The standard relationship between choice frequency and choice time is violated in multi-attribute preferential choice

Many theories of preferential choice assume a latent valuation function drives a decision process that terminates upon meeting a decision rule. This surprisingly simple description encompasses many perspectives that cut across psychology, behavioural economics, marketing, and neuroscience. Take, for instance – in the spirit of Prof Tony Marley – random utility theory. The latent valuation function in random utility theory assumes the agent has a subjective value they place on each option in a set of available options. The utility of any of those options, at any moment in time, is a combination of its specific subjective value and unknown random factors. The decision rule in random utility theory compares the utility of all options, at the moment of choice, and selects the option with greatest utility.

A subset of models derived from this broader class of ‘latent valuation function’ theories are based on sequentially sampling decision-relevant information from the stimulus environment (for reviews, see Forstmann et al., 2016, Ratcliff et al., 2016). The principle of sequential sampling theories is that pieces of decision-relevant information are sampled from the environment, where each piece of information may favour one or another response. This process is repeated sequentially, over time, until we have collected, or accumulated, enough pieces of information, or evidence, to inform a choice. The theories naturally predict the choice (which option had more evidence?) and the choice time (how many samples were collected?). Viewed this way, sequential sampling theories such as random walk and diffusion models generalise static signal detection theory (which predicts choices) into the time domain (to predict the joint distribution of choices and choice times).

Sequential sampling theories are most often studied in the domain of ‘objective’ or veridical choice, where one response option is objectively more closely aligned with the target stimulus quality, such as detecting coherent motion or judging line lengths. Sequential sampling models have been studied in great detail for such ‘low-level’ decisions — for example, those pertaining to perceptual and memory-based representations  (e.g., Ratcliff and McKoon, 2008, Shadlen and Shohamy, 2016). This has advanced psychological theorising by decomposing observed choices and latencies into latent components of processing including the speed of information processing, cautiousness, and biases, among others. A critical aspect of the use of sequential sampling models is the application of these theories to choice and response time data. Many of these components of processing cannot be decomposed on the basis of choice data alone.

More recently, the principle of sequentially sampling decision-relevant information has been applied to ‘higher-level’ decisions – those pertaining to attitudinal and preferential options – often referred to as value-based decisions. One can think of this as generalising random utility theory (which predicts choices) into the time domain, which leads to a set of sequential sampling models known as ‘race’ models, where each response option is represented with a different node or ‘accumulator’ (to predict the joint distribution of choices and choice times, for N≥2 options). Tony Marley, in collaboration with Hans Colonius, formalised the link between the choice and time domains in random utility theory (e.g., Marley, 1989, Marley & Colonius, 1992, Colonius & Marley, 2015;for more recent treatments see, e.g., Webb, 2019).

Generalising the theories to new domains of decision making has been fruitful in many ways. Theoretically, it permits testing the principle of sequential sampling as a foundational unit of the decision process — units that can be pieced together much like atoms making up a molecule. Such work aims to provide a deeper understanding of the decision processes involved in higher-level decisions, working toward theoretical consensus on the principles driving decisions under different conditions and contexts.

Sequential sampling models have been studied in ‘simple’ value-based or preferential decisions in considerable detail. ‘Simple’ in this context pertains to the stimulus structure: choices for value-based options that implicitly have multiple features or attributes but the stimulus structure does not specify values for those attributes. For example, a choice between crisps and a chocolate bar, where information about the options is communicated with images that do not explicitly describe features such as the price, weight, caloric content, and so on. In this context, the primary predictions of the models are borne out in data (e.g., Fisher, 2021; Gluth, Kern, Kortmann, & Vitali, 2020; Krajbich, Armel, & Rangel, 2010; Krajbich & Rangel, 2011; and Smith & Krajbich, 2018; for review, see Clithero, 2018). This supports the hypothesis that sequential sampling may be a principle underlying foundational unit of the decision process.

Value-based decisions are also widely studied with explicit multi-attribute structures, where the available options have specified values for a set of product attributes (for reference sources, see Louviere et al., 2015, Louviere et al., 2000). For instance, a choice between a chocolate bar that costs $2.20, weighs 50 g and contains 780kj of energy and a bag of crisps that cost $2.60, weighs 60 g and contains 1400kj of energy. Throughout the manuscript, we refer to the class of multi-attribute stimuli with explicitly specified attribute values as multi-attribute options with defined features.

