1 Introduction
On his website http://theburninghouse.com designer Foster Huntington invites people to post a picture of the things they would save from their house if it were to be on fire. About the project he writes: “If your house was burning, what would you take with you? It’s a conflict between what’s practical, valuable and sentimental. What you would take reflects your interests, background and priorities. Think of it as an interview condensed into one question.” His introduction captures a number of intuitions about how one would select objects to save from a fire: (i) Multiple considerations will probably go into the decision. (ii) There are likely to be important differences between individuals in the objects they select. (iii) The selection of objects might reveal information about an individual that is otherwise hard to obtain.
The pictures that respondents provide on the website appear to support the above intuitions. An individual’s picture generally contains a set of diverse objects, some of which are functional and some of which are emotionally or financially valuable. Pictures by different individuals contain different numbers of functional versus valuable objects and tend to differ in the specific instantiations of valued objects. It is certainly the case that the pictures provide a peek into the life of the respondents, highlighting those objects they value the most. But what can we infer from a specific set of objects about the considerations that went into their selection? Do these choices of material items convey anything about the purposes and desires of individuals who face the loss of their furnishings? And are individuals really as different as their seemingly idiosyncratic choices might lead us to suspect, or do they reflect more general inclinations that are shared by many?
These are the kinds of questions we would like to answer in this paper. They pertain to the possibility of inferring latent criteria from overt selection decisions and the nature of the inter-individual selection differences. In what follows we will first introduce the terminology that we will use in treating these questions. Then, we will introduce the formal framework that will allow us to answer the above questions. When finally we apply the framework to empirically obtained selection data, the intuitions that Foster Huntington formulated for the category of things you rescue from a burning house will be shown to hold for many other categories as well. We conclude the paper by discussing how the formal framework may be employed to answer substantial questions in the judgment and decision making literature.
2 Terminology
If one abstracts away from the unusual premise that a burning house is involved, the above questions can be recognized as recurring ones in the many disciplines of cognitive science that study human judgments. They pertain to individual differences in the criteria that are used, the number of criteria that are used, the order in which they are considered, the weights that are attached to them, and the manner a judgment is derived from them (Juslin et al. [2003], Pachur and Bröder [2013], Van Ravenzwaaij et al. [2014]). What differs between disciplines are the names for the criteria (attributes, cues, dimensions, features, …) and the judgments (categorization, choice, decision, induction, inference, selection, …) that are employed. One example is categorization, where individuals may rely on different apparent dimensions to arrive at an externally defined correct classification (Bartlema et al. [2014]) or abstract features from their environment to arrive at a conventional classification (Verheyen and Storms [2013]). Multi-attribute decision making is another example. Depending on whether individuals rely upon objectifiable or more subjective attributes, the problem of determining the criteria individuals employ goes under the name probabilistic inference or preferential choice (Pachur and Bröder [2013], Söllner et al. [2014], Weber and Johnson [2009]).
In some disciplines the use of criteria is not the main topic of interest, but considered to be merely indicative of that what individuals strive for. Depending on the discipline these intended end states are referred to as desires, goals, interests, or purposes (Austin and Vancouver [1996], Graff [2000]). A case in point are so-called ad hoc categories like things you rescue from a burning house. Ad hoc categories are constructed on the fly to serve a specific goal such as the minimization of financial loss or the preservation of precious souvenirs (Barsalou [1985]). Selection is important to attain a goal (Barsalou [1991], Barsalou [2003]). One needs to identify those objects that are most instrumental to attain the goal (Austin and Vancouver [1996], Förster et al. [2005]). In the case of an individual whose house is on fire and is willing to risk his life to minimize financial loss, this amounts to identifying and carrying out the objects that are highest in monetary value in the limited time s/he has available. Since this favors the selection of objects with an extreme value on the relevant criterion, the criterion is sometimes called an ideal (Barsalou [1985], Lynch et al. [2000]). The extent to which an object meets the choice criterion determines its idealness and corresponding likelihood of being included in the category.
Regardless of whether the choice criteria are being retrieved from memory, identified in the environment, or a combination of both (Bröder and Schiffer [2003], Gigerenzer and Todd [1999]), the question of how to confine the set of potential criteria pervades all described domains (Glöckner and Betsch [2011], Marewski and Schooler [2011], Scheibehenne et al. [2013], Verheyen and Storms [2013]). The question is perhaps most pressing for theories that adopt constructs such as goals and would like to determine the particular goals that drive individuals (Austin and Vancouver [1996], Ford and Nichols [1987], Kuncel and Kuncel [1995]). By definition goals are internally represented, private constructs that one need not necessarily be able to verbalize or even consciously experience. As a result, most of the research involves artificial laboratory tasks with a limited number of salient criteria. This is true both for categorization (Smits et al. [2002]) and for multi-attribute decision making (Lipshitz [2000]). Similarly, the research into goals has been focusing on a limited set of specific goals (Deci and Ryan [1985], Ryan [1992]). The modus operandi in the field has been to look into goals that are salient in natural settings (e.g., Medin et al. [1996], Ratneshwar et al. [2001], Simon [1994]) or to experimentally induce them in laboratory settings (e.g., Förster et al. [2005], Jee and Wiley [2007], Locke and Latham [1990]) and to investigate whether individuals’ selection decisions differ as a result of the known differences in goals.
Contrary to these customs, our approach will allow the criteria to be uncovered from the selection decisions. We introduce a formal framework that relates the overt decisions to latent constructs that allow one to infer what considerations underlay the selection decisions. This is established by positioning the candidate objects along a dimension according to their likelihood of being selected. Assuming that the objects that are chosen foremost are the ones that best meet the choice criteria, it is only a matter of interpreting the dimension to determine the considerations that went into the selection decisions. If the choices were to pertain to an ad hoc category such as things you rescue from a burning house and the objects that are listed according to frequency of selection were to follow the objects’ monetary value, it is likely that monetary value was the ideal and minimization of financial loss was the goal underlying the selection decisions.
The ability to organize objects according to the likelihood of selection presumes individual differences in selection. If everyone were to select the same objects, this would be an impossible endeavour. We hypothesize that these individual differences come in two kinds: differences in the criteria for selection and differences in the standards that are imposed on these criteria. Both types of individual differences are incorporated in so-called mixture IRT (item-response theory) models (Mislevy and Verhelst [1990], Rost [1990], Verheyen and Storms [2013]), a class of models from the psychometric literature that are generally used to identify differences among individuals in how they solve tests, both with respect to strategy and ability. Before we turn to a discussion of how we intend to use mixture IRT models to infer choice criteria, we elaborate on the inter-individual selection differences we presume. Since the empirical demonstration we offer will involve ad hoc categories and goal-derived categories (i.e., ad hoc categories that have become well-established in memory, for instance, through frequent use; Barsalou [1985]), we will frame both the discussion of these individual differences and the models in terms of goals and ideals. The models can, however, just as well be applied to situations in which one is interested in mere individual differences in objective choice criteria, without reference to more remote constructs.
