Logistic analysis of choice data: A primer

Abstract

Logistic regressions were developed in economics to model individual choice behavior. In recent years, they have become an important tool in decision neuroscience. Here I describe and discuss different logistic models, emphasizing the underlying assumptions and possible interpretations. Logistic models may be used to quantify a variety of behavioral traits, including the relative subjective value of different goods, the choice accuracy, risk attitudes, and choice biases. More complex logistic models can be used for choices between good bundles, in cases of non-linear value functions, and for choices between multiple options. Finally, logistic models can quantify the explanatory power of neuronal activity on choices, thus providing a valid alternative to receiver operating characteristic (ROC) analyses.

Introduction

A growing number of neuroscience studies focuses on the neural underpinnings of economic choices. In many experiments, human or animal subjects choose between different goods that vary on two or more dimensions. The subjective values of the goods are inferred from choices and used to interpret neural activity. The analytical tools used to infer subjective values vary across studies, but they often involve some form of logistic regression – an approach first developed in the economics literature (McFadden, 1974; Cramer, 2002). Logistic models can quantify a variety of effects such as choice accuracy (or its reciprocal, choice variability), risk attitudes, temporal discounting, and choice biases. Logistic models can also be used for more complex choices such as choices between good bundles and ternary choices. Furthermore, logistic models can quantify the (marginal) explanatory power of neuronal activity. The proliferation of logistic models and the rationale for adopting a particular model for a particular set of choices might seem unclear. Importantly, while using any logistic model, it is important to keep in mind the underlying assumptions and recognize the possible interpretations. The purpose of this primer is to provide a conceptual and practical tool for neuroscience researchers analyzing choice data. We present a unified formalism and an interpretation for logistic models that may be used in a variety of experimental conditions, and we indicate relevant examples in the literature. We illustrate the practical implications of common assumptions and discuss the interpretation of specific models. An exhaustive review of the literature on choice behavior is beyond our purposes. The examples discussed in the article are drawn from experiments in which nonhuman primates chose between different types of juices. However, all the models discussed here apply to a broad class of experimental designs and protocols.

Choices between two goods offered in variable quantities

Consider the experiment illustrated in Fig.1. In this session, a rhesus monkey chose between two juices, namely cherry juice and hawaiian punch. Offers were represented by circles on a computer monitor. Each juice was associated with a color, and the quantity offered on any given trial was represented by the radius of the corresponding circle. The animal indicated its choice with an eye movement, and the chosen juice was delivered at the end of each trial. Offered quantities varied randomly from trial to trial between 50 and 300 μl and between 100 and 600 μl, respectively. Fig.1B illustrates the choice the animal made in each trial. In this session, the animal preferred cherry juice, meaning that when the two juices were offered in equal quantities the animal consistently chose cherry juice. Henceforth, we will refer to the preferred juice as juice A and to the other juice as juice B. We will indicate with q_A and q_B the juice quantities offered on any given trial.

Figure 1.

Choices between two juices offered in variable amounts. A. Experimental design. A monkey sat in front of a computer monitor with the head restrained. Offers appeared on the monitor and the gaze direction was monitored by a video camera. Two juice lines were placed in front of the animal’s mouth. At the beginning of each trial, the monkey fixated a dot in the center of the monitor. Two offers, represented by two color circles, appeared on the two sides of the fixation point. For each offer, the color indicated the juice type and the radius indicated the juice amount. The animal maintained center fixation for a randomly variable delay (1–2 s), at the end of which the center fixation point dimmed and the offers were replaced by a two saccade targets (go signal). The animal indicated its choice by gazing to one of the targets; it maintained target fixation for 0.75 s, at the end of which the chosen juice was delivered directly to the mouth. B. Example session, joint distribution of offers and choices. The y- and x-axis represent the offered quantities q_A and q_B. Each symbol represents one trial and the color indicates the chosen juice (red or blue for A or B, respectively). All juice quantities are expressed in units of a conventional quantum q = 100 μl. The session included 560 trials and joint distribution of offers was uniform within a domain. The logistic regression provided the measures ρ = 2.4 and η = 5.1. The black line indicates the indifference line ρ q_A = q_B. C. Choice frequency. The panel illustrates the same data as in B binning trials. For q_A (y-axis) we used a bin size = 1/2; for q_B (x-axis) we used a bin size = ρ/2. Thus in value units the bin size was the same for bot axes. Gray shades indicate the fraction of trials in which the animal chose juice B. D. Fitted sigmoid. We computed the log quantity ratio for each trials and we binned trials (bin size = 0.5; x-axis). For each bin, we computed the fraction of trials in which the animal chose juice B (y-axis). Black dots represent the fraction of B choices for each bin, and the gray sigmoid is that obtained from the logistic fit. The green dotted line highlights the flex point (log(q_B/q_A) = log(ρ)); the yellow dotted line highlights the derivative of the sigmoid through the flex point (η/4).

