Response-time data provide critical constraints on dynamic models of multi-alternative, multi-attribute choice

Nathan J. Evans; William R. Holmes; Jennifer S. Trueblood

doi:10.3758/s13423-018-1557-z

Psychonomic Bulletin & Review

pp 1–33 | Cite as

Response-time data provide critical constraints on dynamic models of multi-alternative, multi-attribute choice

Authors
Authors and affiliations

Nathan J. Evans
William R. Holmes
Jennifer S. Trueblood

Theoretical Review

First Online: 08 February 2019

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Abstract

Understanding the cognitive processes involved in multi-alternative, multi-attribute choice is of interest to a wide range of fields including psychology, neuroscience, and economics. Prior investigations in this domain have relied primarily on choice data to compare different theories. Despite numerous such studies, results have largely been inconclusive. Our study uses state-of-the-art response-time modeling and data from 12 different experiments appearing in six different published studies to compare four previously proposed theories/models of these effects: multi-alternative decision field theory (MDFT), the leaky-competing accumulator (LCA), the multi-attribute linear ballistic accumulator (MLBA), and the associative accumulation model (AAM). All four models are, by design, dynamic process models and thus a comprehensive evaluation of their theoretical properties requires quantitative evaluation with both choice and response-time data. Our results show that response-time data is critical at distinguishing among these models and that using choice data alone can lead to inconclusive results for some datasets. In conclusion, we encourage future research to include response-time data in the evaluation of these models.

Keywords

Decision-making Multi-attribute choice Context effects Bayesian methods

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Notes

Acknowledgements

The authors would like to thank Audrey Parrish, Michael Beran, and George Farmer for sharing their data. The authors would also like to thank Jerome Busemeyer for his comments on extending MDFT to SDE formalism. All authors were supported by NSF grant SES-1556415. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the funding agency.

Supplementary material

13423_2018_1557_MOESM1_ESM.pdf (904 kb)

(PDF 903 KB)

Appendix A: Converting random walks to stochastic differential equations

Here we provide a very brief general description of how we converted the discrete time, random walk models versions of MDFT and LCA to stochastic differential equations (SDE’s). We are not attempting to re-derive the link between random walks and SDE’s. Rather our intent is to provide the interested reader with a short primer. We thus use formal rather than rigorous arguments and language.

Discrete time random walks, which are commonly used to describe the accumulation of evidence or time integration of stimuli, are typically described by specifying the form of a stochastic increment (size of a step) that advances a system from one “time step” to the next

$$ A_{n + 1} - A_{n} = r f(A_{n}) + Noise, $$

(7)

where “r” is the size of the step taken and “Noise” represents a generic stochastic term. We highlight the phrase “time step” here since this really refers to an indexed counter rather than a true time increment. While there are a number of ways to associate real time with random walks, the most common way of doing so when dealing with random walks is to convert them to a stochastic differential equation framework.

To do so, one has to associate each time increment “n” with a real time t_n. This introduces a time increment into the problem, namely Δt = t_n+ 1 − t_n. In order to ensure that the statistical properties of the random walk at a fixed future point in time do not depend on the size of the time increment chosen, it is commonly assumed that r = kΔt, so that halving the size of the time step halves the size of the step. In other words, taking two steps of size Δt/2 yields similar results as taking one step of size Δt. With this assumption, the resulting stochastic differential equation obtained by taking Δt → 0 is

$$ dA = k f(A) dt + \sigma dw. $$

(8)

Since the random walk function f typically has scaling parameters of its own, k can often be folded into that function and omitted for brevity. The random walks formulations of MDFT and LCA can each be reformulated in this way.

Appendix B: MDFT

Multi-alternative decision field theory (MDFT) models the preferences for different alternatives of a multi-attribute choice over decision time through three key components: the valence for each alternative, the lateral inhibition between alternatives, and the evidence leakage within each alternative, and some stochastic noise within the process. Formally, the preference for the vector of alternatives P (e.g., for the choice alternatives X, Y, and Z, P = [P_X, P_Y, P_Z]’) can be written as a stochastic differential equation, with the form:

$$\begin{array}{@{}rcl@{}} dP = \mu \times dT + \sigma \times dW \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} \mu = [S \times P(t) + V(t)] \end{array} $$

(10)

where W is the Wiener process. In terms of the model components, S is an n × n matrix (where n is the number of alternatives) that contains both the amount of lateral inhibition between the alternatives based upon their psychological distances, and the rate of evidence leakage for each alternative. P(t) is the preference for the vector of alternatives at the time t, and V (t) is the vector of valences for each alternative at time t based upon the attribute being attended to. σ is the standard deviation of the noise process, which is a free parameter of the model. This can be written as a discrete process of the form:

$$ P(t + {\Delta} t) = P(t) + S \times P(t) \times {\Delta} t + V(t) \times {\Delta} t + \sigma \times \sqrt{{\Delta} t} \times \eta $$

(11)

where Δt is the time step, η is the standard normal distribution, and P(t + Δt) is the preference for the vector of alternatives at the next point in time. The preferences for each alternative continue to evolve, until one of the preferences passes some threshold level (a).