3 Inter-individual selection differences
When it comes to satisfying a goal, it is important to acknowledge that not all means are equivalent. If one’s goal is to minimize the financial losses due to a fire, one is better off saving the television from the flames than a stuffed animal. However, if one is more intent on rescuing valuable souvenirs, a treasured stuffed animal will be the better choice. Objects differ in their ability to fulfill a particular goal (Barsalou [1991], Garbarino and Johnson [2001]) and people are sensitive to these differences (Barsalou [1985]). In light of these differences, selection serves an important function (Barsalou [1991], Barsalou [2003]).
The example of things you rescue from a burning house allows for the easy identification of two sources of individual differences in the decision to include an object in the category or not. First, individuals can have different goals when confronted with their burning house. Some may want to minimize financial loss, while others may want to preserve as many souvenirs as possible. Depending on one’s goal, the properties that are desirable for objects to be included will differ. Individuals intent on minimizing financial loss will want to save objects of high financial value, while individuals intent on preserving as many souvenirs as possible will want to save objects of high emotional significance. These ideals determine the relative likelihood with which objects will be selected. The likelihood of selection increases with idealness. Among individuals who want to preserve souvenirs, the likelihood of rescue will increase with the emotional value of the object. The same objects will have a different likelihood of being selected by individuals who want to minimize financial loss. Among these individuals the likelihood of rescue will increase with the financial value of the object.
Second, individuals that have a similar goal may impose different standards for including objects in their selection (Barsalou [1985], Barsalou [1991], Graff [2000]). While two individuals may both be intent on minimizing the financial losses due to the fire, the first may require objects to be at least $500 to risk her life for, while the other may require them to be at least $1,000. Whether an object will actually be included in the category things you rescue from a burning house will thus also depend on the cut-off for inclusion an individual imposes on the ideal. Put differently, individuals may agree on how a particular property makes one object more suitable to be included than another, but still differ in opinion about the extent to which objects have to display the property to actually be included. The higher the standard one imposes, the fewer the objects that will be included.
4 The formal framework
To introduce the formal framework let us start off with a hypothetical problem. Imagine that we present a group of people with a collection of objects that are commonly found in houses and ask them to indicate which of these objects they would save from their own house if it were to be on fire. For every individual-object-combination we would then obtain a decision Y io, either taking value 1 when individual i decides that object o would be saved or taking value 0 when i decides that o would not be saved. Let us further assume that we know (i) all respondents to share the same goal and (ii) any individual differences in selection to be due to the use of different standards. Having only the selection decisions Y io at one’s disposal, how could one identify the contents of the goal that underlies all respondents’ decisions?
A straightforward manner to accomplish this would be to determine for every object the proportion of individuals from the group who decided to save it. Since we assumed our hypothetical individuals not to pick out the same objects, but to select different numbers because of differences in the standard they impose on the properties relevant to their goal, objects are likely to differ considerably in selection proportion. The proportion for every object can then be identified with its idealness, provided the assumption holds that the objects that are chosen foremost are the ones best able to satisfy the goal. Arranging the objects according to the proportion of selection yields a dimension of variation (i.e., the presumed ideal). Determining the contents of this ideal involves the interpretation of the dimension.
It is clear in this hypothetical example that individuals’ response patterns are informative. Notably, the responses of any individual would follow a Guttman structure if they were listed in the order of the objects’ frequency of selection (across individuals). A Guttman structure with n entries consists of a series of k zeros (not selected), followed by a series of n-k ones (selected, e.g., {0,0,0,…,1,1}). The order of objects is invariant across individuals, but the value of k may differ between individuals (e.g., patterns {0,0,1,…,1,1} and {0,1,1,…,1,1} would indicate that the first respondent imposes a higher standard than the second respondent does). Such patterns suggest that all individuals employ a common ideal to decide whether to select an object or not, with a higher probability of being selected, the higher an object’s idealness.
Real response patterns, however, rarely conform to this ideal scenario (pun intended). As we already indicated in the introduction, respondents do not necessarily share a common goal. Whenever goals have been elicited with respect to a particular domain, several goals usually exist, and their contents may be quite diverse (Borkenau [1991], Loken and Ward [1990], Voorspoels et al. [2013]). One would expect that individuals with different goals display different selection behavior, as the objects that are considered ideal for one goal are not necessarily those that are considered ideal for other goals. Candidate objects would therefore have a different likelihood of being selected depending on the goal of the individual who is responsible for the selection. Our approach will therefore attempt to identify a number of latent groups g among the individuals, with the understanding that individuals within a group display consistent selection behavior (i.e., share a similar goal) that is different from the selection behavior of other groups (i.e., they have different goals). That is, arranging the candidate objects according to selection proportions is likely to yield a different order and interpretation in different groups.
The purpose of the modeling exercise is to explain the systematicity in the selection differences. Idiosyncratic response patterns are in all likelihood not informative for our purpose. If one were to accommodate any minor deviation with a new group with separate Guttman pattern, this would likely result in an infeasible, uninformative number of groups. We therefore argue for a probabilistic approach in which it suffices that individuals’ response patterns tend toward a Guttman pattern. It comes in the form of a mixture IRT model (Mislevy and Verhelst [1990], Rost [1990]) that considers every selection decision the outcome of a Bernoulli trial with the probability of a positive decision derived as follows:
The model in Equation (1) uses the information that is contained in the individuals’ response patterns to organize both individuals and objects along a latent dimension, much like the procedure that was outlined for our hypothetical example organized objects along a (latent) dimension of variation. The main divergence from the solution to the hypothetical problem is that the current model allows for multiple dimensions of variation, one for each subgroup of respondents the model infers from the data. We will take these dimensions to represent the ideals that serve the respondents’ goals. For each group g of individuals the model organizes the candidate objects along a dimension according to their likelihood of being selected by that group. βgo indicates the position of object o along the dimension for group g. Higher values for βgo indicate objects that are more likely to be selected. It is assumed that individuals in a group share the same goal, and that the organization of the objects can thus be conceived of as reflecting their idealness with respect to the goal. The better an object is at satisfying the goal, the more likely it is to be selected and consequently the higher its βgo estimate.