Given offers q_A and q_B, the animal chose juice A with probability P_A and juice B with probability P_B = 1 − P_A. The goal of the logistic analysis is to model choices – that is, to provide a function that accurately relates P_A and P_B to q_A and q_B. We can indicate with P_J the probability of choosing juice J (J = A or B). The basic logistic model assumes that the probability of choosing juice J on any given trial is proportional to the value of juice J offered in that trial raised to some power. In formulas:

$$P_J=\frac{V_J^{\eta }}{{\sum }_I{V}_I^{\eta }}$$ (1)

where V_J is the value of juice J and I = A, B. The sum at the denominator is a normalization ensuring that probabilities add up to 1. The parameter η, which will be fitted, captures the choice accuracy (or choice consistency). To appreciate this point, consider how probabilities P_A and P_B change with η, given two values V_A and V_B. For example, consider the case in which V_A is slightly larger than V_B. When η becomes large, the term $V_A^{\eta }$ in the denominator becomes increasingly dominant. Hence, P_A becomes close to 1 and P_B becomes close to 0. Conversely, when η is small, choices are more evenly split. Thus the larger η, the more the animal is likely to choose the higher value.

We can now re-write the probability of choosing juice B as follows:

$$P_B=\frac{V_B^{\eta }}{V_A^{\eta }+V_B^{\eta }}=\frac{e^{\eta \text{ log}\left(V_B\right)}}{e^{\eta \text{ log}\left(V_A\right)}+e^{\eta \text{ log}\left(V_B\right)}}=\frac{1}{1+e^{-\eta \text{ log}\left(V_B/V_A\right)}}$$ (2)

where log indicates the natural logarithm. The next step is to specify a value function. We assume the following function:

$$V_J={\rho }_J{q}_J$$ (3)

In essence, the value V_J equals the quantity q_J multiplied by a factor ρ_J that expresses the subjective value of one unit volume of juice J. Notably, that the animal prefers juice A to juice B means ρ_A > ρ_B. Parameters ρ_A and ρ_B are to be fitted. However, Eqs.2–3 reveal that the logistic model only depends on the ratio ρ = ρ_A/ρ_B. Ultimately, this means that any value can only be measured relative to some other value (i.e., in some value units). Without losing generality, we may measure values in units of juice B and thus set ρ_B = 1. The parameter ρ is referred to as the “relative value”.

The logistic model has two free parameters, namely η and ρ. Substituting Eq.3 in Eq.2, we obtain the standard expression for a logistic model:

$$P_B=\frac{1}{1+e^{-X}}$$ (4)

$$X=a_0+a_1\text{ log}\left(q_B/q_A\right)$$ (5)

where we defined:

$$a_0=-\eta \text{ log}(\rho )$$ (6)

$$a_1=\eta$$ (7)

Inverting Eq.4 reveals that

$$X=\text{log}\left(P_B/\left(1-P_B\right)\right)=\text{log}\left(P_B/P_A\right)$$ (8)

The probability ratio P_B/P_A is termed “odds” and X is referred to as the log odds.

The goal of this analysis is to represent the data as accurately as possible, given the assumptions of the model. To do so, one performs a maximum likelihood estimation, defining the objective function

$$\mathcal{F}={\prod }_{i=1\dots N}{P}_{\mathit{\text{chosen juice}}}^i$$ (9)

where N is the number of trials in the session and $P_{\mathit{\text{chosen juice}}}^i$ is the probability that on trial i the animal would choose the juice it actually chose. This probability is computed using Eqs.4–5. To do a logistic regression means to identify values $\widehat{a_0}$ and $\widehat{a_1}$ that maximize $\mathcal{F}$.