In terms of the three key components, the first, S, is the matrix of the lateral inhibition between the alternatives, and the evidence leakage within alternatives. S is an n × n matrix (where n is the number of alternatives):

$$ S_{ij} = - \phi_{2} \times exp(- \phi_{1} \times Dist_{ij}^{4}) $$

(12)

where i and j index alternatives (1, 2,...n) and ϕ₁ and ϕ₂ are parameters. More specifically, ϕ₂ is a free parameter that gives the amount of evidence leakage, and ϕ₁ is a free parameter that controls the overall strength of the lateral inhibition. Dist_ij is the psychological distance between the attributes of alternatives i and j (which is 0 when i = j). To obtain the psychological distance between the attributes of the two alternatives, one obtains the Euclidean distance between the attributes based on the dimensions of indifference (i.e., where alternatives are equally as good) and dominance (i.e., where one alternative dominates another):

$$ Dist_{ij} = \sqrt{ ({\Delta} I_{ij})^{2} + \beta \times ({\Delta} D_{ij})^{2}} $$

(13)

where ΔI_ij gives the distance between the attributes of the alternatives on the dimension of indifference, ΔD_ij gives the distance between the attributes of the alternatives on the dimension of dominance. β is a free parameter that allows the dominance and indifference dimensions to be weighted differently, where β > 1 has a stronger weight to the dominance dimension, β < 1 has a stronger weight to the indifference dimension, and β = 1 gives both dimensions equal weighting. Also note that this formulation of distance is only possible when only two attributes are present in each stimulus. Although Berkowitsch et al., (2014, 2015) developed a formulation that works for more than two alternatives, that version reduces to this one in the specific case of two alternatives. Therefore, we choose to write the mathematically simpler version here.

To define the distance on each of the dimensions, one takes the two subjective attributes values for alternatives i and j, with the first attributed denoted as A and the second denoted as B, and calculates their distance on the unit plane:

$$\begin{array}{@{}rcl@{}} {\Delta} I_{ij} = \frac{1}{\sqrt{2}} \times ((B_{i} - B_{j}) - (A_{i} - A_{j}) ) \end{array} $$

(14)

$$\begin{array}{@{}rcl@{}} {\Delta} D_{ij} = \frac{1}{\sqrt{2}} \times ((B_{i} - B_{j}) + (A_{i} - A_{j}) ) \end{array} $$

(15)

Note that in our application, we constrain the subjective values of the stimulus to equal the objective values, meaning that these values for attributes A and B for alternatives i and j are merely the actual stimulus attributes for these alternatives.

The second key component, V (t), is the vector of valences for each alternative at time t. V (t) is defined as:

$$ V(t) = C \times [M \times W(t)] \times \gamma $$

(16)

where γ is an addition that we made to the model, which is a free parameter that scales the valences. This parameter allows the model to create drift rates of different magnitude, which is critical when modeling the choice response-time distributions. C is an n × n matrix that transforms the absolute valence values into relative advantages/disadvantages, given by:

$$ C = \left[\begin{array}{ccc} 1 & \frac{-1}{n-1} & \frac{-1}{n-1} \\ \frac{-1}{n-1} & 1 & \frac{-1}{n-1} \\ \frac{-1}{n-1} & \frac{-1}{n-1} & 1 \end{array}\right] $$

(17)

M is an n × m matrix, where m is the number of attributes, which contains the subjective values for each attribute of each alternative, and in the case of our two-attribute design would be:

$$ M = \left[\begin{array}{cc} A_{1} & B_{1} \\ A_{2} & B_{2} \\ A_{3} & B_{3} \end{array}\right] $$

(18)

W(t) is a binary vector of length m, where all elements are set to 0, except for the attribute of the matrix that is being attended to at time t (either A or B), which is set to 1. For example, if attribute A were being attended to at time t, then W(t) = W_A = [1,0]’, and if attribute B were being attended to, then W(t) = W_B = [0,1]’. To determine which attribute would be attended to at time t + Δt, we implemented an exponential switching function, where the probability of switching between attended attributes at any point in time was given by:

$$\begin{array}{@{}rcl@{}} P(W_{A} \mapsto W_{B}, {\Delta} t) = 1-exp(-k_{A} \times {\Delta} t) \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} P(W_{B} \mapsto W_{A}, {\Delta} t) = 1-exp(-k_{B} \times {\Delta} t) \end{array} $$

(20)

where k_A and k_B are free parameters, separate for attributes A and B to allow bias towards a single attribute, which control the switching probability, with higher values giving larger switching probabilities. The exponential switching function is another addition that we made to the model. Previous iterations of MDFT operated as a random walk where attention would switch on every step, serving as a “flicker” between attributes. However, this flicker process does not fit the stochastic differential equation version of the model for two key reasons. Firstly, from a psychological perspective, it seems unrealistic that participants would be changing their attentional focus on every time step, when the time step is in the order of several milliseconds. Secondly, the flicker process used in the previous iterations of MDFT would result in a stochastic differential equation where the number of switches occurring within a specified time period is dependent on the time step used, meaning that different results would potentially be obtained from the model on the basis of the time step used. Our exponential switching function serves as a solution to both of these issues, possessing a constant hazard function that results in the probability of switching after a specified time period to be independent of the time step used, and more generally allowing the switching time frame to take on a value more realistic for human attention processes.

