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A rule-based lens model
ARTICLE IN PRESS
International Journal of Industrial Ergonomics 36 (2006) 499–509 www.elsevier.com/locate/ergon
A rule-based lens model Jing Yin, Ling Rothrock The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, USA Available online 9 March 2006
Abstract The purpose of this paper is to introduce a novel approach, called the rule-based lens model (RLM) to model human judgment of a probabilistic criterion. Our method is motivated by a shortcoming of an existing linear-additive model of judgment based on the lens model equation (LME) to adequately represent all rule-based relationships. Through the use of a simple example, we demonstrate the shortcoming of the additive model and set the context for our generalized rule-based formulation of Brunswik’s conceptual lens model. To investigate the behavior of our model and the relationship to the traditional lens model based on the LME, we simulate human judgments and criterion values in a ‘‘drosophila’’ domain where the parameters of the problem in terms of the number of cues, organizing principle of the criterion, organizing principle of the judge and the extent of uncertainty within the system can be systematically varied. Our efforts represent a first step toward the formulation of a generalized lens model framework. Relevance to industry The findings of the proposed research would provide theoretical basis toward the design of decision-aiding and decision-training systems that are adapted to human decision strategies. r 2006 Elsevier B.V. All rights reserved. Keywords: Decision making; Human factors; Lens model
1. Introduction and literature review The purpose of this paper is to introduce a novel approach to model human judgment of a probabilistic criterion. More specifically, we seek to establish a novel method to model the judgment of a human (e.g., a weather forecaster) who selects a criterion value (e.g., if it will rain or not) based on a set of probabilistic cues (e.g., air fronts, dew point temperature and temperature). Our approach examines the extension of the lens model (Brunswik, 1956) so that contingent decision behavior can be incorporated. The lens model, depicted in Fig. 1, represents the decisionmaking system as a symmetrical structure. The task environment, or ecology, is represented in the left half of the figure and the human judge is represented on the right half. The symmetry inherent in this representation allows Corresponding author.
E-mail address: [email protected] (L. Rothrock). 0169-8141/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ergon.2006.01.012
one to measure the degree of adaptation or ‘‘fit’’ between the judge and the demands of the judgment task. We submit that, while the conceptual model framed by Brunswik (1955) is broadly applicable, existing analytical models of the lens model (Hursch et al., 1964; Tucker, 1964) focus primarily on compensatory judgment behavior. Under a compensatory mode of decision making, information is processed exhaustively and trade-offs need to be made between attribute values. Compensatory modes of decision strategies are contrasted against noncompensatory modes (Payne, 1976; Einhorn, 1970; Svenson, 1979) where, typically, not all available information is used and, therefore, trade-offs are often ignored. Common noncompensatory strategies include:
Conjunctive: the decision maker sets thresholds on each attribute so that alternatives not meeting all thresholds are eliminated; Disjunctive: the decision maker accepts an alternative if one or more attributes are acceptable;
ARTICLE IN PRESS J. Yin, L. Rothrock / International Journal of Industrial Ergonomics 36 (2006) 499–509
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Cues
Criterion
Judgment
Fig. 1. Brunswik’s lens model.
Restaurant
Local Reputation
Newspaper Review
Desirability
A B C D
5.0 4.7 4.1 2.0
5.0 3.0 3.5 1.0
0.95 0.75 0.70 0.20
where Desirability represents the attractiveness of the restaurant to the decision maker. As a compensatory model, we can represent the restaurant choice problem as Desirability ¼ 0:17 þ 0:15 Local_Reputation
Elimination by aspects: at each stage of the process, the decision maker selects an attribute and eliminates all alternatives that do not include a specified aspect (i.e., attribute value) until only one alternative remains; and Take the best (Gigerenzer and Goldstein, 1996): the decision maker uses a sequence of rules to choose an alternative in the face of uncertain knowledge.
Research on contingent decision making suggests that decision makers respond adaptively to the variations in the environment through use of different strategies (Einhorn, 1970, 1971; Payne, 1982; Payne et al., 1993; Abelson and Levi, 1985). Research has shown that decision makers are sensitive to the increase of problem size and generally shift to noncompensatory evaluation strategies that save effort but are less accurate (Payne et al., 1993). Time pressure is another reason that the use of a compensatory strategy may be less preferable (Simon, 1981). However, when the task environment does not pose severe constraints in terms of time and task load, people tend to use compensatory strategies so that high decision accuracy can be obtained (Beach and Mitchell, 1978; Payne et al., 1993). We provide an example of restaurant choice problem to illustrate the mathematical differences between compensatory and noncompensatory models. We P represent the compensatory model, fc, as f c ðX Þ ¼ b0 þ ki¼1 bi X i where bi is the weight value and Xi is the cue value for the ith cue. Moreover, we represent a noncompensatory model, fn, as f n ðX Þ ¼ _ð^ X i Þ where 3 is Boolean disjunction over s, s k the set containing all cues, and 4 is Boolean conjunction. We intentionally represent fn in disjunctive normal form (DNF) because it has been shown that any strategy consisting of basic logical operators can be reduced to it (Mendelson, 1997). Hence, the outcomes of strategies ranging from simple conjunctive and disjunctive rules (Einhorn, 1970) to the process-dependent fast and frugal heuristics (Gigerenzer and Goldstein, 1996) can be reduced to DNF (see Rothrock and Kirlik, in press). Consider an example of choosing a restaurant for an evening out. We assume that each prospective restaurant has been reviewed by a local newspaper food critic and that each has established a local reputation. We consider the following four hypothetical restaurants:
þ 0:08 Newspaper_Review: We see that the desirability of a restaurant is determined by weighing all the available information. Now consider the case where the decision maker is under time pressure so that he must choose a restaurant, eat quickly and return to a late business meeting. Therefore, while he is interested in eating well, he is more interested in finishing on time. Revisiting the four restaurants, we see the following: Restaurant
Seating line
Occupancy
Desirability
A B C D
Full Full Empty Empty
Full Seat available Full Seat available
0 1 1 0
Whereas Restaurant A was highly desirable in the compensatory case, in the noncompensatory case, Restaurant A is undesirable. Moreover, because an empty restaurant is a likely indicator of the quality of the food, the decision maker is not interested in Restaurant D either. Therefore, as a noncompensatory model, we can represent the choice problem as ½ðSeating Line ¼ Full AND Occupancy ¼ FullÞ ! ðDesirability ¼ 0Þ OR ½ðSeating Line ¼ Full AND Occupancy ¼ Seat AvailableÞ ! ðDesirability ¼ 1Þ OR ½ðSeating Line ¼ Empty AND Occupancy ¼ FullÞ ! ðDesirability ¼ 1Þ. OR ½ðSeating Line ¼ Empty AND Occupancy ¼ Seat AvailableÞ ! ðDesirability ¼ 0Þ The noncompensatory restaurant choice model is an example of an exclusive-OR relationship which cannot be modeled using linear methods. The best linear model results in the following non-informative solution Desirability ¼ 0:5 þ 0 Seating_Line þ 0 Occupancy:
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Our example illustrates a key difference between compensatory and noncompensatory, or more specifically, rule-based models of decision making. While linear additive models often serve as adequate approximations of nonlinear strategies they are sometimes inadequate representations of rule-based behaviors such as that exhibited in the restaurant choice example. Moreover, existing efforts to model rule-based decision strategies using additive nonlinear formulations (Einhorn, 1970; Ganzach and Czaczkes, 1995; Elrod et al., 2004) lack the robustness to accommodate multiple rule-based strategies. 2. Mathematical representation of the lens model The standard representation of the lens model is given in Fig. 2 where subjects make judgments, denoted as Ys, on an environmental variable measured by the criterion, denoted as Ye, by using a set of cues X (Cooksey, 1996). The predicted values of the environmental criterion and human judgments, Y^ e and Y^ s , respectively, are generated by corresponding linear regression equations, which takes the form of fc. Hursch et al. (1964) first introduced the formulation of the lens model using Ys, Ye and X as random variables. Lens model parameters ra, Re, Rs, G, C are calculated as Pearson’s correlation coefficient, r, such that Re ¼ rðY e ; Y^ e Þ; Rs ¼ rðY s ; Y^ s Þ; G ¼ rðY e ; Y s Þ, and C ¼ r ðY e Y^ e Þ; ðY s Y^ s Þ . Tucker (1964) later modified the Hursch model into the following form now known as the LME: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) ra ¼ GRe Rs þ C 1 R2e 1 R2s , where the achievement, ra, represents how well human decisions adapt to the actual value of the environmental criterion. Re represents the environmental predictability and measures how well the environmental model predicts the actual criterion value. Rs is labeled as human control and indicates how the linear policy model captures the
501
actual human judgments. Linear knowledge, denoted as G, is designed to estimate how well the linear prediction model of the environment maps onto the predicted policy model of the human judgment. Unmodeled knowledge, denoted as C, measures how well the two models (one for the environment and the other for the human judgment) share the common points that are not captured in the corresponding linearly based model. We shall call the LMEbased formulation the compensatory lens model (CLM) to distinguish it from our proposed rule-based formulation. The CLM relies heavily on multiple linear regression models to determine the fitness of human judgments relative to constraints in the task environment. While this formulation has been widely used, its success is tempered by occasions in which judgment is driven by noncompensatory strategies. Although some have attempted to extend the LME to account for noncompensatory strategies (Einhorn, 1970; Cooksey, 1996), a framework which is able to fully model both compensatory and rule-based decisions, such as the two divergent strategies of the restaurant choice problem, has not been developed. The rest of this paper is organized as follows. First, we propose an integration of a rule-based formulation into the conceptual lens model framework. In the second section, we introduce a simulation study which enables us to investigate the behavior of our rule-based model. Section three focuses on one particular simulation finding that, we hypothesize, provides insights into factors which delineate the CLM from the rule-based lens model (RLM). Finally, some conclusions and implications of our research are discussed. 3. Formulation of RLM Rothrock and Kirlik (2003a) proposed a technique called genetic-based policy capturing (GBPC) to infer noncompensatory judgment strategies from human decision data
Fig. 2. Compensatory lens model with labeled statistical parameters.
