The Effect of Set Composition on Item Choice: Separating Attraction, Edge Aversion, and Substitution Effects

Joel Huber, Duke University
[ to cite ]:
Joel Huber (1983) ,"The Effect of Set Composition on Item Choice: Separating Attraction, Edge Aversion, and Substitution Effects", in NA - Advances in Consumer Research Volume 10, eds. Richard P. Bagozzi and Alice M. Tybout, Ann Abor, MI : Association for Consumer Research, Pages: 298-304.

Advances in Consumer Research Volume 10, 1983      Pages 298-304

THE EFFECT OF SET COMPOSITION ON ITEM CHOICE: SEPARATING ATTRACTION, EDGE AVERSION, AND SUBSTITUTION EFFECTS

Joel Huber, Duke University

When a new item is added to a choice set, there are several things that happen to the probability of choosing items in the set. First, there is a substitution effect, the tendency of the new items to take share from the closest items. This substitution effect has been well documented (Becker, DeGroot and Marshak 1963, Tversky 1967). A number of models have been offered which quantitatively relate the share taken by the new alternative to item similarity in a perceptual or competitive space (Urban 1975, Batsell 1980, Huber and Sewall 1982), but the similarity coefficients in these models, while in the expected direction, accounted for very little variation.

Two other effects, attraction and edge-aversion, are less well-known and might be considered biases. An attraction effect is the tendency of a new item to draw choices to items similar to itself--a kind of reverse substitutability effect. Edge-aversion is the tendency to avoid choosing items that are extreme on one dimension or another. These effects became apparent in a number of experiments done by my colleagues and myself at Duke, and can best be understood by a brief review

In the first set of experiments (Huber, Payne and Puto 1982), subjects were asked to make choices from among sets to which dominated alternatives were added. That is, alternatives were defined on two dimensions to . which items were added from the area indicated on Figure 1. It was found these items did not get significant share (being dominated), but they did shift preferences in their direction. Thus one could increase the shares of an item by adding a similar, but dominated, item to the set.

FIGURE 1

GEOMETRIC DESIGN OF FIRST TWO STUDIES

The second set of experiments involved measuring the effect of adding items to the right of those added in the first study. These items were not dominated but represented a relatively worse trade-off on the two dimensions than was found in the core set. As the relative value of the added item increased, the attraction shift appeared to remain constant while the substitution effect, taking share from similar items, increased (Huber and Puto 1982).

In both of the-studies there appeared, both in the data and in debriefing comments from subjects, a reluctance to choose the extreme items in the set. This edge-aversion bad to remain tentative since in these studies it was confounded with the attraction effect.

FIGURE 2

GEOMETRIC DESIGN OF STIMULI IN LINEAR TRADE-OFF STUDY

The study that we present permits a further explanation of some of the issues raised in the first two. It is called a "linear trade-off" study since, as shown in Figure 2, all alternatives offered represent fixed linear combinations of levels on two dimensions. As the alternatives are collinear, their characteristics can be represented by item numbers. Items #2, #4 and #6 represent the core set; they were available in every choice set. The experiment consisted of comparing choice frequencies among groups that chose from only the core set with those who chose from among the core set plus one of the other alternatives. Items 81 and #7 are categorized as extreme alternatives since they extend one of the dimensions. If an attraction effect is operant, they should draw choices in opposite directions. The additions (#3 and #5) lead one to predict a moderate attractiOn effect but rather substantial cannibalism of adjacent stimuli. The dimensional structure of these choices is repeated across six product categories so that the resultant effects could be tested across Product categories.

TABLE 2

PROPORTION CHOOSING STIMULI AGGREGATED ACROSS PRODUCT CLASSES AND EXPERIMENTAL GROUPS

This design permits an examination of several important issues. First, since there is only one effective dimension on which items differ and the differences are in relatively fixed units, fairly simple models may be constructed to account for the probabilities found. Second, it avoids the issue of range effects. Since the new items extend the range of both dimensions proportionately, no range effect is predicted. Thus, any effects found are independent of such distortions.

Finally, three changes in the design permit a test or a global attraction effect under conditions relatively unconfounded by an alternative explanation--edge-aversion. First, there are three items in the core set, so a change in edge position due to adding the extreme alternative only affects one or these three items. Second, in the case where items are added to the interior of the set, it is possible to measure an attraction effect independent of an edge effect, since the same members of the core set are still at the edge. Finally, the simplicity of tie structure of the stimulus sets enables a test of edge-aversion relative to two different parametric models of choice.

