Evaluating internal consistency in cognitive models for repeated choice tasks

Transcription

1 Evaluating internal consistency in cognitive models for repeated choice tasks Manuscript submitted on February 13, 2007 Abstract: The present paper proposes and compares two new methods for evaluating internal consistency in cognitive models for repeated choice tasks. In the first method the task is divided into two parts (e.g., odd versus even trials), and the association between parameters estimated on the two parts is examined. This method is similar to the traditional Split-Half Reliability (SHR). The second method is based on the average consistency between random task subcomponents (similar to Cronbach s Alpha method). In both methods, non-modeled trials are used to enter data into the model but not for estimating model parameters. To assess the two methods, a study was conducted in which 65 participants performed three repeated choice tasks: Two versions of the High Variance task (Thaler et al., 1991), and a dynamic version of the Iowa Gambling task (Bechara et al., 1999). The current methods were used to examine the parameters of the Expectancy Valence cognitive model (Busemeyer & Stout, 2002). The results using the two methods converged in terms of the ranked consistency of model parameters and alternative choice rules (trial dependent and independent choice consistency) in the two tasks. Key words: Decision Making, math modeling and model selection, learning, individual differences.

2 Internal consistency in cognitive models 2 In the last two decades, repeated choice tasks have become a popular test for evaluating individual differences in impulsivity and risk taking. In these tasks, the decision maker chooses repeatedly from multiple alternatives and receives immediate feedback after each choice without prior information concerning the alternatives payoff distribution. Three of the most common recent tasks involving repeated choices are the Iowa Gambling task (Bechara, Damasio, Damsio & Anderson, 1994), the Go-No Go Discrimination task (Helmers, Young & Pihl, 1995; Newman, Widom & Nathan, 1985), and the Balloon Analogue Risk task (Lejuez et al., 2002). In these tasks, individual differences in the ability of performers to learn to make the best choice are the basis for evaluating their decision making style. The advantage of using repeated choice tasks is that they are relatively similar to real-world situations involving learning from experience. Yet with the benefit of using such tasks comes a challenge of disentangling the different factors that influence choice behavior. More recently, cognitive models have been proposed for identifying the different component processes that underlie poor performance in the Iowa Gambling task (Busemeyer & Stout, 2002; Yechiam, Busemeyer, Stout & Bechara, 2005) and in other choice tasks (see e.g., Wallsten, Pleskac & Lezuez, 2005; Yechiam et al., 2006). Cognitive modeling is an approach that explains intelligent (human or animal) behavior by formally simulating behavior (often on a computer). The model outputs are parameters that help to interpret the underlying sources of performance deficits 1. However, while previous authors have stressed the need for rigorous psychometric treatment of cognitive model parameters, especially with regards to 1 For example, the Expectancy Valence model (Busemeyer & Stout, 2002) for the Iowa gambling task (Bechara et al., 1994) provides parameters (e.g., the weighting to gains versus losses) which denote alternative subcomponents that can lead to disadvantageous choices. The model can therefore help in identifying the exact parameters distinguishing target populations from controls.

3 Internal consistency in cognitive models 3 process models (Carter, Neufeld & Benn, 1988; see also Batchelder, 1998) relatively little emphasis has been given to the psychometric properties of cognitive models for repeated choice tasks (e.g., Busemeyer & Stout, 2002; Wallsten et al., 2005; Yechiam et al., 2006). Here it is proposed that the cognitive modeling approach can provide indices for the internal consistency of model parameters. I propose two methods for evaluating the internal consistency of model parameters across different choice trials. One method is called Split-Half Reliability in Learning (SHRL), and is similar to the idea of Split-Half Reliability (SHR; Brown, 1910; Spearman, 1910). Under this method the task is divided into two parts (odd versus even trials), and the consistency between these two parts is examined. The second method is called Internal Consistency in Learning (ICL), and is similar to the Cronbach Alpha method (Cronbach, 1951; See also Guttman, 1945) in that multiple associations between random halves of the task are examined. The main difference between these two new methods and the classical methods is that the process that is truncated into different parts is a learning process rather than a set of independent items. To address the learning context, we can rely on the latent variables used in cognitive modeling. In the current cognitive models the different choice alternatives are continuously represented and can be thought of as the modeled knowledge of the decision maker. These can be distinguished from the parameters of the model, which reflect the cognitive style of the participant (i.e., their personal decision making profile), and are estimated based on the model predictions of future choices. The model predictions can be conceived as test items and the accuracy of the model can be therefore thought of as a score in a conventional test.