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which explores rule-based modeling using propositional logic. This paper serves as a complement to their study. While Rothrock and Kirlik focused on the development of an inductive inference technique to capture judgment policies, the current work seeks to develop a rule-based analytical model of the conceptual lens model as conceived by Brunswik. The role of GBPC within the RLM is to generate Y^ e and Y^ s . We intend for this rule-based model to complement the multiple linear regression models in CLM. In building the functional relationship between human judgment and the task environment for compensatory and rule-based strategies, we note two key differences. First, the prediction mechanisms for compensatory strategies are multiple linear regression models whereas the mechanisms for rule-based strategies are logical propositions. Second, compensatory strategies operate on an interval scale whereas rule-based strategies require only a nominal scale. To use LME in an analysis of compensatory judgment strategies, the dependent variable should be measured on an interval scale (Schneider and Selling, 1996). The scale requirement is noted since linear regression serves as the basis for traditional lens model analysis (Hursch et al., 1964; Tucker, 1964). When modeling rule-based strategies and ecologies, on the other hand, the data used to represent environmental cues, judgments and criteria are discrete and categorical (Rothrock and Kirlik, 2003a). As such, the numbers assigned to them are on a nominal scale. This means that the assignment is arbitrary, as the numbers are simply labels (Lordahl, 1967). Therefore, interval or ratio scale operations can no longer be rigorously applied toward rule-based strategies. To create the RLM, we develop a new framework to compare the results of fitted models of both judgment strategies ðY s Y^ s Þ and the task ecology ðY e Y^ e Þ. We begin by providing a definition of error for rule-based strategies. Whereas compensatory model error is measured using the least-squares estimator, we propose that rulebased model error be measured as the number of mismatches between two sets of categorical data. Table 1 shows a simple example to illustrate the idea. In the example, there are five judgment instances in which three are matched. We call the ratio denoting the number of match cases out of the total number of instances as the match rate: m ¼ 3=5 ¼ 0:6. Using the match rate as the basis of our noncompensatory model, we construct an RLM that accounts for noncompensatory strategies and task environments. We suggest that the fitted parameters Table 1 Two categorical data sets and encoding (Eat—1, and Do Not Eat—0) Y1
Y2
Y1_encoded
Y2_encoded
Eat Do Not Eat Eat Eat Do Not Eat
Do Not Eat Eat Eat Eat Do Not Eat
1 0 1 1 0
0 1 1 1 0
NOT match NOT match Match Match Match
(all bivariate correlations in the CLM) Re, Rs, G, C, ra of the RLM be calculated instead as match frequencies. The operators of RLM are developed in following section. Assuming the ecological criterion to be judged is Y, which takes the categorical values on nominal scale. Let fY 1 ; Y 2 ; Y 3 ; . . . ; Y p g be the set of discrete values that Y can take. For instance i, 1pipn, where n is the number of judgments. Let Yei represents the actual environmental criterion value; Y^ ei represents the predicted criterion value from the environmental rule-based model; Ysi represents the human judgment value and Y^ si represents the predicted judgment value from the judgment rule-based model. We define the following indicator functions to calculate RLM relationships: ( 1 if Y ei ¼ Y^ ei ; I ei ¼ 0 otherwise; ( I si ¼
1 if Y si ¼ Y^ si ; 0 otherwise;
( I ri ¼
1 if Y ei ¼ Y si ; 0 otherwise;
( I Gi ¼
0 otherwise; and
I Ci ¼
1 if Y^ ei ¼ Y^ si ;
1 if I ei ¼ I si ¼ 0; 0 otherwise:
The corresponding RLM parameters are calculated as follows: Pn I ei (2) Re ¼ i¼1 ; n Pn I si Rs ¼ i¼1 ; (3) n Pn I ri ra ¼ i¼1 ; (4) n Pn I Gi G ¼ i¼1 ; and (5) n Pn I Ci C ¼ i¼1 . (6) n The interpretation of ICi is given in Table 2. Fig. 3 gives the RLM framework. The interpretation of the lens model parameters for the rule-based case is analogous to the linear case. Achievement, ra, represents the correspondence between human judgments and the actual value of the environmental criterion. Re, the environmental predictability, measures how well the noncompensatory environmental model can be used to predict the criterion value while Rs, labeled as human
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control, indicates how well the noncompensatory judgment model captures actual human judgments. Instead of representing linear knowledge in the linear-based lens model, in the RLM, the parameter G represents how well the noncompensatory model of the environment maps onto the noncompensatory model of the human judgments strategy. Now, the C parameter captures systematic regularities between the errors of the noncompensatory ecological and judgment models. In the RLM, the range of parameters ra Re Rs and G is [0,1]. The closer these values are to 1, the better the achievement, environmental predictability, human control and modeled knowledge, respectively. For C, a high value reveals a high degree of unmodeled knowledge. In the case of CLM analysis, C measures nonlinear relationships not captured by the linear-based model—which is consistent with cue usage in the rule-based model. In a corresponding manner, we suspect that a high value of C revealed through RLM analysis may represent a high degree of compensatory cue usage. To explore this potential relationship, a simulation study will be presented in the following section. Table 2 Interpretation of relationship existing between ICi and Iei, Isi Iei
Isi
ICi
Interpretation
0
0
1
Both environmental criteria and human judgments are not captured by the corresponding rule-based models. Mapping exists between the disregularities of the two systems (ecology and human judgment system). Either environmental criteria or human judgments are correctly captured by the corresponding rule-based models. System disregularities do not occur simultaneously; thus no mapping exists.
1 0 1
1 1 0
0 0 0
503
4. Simulation study To investigate the behavior of our model and the relationship to the traditional lens model based on the LME, we simulated human judgments and criterion values in a ‘‘drosophila’’ domain where the parameters of the problem (i.e., number of cues, organizing principle of the criterion, organizing principle of the judge and the extent of uncertainty within the system) can be systematically varied. Our efforts represent a first step toward addressing the relationship between a lens model which can account for rule-based decisions, with a basis in the language of logical propositions, and the traditional lens model, which was developed in the language of statistical correlation. To conduct our exploration, we used a framework proposed by Rothrock and Kirlik (2003b) for investigating judgments made under different environmental cue structures and decision strategies (see Fig. 4). For the environment, Cases I and IV represent a linear relationship between the cues and the environment criterion while Cases II and III represent a nonlinear organizing principle. For decisions, Cases I and III characterize a compensatory decision strategy while Cases II and IV reflect a rule-based strategy.