It will be shown that adding an alternative produces a reweighting or the dimensions to favor those alternatives close to the added alternative. First the experimental procedure is detailed, then the results aggregated across all product classes are discussed. Finally, attraction, substitution, and edge effects are examined in the context of two parametric choice models.

Experimental Procedure

The sample consisted of 110 adults interviewed in their homes. Each subject was given a packet of papers describing choices for six different product classes. The product classes were luxury restaurants, cars, flashlight batteries, supermarkets, fast food restaurants and instance developing film. The dimensions and levels used are given in the Appendix. Within each product class, there were six possible choice sets. Two or these contained only the core set but with stimulus order reversed, two added an extreme alternative (#1 or #7), and two added a moderate alternative (#3 or #5). Each respondent was randomly assigned to a group which made choices across all product categories and all experimental conditions according to the design given in Table 2. Additionally, the order or the six sheets was randomized within the packet. Thus the procedure permitted an estimate of the effect or adding both moderate and extreme alternatives while control ling for product classes, their order, and the dimension on which the extreme or moderate addition is defined.

TABLE 1

EXPERIMENTAL DESIGN: DEFINITION OF PRODUCT ADDED

Table 2 gives the aggregate proportion choosing each category across the experimental conditions. This table illustrates both substitution and attraction effects. The attraction effect is evident in that the new extreme alternative (either #1 or #7) gets about as much share as its neighbor (#6 and #9 respectively) had in the core set. Thus, it is as if the locus or choices has moved towards the new item. Notice, however, that the new item takes share mainly from its neighbor, indicating substantial substitution effects. The two moderate additions appear to take primarily from their adjacent neighbors and give weak evidence of an attraction effect. Finally, there is no obvious evidence of an edge effect--indeed those items at the edge of the space appear to gather more shares than one might expect given stability or individual preferences.

The preceding comments must remain intuitive without a parametric model to provide a base from which an attraction or edge effect can be measured. Furthermore, the effects shown in Table 2 are aggregated across six product classes whose differences may account for the results. In the next sections two models will be used to provide that base. The first involves a generalized Luce model (1959) and thus resembles many current models in use today. The second is a pure substitution model formed by modeling the cumulative probability distribution. This second model, while less well known (Benson 1966), has the conceptual advantage or separating attraction from substitution effects. In both models the parameters adjust for reactions to the different product classes and permit tests of consistent attraction and edge effects.

A Modified Luce Model

The original Luce model (1959) has been adapted to fit a number of marketing applications. Some of these have retained the concept of proportionality (Pessemier, Burger, Teach and Tigert 1971, McFadden 1976, Batsell and Lodish 1981, Punj and Staelin 1978), while others have attempted to adjust utilities for item similarities (Silk and Urban 1978, Batsell 1980, Strauss 1981, Huber and Sewall 1982). That is, these latter models have added parametric terms so that a new item takes disproportionality more share from those similar over those that are dissimilar.

In the present application we begin with the proportionality model and include terms to test for attraction and edge effects. The model begins with the standard assumption that the probability of choosing item i in a set S is its utility, Ui, divided by the sum of the utilities of the items in the set, or

EQUATION    (1)

Then, in the present case, the utility of each item is assumed to follow a quadratic function of the stimulus value (xi = 1, 7) or,

EQUATION    (2)

This formulation posits that the utilities will be a smooth, single-valleyed or single peaked, function of stimulus values. Notice only one dimension is present since the two dimensions in Figure 1 are completely collinear.

To model the attraction effect we introduce two variables MOD and EXT to reflect a directional shift in the utility due to adding the moderate and extreme items. respectively.

That is:

EQUATION

These items interact with the linear coefficients in Equation (2) as:

a1 = a10 + a11.MOD + a12.EXT.    (3)

An attraction effect in the predicted direction will result in a positive coefficient.

The edge-aversion effect is modeled by including a negative term whenever an item is at the edge of a set.

Thus

EQUATION

A positive coefficient indicates that there is a consistent decrement to the utility of an item when it is the most extreme member of either dimension.

The different product classes were modeled using effects coding. This coding defines 5 variables for the product classes. For the first 5 product classes the variable was assigned a value of 1. O if its variable number corresponded to the product class and 0.0 otherwise. For the last product class all 5 variables were coded -1. These five variables were then multiplied by the linear and quadratic stimulus values to permit different quadratic choice surfaces for each product category. This kind of coding was used instead of the more common dummy (0-1) coding so that the sum of the product-by-linear terms and product-by-square terms across all six categories would be zero (by virtue of the fact that the sixth coefficient is by definition the negative of the sum of the others). The coefficients of the raw terms (xi, x2i) then reflected the average solution across all product classes.