Overall, our definition of MDFT gives the model eight free parameters: a, k_A, k_B, ϕ₁, ϕ₂, β, σ, and γ.

Appendix C: LCA

We first briefly describe the original formulation of LCA provided in Usher and McClelland (2004). Due to scaling issues, however, this is not the model used in the paper here. We thus subsequently describe our augmentation of the version used.

Random walk LCA

The original version of LCA posited that the activation level associated with alternative i is governed by the following random walk

$$ A^{i}_{t + 1} - {A^{i}_{t}} = (\lambda -1){A^{i}_{t}} + (1 - \lambda) \left( I_{i}(t) - \beta \displaystyle \sum\limits_{j \neq i} {A^{j}_{t}} + Noise \right) . $$

(21)

When converted to a SDE in the standard way, this model would take the form

$$ dA^{i} = (1-\lambda) \left[ -A^{i} + I_{i}(t) - \beta \displaystyle \sum\limits_{j \neq i} {A^{j}_{t}} \right] + \sigma dW $$

(22)

This introduces two issues. The first is that (1 − λ) scales the entire drift component of this SDE. Thus it will trade off with response thresholds and yield a parameter degeneracy. We attempted data/parameter recovery and found that this parameter degeneracy caused serious issues and the model could not recover its own data. The second issue is the interpretation of the parameter λ. In Usher and McClelland (2004), this was described as the leakage rate parameter. However, if you look at either the original random walk (21) or its associated SDE (22), it is plain to see that λ does not act as leakage rate. Due to these two issues, we have constructed a slightly adjusted version of the original LCA model (in SDE form), that maintains all the inhibition and loss aversion features of the original in their exact mathematical form, but more appropriately describes leakage (in a manner that reflects the definition of the LCA in Usher & McClelland, 2001).

Adjusted SDE version of LCA

The leaky computing accumulator (LCA) models the activation for different alternates of multi-attribute choice over decision time through three key components: the activation input for each alternative, the global amount lateral inhibition between alternatives and leakage within each alternative, and some stochastic noise within the process. Formally, the amount of activation for the vector of alternatives A (e.g., for the choice alternatives X, Y, and Z, A = [A_X, A_Y, A_Z]’) can be written as a stochastic differential equation, with the form:

$$\begin{array}{@{}rcl@{}} dA = \mu \times dT + \sigma \times dW \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} \mu = [S \times A(t) + I(t)] \end{array} $$

(24)

where W is the Wiener process. In terms of the model components, S is an n × n matrix (where n is the number of alternatives) that contains both the amount of lateral inhibition between the alternatives based upon their psychological distances, and the rate of evidence leakage for each alternative. A(t) is the activation for the vector of alternatives at the time t, and I(t) is the vector of inputs for each alternative at time t based upon the attribute being attended to. σ is the standard deviation of the noise process, which is a free parameter of the model. This can be written as a discrete process of the form:

$$ A(t + {\Delta} t) = A(t) + S \times A(t) \times {\Delta} t + I(t) \times {\Delta} t + \sigma \times \sqrt{{\Delta} t} \times \eta $$

(25)

where Δt is the time step, η is the standard normal distribution, and A(t + Δt) is the activation for the vector of alternatives at the next point in time. Additionally, the level of activation is truncated at 0, meaning that:

$$ [A_{i}(t + {\Delta} t) \le 0] = 0 $$

(26)

where i indexes an individual alternative. The activation for each alternative continues to evolve, until one of the alternatives’ activation passes some threshold level (a).

In terms of the three key components, the first, the matrix of the lateral inhibition between the alternatives, and the evidence leakage within alternatives, S, is an n × n matrix (where n is the number of alternatives), defined as:

$$\begin{array}{@{}rcl@{}} S = \left[\begin{array}{ccc} -\lambda & - \beta & - \beta \\ - \beta & -\lambda & - \beta \\ - \beta & - \beta & -\lambda \end{array}\right] \end{array} $$

(27)

where β is a free parameter of the model that controls the amount of global inhibition, and λ is a free parameter of the model that controls the amount of leakage.