Cue Structure Linear
Nonlinear
Compensatory
Case I
Case III
Rule-based
Case IV
Case II
Decision Strategy
Fig. 4. Framework for investigating decision-making models.
Fig. 3. Rule-based lens model framework.
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For the study, we focus on the linear environmental structure to examine the relationship between compensatory (Case I) and rule-based (Case IV) decision strategies. This approach requires that we generate criterion values, Ye, that are governed by a linear structure and judgment values, Ys, that are both compensatory and rule-based. We submit that this investigation is not comprehensive, but is a preliminary examination of the relationship between the structure of cues and the decision strategy that is utilized by a decision maker.
Unlike actual cases where human judgments and environmental criteria are collected to produce CLM models, we used linear-based generators (LGs and LGe) and rule-based generators (RGs) to simulate instances of judgment and criteria. 4.1.1. Linear-based generators For the simulation, we used two linear-based generators. One builds the linear relationship between cues in the domain and the criterion to be judged by generating Ye. The generator, LGe, takes the form of Y e ¼ 0:6x1 þ 0:25x2 þ 0 x3 þ 0 x4 þ 0:15up ,
4.1. Elements of the simulation To execute the simulation, we made a key assumption regarding the achievement of the judge (ra) and environmental probability. We agree with Brunswik (1943) that the ambiguity in the environment as experienced by a judge results from a lack of control over all conditions that contribute to the criterion. Given that the cues available to the judge represent only a subset of all conditions, he could at best achieve only a partial knowledge of the criterion. We further submit that, to create criterion values associated with each judgment, it is possible to assume full access to all conditions bearing on the cause of the criterion by including cues which are perceivable as well as those which are not. The criterion ‘‘generator’’, therefore, is deterministic. Similarly, we assumed that the judge is purposive and does not randomly make judgments. Similar to our approach toward the criterion generator, we also accounted for perceivable and non-perceivable cues that are utilized for each judgment. Hence, the judgment ‘‘generator’’ is also deterministic. As a starting point, we simulated a system with four environmental cues represented by x1, x2, x3 and x4 with cue values created by a uniform pseudorandom number generator with a range from 0 to 100 (Banks et al., 2001). We chose four cues because the system provides a nontrivial and yet tractable example to investigate the possible relationship between the RLM and the CLM. Table 3 shows how the components used for calculating lens model parameters are produced under Case I and Case IV conditions.
(7)
where up represents the non-perceivable cue. The linear relationship between cues and human judgments (i.e., for Case I) were created by a linear-based generator, LGs, in the form Y s ¼ 0:55x1 þ 0:3x2 þ 0x3 þ 0x4 þ 0:15cf
(8)
where cf represents the unavailable cue which completes the judgment process for the simulated human. LGs indicates that human does not consider cue x3 and x4 when they make compensatory decisions. 4.1.2. Rule-based generator The rule-based generator, RGs, generated the categorical data for the values of human decisions (i.e., for Case IV) by the prespecified rule sets based on the categorical cue values. The rule set is represented in disjunctive normal form as If ðx3 ¼ x4 Þ And ðcf ¼ C 2 Þ Then ðY s ¼ Y 1 Þ OR If ðx3 ¼ x4 Þ And ðcf ¼ C 3 Þ ThenðY s ¼ Y 1 Þ OR RGs : If ðx3 ax4 Þ And ðcf ¼ C 2 Þ Then ðY s ¼ Y 3 Þ OR If ðx3 ax4 Þ And ðcf ¼ C 3 Þ Then ðY s ¼ Y 3 Þ OR If ðcf ¼ C 1 Þ Then ðY s ¼ Y 3 Þ; (9) where {X 1 ; X 2 ; . . . ; X p } represent the set of discrete values that xi can take, C Fl 2 fC 1 ; C 2 ; . . . ; C q g for l ¼ 1; 2; . . . ; m where {C 1 ; C 2 ; . . . ; C q } represent the set of discrete values that cf can take. Y Sk 2 fY 1 ; Y 2 ; . . . ; Y r g for k ¼ 1; 2; . . . ; m and {Y 1 ; Y 2 ; . . . ; Y r } represents the set of discrete values
Table 3 Components used for calculating lens model parameters under case I and case IV conditions x1,x2,x3,x4
Ys
Ye
Y^ s
Case I
U(0,100)
LGs
LGe
CLM RLM
Linear regression RSs
CLM RLM
Linear regression RSe
Case IV
U(0,100)
RGs
LGe
CLM RLM
Linear regression RSs
CLM RLM
Linear regression RSe
LGs—Linsai-based generator for decision values. LGe—Linsai-based generator for criterion values. RGs—rule-based generator for decision values. RSs—rule-based model for predicting decision values. RSe—rule-based model for predicting criterion values.
Y^ e
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0.8 0.6 0.4 0.2 0
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
Rs (Correlation)
Environmental Predictability for RLM 1 0.8 0.6 0.4 0.2 0
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
Human Control for CLM
1 0.8 0.6 0.4 0.2 0 0 -0.2
0.2
0.4
0.6
0.8
1
Rs (Percentage Match)
Re (Correlation)
1
Re (Percentage Match)
Environmental Predictability for CLM
Human Control for RLM 1 0.8 0.6 0.4 0.2 0
p (Proportion of Judgments Generated by Linear Generator)
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
G (Correlation)
0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
-0.2
G (Percentage Match)
Modeled Knowledge for CLM 1
Modeled Knowledge for RLM 1 0.8 0.6 0.4 0.2 0
p (Proportion of Judgments Generated by Linear Generator)
1 0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated by Linear Generator)
0
C (Correlation)
0.8 0.6 0.4 0.2 0 -0.2
0
0.2
0.4
0.6
0.8
1
C (Percentage Match)
Unmodeled Knowledge for CLM 1
Unmodeled Knowledge for RLM 1 0.8 0.6 0.4 0.2 0
p (Proportion of Judgments Generated by Linear Generator)
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
ra (Correlation)
0.7 0.5 0.3 0.1 -0.10
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
ra (Percentage Match)
Achievement for CLM 0.9
Achievement for RLM 1 0.8 0.6 0.4 0.2 0
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
Fig. 5. Simulation output.