Groups with different goals will value different properties in objects, which in turn will affect the relative likelihood with which various objects are selected. The model therefore identifies subgroups that require separate βo estimates. An object o that is ideal for the goal of group g will often be selected by the members of g, resulting in a high βgo estimate. The same object might be anything but ideal to satisfy the goal of a different group g’. As o will then not be selected by the members of g’ the estimate of βg′ o will be low. That is, contrary to the single dimension of object variation in our initial hypothetical example, there now are several dimensions, one for each of the identified groups.
Individuals who share a similar goal may still differ regarding the number of objects that make up their selection, depending on the cut-offs for inclusion (or standards) they impose on the ideal that is relevant with regard to their goal. They may select a large or small number of objects, depending on whether they require objects to possess the ideal property to a small or to a large extent, respectively. Above, we identified the latent dimension with the ideal and the positions of objects along the dimension with their idealness. In a similar vein, individuals are awarded a position along the dimension, indicating the idealness they require objects to display in order to be selected. In Equation (1) θi indicates the position of individual i along the dimension for the group the individual is placed in. With the positions of the objects fixed for all individuals that belong to the same group, high θi estimates (i.e., high standards) correspond to small selections, while low θi estimates (i.e., low standards) correspond to large selections.
In a sense, θi acts as a threshold, separating objects that are sufficiently able to fulfill individual i’s goal from those that are not. However, it does not do so in a rigid manner. Rather, in Equation (1) a selection decision is considered the outcome of a Bernoulli trial, with the likelihood of selection increasing with the extent an object surpasses the standard θi and decreasing the more an object falls short of it. Hence, an object to the right of the standard is not necessarily selected, nor does an object to the left of the standard necessarily remain unselected. It is the case, however, that an object is more likely to be selected than not when it is positioned to the right of the standard. The reverse holds for objects that are positioned to the left of the standard. That is, across respondents the probability of selection increases from left to right. The probabilistic nature of the decisions accommodates the issue of the imperfect Guttman patterns, in that it allows deviations for individual respondents to occur.
A separate αg for each group determines the shape of the response function that relates the unbounded extent to which an object surpasses/falls short of the standard (βgo−θi) to the probability of selection (bounded between 0 and 1). Unlike the βgo’s and the θi’s, the αg’s in Equation (1) can only take on positive values.
5 Demonstration
To demonstrate the merits of the formal framework we will apply it to selection data for 10 ad hoc and goal-derived categories. Although it has been acknowledged that there might exist individual differences with respect to the goals that underlie these categories (e.g., Barsalou [1991]), this has not been empirically demonstrated. Therefore, these categories make for an interesting test case. An analysis of the selection data with the model in Equation (1) can elegantly test whether individual differences in goals exist, by examining whether more than one subgroup of respondents is identified.
In addition to determining the number of groups, we will try to infer the contents of the corresponding ideals. The model infers ideals from the selection data by awarding objects a position on one or more dimensions (depending on the number of groups that are retained). We will compare these βo’s to independently obtained measures of idealness (i.e., judgments of the extent to which the objects satisfy a number of ideals that were generated for the category). Earlier studies have found that the representativeness of instances of ad hoc and goal-derived categories increases with idealness (e.g., Barsalou [1985], Voorspoels et al. [2013]). These studies treated all respondents alike, however, without regard to possible individual differences. We will investigate whether this relationship also holds in subgroups of respondents that are identified from the data.
5.1 Materials
Categories and candidate objects were taken from Voorspoels et al. [2010]b. They had 80 undergraduate students generate instances of 10 different ad hoc and goal-derived categories as part of a course requirement. For each category, 20 or 25 instances were selected for further study, spanning the range of generation frequency for that category. Eight categories included 20 objects each ( things to put in your car, things you rescue from a burning house, things you use to bake an apple pie, things you take to the beach, means of transport between Brussels and London, properties and actions that make you win the election, weapons used for hunting, tools used when gardening) and two categories included 25 each ( things not to eat/drink when on a diet and wedding gifts). Categories and objects are listed in the Supplemental Materials. Throughout the text, we will employ an italic typeface to denote categories and an italic capital typeface to denote objects.
5.2 Ideal generation
The ideals were taken from Voorspoels et al. [2013]. They had 25 undergraduate students participate in an ideal generation task for course credit. Each participant received a booklet containing a short introduction and instructions to the task. For each of the 10 categories they were asked to generate characteristics or qualities that members ideally display. (Only the category descriptions were presented. No actual members were shown.) Participants could write down up to seven characteristics for each category. Voorspoels et al. [2013] considered ideals that were generated more than three times for inclusion in an idealness judgment task (see below). The resulting number of ideals per category ranged from 3 to 13 ( M=6). They are listed in the Supplemental Materials. Throughout the text ideals will be printed between triangular brackets in an italic typeface.
5.3 Idealness judgments
The idealness judgments were taken from Voorspoels et al. [2013] as well. The degree to which the objects in each category display an ideal property was indicated by 216 undergraduate students in return for course credit. Each participant judged the idealness of each object in an object set relative to one ideal for five categories (a different ideal for each category), yielding 15 participant judgments for each ideal. Participants were instructed to indicate on a 7-point Likert scale to what extent each object (i.e., the instances of the category for which the ideal was generated) possessed the quality. The estimated reliability of the judgments ranged from .71 to .98, with an average of .89. The judgments were averaged across participants and standardized using z-scores, resulting in a single score for each object on each relevant ideal.
5.4 Category selection
The selection data were obtained for the purpose of this study. Two hundred and fifty-four undergraduate students participated as part of a course requirement. They were asked to carefully read through the object set for each category and to select from the set the objects they considered to belong to the category. It was emphasized that there were no right or wrong answers, but that we were interested in their personal opinions. Four different orders of category administration were combined with two different orders of object administration, resulting in eight different forms. These were randomly distributed among participants.
6 Results
We present our findings in two sections. First, we will provide details concerning the model-based analyses. This section comprises inferences regarding the number of latent groups in the participant sample, and the quality of data fit the model achieves. Both aspects are evaluated solely on the basis of the object selection data. Second, we will go a step further and evaluate whether the model provides solutions that are interpretable, that is to say, whether the dimensions of object variation that the model reveals can be related to actual ideals that people conceive of in the particular contexts under consideration.