The model thus permits each product class to have an idiosyncratic quadratic utility surface while testing across product classes for consistent attraction and edge effects. It was estimated using a modification of a maximum likelihood multinominal logit program developed by Cambridge Systematics (1980). The routine uses Newton's method to search for the solution. Any twice-differentiable utility function can serve to define the parameters. The routine estimates the log-likelihood of the actual choices given the parameters of the models and produces asymptotic t-tests on the parameters (Wilks 1962).

Results of the Modified Luce Model

Table 3 gives the results of the multinominal logistic regression. All of the equations were strongly significant at a p < .01 level. The small values of the general linear and square terms can be contrasted with the strength of these terms for the product-class adjustments. Thus, although the average utility function was relatively flat, individual product classes were strongly curvilinear. For example, product class 1 (Luxury Restaurant) has a concave surface reaching a peak just before stimulus #5 (food = ***1/2; atmosphere = **1/2). By contrast, product class 3 (Flashlight Batteries) has a convex utility surface with the lowest utility occurring at stimulus #5.

The edge-aversion parameters were not significant, and in two of the runs, they reversed direction. The attraction hypothesis was upheld for the addition of the extreme alternatives but was not significant for the moderate additions, i.e., those within the range of the original variables. This last result makes intuitive sense. An attraction effect could be expected to be strongest if the new alternative is far from the center of the original observations.

Thus, the adjusted Luce model indicates that there is no significant edge-aversion and that there is a shifting of the utility surfaces as extreme alternative is added to the set. As is discussed later, it may be that these effects were not strong since the Luce model already encompasses aspects of these functions in the assumption of proportionality. This issue will be clarified as we examine a model that begins with the assumption of strict substitutability--that items only take from proximate neighbors.

TABLE 3

ESTIMATION RESULTS ON MODIFIED LUCE MODEL

A Strict Substitutability Model

The strict substitutability model can be easily derived by assuming that choice is non-stochastic and the item chosen is the one closest to an ideal mixture. In the context here of the two-dimensional stimuli, the ideal is the best mixture of the dimensions subject to the constraint that their levels remain linearly related. We then assume that there is a distribution of such ideal mixtures across the population and that can be approximated by the logistic probability density function. A person would switch to a new mixture only if it was closer to the ideal than before. Graphically, the model can be portrayed as in Figure 3. The shares of each item arise from its portion of the density function. This results in a strict substitutability model since the addition of a new alternative takes share only from adjacent mixtures.

Such a model can be operationalized on the linear trade-off data. First the cut points for the choice alternatives need to be defined. If the items in a choice set are ordered in ascending stimulus values (xi's), then the cut point (zi) for the ith stimuli is

zi = 1/2 (xi + xi+1).   (4)

The cumulative distribution on zi, F(z), is then the probability that items with cut point of z or lower are chosen. Taking the cumulative distribution of ideal mixtures as approximated by a logistic cumulative density function results in

F(zi) = (1 + exp-(bo + b1zi))-1    (5)

This function is estimated by defining a cumulative choice variable Pi where

EQUATION

Equation 5 is a rather severe model, but by adding attraction and edge terms its realism can be increased. The attraction term represents a shift in the cumulative distribution toward the new item. MOD and EXT are defined as before but function as constraints for moderate and extreme additions to the set.

The edge effect is somewhat more problematic to model since a cumulative function includes stimuli that are at the edge and those that are not. .N reasonable way to proceed is to assign an edge-penalty to those at the edges and a center bonus for those not and then aggregate the effects. For three and four item sets these are:

TABLE

This variable effectively defines a new variable CUMEDGE which penalizes those cumulative sets with a greater proportion of edge members.

Equation 5, was parameterized with effects-coded dummy variables for each product class. The independent variables (zi) were sum-centered for ease of interpretation. In addition, constants were added for the moderate and extreme attraction effects and for cumulative edge-aversion. The estimation was performed using a logistic regression package by Frank Harrell (1980). This routine fits the multiple logistic regression model to a single binary (0 or 1) dependent variable. The dependent variable in this case is Pi, the probability that item i, or one with a lower index, is chosen.