The second key component, the vector of activation inputs for each alternative at time t, I(t), is defined as:

$$ I_{i}(t) = I_{0} + \gamma \times \sum\limits_{j \neq i}^{n} V(d_{ij}) $$

(28)

where I₀ is a free parameter of the model that is the baseline amount of input, V is a function where:

$$\begin{array}{@{}rcl@{}} V(x) &=& log(1+x), x \geq 0 \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} V(x) &=& - [log(1+|x|) + log(1+|x|)^{2}], x < 0 \end{array} $$

(30)

d_ij is given by:

$$ d_{ij} = (M \times W(t))_{i} - (M \times W(t))_{j} $$

(31)

where M is an n × m matrix, m being the number of attributes, which contains the values for each attribute of each alternative. In the case of our two-attribute design, where the first attribute is denoted F and second attribute is denoted G, M would be:

$$ M = \left[\begin{array}{cc} F_{1} & G_{1} \\ F_{2} & G_{2} \\ F_{3} & G_{3} \end{array}\right] $$

(32)

W(t) is a binary vector of length m, where all elements are set to 0, except for the attribute of matrix that is being attended to at time t (either F or G), which is set to 1. For example, if attribute F were being attended to at time t, then W(t) = W_F = [1,0]’, and if attribute G were being attended to, then W(t) = W_G = [0,1]’. To determine which attribute would be attended to at time t + Δt, we implemented an exponential switching function, where the probability of switching between attended attributes at any point in time was given by:

$$\begin{array}{@{}rcl@{}} P(W_{F} \mapsto W_{G}, {\Delta} t) = 1-exp(-k_{F} \times {\Delta} t) \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} P(W_{G} \mapsto W_{F}, {\Delta} t) = 1-exp(-k_{G} \times {\Delta} t) \end{array} $$

(34)

where k_F and k_G are free parameters, separate for attributes F and G to allow bias towards a single attribute, which control the switching probability, with higher values giving larger switching probabilities. The exponential switching function is another addition that we made to the model. Previous iterations of LCA operated as a random walk where attention would switch on every step, serving as a “flicker” between attributes. However, this flicker process does not fit the stochastic differential equation version of the model for two key reasons. Firstly, from a psychological perspective, it seems unrealistic that participants would be changing their attentional focus on every time step, when the time step is in the order of several milliseconds. Secondly, the flicker process used in the previous iterations of LCA would result in a stochastic differential equation where the number of switches occurring within a specified time period is dependent on the time step used, meaning that different results would potentially be obtained from the model on the basis of the time step used. Our exponential switching function serves as a solution to both of these issues, possessing a constant hazard function that results in the probability of switching after a specified time period to be independent of the time step used, and more generally allowing the switching time frame to take on a value more realistic for human attention processes.

Overall, our definition of the LCA gives the model eight free parameters: a, k_F, k_G, I₀, λ, β, σ, and γ.

Appendix D: MLBA

The multi-attribute linear ballistic accumulator (MLBA) models the activation for different alternates of multi-attribute choice over decision time through three key components: the pairwise alternative comparison, the decaying comparison weights over increasing attribute differences (i.e., similarity-based attention), and the subjective attribute values. Specifically, the MLBA serves as a front-end extension of the linear ballistic accumulator (LBA; Brown & Heathcote, 2008), where the front-end serves to determine the drift rates based on some function of the attribute values. The linear, ballistic format of the LBA framework makes the MLBA more tractable than the MDFT and LCA, having an analytically solvable probability density function. The MLBA can be expressed as:

$$\begin{array}{@{}rcl@{}} dA = \mu \times dT \end{array} $$

(35)

$$\begin{array}{@{}rcl@{}} \mu \sim TN(d,s,0,Inf) \end{array} $$

(36)

where TN is the normal distribution truncated to values between 0 and positive infinite, d is the mean drift rate between trials, and s is the standard deviation in drift rate between trials. With the MLBA, the standard deviation in drift rate is fixed to 1, to satisfy a scaling property of the model (Donkin, Brown, Heathcote, & Wagenmakers, 2011). Alternatives accumulate evidence independently from on another, with the starting evidence distributed as U[0,A], until the evidence for one alternative reaches a threshold, b, with A and b being free parameters of the model.

The mean drift rate (commonly referred to as just the “drift rate”) for each alternatives was determined by pairwise comparisons between the attributes of that alternative and the other alternatives:

$$ d_{i} = I_{0} + \gamma \times \sum\limits_{j \neq i}^{n} V_{i,j} $$

(37)

where I₀ is the baseline drift rate (a free parameter of the model), n is the number of alternatives, and:

$$ V_{i,j} = W_{P,i,j} \times (u_{P,i} - u_{P,j}) + W_{Q,i,j} \times (u_{Q,i} - u_{Q,j}), $$

(38)

where W_P,i,j refers to the weight given to the difference between alternatives i and j for attribute P, given by:

$$\begin{array}{@{}rcl@{}} W_{P,i,j} = exp(-\lambda \times |u_{P,i}-u_{P,j}|) \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} \lambda = \lambda_{1}, u_{P,i}-u_{P,j} \geq 0 \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} \lambda = \lambda_{2}, u_{P,i}-u_{P,j} < 0 \end{array} $$