4 Internal consistency in cognitive models 4 Under the SHRL method the trials are divided into two halves in a way that is not associated with the learning periods of the tasks (see Bechara, Damasio, Tranel & Damasio, 1997). Thus, for example, the two halves can be the odd and even trials. The model is then run twice. In each of the two runs the model gains knowledge and provides predictions for the designated half of the trials (i.e., the odd trials in one run and the even trials in the other). In the other half the model simply gains knowledge without providing predictions. The reliability of the model parameters is examined by calculating the correlation between the parameters estimated in the two runs. If the task is a homogeneous measure, then the parameters estimated for an individual decision maker using the different halves of the task should be consistent. The second method proposed here, the ICL method, is identical to the SHRL method with exception that multiple random partitions of the trials are used. The present paper is organized as follows. Section 1 formally presents the current methods. Section 2 presents an experimental evaluation of three repeated choice tasks with the current methods, using two versions of the Expectancy Valence model (Busemeyer & Stout, 2002). The discussion section summarizes the value and limitations of the methods. 1. Evaluating the internal consistency of model parameters The present study focuses on cognitive models which are essentially learning models applied to predict the next step ahead in repeated choice tasks. This modeling approach enables the estimation of parameters at the individual level (that is, for each player). Under most learning models, performers are assumed to form expectancies for each choice alternative, denoting the anticipated consequences of choosing an alternative (this could be thought of as an attraction score for each alternative). A

5 Internal consistency in cognitive models 5 general linear model can be used to represent many of the previously proposed models (see Bush & Mosteller, 1955; Estes & Burke, 1953; Yechiam & Busemeyer, 2005). In this model, the expectancy E j for choice alternative j is updated as a function of its value in the previous trial (which reflects the past experience), as well as on the basis of new payoffs, as follows: E j (t)= α jt E j (t-1) + β jt u(t) (1) Where α jt and β jt denote the weights given to the prior experience E j (t-1) versus the value of the new information obtained in the present trial u(t). Different assumptions about the learning weights result in different specific learning models. A learning model therefore internalizes each performer s past experience in the expectancies of the alternatives, which represent the learned preference in any stage of the choice task. Under the Split-Half Reliability in Learning (SHRL) method the model is run twice for each task, but in each run it makes its predictions only for half of the trials. For example, the two halves could be odd and even trials or two random series of different trials. In both runs, the model gains the knowledge regarding the previous choice outcomes in each trial and updates its expectancies accordingly, but it only makes predictions (concerning the choice in the next trial ahead) for the designated half of the trial. That is, for example, in one run it makes predictions only for the odd trials and in the other run it makes predictions only for the even trials. In addition to expectancies, the model includes parameters (e.g., α jt,β jt ) that are estimated independently for each model. The parameter estimation process is independent in the two runs because each run predicts the choices made in different

6 Internal consistency in cognitive models 6 trials. The association between parameters therefore reflects their internal consistency, or their replicability in the two runs. The final consistency can be calculated using the Spearman-Brown formula 2. One caveat in the use of the SHRL model is that there may be dependencies between any given two partitions of the data. For example, there may be an alternation process which induces different responses in odd and even trials. This can be resolved by the use of a more complex method, the Internal Consistency in Learning (ICL) method in which (a) random partitions of the trials are used, and (b) multiple partitions are used. For each partition, the model is run twice, and the consistency can be calculated. The final consistency is the average of the split half reliabilities obtained for each partition, which is formally equivalent to Cronbach s (1951) alpha. 2. An experimental study The reliability of model parameters was examined in three repeated choice tasks using the current methods. Each Participant performed two versions of the twoalternative High Variance task (Thaler, Tversky, Kahneman & Schwartz, 1997; see also Barron & Erev, 2003; Yechiam & Busemeyer, 2005). In this task the performer selects between a High expected-value high-variability (H) option and a Low expected-value low variability option (L), as follows: 2 r sh = 2 r / (1 + r), where r is the correlation between the two parts and r sh is the split half reliability.