6.1 Model analyses
6.1.1 Discovering latent groups
Each category’s selection data were analyzed separately using the model in Equation (1). For every category solutions with 1, 2, 3, 4, and 5 latent subgroups were obtained. This was done using WinBUGS (Lunn et al. [2000]) following the procedures for the Bayesian estimation of mixture IRT models that were outlined by Li et al. [2009]. (See Appendix A for WinBUGS example code.) We followed Cho et al. [2013] in our specification of the priors for the model parameters:
with G the number of latent groups (1 to 5), O the number of candidate objects (20 or 25, depending on the category) and I the number of individuals (254 for each category). µg is the mean group standard of group g. z i is the latent variable that does the group assignment. Normal priors were chosen for the distributions of βgo and θi because this has been found to improve the stability of the estimation process (Cho et al. [2013]). Latent group membership was parameterized as a multinomially distributed random variable with πg reflecting the probability of membership in subgroup g. Both a Dirichlet prior and a Dirichlet process with stick-breaking prior have been described as priors for the membership probabilities. In a series of simulations Cho et al. [2013] have established that the latter choice is not substantial. We ran 3 chains of 10,000 samples each, with a burn-in of 4,000 samples. The chains were checked for convergence and label switching. All reported values are posterior means, except for group membership which is based on the posterior mode of z i.
To determine the suitable number of latent groups we relied on the Bayesian Information Criterion (BIC, Schwarz [1978]) because of extensive simulations by Li et al. [2009] that showed that the BIC outperforms the AIC, the DIC, the pseudo-Bayes factor, and posterior predictive model checks in terms of selecting the generating mixture IRT model. (See Appendix B for additional simulations.) The BIC provides an indication of the balance between goodness-of-fit and model complexity for every solution. The solution to be preferred is that with the lowest BIC. In accordance with the procedure described by Li et al. [2009] every αg, βgo, and µg was counted as a parameter, along with all but one πg (because the different πg sum to 1). This means that the number of parameters that enter the BIC equals G×( O+3)−1.
Table 1: BIC values for five partitions of the selection data
Table 1 holds for every category five BIC values, corresponding to five partitions of increasing complexity. For each category the lowest BIC is set in bold typeface. There were four categories for which the BIC indicated that a one-group solution was to be preferred. This was the case for things you use to bake an apple pie, things you take to the beach, properties and actions that make you win the election and tools used when gardening. For these categories the solution that provided the best account of the selection data (taking into account both fit and complexity) involved the extraction of a single set of βo estimates for all 254 respondents. Any individual selection differences were accounted for in terms of differences in θi estimates.
For the remainder of the categories the BIC indicated that multiple groups were to be discerned among the respondents. In the case of things to put in your car, things you rescue from a burning house, things not to eat/drink when on a diet, means of transport between Brussels and London, and weapons used for hunting the BIC suggested there were two such groups. In the case of wedding gifts the BIC suggested there were three. The individual selection differences in these categories could not be accounted for merely by different θi estimates. They also required the extraction of multiple sets of βo estimates, one for each subgroup that was discerned. Whenever multiple sets of βo estimates were required to account for the selection data, this constituted evidence that respondents employed different choice criteria.
6.1.2 Model fit
The BIC is a relative measure of fit. For a given data set it indicates which model from of a set of candidate models is to be preferred. The BIC is not an absolute measure of fit, however. It does not indicate whether the preferred model adequately describes the data it was fitted to. We used the posterior predictive distribution to see whether this was the case. The posterior predictive distribution represents the relative probability of different observable outcomes after the model has been fitted to the data. It allows us to assess whether the solutions the BIC prefers fit the selection data in absolute terms.
First, we consider the categories for which the BIC revealed only one group. As an illustrative case, Figure 1 depicts the data and posterior predictive distributions for the things you use to bake an apple pie category. For every object it contains a filled gray square, representing the proportion of respondents who selected it. The objects are ordered along the horizontal axis in increasing order of selection to facilitate inspection. Object 1 ( MICROWAVE) is the object that was least selected: less than 20% of respondents chose to include it in the category. Object 20 ( BAKING TIN) is the object that was most selected: all respondents except one chose to include it. The remaining objects are in between in terms of selection proportion. Object 2 ( LADLE), for instance, was chosen by about half of the respondents. Object 3 ( FOOD PROCESSOR) was chosen somewhat more often, etc. Figure 1 also contains for every object outlines of squares, representing the posterior predictive distribution for the corresponding selection probability. The size of the squares’ outlines is proportional to the posterior mass that was given to the various selection probabilities.
The posterior predictive distributions indicate that the one-group model provided a decent fit to the selection data. The distributions are centered on the objects’ selection proportions and drop off pretty quickly from there. In this manner, they capture the relative selection differences that exist between the objects: The posterior predictive distributions follow the rising pattern that the empirical data show.Footnote 1 A similar pattern was observed for the other one-group categories.
Figure 1: Posterior predictive distribution of the one-group model for the things you use to bake an apple pie selection data. Filled gray squares show per object the proportion of respondents who selected it for inclusion in the category. Objects are ordered along the horizontal axes according to the proportion of selection. Outlines of squares represent the posterior predictive distribution of selection decisions. The size of these outlines is proportional to the posterior mass that is given to the various selection probabilities.
We now turn to the categories for which the framework identified two or more latent participant groups. The results for the meanwhile familiar category of things you rescue from a burning house provide an exemplary case. The BIC indicated that a two-groups solution was to be preferred for this category.
In Figure 2 the category’s 20 candidate objects are ordered along the horizontal axes according to the selection proportion in the larger of the two groups.Footnote 2 For each object a filled gray square represents the proportion of respondents from the dominant group who selected it for inclusion in the category. A filled black circle represents for each object the proportion of respondents from the smaller group who selected it for inclusion. The two panels in Figure 2 are identical with respect to these data. Whether an object was likely to be selected or not, depends on the subgroup. Objects 12 ( LETTERS), 17 ( PICTURES), and 18 ( HEIRLOOMS), for instance, were selected more often by members of the dominant group (gray squares) than they were by members of the smaller group (black circles). The reverse holds for objects 7 ( CLOTHING), 13 ( CAR KEYS), and 15 ( CELLULAR PHONE). These selection differences support the division the BIC suggested.
Figure 2: Posterior predictive distribution of the one-group model (upper panel) and the two-groups model (lower panel) for the things you rescue from a burning house selection data. Filled gray squares show per object the proportion of respondents from the larger group who selected it for inclusion in the category. Filled black circles show per object the proportion of respondents from the smaller group who selected it for inclusion in the category. Objects are ordered along the horizontal axes according to the proportion of selection in the larger group. Outlines of squares and circles represent the posterior predictive distributions of selection decisions for the larger and smaller group, respectively. The size of these outlines is proportional to the posterior mass that is given to the various selection probabilities.