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that Ys can take. The rule-based generator was designed so that achievement, ra, when compared with the linearly organized environmental criterion is established at 0.25. RGs suggests when humans make noncompensatory decisions, cue x1 and x2 are not utilized. 4.1.3. CLM analysis For the continuous data set, statistical regression method was used to predict the values of environmental criteria, Y^ e , and human judgments, Y^ s . The LME parameters were calculated as the correlation coefficients between the corresponding data set. To simulate ambiguity in the environment, we removed up (see Eq. (7)) so that the regression model of the predictor becomes Y^ e ¼ a0 þ a1 x1 þ a2 x2 þ a3 x3 þ a4 x4 ,
(10)
where a0, a1, a2, a3, a4 represent the coefficients for the environmental criteria regression model. Correspondingly, we simulated uncertainty in modeling the judge by removing cf (see Eq. (8)) so that the regression model becomes Y^ s ¼ b0 þ b1 x1 þ b2 x2 þ b3 x3 þ b4 x4 ,
(11)
where b0, b1, b2, b3, b4 represent the coefficients for the human judgments regression model. 4.1.4. RLM analysis Tools such as GBPC (Rothrock and Kirlik, 2003a) would normally be employed to infer the rule sets to predict judgments and criteria values for the categorical data set. For the present analysis, however, we simplified our evaluation by using the rule-based generator (Eq. (9)) as the basis for establishing a prediction mechanism for human judgments. To simulate judgment ambiguity, we removed cf (see Eq. (9)) so that the rule-based model of the predictor, RSs, becomes RSs :
If ðx3 ¼ x4 Þ Then ðY s ¼ Y 1 Þ OR If ðx3 ax4 Þ Then ðY s ¼ Y 3 Þ:
(12)
In establishing a predictor for the environmental criteria, we empirically found the following prediction model that correlates well with LGe (see Eq. (7)): If ðx1 ¼ X 3 Þ Then ðY e ¼ Y 3 Þ OR If ðx1 ¼ X 2 Þ Then ðY e ¼ Y 2 Þ OR RSe : If ðx1 ¼ X 1 Þ And ðx2 ¼ X 1 Þ Then ðY e ¼ Y 1 Þ OR If ðx1 ¼ X 1 Þ And ðx2 ¼ X 2 Þ Then ðY e ¼ Y 2 Þ OR If ðx1 ¼ X 1 Þ And ðx2 ¼ X 3 Þ Then ðY e ¼ Y 2 Þ: (13)
scenarios in which, for each scenario, we simulated 1000 judgments. Each scenario represents a different probability, p, for inducing a compensatory judgment strategy. Each scenario was realized through creating a proportion of compensatory judgments ranging from completely compensatory (p ¼ 1) to entirely rule based (p ¼ 0). Within each scenario, 1000p of the decisions are made under compensatory strategy while the other 1000(1p) decisions are made by the rule-based strategy. 4.3. Simulation results The main simulation process began with creating instances of task scenarios ði:e:; x1 ; x2 ; x3 ; x4 ; up ; cf Þ in which judgments, Ys, and criteria, Ye, are needed. We then applied both RLM and CLM analyses to get two sets of lens model parameters for each task scenario. Simulation output in terms of the lens model parameters is shown in Fig. 5. The abscissa represents p, the proportion of compensatory strategies contained in the task scenario while the ordinate represents the value of the lens model parameter. We caution the reader from integrating both analyses into a single interpretation at this time (e.g., looking at points of intersection) since the lens model parameters are taken from two different measurement scales (i.e., one based on correlation and the other based on percentage match). For now, we highlight the results in the CLM case. Because human judgment is the element which varies as p changes, we see that human control and modeled knowledge improves as p approaches 1. As unmodeled knowledge remains low, the overall achievement improves as p approaches 1. In the next section of this paper, we will discuss our preliminary results and ways of deriving a juxtaposition of both outcomes. 5. Discussion Based on Fig. 5, two particular lens model parameters become of interest to us in the comparison of CLM and RLM: human control, Rs and unmodeled knowledge, C. We expect that human control in the CLM case would improve and in the RLM case would deteriorate as p ! 1. Moreover, since unmodeled knowledge in a linear sense represents a nonlinear relationship, we hypothesized that unmodeled knowledge in a rule-based context would indicate a linear relationship. In fact, we found that there exists a high positive correlation (R2 1) between C under RLM analysis and the proportion of compensatory judgments.
4.2. Simulation procedure 5.1. Analysis of CLM and RLM parameters To evaluate the performance of the lens model under Case I and Case IV (see Table 3) we simulated conditions in which the environmental criteria are linearly organized and that judgment strategy ranges from compensatory (Case I) to rule-based (Case IV). To do so, we created 1000
Recall that, in CLM, C is calculated as the linear correlation between residuals of environmental criteria value and human judgment which cannot be captured by the two linear models representing linear cue-criteria
ARTICLE IN PRESS J. Yin, L. Rothrock / International Journal of Industrial Ergonomics 36 (2006) 499–509
structures and compensatory decision strategies. Alternatively, in the RLM, all the values are discretized so that percentage match is used to calculate the value of C. Therefore, the results, shown in the simulation output, are not measured on the same scale.
To derive a meaningful relationship between the RLM and CLM parameters, we seek to calibrate the lens model parameters so that results from both analyses are measured on the same scale. To do this, we recomputed the values in CLM in the same manner that it was calculated under Human Control for Both CLM and RLM
Environmental Predictability for Both CLM and RLM
0.8
RLM
0.6
0.4
0.4
0.2
0.2
0 0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated
RLM
0.6
0.6
0.4
0.4
0.2
0.2
1
0 0
0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated
by Linear Generator) Modeled Knowledge for Both CLM and RLM
Unmodeled Knowledge for Both CLM and RLM
0.4 RLM
0.2
0.2
0
0 0.2
0.4
0.6
0.8
0.8
0.8
0.6
0.6
0.4
0.4
0.2
RLM
CLM
0
0 0
1
p (Proportion of Judgments Generated by Linear Generator)
0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated by Linear Generator)
Achievement for Both CLM and RLM
1
1 0.8
0.8 0.6
RLM
0.6
0.4
CLM
0.4 0.2
0.2
0
0 0
0.2
0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated by Linear Generator) Fig. 6. Transformed simulation output.
1
1
RLM C (Percentage Match)
0.4
CLM C (Percentage Match)
0.6
RLM G (Percentage Match)
0.6
1
1
0.8
CLM
CLM ra (Parcentage Match)
CLM G (Percentage Match)
1
0
1
by Linear Generator)
1 0.8
0.8
CLM
0
0 0
0.8
RLM ra (Percentage Match)
0.6
CLM Rs (Percentage Match)
CLM
0.8
1
1
RLM Rs (Percentage Match)
1 RLM Re (Percentage Match)
CLM Re (Percentage Match)
1
507
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RLM analysis (see Eqs. (2)–(6) and use percentage match as the underlying formulation for comparing unmodeled knowledge resulting from RLM and CLM analysis. The transformed lens model parameter values are shown in Fig. 6. Results in human control are consistent with our expectations. As the proportion of linear judgments increased (p ! 1), CLM-based control increased while RLM-based control decreased. It is unmodeled knowledge that provides a more interesting outcome. We hypothesize that the intersection point of the lines for compensatory and noncompensatory C represents the location where the same amount of unmodeled knowledge is captured by both CLM and RLM. Through our simulation, we empirically show that the intersection point of C for CLM and RLM analyses could be used as the reference point for judgment strategies along a potential compensatory–noncompensatory spectrum. We submit this result as a very preliminary benchmark for the further development of technologies to differentiate compensatory from rule-based strategies. 6. Conclusion Our research focused on an approach of using the conceptual lens model framework to investigate contingent human decision. The study is based on past research evidence which suggests that humans alter decision strategies in order to cope with tasks under time pressure and high cognitive workload. Our intent is to provide a technique, called the Rule-based Lens Model, which enables practitioners to explore various decision behaviors. As our starting point, we used a simulation study to illustrate the performance of the RLM and CLM under varying conditions. By integrating the rule-based formulation into the lens model framework as conceptualized by Brunswik (1956), we are able to calculate the lens model parameters for both CLM and RLM analysis under the notion of percentage match. While our current results suggest that C could serve as an indicator of the strategy being employed by a judge, we submit that further analysis is needed to evaluate the sensitivity of C under a varying environmental and judgment conditions. Furthermore, validation of the proposed model with actual human performance data will eventually be required. Nevertheless, we suggest that this research serves as a precursor toward the development of the generalized formulation of the lens model. We propose that a generalized lens model which systematically incorporates both compensatory and rulebased components could be used as a real-time quantitative analyzing tool to infer the judgment strategies under various environment conditions (i.e. different time pressures or cognitive workloads). Research toward the development of such a model can serve as an important reference point for the development of decision-support systems and more effective training methodologies. More-