7 Internal consistency in cognitive models 7 Original High Variance task (High-Loss version): H A draw from a normal distribution with a mean of 100 and standard deviation of 354. L A draw from a truncated (above zero) normal distribution with a mean of 25 and standard deviation of 17.7 (Implied mean of 25.63). Revised Low-Loss version H A draw from a normal distribution with a mean of 28 and standard deviation of L A draw from a normal distribution with a mean of 24 and standard deviation of 3. A cognitive model called the Expectancy Valence model (Busemeyer & Stout, 2002) was used for analyzing the results. This model s parameters evaluate (a) the weighting of gains compared to losses, (b) the weighting of recent payoffs compared to payoffs experienced in the more distant past, and (c) the continuum between erratic and deterministic choices (choice consistency). Notice that the two task versions enable to examine the sensitivity of the method to predicted changes in the reliability of the weighting to gains/losses parameter. In the High-Loss version the risky alternative produces frequent high losses (40% to lose 247 points on average). In contrast, in the Low-Loss version it produces only infrequent negligible losses (6% to lose 7.5 points on average). Accordingly, because in the Low-Loss version the effect of past losses on future choices is less predictable, the reliability of the model parameter determining the relative weight of gains compared to losses was expected to diminish. In addition to the two versions of the High Variance task, participants also performed the dynamic version of the Iowa Gambling task (Bechara et al., 2001; Lovallo et al., 2006). This more complex task is commonly employed for assessing impulsive decision making, and was studied as a benchmark.

8 Internal consistency in cognitive models 8 Task performance was analyzed with the Expectancy Valence model (Busemeyer & Stout, 2002), a simple model of repeated choice behavior originally proposed for the Iowa Gambling task. This cognitive model is comprised of three basic procedures: First, a utility function represents the evaluation of gains and losses; second, a learning rule forms an expectancy for each choice alternative; third, a choice rule compares the expectancies of the different choice option to make a prediction. The learning rule of the model has been evaluated previously (see Yechiam & Busemeyer, 2005; 2007). Here I evaluate two alternative choice rules, one assuming trial dependence and one trial independence (as explained below). In addition to examining model fit, the current methods allow the examination of the consistency of the model s estimated parameters. Participants: Sixty-five students from the Technion Israel Institute of Technology, (31 males and 34 females), participated in the experiment. Most of the students (91%) were from the Faculty of Industrial Engineering and Management. The majority of the students were sophomores (83%) and the rest were freshmen and seniors. All participants were paid in cash 20 NIS (about $4.50) per hour and whatever monetary bonuses they had earned in association with their performance. Measures: High Variance task: The decision task included two buttons (the size of each was 4 by 6.5 cubic cm). Payoffs were contingent upon the button chosen (H or L) and were calculated independently in each trial as indicated above. The location of the H and L options was randomized for each participant. The minimum inter-trial interval