The upper panel in Figure 2 shows the posterior predictive distributions of selection probabilities that result from the one-group model. The lower panel shows the posterior predictive distributions that result from the two-groups model. For every object the panels include a separate distribution for each subgroup (square outlines for the larger group; circular outlines for the smaller group). The size of the plot symbols is proportional to the posterior mass given to the various selection probabilities.
Contrary to the one-group model, the two-groups model can yield different model predictions due to separate βo estimates for each group. In the lower panel of Figure 2 the posterior predictive distributions for the two groups are quite different when this is required. In the case of object 15 ( CELLULAR PHONE), for instance, a positive selection response is predicted for members of the smaller group, while the members of the dominant group are deemed undecided with the posterior predictive distribution centering on .50. The posterior predictive distributions that are due to the two-groups model (lower panel) are clearly different for the two groups, while the posterior predictive distributions that are due to the one-group model (upper panel) are not. Figure 2 thus shows that for things you rescue from a burning house the two-groups model provides a better fit to the selection data than the one-group model does and that its extra complexity is justified.
The results for the things you rescue from a burning house category are representative for things not to eat/drink when on a diet, weapons used for hunting and means of transport between Brussels and London. The respondents fall into distinct groups, whose members employ different choice criteria. That is, between groups different objects are likely to be selected for inclusion in the category. The model is able to account for these differences by extracting a separate set of βo estimates for every group. Within each group, the individuals use the same choice criteria. That is, by combining different θi estimates with a single set of βo estimates for the individuals within a group, the model is able to account for the subgroup’s selection data. The categories things to put in your car and wedding gifts are different in this respect. They warrant a separate treatment.
The BIC indicated that for things to put in your car two-groups were to be discerned among the respondents. Figure 3 presents the corresponding selection proportions in a similar manner as Figure 2 did. Both panels contain for every object a gray square that represents the proportion of respondents from the dominant group who selected the object and a black circle that represents the proportion of respondents from the small group who selected it. As before, objects are ordered along the horizontal axes according to the selection proportion in the dominant group. This allows for the identification of objects that were not as likely to be selected in one group as they were in the other. Object 1 ( DECK OF CARDS), for instance, was hardly selected by members of the dominant group, but selected by the majority of the smaller group members. Selection differences like these again support the division the BIC suggested.
The inter-object selection proportions are pronounced in the dominant group. The selection proportions start off small for objects like DECK OF CARDS (object 1) for which the majority in the dominant group agrees that they are not generally kept in cars. They then gradually increase until high selection proportions are attained for an object like PARKING DISC (object 20), which almost everyone keeps in his or her car. The corresponding posterior predictive distributions closely resemble those we saw in Figure 1 for things you use to bake an apple pie and the lower panel of Figure 2 for things you rescue from a burning house: The distributions follow the rising pattern that the empirical data show, centered as they are on the objects’ selection proportions and dropping off rapidly from there. This is true, both for the posterior predictive distributions that are due to the two-groups model and the ones that are due to the one-group model. The latter’s ability to account for the empirical data of the dominant group is not that surprising given that the dominant group comprises the vast majority of the respondents (91%) and as such counts heavily towards the estimation of the model.
While the two-groups model accounts well for the data of the dominant group, it does not appear to fit the data of the smaller group. The corresponding posterior predictive distributions are not centered on the empirical selection proportions, nor are they very specific. For the distributions that are due to the one-group model (upper panel), this lack of fit might be attributed to the model’s inability to account for pronounced between-group selection differences with a single set of βo estimates, but this is hardly an explanation for the two-groups model’s failure to fit the empirical data. After all, the two-groups model’s estimation is intimately tied to the identification of the two latent groups. The broad posterior predictive distributions that are due to the two-groups model indicate that the set of parameter estimates obtained for the smaller group does not allow for predictions that closely mirror the group’s selection data. The smaller group’s response patterns do not appear to carry sufficient information to allow accurate prediction, perhaps because there are few response patterns to go on (the smaller group is only comprised of 9% of the respondents), there is little variability in the response patterns (the objects’ selection proportions are almost invariably high), or the variability that is contained in the response patterns is not consistent (individuals decide to leave different objects out of the selection).
Whatever the reason may be, the result is a division of the respondents that entails the identification of one group of individuals who behave consistently (the dominant group) and that of a “rest” group of individuals who behave differently (the smaller group). The βo estimates for this smaller group do not allow one to specify what it means to be in this second group, besides not being in the first, dominant group. Indeed, the BIC indicated that it is beneficial (in terms of fit) to retain these individuals in a separate group, but the posterior predictive distributions indicate that the parameter estimates for that group are not a reliable source to characterize its members. All that can be said about the smaller group’s members is that their response patterns are so different from those of the dominant group that it is not tenable to assume they have the same origin. Note that the resulting division is still a sensible one, as it is better to discern the individuals that select objects in one way from those that do so differently (whatever that may mean) than to treat all of them (erroneously) as behaving the same way. (See Appendix B for a simulation study that supports this interpretation.)
Figure 3: Posterior predictive distribution of the one-group model (upper panel) and the two-groups model (lower panel) for the things to put in your car selection data. Filled gray squares show per object the proportion of respondents from the larger group who selected it for inclusion in the category. Filled black circles show per object the proportion of respondents from the smaller group who selected it for inclusion in the category. Items are ordered along the horizontal axes according to the proportion of selection in the larger group. Outlines of squares and circles represent the posterior predictive distributions of selection decisions for the larger and smaller group, respectively. The size of these outlines is proportional to the posterior mass that is given to the various selection probabilities.
A similar pattern was observed for wedding gifts: The BIC indicated that three sets of βo estimates were to be retained. The parameter estimates for the largest and the smallest group suffered from the same problem as the parameter estimates for the smaller group for the things to put in your car data did. They were not very informative when it comes to identifying the considerations that underlay the selection decisions of the individuals in those groups. That is, it is just not the case that the largest (smallest) and the intermediate group employed different choice criteria. The members of the largest (smallest) group did not employ the same choice criterion either, so there is no use in trying to determine it. This conclusion, of course, has implications for the analyses that follow. One should refrain from interpreting the uninformative estimates through regression analyses.