Many existing theories of choice between value-based options with defined features either have no role for choice times (e.g., the family of logit and probit models derived from random utility theory; Train, 2009) or they have a role for choice times though they are rarely used to test model adequacy (e.g., sequential sampling models; Otter, Johnson, Rieskamp, Allenby, Brazell, Diederich, Hutchinson, MacEachern, Ruan, & Townsend, 2008 — see below for some exceptions). Such sequential sampling models naturally accommodate options with defined features, and can generate predictions for observed patterns of choices marginalised over the predicted distribution of choice times. This permits a test of the models’ predictive scope – the capacity to explain observed choices for options with defined features. This has been a useful test because latency data are rarely collected in multi-attribute preferential choice contexts, and when they are collected they are less frequently analysed and modelled. When they have been collected, there are typically far fewer trials than are considered in conventional applications of sequential sampling models in ‘low-level’ decision contexts, or even simple value-based decisions, which reduces the information value of the choice times, at least to some extent.

Nevertheless, applying sequential sampling models to choice data (without response times) is a limited test of the theories. The models predict a joint distribution of choice outcomes and choice times, so a test of the models based on just one dimension of the joint distribution has limited capacity to discriminate between competing theories. It is also possible the class of models mispredicts important features of the observed choice times, which has the potential to falsify the class of models as a proposed psychological process for preferential decision making.

Exceptions exist, of course, and sequential sampling models have been evaluated against choice frequency and choice times for preferential options with defined features (e.g., Cataldo and Cohen, 2021, Cohen et al., 2017, Evans et al., 2019, Hawkins et al., 2014b, Jones et al., 2015); we restrict our focus to consumer-like decisions and do not consider further other types of preferential options, such as those studied in paradigms of risky gambles or delay discounting. While the two outcome measures have been jointly used to constrain the theories (i.e., estimate parameters) in previous studies, models have tended to be evaluated against marginal distributions of choice outcomes and choice times (i.e., for measures of goodness of fit to data). For example, Evans et al. (2019) compared choice frequency and choice time predictions between four of the major cognitive process models of multi-attribute choice: multi-alternative decision field theory (Roe, Busemeyer, & Townsend, 2001), the multi-attribute linear ballistic accumulator (Trueblood, Brown, & Heathcote, 2014), the leaky competing accumulator (Usher & McClelland, 2004), and the associative accumulation model (Bhatia, 2013). They found that analysing the models against the joint distribution of choices and choice times led to different model preference compared to analyses that only assessed the models against the choice outcomes, such as in Trueblood et al. (2014) and Turner, Schley, Muller, and Tsetsos (2018).

It is rare to consider the joint distribution of choice frequency and choice times in multi-attribute choice for options with defined features – evaluating choice times as a function of choice outcomes, or vice versa. This is commonplace in low-level decisions, for instance, with visualisation including latency-probability and quantile-probability (QP) plots, or defective CDFs (e.g., Donkin and Brown, 2018, Ratcliff et al., 2016). When the joint distribution of multi-attribute preferential choice frequency and choice times are inspected in this way an interesting pattern emerges that, to our knowledge, has not previously been documented. The pattern is important as it places critical constraints on the capacity of sequential sampling models to explain the cognitive processes driving multi-attribute preferential choice.

Here, we study a pattern in multi-attribute choice frequency and choice time data, for options with defined features, that violates a qualitative trend observed in countless studies of choices and times in speeded low-level decisions  (for  reviews,  see Forstmann et al., 2016, Ratcliff and McKoon, 2008, Ratcliff et al., 2016). In low-level decisions, we almost always observe a monotonically decreasing relationship between choice frequency and choice times, conditioned on a given threshold/boundary. That is, as a stimulus dimension becomes more discriminable (e.g., increasing brightness in a brightness discrimination task) it is increasingly likely to be chosen and those choices are made more rapidly. This trend is typically observed in simple value-based decisions, too (for review, see Clithero, 2018). In the value-based context, option discriminability refers to the difference in value of the available options: there is lower discriminability when the set of available options are of similar value and higher discriminability when the set of options differ markedly in value.

Sequential sampling models predict this relationship with a change in the parameter corresponding to the rate of accumulation – the drift rate. As decision-relevant information is acquired more rapidly (greater drift rate), the decision process reaches threshold sooner (faster response) with increasing likelihood (greater choice probability). This relationship is conditioned on a given value of the threshold/boundary parameter – the standard assumption in the decision making literature when pre-stimulus information about the upcoming decision context is not available. We maintain this standard fixed-threshold assumption throughout.

