universal set has 20 brands, the com- pensatory two-stage model does not predict better than logit, but the noncompensatory models average 3.72% more cor- rect predictions than logit. Thus, the logit model is more capable of accommodating choice simplification effects when the universal set is small.

Perhaps the most surprising finding of this simulation, though, is that a noncompensatory model makes 2.64% more correct predictions than logit, even when the choice set size is three and the universal set size is only five. Thus, there are potential gains from modeling choice simplifica- tion effects even when as few as two brands are available but not considered. Note that the maximum gains from mod- eling choice simplification effects are smaller in this simu- lation than in the previous two simulations (5.96% and 8.21%), because 50% of consumers make choices from the universal set in this simulation, whereas only 25% of con- sumers make choices from the universal set in the other sim- ulations. Two simplification strategies were used in this simulation (compared with four in the other simulations) to

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Simulation Experiments in Choice Simplification

keep choice set size constant across universal set size con- ditions; no augmented sets were allowed.

Number of observations per consumer. H4 argues that be- cause two-stage models appear to use purchase history in- formation more efficiently than the logit model does, the lat- ter model will be at a disadvantage when more purchase his- tory information is available for each consumer. The results of Simulation 4 support this hypothesis.

Differences in "model type" cell means within levels of the other factor are significant if the difference is .0055 or more. In the two-observation condition, the differences between the two-stage models and the logit model are sta- tistically significant, but the difference in forecasting accu- racy is small, practically speaking (1.13%-1.56%). The choice sets defined by prior purchases are incomplete because the purchase histories are truncated, and the mis- specification of the choice sets in the two-stage models results in a bias toward the universal set. The models then produce forecasts similar to those of the logit model. In the ten-observation condition, the forecasting accuracy advan- tage for the two-stage models ranges from 2.02% to 3.14%, which reflects that the two-stage models use the extra pur- chase information more efficiently than the logit model does. The average probability estimated for the full univer- sal set for the promotion expansion model (i.e., 04 in Equa- tion 10) was .8092 in the two-observation condition but .2290 in the ten-observation condition, which supports our explanation for H4. The "true" probability assigned to the full universal set was approximately .25 for both conditions, because the consumer's screening strategy was picked ran- domly from four possible strategies.

Even in the ten-observation condition the forecasting ad- vantage for two-stage models is relatively small compared with the results of other simulations. Simulations I and 2 showed maximum forecasting gains of 5.96% and 8.21% for the two-stage models relative to logit, whereas the current simulation produced a maximum of 3.14% more correct pre- dictions. In Simulation 4, consumers purchased an average of 6 of the 12 brands as a result of having 10 initialization purchases, so the choice set is larger than in the other ex- periments. The smaller forecasting advantage of the two- stage models in Simulation 4 is due to the larger choice set size.

IMPLICATIONS AND CONCLUSIONS

Collectively, the simulations suggest that two-stage mod- els perform better than logit because the logit model is not capable of mimicking the complex, noncompensatory roles of screening attributes and purchase histories in the EBA and promotion expansion strategies, regardless of universal set size. The logit model gives up only about 1% correct pre- dictions to the noncompensatory models when the choices are made using the relatively simple EBA-type screening, but an investigation of diagnostic validity shows that a sig- nificant amount of parameter bias was induced by the mis- specification. Specifically, the coefficients of screening variables and the brand loyalty variable were overestimated as the logit model attempted to mimic the simplification effects. The logit model gave up nearly 6% correct predic- tions to the noncompensatory models when the data were generated using the more complicated promotion expansion strategy, and the performance of logit did not improve when

attributes used exclusively for screening were added to the logit model utility functions. That being the case, the deci- sion to model choice simplification strategies should not be affected by whether consumers are thought to use the screening attribute in evaluating brands in the choice set.

Whether a consumer uses a screening attribute in the evaluation of considered brands can depend in large part on the type of screening strategy the consumer uses. If the con- sumer uses an EBA-type strategy, then it might not be pos- sible for the consumer to use the screening attribute again in evaluating the brands in the choice set. When the consumer considers only promoted brands, for example, then all brands in the choice set share the attribute, and it must be ir- relevant to the final choice. However, if consumers use screening attributes such as promotion to augment the set of previously purchased brands (Siddarth, Bucklin, and Morri- son 1995), then the screening attribute can be used again in the selection of a brand from the choice set.