over, it can give us deeper insights on why humans change judgment strategies to cope with changes in the environments. Acknowledgements This research was supported by the National Science Foundation under Grant No. SES-0452416. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. We wish to thank Alex Kirlik, Ann Bisantz, and the anonymous reviewers for their valuable comments. References Abelson, R.P., Levi, A., 1985. Decision making and decision theory. In: Lindzey, G., Aronson, E. (Eds.), The Handbook of Social Psychology, First ed. Random House, New York. Banks, J., Carson, J.S., Nelson, B.L., Nicol, D.M., 2001. Discrete-Event System Simulation, third ed. Prentice-Hall, Inc., Upper Saddle River, NJ. Beach, L.R., Mitchell, T.R., 1978. A contingency model for the selection of decision strategies. Academy of Management Review 3 (4), 39–49. Brunswik, E., 1943. Organismic achievement and environmental probability. Psychological Review 50, 255–272. Brunswik, E., 1955. Representative design and probabilistic theory in a functional psychology. Psychological Review 62, 193–217. Brunswik, E., 1956. Perception and the Representative Design of Psychological Experiments. University of California Press, Berkeley, CA. Cooksey, R.W., 1996. Judgment Analysis: Theory, Methods, and Applications. Academic Press, San Diego, CA. Einhorn, H.J., 1970. Use of nonlinear, noncompensatory models in decision making. Psychological Bulletin 73 (2), 21–30. Einhorn, H.J., 1971. Use of nonlinear, noncompensatory models as a function of task and amount of information. Organizational Behavior and Human Performance 6, 1–27. Elrod, T., Johnson, R.D., White, J., 2004. A new integrated model of noncompensatory and compensatory decision strategies. Organizational Behavior and Human Decision Processes 95 (1), 1–19. Ganzach, Y., Czaczkes, B., 1995. On detecting nonlinear noncompensatory judgment strategies: comparison of alternative regression models. Organizational Behavior and Human Decision Processes 61 (2), 168–176. Gigerenzer, G., Goldstein, D.G., 1996. Reasoning the fast and frugal way: models of bounded rationality. Psychological Review 103, 650–669. Hursch, C.J., Hammond, K.R., Hursch, J.L., 1964. Some methodological considerations in multiple-cue probability studies. Psychological Review 71 (1), 42–60. Lordahl, D.S., 1967. Modern Statistics for Behavioral Sciences. Ronald Press, New York. Mendelson, E., 1997. Introduction to Mathematical Logic, fourth ed. Chapman and Hall, London. Payne, J.W., 1976. Task complexity and contingent processing in decision making: an information search and protocol analysis. Organizational Behavior and Human Performance 16, 366–387. Payne, J.W., 1982. Contingent decision behavior. Psychological Bulletin 92, 382–402. Payne, J.W., Bettman, J.R., Johnson, E.J., 1993. The adaptive decision maker: effort and accuracy in choice. In: Hogarth, R.M. (Ed.), Insights in Decision Making: A Tribute to Hillel J. Einhorn. University of Chicago Press, Chicago, IL, pp. 262–279. Rothrock, L., Kirlik, A., 2003a. Inferring rule-based strategies in dynamic judgment tasks: toward a noncompensatory formulation of the lens
ARTICLE IN PRESS J. Yin, L. Rothrock / International Journal of Industrial Ergonomics 36 (2006) 499–509 model. IEEE Transactions on Systems, Man, and Cybernetics, Part A 33 (1), 58–72. Rothrock, L., Kirlik, A., 2003b. Explorations in modeling human decision making in dynamic contexts. Paper presented at the Fifth International Conference on Engineering Psychology and Cognitive Ergonomics, Crete, Greece. Rothrock, L., Kirlik, A. Inferring fast and frugal heuristics from human judgment data. In: A. Kirlik (Ed.), Adaptive Perspectives on HumanTechnology Interaction: Methods and Models for Cognitive Engineering and Human-Computer Interaction. Oxford University Press, Oxford (in press).
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Schneider, L.G., Selling, T.I., 1996. A comparison of compensatory and noncompensatory models of judgment: effects of task predictability and degree of freedom. Accounting, Organizations and Society 21 (1), 3–22. Simon, H.A., 1981. The Sciences of the Artificial, second ed. MIT Press, Cambridge, MA. Svenson, O., 1979. Process descriptions of decision making. Organizational Behavior and Human Performance 23, 86–112. Tucker, L.R., 1964. Suggested alternative formulation in the developments by Hursch, Hammond, and Hursch, and by Hammond, Hursch, and Todd. Psychological Review 71 (6), 528–530.
International Journal of Industrial Ergonomics 36 (2006) 499–509 www.elsevier.com/locate/ergon
A rule-based lens model Jing Yin, Ling Rothrock The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, USA Available online 9 March 2006
Abstract The purpose of this paper is to introduce a novel approach, called the rule-based lens model (RLM) to model human judgment of a probabilistic criterion. Our method is motivated by a shortcoming of an existing linear-additive model of judgment based on the lens model equation (LME) to adequately represent all rule-based relationships. Through the use of a simple example, we demonstrate the shortcoming of the additive model and set the context for our generalized rule-based formulation of Brunswik’s conceptual lens model. To investigate the behavior of our model and the relationship to the traditional lens model based on the LME, we simulate human judgments and criterion values in a ‘‘drosophila’’ domain where the parameters of the problem in terms of the number of cues, organizing principle of the criterion, organizing principle of the judge and the extent of uncertainty within the system can be systematically varied. Our efforts represent a first step toward the formulation of a generalized lens model framework. Relevance to industry The findings of the proposed research would provide theoretical basis toward the design of decision-aiding and decision-training systems that are adapted to human decision strategies. r 2006 Elsevier B.V. All rights reserved. Keywords: Decision making; Human factors; Lens model
1. Introduction and literature review The purpose of this paper is to introduce a novel approach to model human judgment of a probabilistic criterion. More specifically, we seek to establish a novel method to model the judgment of a human (e.g., a weather forecaster) who selects a criterion value (e.g., if it will rain or not) based on a set of probabilistic cues (e.g., air fronts, dew point temperature and temperature). Our approach examines the extension of the lens model (Brunswik, 1956) so that contingent decision behavior can be incorporated. The lens model, depicted in Fig. 1, represents the decisionmaking system as a symmetrical structure. The task environment, or ecology, is represented in the left half of the figure and the human judge is represented on the right half. The symmetry inherent in this representation allows Corresponding author.
E-mail address: [email protected] (L. Rothrock). 0169-8141/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ergon.2006.01.012
one to measure the degree of adaptation or ‘‘fit’’ between the judge and the demands of the judgment task. We submit that, while the conceptual model framed by Brunswik (1955) is broadly applicable, existing analytical models of the lens model (Hursch et al., 1964; Tucker, 1964) focus primarily on compensatory judgment behavior. Under a compensatory mode of decision making, information is processed exhaustively and trade-offs need to be made between attribute values. Compensatory modes of decision strategies are contrasted against noncompensatory modes (Payne, 1976; Einhorn, 1970; Svenson, 1979) where, typically, not all available information is used and, therefore, trade-offs are often ignored. Common noncompensatory strategies include:
Conjunctive: the decision maker sets thresholds on each attribute so that alternatives not meeting all thresholds are eliminated; Disjunctive: the decision maker accepts an alternative if one or more attributes are acceptable;
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Cues
Criterion
Judgment
Fig. 1. Brunswik’s lens model.
Restaurant
Local Reputation
Newspaper Review
Desirability
A B C D
5.0 4.7 4.1 2.0
5.0 3.0 3.5 1.0
0.95 0.75 0.70 0.20
where Desirability represents the attractiveness of the restaurant to the decision maker. As a compensatory model, we can represent the restaurant choice problem as Desirability ¼ 0:17 þ 0:15 Local_Reputation
Elimination by aspects: at each stage of the process, the decision maker selects an attribute and eliminates all alternatives that do not include a specified aspect (i.e., attribute value) until only one alternative remains; and Take the best (Gigerenzer and Goldstein, 1996): the decision maker uses a sequence of rules to choose an alternative in the face of uncertain knowledge.