9 Internal consistency in cognitive models 9 was set to 1 second. Two types of feedback immediately followed each choice: (1) The basic payoff for the choice, which appeared on the selected button for one second and then re-appeared on the caption below the buttons; (2) an accumulating payoff counter, which was displayed constantly. Participants were given the following written instructions: In each trial you will have to click a button, A or B. You can press a button several times (as many times as you like) and you can switch between buttons (as much as you like). The payoff for your selection will appear on the button that you choose. We cannot tell you in advance what will be the payoffs, just that they may sometimes include gains and sometimes losses. You receive 10 Israeli Ag. for every 100 game points. Good luck. Iowa Gambling task: In the Iowa Gambling task (Bechara et al., 1994), participants are presented with four card decks (labeled A, B, C, D), and are told to accumulate as much (real) money as possible by picking cards from the decks. Decks differ with respect to the size and frequency of payoffs produced by each card selection. Selecting any card from deck A or deck B yields $100 while selecting card from deck C or deck D yields $50 in game money. However, in some selections the participant loses money as well (in these cases, the participant are shown separately the amounts won and lost). The average gains and losses in the first block are described in Table 1. In addition, in the dynamic version of the task (Bechara et al., 1999; see also Lovallo et al., 2006) differences between decks increase over time. Specifically, the average positive outcome in decks A and B is increased by $10 on each block of 10 trials while the average loss is increased by $25. For the advantageous decks the average gain is improved by $5 every 10 trials compared to a $2.5 increase in the

10 Internal consistency in cognitive models 10 average loss. The probability of loss in decks A and C is increased by 0.1 on each block as well. The experiment used a computer-simulated version of the task, described previously (Bechara et al., 1999). The minimum inter-trial interval was set to 0.5 seconds, and the game included 100 trials. Participants were given written instructions identical to those given in Bechara et al. (1999), translated into Hebrew. Briefly, participants were told that some decks are worse than others are, and they should avoid those decks to win money. They were not given any information about the expected amounts and proportions of gains and losses. There were some differences from the original task: (1) The experiment used Israeli currency (NIS) and started from an initial positive account of 2,500 NIS. (2) Gains and losses appeared simultaneously in the same feedback message. (3) The position of the advantageous and disadvantageous decks was controlled (for half of the participants the advantageous alternatives were on the left and for the other half they were on the right). (4). In the task used in Bechara et al. (1999) performers who at one point in the game lost more than the original loan were given another loan of $20. As the effect of this manipulation was difficult to assess, in the present version the task was stopped at this point. In reality, this occurred for 13 performers in the first 100 trials. Of these, only one performer reached this point before trial 60; so it was decided to keep these 13 participants. Cognitive modeling of the tasks: The Expectancy Valence model was described previously (see e.g., Busmeyer & Stout, 2002), and it is briefly summarized here. The model is comprised of three basic components:

11 Internal consistency in cognitive models 11 Weighting of gains and/or losses. The evaluation of gains and losses experienced after making a choice is called a valence, and is represented by a prospect theory type of utility function (Kahneman & Tversky, 1979). The utility is denoted u(t), and is calculated as a weighted average of the gains (win) and losses (loss) obtained in trial t. u(t) = W win(t) (1-W) loss(t) (2) Where W is a parameter that indicates the weights to gains versus losses. This parameter assesses the motivational difference in the weighting to gains and losses, and is used to identify performers who are willing to risk losing money for the sake of winning. The parameter ranges from 0, denoting attention to loses only, to 1, denoting attention to gains only. Values between 0 and 1 indicate the relative weight of losses and gains. Weighting of recent outcomes. A Delta model is employed for updating the expectancies (see Busemeyer & Myung, 1992; Gluck & Bower, 1988; Rumelhart & McClelland, 1986; Sutton & Barto, 1998). According to this learning model, the two learning weights (in Equation 1) are defined as follows: α jt = 1-δ j (t) φ (3) β jt = δ j (t) φ (4) where δ j (t) equals 1 if payoff information about option j is presented on trial t, and 0 otherwise. Putting these values back into equation 2, the expectancy updating process can be expressed as:

12 Internal consistency in cognitive models 12 E j (t)= E j (t-1) + φ δ j(t) [u(t) E j (t-1)] (5) It can be seen that the expectancy is changed in the direction of the prediction error given by [u(t) E j (t)]. The parameter φ is the recency (or learning rate) parameter. It dictates how much of the expectancy is changed by the prediction error. According to this model, if 0 <φ < 1, then the effect of a payoff on the expectancy for an alternative decreases exponentially as a function of the number of times a particular alternative is chosen. Thus, recently experienced payoffs have larger effects on the current expectancy as compared to payoffs that were experienced in the distant past. Reliability of choice behavior. The expectancies of each alternative are translated into the model s prediction about the next choice ahead. The prediction depends on the reliability with which decision makers apply the expectancies when making their selection. Formally, the probability of choosing an alternative is a strength ratio of the expectancy of that alternative relative to the sum of the strengths of all expectancies 1 to k (Luce, 1959): Pr[ E j ( t ) e G j ( t)] = θ Ek ( t) e k θ (6) Where θ controls the consistency of the choice probabilities and the expectancies. When the value of θ is very high, choices converge to the alternative with maximum expectancy. When the value of θ is close to zero, choices are inconsistent, random, erratic, and independent of the expectancies. In the Expectancy-Valence model the choice-consistency may change as a function of experience, and is therefore trial dependent. This is formalized by a power function for the sensitivity change over trials: θ (t) = (t/10) c. However, this time

13 Internal consistency in cognitive models 13 dependence may be problematic in choice problems with an asymmetric expected value of gains or losses. In these problems, the weight to gains vs. losses parameter (W) may reflects two separate factors: (a) The sensitivity to gains relative to losses (as intended), and (b) the sensitivity to payoff magnitude within the same valence (i.e., gains or losses). The importance of the second factor is presumably increased with the asymmetry in the expected value of gains and losses in the task. This can be resolved by using a fixed (or trial-independent) choice-consistency θ parameter, which controls for the second factor above. The present study examined the current model under both assumptions 3. Notice that the Low-Loss version of the High Variance task has asymmetry in the expected value of gains and losses, having only rare and small losses. Therefore it is predicted that in this task version, the reliability of the parameters under the trialdependent choice rule would be reduced (compared to a rule assuming trial independence). To keep the value of c within reasonable bounds, in the trial dependent choice rule, the parameter c was bounded between -5 and 5 (as in Busemeyer and Stout, 2002). In the trial-independent choice rule we set θ (t) to be 5 c 1, and c was constrained between 0 and 10. Both cases cover the full range between a random (θ (t) 0) and a highly deterministic (θ (t) > 700) choice. Increasing the bounds beyond these values does not change the results reported below. 3 In addition, I examined a model where θ (t) = (1+ t/10) c. In the original EV model the consistency parameter has an opposite effect in the first 10 trials (c > 1 increases exploration and c < 1 decreases exploration) and in the next trials (c > 1 decreases exploration and c < 1 increases exploration). In this revised model the consistency parameter has a monotonic effect on the expectancies. However, as this rule came midway in terms of both fit and reliability, it is not included in the final analysis.

14 Internal consistency in cognitive models 14 Modeling proceeds as follows. First, the model parameters are estimated to increase the fit of model predictions to the data from an individual player (using maximum likelihood). Second, the model fits are compared to those of a Bernoulli baseline model. The baseline model s predictions are based on the optimized proportion of the choices of the different alternatives (i.e., without the influence of learning). The relative fit is assessed by measuring the improvement in the fit of the decision model over the baseline model. The statistical test of this improvement is G 2, which is a model fit statistic analogous to the chi-square (= 2 log likelihood difference; see Busemeyer & Wang, 2000). Positive values of the G 2 statistic indicate that the cognitive model performs better than the baseline model, whereas negative values indicate the reverse. Because in the current two-alternative task the baseline model has only one parameter, while the learning models have three parameters, I adjusted the G 2 score for the difference in number of parameters using the Bayesian Information Criterion (BIC; Schwartz, 1978). The BIC is a model fit correction that penalizes models with additional parameters, as follows: G 2 = G 2 - k ln(n), where k equals the difference in number parameters and N equals the number of observations. In the High Variance task, we have k = 2 (two additional parameters in the learning model compared to the baseline model) and N = 100. Thus, 2 ln(100) 9.2. Positive values of the corrected G 2 indicate that a learning model performs better than the baseline model. In addition to assessing the fit statistics, the model equations are solved to generate values for the model s three parameters for each participant. In the present study, the internal consistency of the parameters was also calculated using the SHRL and ICL methods. For the SHRL method the two halves of the task were the odd and