TABLE 4

ESTIMATION RESULTS ON CUMULATIVE LOGISTIC MODEL

FIGURE 3

EFFECT OF ADDING AN ALTERNATIVE ON CUMULATIVE LOGIT MODEL

Table 4 gives the results. Notice once again that there are considerable shifts in the assumed logistic PDF of ideal mixtures across product classes. Both the cumulative edge-aversion variable and the two attraction variables resulted in substantial improvements in the likelihood ratios. Observe that the cumulative edge-aversion variable makes the average linear term go to zero. This is a function of the fact that the two variables have a correlation of 0.78. The z. term reflects the spacing of the items in the set, white CUMEDGE is only a function of the rank order of the variables in the see. Thus the CUMEDGE variable assigns the same cumulative probability to the four items regardless of the position of the added variable. Intuitively, this means that it is assuming a shift in the function so that the predictions for the set (#1, #2, #4, #6) are the same as those for (#2, #4, #6, #7). Thus, the edge variable acts as a combined linear and attraction variable in addition to a penalty function for being at the edge.

A comparison of the gain in the log likelihood gives some indication of whether the edge-aversion is more important than the attraction effect. The gains from these competing explanations are quite similar. Furthermore, in the combined model both edge-aversion and the attraction due to adding the extreme alternative are significant, indicating that they contribute to different, but approximately equal ways to explaining choice. Notice additionally that the moderate attraction variable is only significant when the edge variable is not present, indicating a rather weak effect if it exists at all.

Discussion

The initial intent of this paper was to measure the extent of changes in choice due to adding alternatives to the set. It was found, in both parameterizing models, that adding an extreme alternative "drew" choices in its direction. The attraction effect of adding an alternative nearer the middle of the set was statistically significant only when a pure substitution model served as a base, indicating that additions to the interior of a set have much less attraction effect than those to the exterior and may indeed be negligible. Similarly, the edge-aversion term had conflicting signs in the modified Luce model but was significant and in the expected direction using the pure substitution model. Since the results depend on the base model, it is appropriate to consider reasons for choosing one model over the other.

One way to discriminate between the models is in terms of indices of fit. The likelihood ratios given are not comparable since the cumulative model uses n-1 observations out of each choice set while the modified Luce modes uses all n. Furthermore, the observations in both models are correlated in different ways. The observations in the modified Luce model were constrained to sum to 1.0 while in the cumulative model the value of Pi is strongly dependent on the value of Pi-1. To make them comparable, the predictions for both models were converted into raw predicted probabilities of choosing items in the set. A square scoring rule, (Pi-Pi)2, was used to compare the models. Controlling for the numbers of parameters used in the regressions, the cumulative model did slightly better, but the difference was not statistically significant. This was surprising given the fundamental differences in the formulation of the models, but indicates that the models appear to have compensating strengths and weaknesses.

The modified Luce model has the advantage of permitting r U-shaped utility surfaces, while the cumulative model is constrained to single peaked distributions. Furthermore, the assumption of proportionality already contains certain edge-aversion and attraction properties relative to the strict substitutability model. To illustrate, suppose the new alternative is extreme (a #7) and very popular. Under the assumption of proportionality the new item takes share from all stimuli in proportion to the original shares. Thus, the most distant stimuli lose proportionality as much of the most proximate, resulting in a distribution of individual ideal mixtures that appears to have moved towards the new alternative.

For its part, the cumulative model will clearly be at an advantage where new items added are very similar to one another. As an extreme case, suppose that a large number of alternatives are added to the set near one another. The Luce model, with its utilities constrained to follow a smooth function of item characteristics, would assign the same utility regardless of how many items were added and, thus, systematically over-predict share for these items. By contrast, the cumulative model would divide share among those added alternatives, so they would take largely from one another as expected.

Since the two models each have their own advantages, it is not appropriate to ask which is generally better. Indeed one could design a study (with many highly substitutable items) where the cumulative model would be t more accurate, or a study (with U-shaped preference i distributions) that would favor the modified Luce model. Clearly, both models are abstractions which fail if their simplifying assumptions are pushed too hard. The issue of which base model (Luce or cumulative) is appropriate then becomes one that depends on the needs of the user. While the deterministic and non-stochastic nature of the cumulative model may appeal to researchers in some contexts, the familiarity and robustness of the Luce framework will be appropriate in others. Perhaps this ambivalence will prompt the development of a new model which includes both a coherent substitution framework as found in the-cumulative model and which permits a robust account of residual variability analogous to that found in Luce model.

APPENDIX

DEFINITION OF THE STIMULUS SETS AND PROPORTIONS OF CHOICES

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