6.2 Regression analyses
To attempt to infer which ideals were used by the respondents, we regressed the βgo estimates upon the various idealness judgments that were obtained for a category. The higher the estimate for an object is, the higher its likelihood of being included in the category. We therefore expect significant positive regression weights for the ideals that are driving the selection decisions. The use of regression analyses allows one to investigate whether more than one ideal drives the selection decisions. To keep the analyses in line with traditional correlational analyses, in which only the best ideal is determined, we opted for a forward selection procedure with a criterion of .05 to determine which ideals are included in the regression equation. This way the ideal with the highest correlation with the βgo estimates is always the first to be included (provided that it is a significant predictor of the βgo estimates).
In case a solution with multiple groups is retained for a category, one can turn to the relation of the respective βgo estimates with the idealness judgments to better understand how the subgroups differ from one another. If individuals select objects with extreme values on a relevant ideal in order to satisfy their goals, groups with different goals are likely to select objects that have extreme values on different ideals.
A separate regression analysis was conducted for all groups determined in the previous section, except for the smallest one for things to put in your car and the largest and the smallest one for wedding gifts. Inspection of the posterior predictive distributions for these groups indicated that the mean βgo estimates were not sufficiently reliable to establish conclusions on regarding the considerations that underlay the selection decisions of the individuals who comprise the groups (see above). Table 2holds the results of the regression analyses. For every group it shows the R 2 and the signs of the regression weights for ideals with a p-value less than .05. Ideals that did not contribute significantly are indicated by dots. The number of the ideals refers to their order in the Supplemental Materials. The first line in Table 2, for instance, conveys that five ideals were withheld for things to put in your car of which ideals 1 (< easy to store away>), 3 (< makes travel more agreeable>) and 4 (< small>) did not enter in the regression equation for the β1 o estimates. The contribution of ideal 2 (< guarantees safety>) was significant and negative, while the contribution of ideal 5 (< useful>) was significant and positive.
Table 2 R 2 and regression weights from the multiple regression analyses with forward selection procedure. The signs of the regression weights with a p-value less than .05 are displayed, others are replaced by a dot
Table 2 shows that the externally obtained idealness judgments account very well for the relative probability with which objects are selected for inclusion in a category. Across the 14 groups retained for interpretation, the squared correlation between the βgo estimates and the best idealness judgments averaged .81.
For several categories more than one ideal was driving the selection decisions. This was the case for things to put in your car (group 1), things you rescue from a burning house (group 1), things you take to the beach, and weapons used for hunting (group 1). Yet, the contribution of ideals over and above the first dominant one, while statistically reliable, was generally rather small, and for the majority of groups only one ideal contributed significantly to the βgo estimates.
In two cases where multiple ideals entered the regression equation, one ideal contributed negatively (contrary to our expectations). For weapons used for hunting the regression analysis for the large group indicated that three ideals (< easy to take with you>, < light>, and < discreet>) yielded a significant contribution. We presume that < discreet> had a negative contribution because some weapons that are suited for hunting are difficult to conceal (e.g., SPEAR, β=1.32), while others that are less suited for hunting are easy to conceal (e.g., ALARM GUN, β=-1.11). In the regression analysis for things to put in your car both < useful> and < guarantees safety> were significant predictors. Here the negative contribution of < guarantees safety> probably reflects the fact that many objects we keep in our car do not benefit safety (e.g., COMPACT DISCS, β=1.30).
The results of the regression analyses support our assertion that, for the four categories with two groups, the criteria that supposedly governed the selection decisions differ from group to group. Either different ideals predicted the βo estimates of the different groups (< important> and < valuable> vs. < necessary> in the case of things you rescue from a burning house and < comfortable> vs. < fast> in the case of means of transport between Brussels and London). Or the regression analyses identified ideals that contributed to one set of βo estimates, but not to the other (< light> and < discreet> in the case of weapons used for hunting). Or the βo estimates of the different groups related to the same ideal in opposing directions. This was the case for the < many calories> ideal in the things not to eat/drink when on a diet category.Footnote 3
6.3 Conclusions and discussion
This paper started with a quote that was taken from theburninghouse.com . It described a number of intuitions regarding the decision which objects to save from one’s burning house. The intuitions were intended to account for the diversity of objects in the pictures that respondents uploaded to the website of the belongings they would save. These intuitions were found to hold across a variety of other ad hoc and goal-derived categories: (iii) The selection decisions revealed information about the participants in the shape of the ideals they used when making their choices. (ii) We established considerable individual differences, both in the employed ideals and the required idealness. (i) Across different groups, but also within a single group, multiple considerations informed the selection decisions. We discuss these findings below.
Goal-derived and ad hoc categories have vague boundaries. Barsalou [1983] already pointed to this when he observed that respondents do not agree about the objects that are to be considered members of a particular ad hoc category. The current results establish that it is unfortunate to denote divergences from the majority opinion as inaccuracies, as is habitually done (e.g., Hough et al. [1997], Sandberg et al. [2012], Sebastian and Kiran [2007]). Rather, these individual differences can be taken to reflect differences of opinion as to which objects meet goal-relative criteria.
In some categories, these individual differences are best explained by assuming that all respondents share the same goal but differ in the standard they impose for inclusion (see also Barsalou [1985], Barsalou [1991], Graff [2000]). That is, although they agree on the properties that objects preferably have (i.e., the ideal), they disagree about the extent to which objects have to display these properties (i.e., the idealness) to be included. For these categories a single dimension of object variation was retained for the entire group of respondents. The only individual differences required to account for the selection differences were differences in θi, the cut-off for inclusion that is imposed on this dimension. The positions of the objects along the single dimension of object variation (i.e., the βo estimates) could reliably be related to (external) idealness judgments (see also Barsalou [1985], Lynch et al. [2000], Voorspoels et al. [2013]).
In other categories, a proper account of the individual differences requires one to abandon the assumption that all respondents share the same goal. Rather, one needs to recognize that there exist subgroups of respondents with different goals. Within each of these subgroups, respondents are still thought to differ with regard to the standard they impose for inclusion. As before, this standard is goal-relative: It pertains to an ideal that serves a particular goal. The contents of the ideal, then, is no longer the same for two individuals when they belong to separate subgroups. In order to account for the selection differences that were observed for these categories, both individual differences in βo and θi were required. One dimension of object variation (i.e., a set of βo estimates) was retained for each subgroup of respondents. For every individual one θi estimate was determined, indicating the cut-off for inclusion s/he imposed on one of these dimensions (depending on the subgroup the individual belongs to).