15 Internal consistency in cognitive models 15 even choice trials. For the ICL method five random partitions were created, and the analysis was repeated for each partition. Procedure: The participants performed the three experimental tasks as well as the Balloon Analogue Risk task (Lejuez et al., 2002) which was run for pilot purposes. The order of tasks was controlled. The entire experiment lasted approximately 40 minutes. In the beginning of the experiment participants were told that they would perform different tasks, each with different payoff rules; and that their payoff would consist of a participation fee of NIS 20 plus whatever bonuses they gained during the task. Only one participant s accumulating payoffs were below zero, and this participant was paid the participation fee. Due to a technical error, five participants did not complete the Low-Payoff task version. In addition, four participants were removed from the analysis because they had made the same exact choices in all 100 trials of the task performed second or third. This implies that they have misunderstood the instructions that each task is different (leaving an N of 56). Results: The first section of the results focuses on the main analysis of the two versions of the High Variance task. The choice proportions in the two versions are presented in the top two panels of Figure 1. Briefly, in the High-Loss version, alternative H was chosen only 38% of the time, on average, replicating previous results (e.g., Barron & Erev, 2003; Thaler et al., 1997). In the Low-Loss variant the proportion of H choice increased to only 43% despite the fact that it produced only small infrequent losses. The modal choice behavior in the two tasks can therefore be explained based on variance aversion, or extreme loss aversion (as postulated by Thaler et al., 1997).

16 Internal consistency in cognitive models 16 Model fit: Table 2 summarizes the fit of the models when they are used to predict the next choice ahead for the entire task (BIC corrected G 2 ). As can be seen, the fit was better for the Low-Loss than for the High-Loss version under both choice rules. In addition, the trial-independent choice rule was superior to the trial dependent rule across both task versions. An all-within analysis of variance showed both effects to be significant (task version: F (1,55) = 5.20, p <.05; choice rule: F (1,55) = 7.00, p <.05). In the Low-Loss version the fit of the learning model was significantly better than the fit of the baseline model under both choice rules (trial dependent: t (55) = 2.93, p <.01); trial independent: t (55) = 4.07, p <.01). Internal consistency of the parameters: Table 3 presents the internal consistency of the parameters in the two task versions using the new SHRL and ICL methods. The table also presents the standard errors of the ICL scores in five task partitions (following Cortina, 1993). As can be seen, the parameter denoting the weight to gains versus losses was the only parameter with adequate internal consistency, with an r sh above 0.7, and this occurred only in the High-Loss version. This result was replicated under both choice rules. A comparison of the difference between the reliability of the parameters in the two task versions can be made by reversing the Spearman-Brown extrapolation and testing for differences between the correlations (Alsawalmeh & Feldt, 1992). In this way, it was found that, in the High-Loss task, the reliability of the of the weight to gains/ losses parameter was significantly higher than any of the other parameters under both methods (SHRL and ICL) and choice rules, with the minimum Z in the four comparisons being 3.21 (p <.01). Also as predicted, the reliability of this