Research on contingent decision making suggests that decision makers respond adaptively to the variations in the environment through use of different strategies (Einhorn, 1970, 1971; Payne, 1982; Payne et al., 1993; Abelson and Levi, 1985). Research has shown that decision makers are sensitive to the increase of problem size and generally shift to noncompensatory evaluation strategies that save effort but are less accurate (Payne et al., 1993). Time pressure is another reason that the use of a compensatory strategy may be less preferable (Simon, 1981). However, when the task environment does not pose severe constraints in terms of time and task load, people tend to use compensatory strategies so that high decision accuracy can be obtained (Beach and Mitchell, 1978; Payne et al., 1993). We provide an example of restaurant choice problem to illustrate the mathematical differences between compensatory and noncompensatory models. We P represent the compensatory model, fc, as f c ðX Þ ¼ b0 þ ki¼1 bi X i where bi is the weight value and Xi is the cue value for the ith cue. Moreover, we represent a noncompensatory model, fn, as f n ðX Þ ¼ _ð^ X i Þ where 3 is Boolean disjunction over s, s k the set containing all cues, and 4 is Boolean conjunction. We intentionally represent fn in disjunctive normal form (DNF) because it has been shown that any strategy consisting of basic logical operators can be reduced to it (Mendelson, 1997). Hence, the outcomes of strategies ranging from simple conjunctive and disjunctive rules (Einhorn, 1970) to the process-dependent fast and frugal heuristics (Gigerenzer and Goldstein, 1996) can be reduced to DNF (see Rothrock and Kirlik, in press). Consider an example of choosing a restaurant for an evening out. We assume that each prospective restaurant has been reviewed by a local newspaper food critic and that each has established a local reputation. We consider the following four hypothetical restaurants:
þ 0:08 Newspaper_Review: We see that the desirability of a restaurant is determined by weighing all the available information. Now consider the case where the decision maker is under time pressure so that he must choose a restaurant, eat quickly and return to a late business meeting. Therefore, while he is interested in eating well, he is more interested in finishing on time. Revisiting the four restaurants, we see the following: Restaurant
Seating line
Occupancy
Desirability
A B C D
Full Full Empty Empty
Full Seat available Full Seat available
0 1 1 0
Whereas Restaurant A was highly desirable in the compensatory case, in the noncompensatory case, Restaurant A is undesirable. Moreover, because an empty restaurant is a likely indicator of the quality of the food, the decision maker is not interested in Restaurant D either. Therefore, as a noncompensatory model, we can represent the choice problem as ½ðSeating Line ¼ Full AND Occupancy ¼ FullÞ ! ðDesirability ¼ 0Þ OR ½ðSeating Line ¼ Full AND Occupancy ¼ Seat AvailableÞ ! ðDesirability ¼ 1Þ OR ½ðSeating Line ¼ Empty AND Occupancy ¼ FullÞ ! ðDesirability ¼ 1Þ. OR ½ðSeating Line ¼ Empty AND Occupancy ¼ Seat AvailableÞ ! ðDesirability ¼ 0Þ The noncompensatory restaurant choice model is an example of an exclusive-OR relationship which cannot be modeled using linear methods. The best linear model results in the following non-informative solution Desirability ¼ 0:5 þ 0 Seating_Line þ 0 Occupancy:
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Our example illustrates a key difference between compensatory and noncompensatory, or more specifically, rule-based models of decision making. While linear additive models often serve as adequate approximations of nonlinear strategies they are sometimes inadequate representations of rule-based behaviors such as that exhibited in the restaurant choice example. Moreover, existing efforts to model rule-based decision strategies using additive nonlinear formulations (Einhorn, 1970; Ganzach and Czaczkes, 1995; Elrod et al., 2004) lack the robustness to accommodate multiple rule-based strategies. 2. Mathematical representation of the lens model The standard representation of the lens model is given in Fig. 2 where subjects make judgments, denoted as Ys, on an environmental variable measured by the criterion, denoted as Ye, by using a set of cues X (Cooksey, 1996). The predicted values of the environmental criterion and human judgments, Y^ e and Y^ s , respectively, are generated by corresponding linear regression equations, which takes the form of fc. Hursch et al. (1964) first introduced the formulation of the lens model using Ys, Ye and X as random variables. Lens model parameters ra, Re, Rs, G, C are calculated as Pearson’s correlation coefficient, r, such that Re ¼ rðY e ; Y^ e Þ; Rs ¼ rðY s ; Y^ s Þ; G ¼ rðY e ; Y s Þ, and C ¼ r ðY e Y^ e Þ; ðY s Y^ s Þ . Tucker (1964) later modified the Hursch model into the following form now known as the LME: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) ra ¼ GRe Rs þ C 1 R2e 1 R2s , where the achievement, ra, represents how well human decisions adapt to the actual value of the environmental criterion. Re represents the environmental predictability and measures how well the environmental model predicts the actual criterion value. Rs is labeled as human control and indicates how the linear policy model captures the
501
actual human judgments. Linear knowledge, denoted as G, is designed to estimate how well the linear prediction model of the environment maps onto the predicted policy model of the human judgment. Unmodeled knowledge, denoted as C, measures how well the two models (one for the environment and the other for the human judgment) share the common points that are not captured in the corresponding linearly based model. We shall call the LMEbased formulation the compensatory lens model (CLM) to distinguish it from our proposed rule-based formulation. The CLM relies heavily on multiple linear regression models to determine the fitness of human judgments relative to constraints in the task environment. While this formulation has been widely used, its success is tempered by occasions in which judgment is driven by noncompensatory strategies. Although some have attempted to extend the LME to account for noncompensatory strategies (Einhorn, 1970; Cooksey, 1996), a framework which is able to fully model both compensatory and rule-based decisions, such as the two divergent strategies of the restaurant choice problem, has not been developed. The rest of this paper is organized as follows. First, we propose an integration of a rule-based formulation into the conceptual lens model framework. In the second section, we introduce a simulation study which enables us to investigate the behavior of our rule-based model. Section three focuses on one particular simulation finding that, we hypothesize, provides insights into factors which delineate the CLM from the rule-based lens model (RLM). Finally, some conclusions and implications of our research are discussed. 3. Formulation of RLM Rothrock and Kirlik (2003a) proposed a technique called genetic-based policy capturing (GBPC) to infer noncompensatory judgment strategies from human decision data
Fig. 2. Compensatory lens model with labeled statistical parameters.
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which explores rule-based modeling using propositional logic. This paper serves as a complement to their study. While Rothrock and Kirlik focused on the development of an inductive inference technique to capture judgment policies, the current work seeks to develop a rule-based analytical model of the conceptual lens model as conceived by Brunswik. The role of GBPC within the RLM is to generate Y^ e and Y^ s . We intend for this rule-based model to complement the multiple linear regression models in CLM. In building the functional relationship between human judgment and the task environment for compensatory and rule-based strategies, we note two key differences. First, the prediction mechanisms for compensatory strategies are multiple linear regression models whereas the mechanisms for rule-based strategies are logical propositions. Second, compensatory strategies operate on an interval scale whereas rule-based strategies require only a nominal scale. To use LME in an analysis of compensatory judgment strategies, the dependent variable should be measured on an interval scale (Schneider and Selling, 1996). The scale requirement is noted since linear regression serves as the basis for traditional lens model analysis (Hursch et al., 1964; Tucker, 1964). When modeling rule-based strategies and ecologies, on the other hand, the data used to represent environmental cues, judgments and criteria are discrete and categorical (Rothrock and Kirlik, 2003a). As such, the numbers assigned to them are on a nominal scale. This means that the assignment is arbitrary, as the numbers are simply labels (Lordahl, 1967). Therefore, interval or ratio scale operations can no longer be rigorously applied toward rule-based strategies. To create the RLM, we develop a new framework to compare the results of fitted models of both judgment strategies ðY s Y^ s Þ and the task ecology ðY e Y^ e Þ. We begin by providing a definition of error for rule-based strategies. Whereas compensatory model error is measured using the least-squares estimator, we propose that rulebased model error be measured as the number of mismatches between two sets of categorical data. Table 1 shows a simple example to illustrate the idea. In the example, there are five judgment instances in which three are matched. We call the ratio denoting the number of match cases out of the total number of instances as the match rate: m ¼ 3=5 ¼ 0:6. Using the match rate as the basis of our noncompensatory model, we construct an RLM that accounts for noncompensatory strategies and task environments. We suggest that the fitted parameters Table 1 Two categorical data sets and encoding (Eat—1, and Do Not Eat—0) Y1
Y2
Y1_encoded
Y2_encoded
Eat Do Not Eat Eat Eat Do Not Eat
Do Not Eat Eat Eat Eat Do Not Eat
1 0 1 1 0
0 1 1 1 0
NOT match NOT match Match Match Match
(all bivariate correlations in the CLM) Re, Rs, G, C, ra of the RLM be calculated instead as match frequencies. The operators of RLM are developed in following section. Assuming the ecological criterion to be judged is Y, which takes the categorical values on nominal scale. Let fY 1 ; Y 2 ; Y 3 ; . . . ; Y p g be the set of discrete values that Y can take. For instance i, 1pipn, where n is the number of judgments. Let Yei represents the actual environmental criterion value; Y^ ei represents the predicted criterion value from the environmental rule-based model; Ysi represents the human judgment value and Y^ si represents the predicted judgment value from the judgment rule-based model. We define the following indicator functions to calculate RLM relationships: ( 1 if Y ei ¼ Y^ ei ; I ei ¼ 0 otherwise; ( I si ¼
1 if Y si ¼ Y^ si ; 0 otherwise;
( I ri ¼
1 if Y ei ¼ Y si ; 0 otherwise;
( I Gi ¼
0 otherwise; and
I Ci ¼
1 if Y^ ei ¼ Y^ si ;
1 if I ei ¼ I si ¼ 0; 0 otherwise:
The corresponding RLM parameters are calculated as follows: Pn I ei (2) Re ¼ i¼1 ; n Pn I si Rs ¼ i¼1 ; (3) n Pn I ri ra ¼ i¼1 ; (4) n Pn I Gi G ¼ i¼1 ; and (5) n Pn I Ci C ¼ i¼1 . (6) n The interpretation of ICi is given in Table 2. Fig. 3 gives the RLM framework. The interpretation of the lens model parameters for the rule-based case is analogous to the linear case. Achievement, ra, represents the correspondence between human judgments and the actual value of the environmental criterion. Re, the environmental predictability, measures how well the noncompensatory environmental model can be used to predict the criterion value while Rs, labeled as human
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control, indicates how well the noncompensatory judgment model captures actual human judgments. Instead of representing linear knowledge in the linear-based lens model, in the RLM, the parameter G represents how well the noncompensatory model of the environment maps onto the noncompensatory model of the human judgments strategy. Now, the C parameter captures systematic regularities between the errors of the noncompensatory ecological and judgment models. In the RLM, the range of parameters ra Re Rs and G is [0,1]. The closer these values are to 1, the better the achievement, environmental predictability, human control and modeled knowledge, respectively. For C, a high value reveals a high degree of unmodeled knowledge. In the case of CLM analysis, C measures nonlinear relationships not captured by the linear-based model—which is consistent with cue usage in the rule-based model. In a corresponding manner, we suspect that a high value of C revealed through RLM analysis may represent a high degree of compensatory cue usage. To explore this potential relationship, a simulation study will be presented in the following section. Table 2 Interpretation of relationship existing between ICi and Iei, Isi Iei
Isi
ICi
Interpretation
0
0
1
Both environmental criteria and human judgments are not captured by the corresponding rule-based models. Mapping exists between the disregularities of the two systems (ecology and human judgment system). Either environmental criteria or human judgments are correctly captured by the corresponding rule-based models. System disregularities do not occur simultaneously; thus no mapping exists.