The results of the regression analyses suggested that different criteria governed the selection of objects in the subgroups of respondents identified by the model. Either the βo estimates of the different groups related to the same ideal in opposing directions; or different ideals correlated with the βo estimates of the different groups; or the regression analyses identified ideals that contributed to one set of βo estimates, but not to the other. Note that the finding that sometimes multiple ideals predicted a group’s βo estimates should not be mistaken for a source of individual differences. The model analysis identified the members of the group as using the same criteria when selecting objects. A regression analysis deeming multiple ideals significant hence suggests that all respondents within that group consider all these ideals when selecting objects.
Other predictors of ad hoc and goal-derived category membership than ideals have been considered in the past (Barsalou [1985], Voorspoels et al. [2013]). We did not include familiarity as a predictor because Barsalou already discarded the variable as a predictor in his seminal 1985 paper. We did not consider central tendency because both Barsalou [1985] and Voorspoels et al. [2013] discarded the variable in favor of ideals. Frequency of instantiation was not included as a predictor because this was the variable that informed the inclusion of candidate objects for study. Barsalou [1991], Barsalou [2003], Barsalou [2010] has noted that when instances of a previously uninstantiated category have to be generated, there are yet other considerations that need to be monitored. Not just any object makes for a genuine instantiation of the category things you rescue from a burning house. Credible instances have to meet particular constraints that reflect everyday knowledge about the world. For this particular category, the objects are to be generally found in houses and should be movable, for instance. Our analyses of the selection data could not pick up these kinds of considerations as all the candidate objects in the selection task came from an exemplar generation task and therefore already adhered to the necessary constraints.
7 General discussion
The premise of this paper is that object selection carries information about the selection criteria that decision makers use. Assuming that most selection criteria are not idiosyncratic, but shared by several individuals, the relative frequency with which particular objects are selected can be used to uncover the common criteria. An object’s selection frequency is likely to reflect the extent to which the object meets the choice criteria, with objects being selected more frequently, the better they meet the choice criteria. From the identification of the selection criteria, it tends to be a small step to the identification of the end states individuals may have been aiming for. For instance, if one observes an individual saving mostly pricey objects from a house that is on fire, the inference that this person’s main goal is to minimize financial losses tends to be justified.
The challenge lies in the identification of individuals who use the same criterion. A particular object might meet one criterion, but not another. The above rationale will thus break down when individuals employ different criteria, because the resulting selection frequencies will reflect a mixture of criteria. The (common) criterion one might infer from such an unreliable source, might not be employed by any of the individuals making the selection decisions. One should exercise care not to discard important individual differences in favor of a nonsensical solution.
To this end we offered a treatment of individual differences in selection data that allows us to infer the criteria that underlay the selection decisions. It recognizes individual differences, both in the criteria and in the extent to which objects are required to meet them. Its usefulness was demonstrated in the context of 10 ad hoc and goal-derived categories. It accounted well for the selection differences that were found for these categories; it allowed for the identification of individuals who used different criteria; and the contents of these criteria could be substantiated. This suggests that our contention about the two kinds of individual differences is a viable one.
The distinction between within-group (standard) differences and between-group (criteria) differences has been made in several different contexts (e.g., Bonnefon et al. [2008], Lee and Wetzels [2010],Zeigenfuse and Lee [2009]). It is tempting to think of this distinction as one involving continuous (quantitative) versus discrete (qualitative) individual differences. However, if one is willing to assume that all potential criteria are originally available to individuals and the groups merely differ regarding the criteria they do not attend or consider important, the between-group differences may also be considered continuous. The situation could then be conceived of as a distribution of positive and zero weights across employed versus unattended or irrelevant criteria, respectively (see Verheyen and Storms [2013], for a discussion). The problem of distinguishing continuous (quantitative) and discrete (qualitative) differences echoes the debate in the decision making literature on the ability to discriminate between single-process and multiple-strategy models (Newell [2005], Newell and Bröder [2008]).
Irrespective of how the debate will be resolved, the two kinds of individual differences can offer a fresh perspective on research that attempts to relate external information about individuals to their decision making. Examples pertain to the effects of personality (Dewberry et al. [2013], Hilbig [2008]), affective state (Hu et al. [2014], Scheibehenne and von Helversen [2015], Shevchenko et al. [2014]), intelligence (Bröder [2003], Bröder and Newell [2008], Mata et al. [2007]) and expertise (Garcia-Retamero and Dhami [2009], Pachur and Marinello [2013]). It would be straightforward to relate variables like these to criteria use (group membership) and/or standard use (see Maij-de Meij et al. [2008], Van den Noortgate and Paek [2004], and Verheyen et al. [2011a], for demonstrations). Alternatively, one could consider selection decisions in various circumstances (e.g., Slovic [1995]) or at various times (Hoeffler and Ariely [1999], Simon et al. [2008]) and look for (in)consistencies in criteria and/or standard use across them (see Tuerlinckx et al. [2014], and Verheyen et al. [2010], for demonstrations).
We believe the above examples testify to the potential of mixture IRT models to answer substantial questions in a variety of judgment and decision making contexts, particularly in those such as multi-attribute decision making, where individual differences are likely to exist in the sources of information that inform decisions. We have presented one particular mixture IRT model. The class of mixture IRT models includes many more, some of which can incorporate guesses (Li et al. [2009]) or can accommodate continuous outcome measures (Maij-de Meij et al. [2008], Von Davier and Yamamoto [2004]) to give just a few possibilities. The applications are thus by no means limited to the choice situations that we have treated here. Mixture IRT models add to the mixture models that are already available in the decision making literature (Lee [2014], Lee and Newell [2011], Scheibehenne et al. [2013], Van Ravenzwaaij et al. [2014]). An important difference with the existing models is that the mixture IRT models do not require one to confine the set of decision criteria beforehand, but rather uncover them as latent sources of individual differences. Selection between models with various numbers of inferred criteria then offers a natural way of dealing with the question of how many criteria comprise the set of actual alternatives (Glöckner and Betsch [2011], Marewski and Schooler [2011], Scheibehenne et al. [2013]). The main challenge for mixture IRT applications may lie in the (post hoc) interpretation of the established latent criteria (but note that a priori candidate interpretations can be made part of the modeling endeavour and tested for suitability; see Janssen et al. [2004], and Verheyen et al. [2011]b).
8 Conclusion
In this paper we have demonstrated how one can infer from selection decisions the considerations that preceded them. We have shown how, from the choice for a specific set of objects, one can infer something about the purposes and desires of the individuals making the choices. We have learned that, despite pronounced selection differences, individuals tend not to be so different after all. The goals they pursue with their choices are generally shared by many others. Perhaps most importantly, we think that even more can be learned if the proposed approach to individual selection differences is combined with other sources of information about the individuals and is applied in other choice or judgment situations as well.