(41)

and u_P,i refers to the subjective attribute value for alternative i on attribute P, given by:

$$\begin{array}{@{}rcl@{}} a &=& b = P_{i} + Q_{i} \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} angle &=& tan^{-1}\left( \frac{Q_{i}}{P_{i}}\right) \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} u_{P,i} &=& \frac{b}{((tan(angle))^{m} + \left( \frac{b}{a}\right)^{m})^{\frac{1}{m}}} \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} u_{Q,i} &=& b \times \left( 1-\left( \frac{u_{P,i}}{a}\right)^{m} \right)^{\frac{1}{m}} \end{array} $$

(45)

Overall, this gives the MLBA nine free parameters: A, b, t₀, γ, β, λ₁, λ₂, I₀, and m.

Appendix E: AAM

The associations and accumulation model (AAM) models the activation for different alternates of multi-attribute choice over decision time through three key components: the activation input for each alternative, the global amount lateral inhibition between alternatives and leakage within each alternative, and some stochastic noise within the process. Formally, the amount of activation for the vector of alternatives A (e.g., for the choice alternatives X, Y, and Z, A = [A_X, A_Y, A_Z]’) can be written as a stochastic differential equation, with the form:

$$\begin{array}{@{}rcl@{}} dA &=& \mu \times dT + \sigma \times dW \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} \mu &=& [S \times A(t) + I(t)] \end{array} $$

(47)

where W is the Wiener process. In terms of the model components, S is an n × n matrix (where n is the number of alternatives) that contains both the amount of lateral inhibition between the alternatives, and the rate of evidence leakage for each alternative. A(t) is the activation for the vector of alternatives at the time t, and I(t) is the vector of inputs for each alternative at time t based upon the attribute being attended to. σ is the standard deviation of the noise process, which is a free parameter of the model. This can be written as a discrete process of the form:

$$ A(t + {\Delta} t) = A(t) + S \times A(t) \times {\Delta} t + I(t) \times {\Delta} t + \sigma \times \sqrt{{\Delta} t} \times \eta $$

(48)

where Δt is the time step, η is the standard normal distribution, and A(t + Δt) is the activation for the vector of alternatives at the next point in time. The activation for each alternative continues to evolve, until one of the alternatives’ activation passes some threshold level (a).

In terms of the three key components, the first, the matrix of the lateral inhibition between the alternatives, and the evidence leakage within alternatives, S, is an n × n matrix (where n is the number of alternatives), defined as:

$$\begin{array}{@{}rcl@{}} S = \left[\begin{array}{ccc} -\lambda & - \beta & - \beta \\ - \beta & -\lambda & - \beta \\ - \beta & - \beta & -\lambda \end{array}\right] \end{array} $$

(49)

where β is a free parameter of the model that controls the amount of global inhibition, and λ is a free parameter of the model that controls the amount of leakage.

The second key component, the vector of activation inputs for each alternative at time t, I(t), is defined as:

$$ I_{i}(t) = \gamma \times V(d_{i}) $$

(50)

where V is a function where:

$$ V(x) = sign(x) |x|^{\alpha} $$

(51)

d_i is given by:

$$ d_{i} = (M \times W(t))_{i} $$

(52)

where M is an n × m matrix, m being the number of attributes, which contains the values for each attribute of each alternative. In the case of our two-attribute design, where the first attribute is denoted F and second attribute is denoted G, M would be:

$$ M = \left[\begin{array}{cc} F_{1} & G_{1} \\ F_{2} & G_{2} \\ F_{3} & G_{3} \end{array}\right] $$

(53)

W(t) is a binary vector of length m, where all elements are set to 0, except for the attribute of matrix that is being attended to at time t (either F or G), which is set to 1. For example, if attribute F were being attended to at time t, then W(t) = W_F = [1,0]’, and if attribute G were being attended to, then W(t) = W_G = [0,1]’. To determine which attribute would be attended to at time t + Δt, we implemented an exponential switching function, where the probability of switching between attended attributes at any point in time was given by:

$$\begin{array}{@{}rcl@{}} P(W_{F} \mapsto W_{G}, {\Delta} t) = 1-exp\left( -\left( k_{F} + \sum\limits_{j = 1}^{n} G_{j}\right) \times k_{scale} \times {\Delta} t\right) \end{array} $$

(54)

$$\begin{array}{@{}rcl@{}} P(W_{G} \mapsto W_{F}, {\Delta} t) = 1-exp\left( -\left( k_{G} + \sum\limits_{j = 1}^{n} F_{j}\right) \times k_{scale} \times {\Delta} t\right) \end{array} $$

(55)

where k_F and k_G are free parameters, separate for attributes F and G to allow bias towards a single attribute, which control the switching probability, with higher values giving larger switching probabilities. The sum of the attribute values over all alternatives for the attribute being switched to represent the attribute’s activation, with attributes that have higher overall values being more highly activated, and therefore, more likely to be attended to. The final parameter, k_scale, is used to re-scale these activation values onto how frequently the switching should occur. The exponential switching function is another addition that we made to the model, as with MDFT and the LCA, to aide the extension to a stochastic differential equation.