Unfortunately, researchers might not be able to determine the exact screening strategy used by consumers, a task made more difficult by a significant amount of heterogeneity in consumers' screening strategies. Regardless of the exact screening strategy used by consumers, our analyses of pre- dictive and diagnostic validity show that the EBA-based model is more robust than the model involving promotion- augmented sets. The EBA model closely estimated the util- ity function coefficients when there were no simplification effects and when the simplification process was based on promotion-augmented sets. Consistent with the findings of Johnson, Meyer, and Ghose (1989), a compensatory screen- ing model did not fit the noncompensatory choices as well as the literature on compensatory decision models suggests, especially when the choices were generated using an EBA- type screening strategy. Analysis of diagnostic validity showed that the bias in the utility function coefficients was moderate to severe, which suggests that the compensatory two-stage model is not suitable for empirical applications unless there is convincing evidence that consumers screen brands using some type of compensatory strategy.

Simulation results also show that choice simplification models are not prone to spurious choice simplification ef- fects. When consumers never use choice simplification strategies prior to choice, the forecasting accuracy and para- meter estimates of noncompensatory two-stage models are nearly identical to those of the logit model. Other than com- putational and human effort associated with two-stage mod- els, there are no penalties for incorrectly assuming that con- sumers use choice simplification strategies.

Both the EBA-based model and the promotion expansion model use purchase event feedback to determine the con- tents of the choice set antd the selection of a brand from the choice set, so they are extracting more information from purchase histories than the logit model is. Simulation 1 showed that logit models use loyalty variables to explain choice simplification effects, which might be yet another ex- planation for the almost universal significance of loyalty variables in empirical logit models. For applied modelers, the simulations suggest that larger amounts of initialization data favor the use of two-stage models, unless the sizes of consumers' choice sets increase as more initialization pur- chases are made. As choice set size approaches universal set size, the benefits of modeling two-stage choice processes

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JOURNAL OF MARKETING RESEARCH, MAY 1998

decrease. By extrapolating the findings of Simulation 4 to the case of cross-sectional data, it might be concluded that there is little benefit from modeling choice simplification ef- fects when there is no temporal element in choices. Certain- ly, most theoretical discussions of choice set formation in- volve some mention of past purchases.

Simulation 3 shows that the logit model is more capable of accommodating choice simplification effects when the universal set is small rather than large. However, a noncom- pensatory two-stage model produced an average of 2.64% more correct predictions than logit, even when the choice set size was three and the universal set size was only five. This finding is even more surprising given that 50% of con- sumers used no choice simplification strategies at all in this simulation. In general, unless the researcher has solid evi- dence (such as survey-based information or verbal proto- cols) that consumers in the product class of interest do not use choice simplification strategies, it is not safe to ignore choice simplification effects, even when the universal set is small.

The forecasting advantages of the two-stage models are at the expense of complexity and estimation time. Numerical optimization procedures found in the Gauss Maxlik applica- tion module were used to find maximum likelihood esti- mates of the models' parameters on a Unix mainframe com- puter. The EBA-based model, compensatory two-stage model, and promotion expansion model required 6.3, 6.7, and 8.6 times the computer resources required by the logit models, respectively, even though only three or four more parameters were required. Given a canned logit program that uses analytical derivatives instead of numerical opti- mization and a Taylor series expansion of the smoothing constant (e.g., Fader, Lattin, and Little 1992), the estimation time for the logit model could be significantly shorter. Cen- tral processing unit time alone does not tell the complete story; the development of the computer programs requires human effort.

Whether the predictive advantage obtained from model- ing choice simplification effects is worth this degree of

complexity and computational burden is a value judgment; the magnitude of the predictive advantage under various task and context conditions should not be. We hope that ap- plied researchers can use the findings of this study to make informed judgments as to whether modeling choice simpli- fication effects is crucial to the objectives of a particular study.

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