Appendix A: WinBUGS code for the two-groups mixture IRT model
Appendix B: Simulation studies
Both Li et al. [2009] and Cho et al. [2013] present simulation studies that elucidate certain aspects of mixture IRT models, including model selection and choice of priors. As suggested by reviewers, we here describe two additional simulation studies. The first simulation study pertains to the behavior of the employed model selection criterion (BIC) when indivduals’ choices are completely independent. The second simulation study is intended to elucidate the results for the categories in which the model selection criterion identified a “rest” group along a group of consistently behaving individuals ( things to put in your car and wedding gifts). We will show that it is plausible to think of this “rest” group as a haphazard group of individuals, just like the individuals from the first simulation study.
Both in simulation study 1 and in simualtion study 2, we simulated choices of 254 participants for 25 objects. The number of simulated participants equals the number of participants in our empirical study. The number of objects equals that of the largest categories in our empirical study ( things not to eat/drink when on a diet and wedding gifts). The data were generated according to the model formula in Equation 1). We set α to 1.5 and varied βo from −3 to 3 in steps of .25. (These values are representative for the ones we observed in our empirical study.) Individual θi’s were drawn from the standard normal distribution. To generate data for independent decision makers, the βo’s were permuted for every new individual. They comprised all 254 participants in simulation study 1 and 54 participants (21%) in simulation study 2. The remaining 200 participants in simulation study 2 were assumed to employ the same criterion for their choices, but to differ regarding the standard they emposed on it. That is, to generate data for the consistent individuals the same βo’s were used (varying between −3 and 3 in steps of .25) and only the θi’s differed. Five simulated data sets were created in this manner for simulation study 1 and for simulation study 2. While the data sets in simulation study 1 are in effect comprised of independent choices (as evidenced by Kappa coefficients close to zero), the data sets in simulation study 2 each comprise a subgroup of heterogeneous decision makers (similar to the study 1 participants) and a subgroup of consistently behaving decision makers.
Each of the ten data sets was analyzed in the same manner as the empirical data sets in the main text. For each of the five simulated data sets in simulation study 1, the BIC favored the one-group solution, with the averages across data sets for the one- to five-groups solutions equaling 8540, 8671, 8784, 8906, and 9036. This result is in line with our intuitive introduction of how the mixture IRT model works. It relies on consistent behavior across participants to abstract one or more latent dimensions. Without common ground on which the decisions are based, the conservative BIC favors the least complex account of the data. The model parameters and the posterior predictive distributions in this case testify to the fact that this group should be considered a haphazard group of individuals. The range of the mean βo’s, for instance, is rather restricted ([−.63,.70] compared to the “empirical” range [−3,3]), yielding selection probabilities close to .50 for all objects. The posterior predictive distributions of the one-group model for the selection proportions resemble the circular outlines in the lower panel of Figure 4 (see text below for details). The fact that these distributions are wide compared to the observed differences between objects should be a red flag as well.
When the participants are comprised of a consistently behaving group and a group of heterogeneous decision makers, the BIC is able to pick up on this. The BIC values in Table 3 favor a two-groups solution for each of the simulation study 2 data sets. The solutions are 99% accurate (1262/1270) in allocating individuals to their respective groups (consistent vs. heterogenous) based on the posterior mode of z i. Only once was an individual belonging to the consistent group placed in the heterogeneous group. On seven occasions an individual from the heterogenous group was placed in the consistent group. While the former misallocation represents a true error, the same does not necessarily hold for the latter ones. The choice pattern of any of the heterogeneous individuals could by chance resemble the choice pattern of the consistent group. The generating βo’s are also recovered well. The correlation between the generating values and the posterior means of the βo’s is greater than .99 for all five data sets.
Table 3: BIC values for simulation study 2 data sets
Figure 4: Posterior predictive distribution of the one-group model (upper panel) and the two-groups model (lower panel) for data set 1 from simulation study 2. Filled black circles show per object the selection proportion for the heterogeneous group. Filled gray squares show per object the selection proportion for the consistent group. Objects are ordered along the horizontal axes according to the generating βo values for the consistent group. Outlines of circles and squares represent the posterior predictive distributions of selection decisions for the heterogeneous and consistent group, respectively. The size of these outlines is proportional to the posterior mass that is given to the various selection probabilities.
Figure 4 presents the posterior predictive distributions for data set 1 from simulation study 2 in a similar manner as Figures Figure 2 and Figure 3 did. Both panels contain for every object a black circle that represents the selection proportion for the heterogeneous group and a gray square that represents the selection proportion for the dominant, consistent group. Unlike the demonstration in the main text, this division of participants is based on known group membership, instead of inferred. Objects are ordered along the horizontal axes according to the generating βo values for the consistent group. In accordance with the manner in which the data were generated, the selection proportions for the rest group are close to .50, while the selection proportions for the consistent group show a steady increase.
The upper panel in Figure 4 shows the posterior predictive distributions of selection probabilities that result from the one-group model. The lower panel shows the posterior predictive distributions that result from the two-groups model. For every object the panels include a separate distribution for each subgroup (circular outlines for the rest group; square outlines for the consistent group). The size of the plot symbols is proportional to the posterior mass given to the various selection probabilities. The larger, consistent group dominates the results for the one-group model. The posterior predictive distributions tend toward the selection proportions of this dominant group but are not really centered on the empirical proportions because the one-group model is trying to accommodate the choices from the heterogeneous group as well. Especially for objects with selection proportions that are considerably smaller or considerably larger than .50, the posterior predictive distributions are being pulled away from the consistent selection proportions toward the heterogeneous group’s selection proportions. The two-groups model, on the other hand, distinguishes between heterogeneous and consistent responses. The posterior predictive distributions for the consistent group are tightly centered around the empirical selection proportions, while the posterior predictive distributions for the heterogeneous group vary more widely around a selection proportion of .50 for all objects. Although the latter distribution is not as wide as in the empirical cases, this pattern is reminiscent of the one observed in the main text for the categories things not to eat/drink when on a diet and wedding gifts. It supports the interpretation that for these categories the mixture IRT model identified a group of heterogeneous decision makers, that is best regarded as not following the same selection principle as the consistent group (a “rest” group). In a more general sense, the simulation results stress the importance of inspecting the posterior predictive distributions before turning to a substantial interpretation of the results.
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