Overall, our definition of the AAM gives the model nine free parameters: a, k_F, k_G, k_scale, α, λ, β, σ, and γ.

Appendix F: Recovery assessment

Here we discuss the efficacy of the aforementioned methods when applied to these models with the type of data available for this study. Our goal is to determine how well each of these models accounts for data in these six studies. Toward this end, we generate data out of each model with pre-determined parameters, apply the fitting methodologies, and determine how well each model recovers the RT distributions for each alternative. Importantly, based on the results of these recoveries we choose to not report the estimated parameter values from fitting any of the models, as based on the recovery analysis, these values are of little meaning.

Importantly, to make robust inferences using a quantitative model, one should first establish its reliability. Without having established this reliability, any inferences that are made about the estimated parameter values, or its ability to fit the data, may be spurious. Two important reliability benchmarks to establish are whether a model is able to recover its own parameters, where fitting a model to data it generated results in estimated parameters that match those used to generate the data, and whether a model is able to recover its own data, where fitting a model to data it generated results in the quantitative predictions of the estimated parameters matched the generated data. In this section, we overview our testing of these reliability benchmarks for the three models, and the full analysis can be found in the Supplementary Materials.

For the recoveries of each models, we generated eight sets of data: four types of effects (attraction only, compromise only, similarity only, all three effects together), each with two different numbers of trials (120, 1000). Data recovery was assessed for each type of effect for both numbers of trials, whereas parameter recovery was assessed for each type of effect on only the highest (1000) number of trials. Note that each data set is generated using a single set of parameters with a single set of stimuli. This is be akin to a simple experiment with just one condition.

To briefly summarize, each of these models appears to have strengths and weaknesses in terms of recovery. In terms of data recovery, MDFT AAM, and MLBA showed perfect performance in the 1000 trials data sets for all effects. Although the performance was not quite perfect in the 120 trials data sets, this appeared to be due to the high amount of noise in the generating data, as the 120 trials data set differed substantially from the 1000 trials data set. Thus, the combination of method and model performs very well at recovering the data generated from these models. However, both models did have issues with parameter identifiability. MLBA exhibited significant correlations between parameters in the function that maps attribute information to drift rates. MDFT had similar issues, with large correlations between the attention switching parameters, and between the threshold, drift rate, and stochastic noise coefficient parameters, as well as containing other unclear problems in the recovery of the inhibition function. For AAM, parameter correlation again appeared to be the key issue in inability to recover parameters, though the precise problematic correlations were less clear.

The LCA had more significant issues that required a slight augmentation of the stochastic differential equation. In terms of data recovery, the standard LCA (i.e., with the “(1 − λ)” term applied to every element of the equation) produced poor recovery for all effects at all numbers of trials, which would mean inferences about the LCA’s ability to fit to empirical data would not be reliable. To fix this issue, so that LCA could be part of the comparison, we altered the functional form to no longer include the “(1 − λ)” term, which is the version that we presented earlier in the “Models and methods” section. When using this altered version, the performance of the LCA was very similar to that of MDFT: data recovery was near perfect on the 1000 trials, and parameter recovery revealed identifiability issues that appeared to apply to every parameter within its functional form to at least some degree. Therefore, we will only discuss the altered version from here onwards (though fitting E1 with the “(1 − λ)” term applied made no qualitative differences to the fit).

Appendix G: Additional modeling details

This section provides the Bayesian hierarchical structure (E1, E2, E3, E4, and E6) and the maximum likelihood boundaries (E5) used for each model across the six analyzed studies.