17 Internal consistency in cognitive models 17 parameter was significantly lower in the Low-Loss version under both methods and choice rules, with a minimum Z of 2.55 (p <.01). A second interesting result is the interaction between the task version and choice rule. Under both methods, the trial-dependent rule was more consistent in the High-Loss task version in all of the parameters (although not significantly). However, in the Low-Loss version the trial-independent rule was more consistent in two of the model parameters, the weighting of gains and losses and recency (but again, not significantly). The decreased internal consistency of the trial-dependent choice rule in the Low-Loss version was predicted because of the asymmetry in the average magnitude of gains and losses. However, this result should be interpreted with caution because of the lack of statistical significance. A general quantitative estimation of the similarity between the two methods (SHRL and ICL) can be obtained by calculating the Mean Square Deviations between the reliabilities indicated for each parameter and task by the two methods, presented on Table 3. The result showed that the average Root Mean Square Deviations (RMSD) was 0.04, denoting a relatively small difference between the results obtained by the two different procedures. Comparison to the Iowa Gambling task (IGT): The choices on the IGT are presented in the bottom panel in Figure 1. The results replicate previous findings, showing that (a) Participants learn to choose the advantageous decks (C and D) in 100 trial repetitions (Bechara et al., 1999). Comparing the two halves of the task, choices from the advantageous decks increased from 50% to 64% (t (54) = 5.45, p <.01). (b) Participants prefer the decks producing gains most of the times along with infrequent losses (B over A and D over C) (Barron and Erev, 2003; Estes, 1976;

18 Internal consistency in cognitive models 18 Yechiam & Busemeyer, 2005). Choices from the infrequent-losses decks increased from 66% to 72% in the two halves of the task (t (54) = 2.27, p <.05). The corrected fit of the learning models was relatively high (see Table 2), significantly higher than that of the baseline model (trial-dependent: t (55) = 3.11, p <.01; trial-independent: t (55) = 4.62, p <.01). However, as in the other two task the fit was higher for the trial-independent choice rule (t (55) = 5.18, p <.01). Given the close similarity between the internal consistency results using the two methods, I present the results with the simpler SHRL method (these were replicated using the ICL method). The results (see Table 4) showed that under both models, the parameter with the highest internal consistency was the weight to gains versus losses parameter. Thus, the pattern found in the High-Loss task version was replicated. The differences between models were less clear, with the trial-independent choice rule achieving better consistency in two of the parameters (weight to gains vs. losses and choice consistency) and the trial-dependent rule having better consistency in the third parameter (recency). Yet the only cases of adequate parameter consistency (r of about 0.7) were for the choice rule assuming trial-independence. 3. General Discussion The present paper presented two novel approaches for studying the internal consistency of model parameters, one employing a single split of the data (the Split Half Reliability in Learning method) and one using multiple random partitions (the Internal Consistency in Learning method). The main result of the current paper is that in an examination of three different tasks using two three-parameter model variants, the results of the two methods converged. The difference between the reliability scores under the two methods was small (RMSD of 0.04 for the two versions of the

19 Internal consistency in cognitive models 19 High Variance task). Moreover, the methods gave the same ranking for different parameters, tasks, and models. The most robust empirical pattern in the present study was that the parameter denoting the weight of gains relative to losses reached the highest degree of internal consistency. This naturally occurred in tasks involving non-trivial losses (the High- Loss version of the High Variance task and the Iowa Gambling task). As predicted, in the absence of substantial losses (in the Low-Loss version) the internal consistency of this parameter was considerably diminished. Note that the latter finding occurred despite the fact that the fit of the model was actually better in the Low-Loss compared to the High-Loss version. It therefore cannot reflect a general decrease in model accuracy in the Low-Loss condition. In another study (Yechiam & Busemeyer, 2007), the weight to gains versus losses parameter was also found to have the highest external consistency when the parameters are separately estimated in two different tasks (the original High Variance task and a two-choice task producing rare negative outcomes). Note that although in the current study and in Yechiam and Busemeyer (2007) this parameter determines the weight of gains compared to losses, it may reflect a more general type of risk taking that involves a tradeoff between the potential to win a large amount and the potential to win a much smaller amount. The division into gains and losses is considered to be a natural cue that ranks the payoffs of a certain alternative into high and low outcomes (Ert, Yechiam & Erev, 2007). A second empirical result involved the internal consistency of the two alternative choice rules. As predicted (although not in a significant manner) the internal consistency under the trial-dependent choice rule was penalized to a greater extent in the move from the High-Loss to the Low-Loss task version. The internal