1 0 1
1 1 0
0 0 0
503
4. Simulation study To investigate the behavior of our model and the relationship to the traditional lens model based on the LME, we simulated human judgments and criterion values in a ‘‘drosophila’’ domain where the parameters of the problem (i.e., number of cues, organizing principle of the criterion, organizing principle of the judge and the extent of uncertainty within the system) can be systematically varied. Our efforts represent a first step toward addressing the relationship between a lens model which can account for rule-based decisions, with a basis in the language of logical propositions, and the traditional lens model, which was developed in the language of statistical correlation. To conduct our exploration, we used a framework proposed by Rothrock and Kirlik (2003b) for investigating judgments made under different environmental cue structures and decision strategies (see Fig. 4). For the environment, Cases I and IV represent a linear relationship between the cues and the environment criterion while Cases II and III represent a nonlinear organizing principle. For decisions, Cases I and III characterize a compensatory decision strategy while Cases II and IV reflect a rule-based strategy.
Cue Structure Linear
Nonlinear
Compensatory
Case I
Case III
Rule-based
Case IV
Case II
Decision Strategy
Fig. 4. Framework for investigating decision-making models.
Fig. 3. Rule-based lens model framework.
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For the study, we focus on the linear environmental structure to examine the relationship between compensatory (Case I) and rule-based (Case IV) decision strategies. This approach requires that we generate criterion values, Ye, that are governed by a linear structure and judgment values, Ys, that are both compensatory and rule-based. We submit that this investigation is not comprehensive, but is a preliminary examination of the relationship between the structure of cues and the decision strategy that is utilized by a decision maker.
Unlike actual cases where human judgments and environmental criteria are collected to produce CLM models, we used linear-based generators (LGs and LGe) and rule-based generators (RGs) to simulate instances of judgment and criteria. 4.1.1. Linear-based generators For the simulation, we used two linear-based generators. One builds the linear relationship between cues in the domain and the criterion to be judged by generating Ye. The generator, LGe, takes the form of Y e ¼ 0:6x1 þ 0:25x2 þ 0 x3 þ 0 x4 þ 0:15up ,
4.1. Elements of the simulation To execute the simulation, we made a key assumption regarding the achievement of the judge (ra) and environmental probability. We agree with Brunswik (1943) that the ambiguity in the environment as experienced by a judge results from a lack of control over all conditions that contribute to the criterion. Given that the cues available to the judge represent only a subset of all conditions, he could at best achieve only a partial knowledge of the criterion. We further submit that, to create criterion values associated with each judgment, it is possible to assume full access to all conditions bearing on the cause of the criterion by including cues which are perceivable as well as those which are not. The criterion ‘‘generator’’, therefore, is deterministic. Similarly, we assumed that the judge is purposive and does not randomly make judgments. Similar to our approach toward the criterion generator, we also accounted for perceivable and non-perceivable cues that are utilized for each judgment. Hence, the judgment ‘‘generator’’ is also deterministic. As a starting point, we simulated a system with four environmental cues represented by x1, x2, x3 and x4 with cue values created by a uniform pseudorandom number generator with a range from 0 to 100 (Banks et al., 2001). We chose four cues because the system provides a nontrivial and yet tractable example to investigate the possible relationship between the RLM and the CLM. Table 3 shows how the components used for calculating lens model parameters are produced under Case I and Case IV conditions.
(7)
where up represents the non-perceivable cue. The linear relationship between cues and human judgments (i.e., for Case I) were created by a linear-based generator, LGs, in the form Y s ¼ 0:55x1 þ 0:3x2 þ 0x3 þ 0x4 þ 0:15cf
(8)
where cf represents the unavailable cue which completes the judgment process for the simulated human. LGs indicates that human does not consider cue x3 and x4 when they make compensatory decisions. 4.1.2. Rule-based generator The rule-based generator, RGs, generated the categorical data for the values of human decisions (i.e., for Case IV) by the prespecified rule sets based on the categorical cue values. The rule set is represented in disjunctive normal form as If ðx3 ¼ x4 Þ And ðcf ¼ C 2 Þ Then ðY s ¼ Y 1 Þ OR If ðx3 ¼ x4 Þ And ðcf ¼ C 3 Þ ThenðY s ¼ Y 1 Þ OR RGs : If ðx3 ax4 Þ And ðcf ¼ C 2 Þ Then ðY s ¼ Y 3 Þ OR If ðx3 ax4 Þ And ðcf ¼ C 3 Þ Then ðY s ¼ Y 3 Þ OR If ðcf ¼ C 1 Þ Then ðY s ¼ Y 3 Þ; (9) where {X 1 ; X 2 ; . . . ; X p } represent the set of discrete values that xi can take, C Fl 2 fC 1 ; C 2 ; . . . ; C q g for l ¼ 1; 2; . . . ; m where {C 1 ; C 2 ; . . . ; C q } represent the set of discrete values that cf can take. Y Sk 2 fY 1 ; Y 2 ; . . . ; Y r g for k ¼ 1; 2; . . . ; m and {Y 1 ; Y 2 ; . . . ; Y r } represents the set of discrete values
Table 3 Components used for calculating lens model parameters under case I and case IV conditions x1,x2,x3,x4
Ys
Ye
Y^ s
Case I
U(0,100)
LGs
LGe
CLM RLM
Linear regression RSs
CLM RLM
Linear regression RSe
Case IV
U(0,100)
RGs
LGe
CLM RLM
Linear regression RSs
CLM RLM
Linear regression RSe
LGs—Linsai-based generator for decision values. LGe—Linsai-based generator for criterion values. RGs—rule-based generator for decision values. RSs—rule-based model for predicting decision values. RSe—rule-based model for predicting criterion values.