Bayesian hierarchical structure

$$\begin{array}{@{}rcl@{}} MDFT \\ Data \hspace{5pt} level: & & \\ (RT_{i},resp_{i}) & \sim & MDFT(a_{i},k_{A,i},k_{B,i}, \\ & & \hspace{20pt} phi_{1,i},phi_{2,i},\beta_{i},\sigma_{i},\gamma_{i}) \\ Group \hspace{5pt} level: & & \\ a_{i} & \sim & TN(\mu_{a},\sigma_{a},0,100)\\ k_{A,i} & \sim & TN(\mu_{k_{A}},\sigma_{k_{A}},0,200)\\ k_{B,i} & \sim & TN(\mu_{k_{B}},\sigma_{k_{B}},0,200)\\ phi_{1,i} & \sim & TN(\mu_{phi_{1}},\sigma_{phi_{1}},0,10)\\ phi_{2,i} & \sim & TN(\mu_{phi_{2}},\sigma_{phi_{2}},0,30)\\ \beta_{i} & \sim & TN(\mu_{\beta},\sigma_{\beta},0,40)\\ \sigma_{i} & \sim & TN(\mu_{\sigma},\sigma_{\sigma},0,40)\\ \gamma_{i} & \sim & TN(\mu_{\gamma},\sigma_{\gamma},0,400)\\ Prior \hspace{5pt} distributions: & & \\ \mu_{a}, \mu_{k_{A}}, \mu_{k_{B}} & \sim & N_{+}(20,20) \\ \mu_{phi_{1}} & \sim & N_{+}(0.3,0.3) \\ \mu_{phi_{2}} & \sim & N_{+}(0.5,0.5) \\ \mu_{\beta} & \sim & N_{+}(10,10) \\ \mu_{\sigma} & \sim & N_{+}(1,3) \\ \mu_{\gamma} & \sim & N_{+}(50,30) \\ \sigma_{a}, \sigma_{k_{A}}, \sigma_{k_{B}}, \sigma_{phi_{1}}, \sigma_{phi_{2}}, \sigma_{\beta}, \sigma_{\sigma}, \sigma_{\gamma} & \sim & {\Gamma}(0.001,0.001) \end{array} $$

$$\begin{array}{@{}rcl@{}} LCA \\ Data \hspace{5pt} level: & & \\ (RT_{i},resp_{i}) & \sim & LCA(a_{i},k_{F,i},k_{G,i}, \\ & & \hspace{20pt} I_{0,i},\lambda_{i},\beta_{i},\sigma_{i},\gamma_{i}) \end{array} $$

$$\begin{array}{@{}rcl@{}} Group \hspace{5pt} level: & & \\ a_{i} & \sim & TN(\mu_{a},\sigma_{a},0,100)\\ k_{F,i} & \sim & TN(\mu_{k_{F}},\sigma_{k_{F}},0,200)\\ k_{G,i} & \sim & TN(\mu_{k_{G}},\sigma_{k_{G}},0,200)\\ I_{0,i} & \sim & TN(\mu_{I_{0}},\sigma_{I_{0}},0,50)\\ \lambda_{i} & \sim & TN(\mu_{\lambda},\sigma_{\lambda},0,50)\\ \beta_{i} & \sim & TN(\mu_{\beta},\sigma_{\beta},0,1)\\ \sigma_{i} & \sim & TN(\mu_{\sigma},\sigma_{\sigma},0,40)\\ \gamma_{i} & \sim & TN(\mu_{\gamma},\sigma_{\gamma},0,400)\\ Prior \hspace{5pt} distributions: & & \\ \mu_{a}, \mu_{k_{F}}, \mu_{k_{G}} & \sim & N_{+}(20,20) \\ \mu_{I_{0}}, \mu_{\sigma} & \sim & N_{+}(1,3) \\ \mu_{\lambda} & \sim & N_{+}(0.8,0.4) \\ \mu_{\beta} & \sim & N_{+}(0.5,0.5) \\ \mu_{\gamma} & \sim & N_{+}(50,30) \\ \sigma_{a}, \sigma_{k_{F}}, \sigma_{k_{G}}, \sigma_{I_{0}}, \sigma_{\lambda}, \sigma_{\beta}, \sigma_{\sigma}, \sigma_{\gamma} & \sim & {\Gamma}(0.001,0.001) \end{array} $$

$$\begin{array}{@{}rcl@{}} MLBA \\ Data \hspace{5pt} level: & & \\ (RT_{i},resp_{i}) & \sim & MLBA(A_{i},b_{i},t_{0,i},I_{0,i}, \\ & & \hspace{20pt} \lambda_{1,i},\lambda_{2,i},\beta_{i},m_{i},\gamma_{i}) \\ Group \hspace{5pt} level: & & \\ A_{i} & \sim & TN(\mu_{A},\sigma_{A},0,Inf)\\ b_{i}-A_{i} & \sim & TN(\mu_{b},\sigma_{b},0,Inf)\\ t_{0,i} & \sim & TN(\mu_{t_{0}},\sigma_{t_{0}},0,Inf)\\ I_{0,i} & \sim & TN(\mu_{I_{0}},\sigma_{I_{0}},0,Inf)\\ \lambda_{1,i} & \sim & TN(\mu_{\lambda_{1}},\sigma_{\lambda_{1}},0,Inf)\\ \lambda_{2,i} & \sim & TN(\mu_{\lambda_{2}},\sigma_{\lambda_{2}},0,Inf)\\ \beta_{i} & \sim & TN(\mu_{\beta},\sigma_{\beta},0,Inf)\\ m_{i} & \sim & TN(\mu_{m},\sigma_{m},0.2,5)\\ \gamma_{i} & \sim & TN(\mu_{\gamma},\sigma_{\gamma},0,Inf)\\ Prior \hspace{5pt} distributions: & & \\ \mu_{A}, \mu_{b} & \sim & N_{+}(2,2) \\ \mu_{I_{0}}, \mu_{\beta} & \sim & N_{+}(1,3) \\ \mu_{\lambda_{1}}, \mu_{\lambda_{2}} & \sim & N_{+}(0.3,0.3) \\ \mu_{t0} & \sim & N_{+}(0.5,0.5) \\ \mu_{m} & \sim & N_{+}(1,0.5) \\ \mu_{\gamma} & \sim & N_{+}(2,3) \\ \sigma_{A}, \sigma_{b}, \sigma_{t_{0}}, \sigma_{I_{0}}, \sigma_{\lambda_{1}}, \sigma_{\lambda_{2}}, \sigma_{\beta}, \sigma_{m}, \sigma_{\gamma} & \sim & {\Gamma}(0.001,0.001) \end{array} $$