20 Internal consistency in cognitive models 20 consistency of the model using the trial-dependent rule may therefore be limited to tasks where the average magnitude of gains and losses is relatively symmetric. The results regarding the value of the two choice rules are therefore mixed. However, they are informative in suggesting that the trial-independent rule could be at least as useful as the trial-dependent rule currently included in the Expectancy Valence model. The fit of the model assuming trial-independence was superior across all of the studied choice tasks. Moreover, in the dynamic version of the IGT this model had adequate internal consistency (r of about 0.7) for two of the model parameters: weight to gains versus losses and choice consistency. In contrast, the model with the trial dependent choice rule did not show adequate internal consistency in this task for any of the parameters. Conclusions: The current study presented two new methods for studying the internal consistency of parameters in cognitive learning models. Clearly, internal consistency is an important psychometric index, and this index has so far been neglected in the evaluation of cognitive models for repeated choice tasks. The current methods are not limited however to the domain of repeated choice tasks, and could be used in other process models as well. There are alternative methods for studying internal consistency that do not rely on task splitting. For example, one can use a parameter recovery method to see if the parameters of the model can be correctly re-estimated from data generated by the model. However in the domain of repeated choice tasks there is a discrepancy between parameters obtained using the prediction of one step ahead method and parameters used for simulating choices (Erev & Haruvy, 2005), which complicates

21 Internal consistency in cognitive models 21 the application of parameter recovery methods. There are also methods based on the administration of multiple tasks, an approach similar to the idea of inter-judge reliability (see Rieskamp, Busemeyer, and Laine, 2003; Yechiam & Busemeyer, 2007). The latter approaches are considered to be complimentary to the current methods, and have the advantage of assessing the consistency beyond a single task. Here I attempted to find a solution for the common application of a single task to study individual differences. Surely, even in this case it is important to present the internal consistency of parameters of cognitive models if one assumes that they present meaningful interpretations of the data (see Wilkinson et al.,1999).

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27 Internal consistency in cognitive models 27 Table 1: The payoffs for the dynamic Iowa Gambling task (first block of trials). Deck Wins Losses Description A 100 every card.5 to loss 250 Disadvantageous: Risky B 100 every card.1 to loss 1,250 Disadvantageous: Risky, rare loss C 50 every card.5 to loss 50 Advantageous: Safe D 50 every card.1 to loss 250 Advantageous: Safe, rare loss

28 Internal consistency in cognitive models 28 Table 2. Model fits: Average and standard deviation (in parentheses) of the corrected G 2 scores for the different tasks and models (trial dependent and independent choice rules). Task Trial Dependent Trial Independent High-Loss 0.77 (16.1) 2.00 (16.3) Low-Loss 7.28 (18.6) (18.5) Iowa Gambling task (26.5) (28.3)

29 Internal consistency in cognitive models 29 Table 3. Reliability results for the two versions of the High Variance task using the Split-Half Reliability in Learning (SHRL) and Internal Consistency in Learning (ICL) methods, for the trial dependent and independent choice rules. Task Model Wgt. gain/loss W Recency φ consistency C SHRL ICL SHRL ICL SHRL ICL High-Loss Trial Dependent (0.02) (0.04) (0.08) Trial Independent (0.02) (0.04) (0.06) Low-Loss Trial Dependent (0.03) (0.03) (0.08) Trial Independent (0.07) (0.03) (0.05) Note: The scores denote r sh s for the SHRL test and average r sh for the ICL test (standard errors appear in parentheses).

30 Internal consistency in cognitive models 30 Table 4. Reliability results for the two Iowa Gambling task using the Split-Half Reliability in Learning (SHRL) method for the trial dependent and independent choice rules. Model Wgt. gain/loss W Recency φ consistency C Trial Dependent Trial Independent Note: The scores denote r sh s.

31 Figure 1: Proportion of choices from the High (H) and Low (L) expected-value alternatives of the High Variance Task (High and Low-Loss versions) and from the four alternatives of the Iowa Gambling Task in 100 trials High Variance task: High-Loss High Variance task: Low-Loss L H L H Trial Trial Iowa Gambling task D C B A Trial