Y^ e
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0.8 0.6 0.4 0.2 0
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
Rs (Correlation)
Environmental Predictability for RLM 1 0.8 0.6 0.4 0.2 0
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
Human Control for CLM
1 0.8 0.6 0.4 0.2 0 0 -0.2
0.2
0.4
0.6
0.8
1
Rs (Percentage Match)
Re (Correlation)
1
Re (Percentage Match)
Environmental Predictability for CLM
Human Control for RLM 1 0.8 0.6 0.4 0.2 0
p (Proportion of Judgments Generated by Linear Generator)
0
0.2 0.4 0.6 0.8 1 p (Proportion of Judgments Generated by Linear Generator)
G (Correlation)
0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
-0.2
G (Percentage Match)
Modeled Knowledge for CLM 1
Modeled Knowledge for RLM 1 0.8 0.6 0.4 0.2 0
p (Proportion of Judgments Generated by Linear Generator)
1 0.2 0.4 0.6 0.8 p (Proportion of Judgments Generated by Linear Generator)
0
C (Correlation)
0.8 0.6 0.4 0.2 0 -0.2
0
0.2
0.4
0.6
0.8
1
C (Percentage Match)
Unmodeled Knowledge for CLM 1
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Fig. 5. Simulation output.
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that Ys can take. The rule-based generator was designed so that achievement, ra, when compared with the linearly organized environmental criterion is established at 0.25. RGs suggests when humans make noncompensatory decisions, cue x1 and x2 are not utilized. 4.1.3. CLM analysis For the continuous data set, statistical regression method was used to predict the values of environmental criteria, Y^ e , and human judgments, Y^ s . The LME parameters were calculated as the correlation coefficients between the corresponding data set. To simulate ambiguity in the environment, we removed up (see Eq. (7)) so that the regression model of the predictor becomes Y^ e ¼ a0 þ a1 x1 þ a2 x2 þ a3 x3 þ a4 x4 ,
(10)
where a0, a1, a2, a3, a4 represent the coefficients for the environmental criteria regression model. Correspondingly, we simulated uncertainty in modeling the judge by removing cf (see Eq. (8)) so that the regression model becomes Y^ s ¼ b0 þ b1 x1 þ b2 x2 þ b3 x3 þ b4 x4 ,
(11)
where b0, b1, b2, b3, b4 represent the coefficients for the human judgments regression model. 4.1.4. RLM analysis Tools such as GBPC (Rothrock and Kirlik, 2003a) would normally be employed to infer the rule sets to predict judgments and criteria values for the categorical data set. For the present analysis, however, we simplified our evaluation by using the rule-based generator (Eq. (9)) as the basis for establishing a prediction mechanism for human judgments. To simulate judgment ambiguity, we removed cf (see Eq. (9)) so that the rule-based model of the predictor, RSs, becomes RSs :
If ðx3 ¼ x4 Þ Then ðY s ¼ Y 1 Þ OR If ðx3 ax4 Þ Then ðY s ¼ Y 3 Þ:
(12)
In establishing a predictor for the environmental criteria, we empirically found the following prediction model that correlates well with LGe (see Eq. (7)): If ðx1 ¼ X 3 Þ Then ðY e ¼ Y 3 Þ OR If ðx1 ¼ X 2 Þ Then ðY e ¼ Y 2 Þ OR RSe : If ðx1 ¼ X 1 Þ And ðx2 ¼ X 1 Þ Then ðY e ¼ Y 1 Þ OR If ðx1 ¼ X 1 Þ And ðx2 ¼ X 2 Þ Then ðY e ¼ Y 2 Þ OR If ðx1 ¼ X 1 Þ And ðx2 ¼ X 3 Þ Then ðY e ¼ Y 2 Þ: (13)
scenarios in which, for each scenario, we simulated 1000 judgments. Each scenario represents a different probability, p, for inducing a compensatory judgment strategy. Each scenario was realized through creating a proportion of compensatory judgments ranging from completely compensatory (p ¼ 1) to entirely rule based (p ¼ 0). Within each scenario, 1000p of the decisions are made under compensatory strategy while the other 1000(1p) decisions are made by the rule-based strategy. 4.3. Simulation results The main simulation process began with creating instances of task scenarios ði:e:; x1 ; x2 ; x3 ; x4 ; up ; cf Þ in which judgments, Ys, and criteria, Ye, are needed. We then applied both RLM and CLM analyses to get two sets of lens model parameters for each task scenario. Simulation output in terms of the lens model parameters is shown in Fig. 5. The abscissa represents p, the proportion of compensatory strategies contained in the task scenario while the ordinate represents the value of the lens model parameter. We caution the reader from integrating both analyses into a single interpretation at this time (e.g., looking at points of intersection) since the lens model parameters are taken from two different measurement scales (i.e., one based on correlation and the other based on percentage match). For now, we highlight the results in the CLM case. Because human judgment is the element which varies as p changes, we see that human control and modeled knowledge improves as p approaches 1. As unmodeled knowledge remains low, the overall achievement improves as p approaches 1. In the next section of this paper, we will discuss our preliminary results and ways of deriving a juxtaposition of both outcomes. 5. Discussion Based on Fig. 5, two particular lens model parameters become of interest to us in the comparison of CLM and RLM: human control, Rs and unmodeled knowledge, C. We expect that human control in the CLM case would improve and in the RLM case would deteriorate as p ! 1. Moreover, since unmodeled knowledge in a linear sense represents a nonlinear relationship, we hypothesized that unmodeled knowledge in a rule-based context would indicate a linear relationship. In fact, we found that there exists a high positive correlation (R2 1) between C under RLM analysis and the proportion of compensatory judgments.
4.2. Simulation procedure 5.1. Analysis of CLM and RLM parameters To evaluate the performance of the lens model under Case I and Case IV (see Table 3) we simulated conditions in which the environmental criteria are linearly organized and that judgment strategy ranges from compensatory (Case I) to rule-based (Case IV). To do so, we created 1000
Recall that, in CLM, C is calculated as the linear correlation between residuals of environmental criteria value and human judgment which cannot be captured by the two linear models representing linear cue-criteria
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structures and compensatory decision strategies. Alternatively, in the RLM, all the values are discretized so that percentage match is used to calculate the value of C. Therefore, the results, shown in the simulation output, are not measured on the same scale.
To derive a meaningful relationship between the RLM and CLM parameters, we seek to calibrate the lens model parameters so that results from both analyses are measured on the same scale. To do this, we recomputed the values in CLM in the same manner that it was calculated under Human Control for Both CLM and RLM
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RLM analysis (see Eqs. (2)–(6) and use percentage match as the underlying formulation for comparing unmodeled knowledge resulting from RLM and CLM analysis. The transformed lens model parameter values are shown in Fig. 6. Results in human control are consistent with our expectations. As the proportion of linear judgments increased (p ! 1), CLM-based control increased while RLM-based control decreased. It is unmodeled knowledge that provides a more interesting outcome. We hypothesize that the intersection point of the lines for compensatory and noncompensatory C represents the location where the same amount of unmodeled knowledge is captured by both CLM and RLM. Through our simulation, we empirically show that the intersection point of C for CLM and RLM analyses could be used as the reference point for judgment strategies along a potential compensatory–noncompensatory spectrum. We submit this result as a very preliminary benchmark for the further development of technologies to differentiate compensatory from rule-based strategies. 6. Conclusion Our research focused on an approach of using the conceptual lens model framework to investigate contingent human decision. The study is based on past research evidence which suggests that humans alter decision strategies in order to cope with tasks under time pressure and high cognitive workload. Our intent is to provide a technique, called the Rule-based Lens Model, which enables practitioners to explore various decision behaviors. As our starting point, we used a simulation study to illustrate the performance of the RLM and CLM under varying conditions. By integrating the rule-based formulation into the lens model framework as conceptualized by Brunswik (1956), we are able to calculate the lens model parameters for both CLM and RLM analysis under the notion of percentage match. While our current results suggest that C could serve as an indicator of the strategy being employed by a judge, we submit that further analysis is needed to evaluate the sensitivity of C under a varying environmental and judgment conditions. Furthermore, validation of the proposed model with actual human performance data will eventually be required. Nevertheless, we suggest that this research serves as a precursor toward the development of the generalized formulation of the lens model. We propose that a generalized lens model which systematically incorporates both compensatory and rulebased components could be used as a real-time quantitative analyzing tool to infer the judgment strategies under various environment conditions (i.e. different time pressures or cognitive workloads). Research toward the development of such a model can serve as an important reference point for the development of decision-support systems and more effective training methodologies. More-
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