$$\begin{array}{@{}rcl@{}} AAM \\ Data \hspace{5pt} level: & & \\ (RT_{i},resp_{i}) & \sim & AAM(a_{i},k_{F,i},k_{G,i}, \\ & & \hspace{20pt} k_{scale,i},\lambda_{i},\beta_{i},\sigma_{i},\alpha_{i},\gamma_{i}) \end{array} $$

$$\begin{array}{@{}rcl@{}} Group \hspace{5pt} level: & & \\ a_{i} & \sim & TN(\mu_{a},\sigma_{a},0,100)\\ k_{F,i} & \sim & TN(\mu_{k_{F}},\sigma_{k_{F}},0,200)\\ k_{G,i} & \sim & TN(\mu_{k_{G}},\sigma_{k_{G}},0,200)\\ k_{scale,i} & \sim & TN(\mu_{k_{scale}},\sigma_{k_{scale}},0,50)\\ \lambda_{i} & \sim & TN(\mu_{\lambda},\sigma_{\lambda},0,50)\\ \beta_{i} & \sim & TN(\mu_{\beta},\sigma_{\beta},0,1)\\ \sigma_{i} & \sim & TN(\mu_{\sigma},\sigma_{\sigma},0,40)\\ \alpha_{i} & \sim & TN(\mu_{\alpha},\sigma_{\alpha},0,50)\\ \gamma_{i} & \sim & TN(\mu_{\gamma},\sigma_{\gamma},0,400)\\ Prior \hspace{5pt} distributions: & & \\ \mu_{a}, \mu_{k_{F}}, \mu_{k_{G}} & \sim & N_{+}(20,20) \\ \mu_{k_{scale}}, \mu_{\sigma} & \sim & N_{+}(1,3) \\ \mu_{\lambda} & \sim & N_{+}(4,4) \\ \mu_{\beta}, \mu_{\alpha} & \sim & N_{+}(0.5,0.5) \\ \mu_{\gamma} & \sim & N_{+}(50,30) \\ \sigma_{a}, \sigma_{k_{F}}, \sigma_{k_{G}}, \sigma_{k_{scale}}, \sigma_{\lambda}, \sigma_{\beta}, \sigma_{\sigma}, \sigma_{\alpha}, \sigma_{\gamma} & \sim & {\Gamma}(0.001,0.001)\end{array} $$

Note that the truncation used in MDFT, LCA and AAM was included to prevent extreme parameter scaling, which is a result of the models being unidentifiable.

Maximum likelihood boundaries

$$\begin{array}{@{}rcl@{}} MDFT & LCA\\ a = [0,100] & a = [0,100]\\ k_{A} = [0,200] & k_{F} = [0,200]\\ k_{B} = [0,200] & k_{G} = [0,200]\\ \phi_{1} = [0,10] & I_{0} = [0,50]\\ \phi_{2} = [0,30] & \lambda = [0,50]\\ \beta = [0,40] & \beta = [0,1]\\ \sigma = [0,40] & \sigma = [0,40]\\ \gamma = [0,400] & \gamma = [0,400]\\ \\ MLBA & AAM\\ A = [0,10] & a = [0,100]\\ b-A = [0,10] & k_{F} = [0,200]\\ t_{0} = [0,1.2] & k_{G} = [0,200]\\ I_{0} = [0,8] & k_{scale} = [0,50]\\ \lambda_{+} = [0,0.8] & \lambda = [0,50]\\ \lambda_{-} = [0,0.8] & \beta = [0,1]\\ \beta = [0,30] & \sigma = [0,40]\\ m = [0.2,5] & \alpha = [0,50]\\ \gamma = [0,30] & \gamma = [0,400]\\ \end{array} $$

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Nathan J. Evans
- 1
- 2
William R. Holmes
- 3
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Jennifer S. Trueblood
- 1
Email author

1.Department of PsychologyVanderbilt UniversityNashvilleUSA
2.Department of PsychologyUniversity of AmsterdamAmsterdamThe Netherlands
3.Department of Physics and Astronomy, Department of MathematicsVanderbilt UniversityNashvilleUSA

Response-time data provide critical constraints on dynamic models of multi-alternative, multi-attribute choice

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