We show that the negative monotonic relationship between choice frequency and choice time is violated for decisions between multi-attribute preferential stimuli with defined features – there is an invariance between the two outcome measures. This result places a critical constraint on the generalisability of the principle of sequential sampling from low-level to high-level decisions – at least for options with defined features.

To our knowledge, two previous results hint at this finding. First, Cohen et al. (2017) report a moderate-to-strong positive correlation between choice proportions observed (in people) and predicted (by the multi-attribute linear ballistic accumulator) in preferential and risky choice contexts, yet a very low correlation between observed and predicted choice times. Given the model predicts a negative monotonic relationship between choice frequency and choice times, and the correlation between observed data and model predictions differed markedly for choice frequency and times, one might conclude the data violate some feature of the predicted relationship. Second, Cataldo and Cohen (2021) found that of the multi-attribute context effects, faster responses tend to be associated with a stronger similarity effect while slower responses tend to be associated with stronger attraction and compromise effects. This empirical result demonstrates choice frequency can depend on choice time, marginalised across stimulus values. We address an orthogonal question: conditioned on stimulus value, what is the relationship between choice frequency and choice time?

In the remainder of the paper, we document experimental evidence for our claim, along with preliminary evidence from standard sequential sampling models. We start with an overview of the generic paradigm studied in much of the decision literature: the discrete choice experiment (DCE).

Many studies of multi-attribute preferential choice with defined features involve a Discrete Choice Experiment (DCE). A multi-attribute DCE presents a set of two or more options that vary across two or more features or attributes, and asks the respondent to select their most preferred option. For instance, when choosing between candidate fast food options, would you prefer an expensive meal that arrives quickly or a cheaper meal with a longer wait time? By manipulating the levels of the attributes, like cheap vs expensive meal and fast vs slow delivery time, the DCE requires participants to make tradeoffs between the multiple attributes that make up an option. By selecting one option over another the respondent reveals information about the psychological weight they place on the attributes of the chosen option relative to the unchosen option/s. Through repeated choices between sets of options with different combinations of attributes and levels, the analyst learns increasingly precise information about the psychological weight a decision maker assigns to the component attributes.

Multi-attribute DCEs in the applied choice field are typically analysed with random utility models. Random utility models share many similarities with standard sequential sampling models: the existence of a latent valuation function, sampling some number of times from the valuation function on presentation of a stimulus, and selecting an option based on some function of the sampled values. Indeed, there is close correspondence between random utility and sequential sampling models formally (e.g., see Colonius and Marley, 2015, Marley and Colonius, 1992) and empirically (e.g., Hawkins, Marley, Heathcote, Flynn, Louviere, & Brown, 2014a). Given similarities in the standard analysis approaches applied across disciplines, one might expect similar relationships to hold in data — such as the joint distribution of choice outcomes and choice times.1

The specific question we address here is whether value-based options in a multi-attribute DCE that are chosen more often are also chosen more quickly. One can address this question at two levels: choosing options (a collection of attribute levels – like a cheap meal with fast delivery) or component attributes of options (the attribute levels considered independently – like cheap meals, or fast delivery). At the level of options, one can map all unique combinations of attribute levels permitted in the experimental design, where some will naturally be more desirable — a cheap meal delivered quickly is better than a slow and expensive meal. We ask if the former is chosen more rapidly than the latter. At the level of attributes, we examine the relationship between the choice frequency for options with a specified attribute level marginalised over all other attributes and options, and how quickly those options are chosen. To continue the earlier example, a negative monotonic relationship at the level of attributes would be that cheap fast food meals are chosen more often and more quickly than expensive fast food meals — independent of the speed of delivery. As another example, and to foreshadow the DCE studied in Experiment 1, we would expect phones that cost $250 are chosen more often and more quickly than phones that cost $500, which in turn would be chosen more often and more quickly than phones that cost $750 — independent of the values of the 4 other attributes that comprise the DCE stimulus. The latter is a powerful test of an effect because it does not rely on specific collections of attribute levels to emerge. We present evidence for our argument using a common multi-attribute DCE context across four experiments: preferences for mobile (cell) phones. In each experiment, participants were presented with 2 or 3 phones per choice set (trial), where each phone was described by different values for each of 4 or 5 attributes. Participants were asked to select their most preferred phone in each choice set. Across studies we assessed ‘free’ choice, the impact of time pressure, whether the attribute structure of the options was correlated or not, and whether the analysis focus was at the level of component attributes or options. Regardless of the study differences, the same pattern emerged in data: the standard relationship between choice frequency and choice time is violated for multi-attribute preferential choice.

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