Of note, $\mathcal{F}$ in Eq.9 includes all trials except forced choices – i.e., trials for which either q_A = 0 or q_B = 0. Forced choices are excluded from the regression because the log in Eq.5 would be infinite.

Consider now Eqs.4–5. First, note that the function relating P_B and log(q_B/q_A) is a sigmoid (Fig.1D). Second, resolving Eq.6, we obtain the estimate for the relative value $\rho =e^{-\widehat{a_0}/\widehat{a_1}}$. Eqs.4–5 reveal that ρ equals the quantity ratio q_B/q_A that makes the animal indifferent between the two offers. To appreciate this point, note that the indifference condition P_B = 0.5 is equivalent to X = 0, which implies $q_B/q_A=e^{-\widehat{a_0}/\widehat{a_1}}$.

We may also compute the first derivative of the sigmoid function (Fig.1D) at the indifference point:

$${\frac{d{P}_B}{d\left(\text{log}\left(q_B/q_A\right)\right)}|}_{X=0}={\frac{1}{{\left(1+e^{-X}\right)}^2}\frac{dX}{d\left(\text{log}\left(q_B/q_A\right)\right)}|}_{X=0}=\widehat{a_1}/4$$ (10)

Thus $\widehat{a_1}/4$ is the steepness of the sigmoid, confirming that $\eta =\widehat{a_1}$ is a measure of choice accuracy (the steeper the sigmoid, the more consistent are choices). Hence, η is inversely related to choice variability. In the language of statistical mechanics, η is referred to as the “inverse temperature”.

For the experimental session in Fig.1, the logistic regression provides the estimates ρ = 2.4 and η = 5.1.

Alternative logistic model: value difference versus log value ratio

A frequently used alternative to Eq.1 is a logistic model in which the probability of choosing juice J on any given trial is proportional to $e^{\eta V_J}$. In other words,

$$P_J=\frac{e^{\eta V_J}}{{\sum }_I{e}^{\eta V_I}}$$ (11)

Again, η is a measure of choice accuracy. Numerous neuroscience studies have used this model (Kim et al., 2008; Amemori and Graybiel, 2012; Hunt et al., 2012; Ferrari-Toniolo et al., 2021). As in the previous section, we may assume the following value function:

$$V_J={\rho }_J{q}_J$$ (12)

The probability of choosing juice B may thus be written as follows:

$$P_B=\frac{e^{\eta V_B}}{e^{\eta V_A+e^{\eta V_B}}}=\frac{1}{1+e^{-\eta \left(V_B-V_A\right)}}$$ (13)

Again, we may measure values in units of juice B and thus set ρ_B = 1. We may also define ρ = ρ_A/ρ_B. We thus obtain the standard expression:

$$P_B=\frac{1}{1+e^{-X}}$$

$$X=a_0{q}_A+a_1{q}_B$$ (14)

where we defined a₀ = −ηρ and a₁ = η. After the logistic regression, we obtain the fitted values for ρ and η. As in the logistic model discussed in the previous section, ρ equals the quantity ratio q_B/q_A that makes the animal indifferent between the two offers and η is proportional to the sigmoid steepness at the indifference point.

We might refer to the model discussed in the previous section as the “log value ratio model” (because of Eq.2) and to the model described in this section as the “value difference model” (because of Eq.13). These two models make different predictions. In particular, referring to the experiment of Fig.1, consider the two sets of quantities [q_A, q_B] such that the animal chooses juice B 25% of the time (set 1) or 75% of the time (set 2). Consider now the plane defined by the two quantities. In the log value ratio model, P_B = const implies q_B/q_A = const. Thus this model predicts that set 1 and set 2 are two straight lines that cross the axes origin and have different slopes (Fig.2A). Conversely, in the value difference model, the condition P_B = const implies q_B − q_A = const. Thus this model predicts that set 1 and set 2 are two straight lines that are parallel to each other and to the indifference line (P_B=0.5) and that do not cross the axes origin (Fig.2B).

Figure 2.

Comparing logistic models (cartoon). A. Log value ratio model. The two axes represent the offer values in units of juice B. The solid line represents the indifference line (P_B=0.5); the two dotted line represent the conditions for which the animal chooses juice B 25% and 75% of the time (P_B=0.25 and P_B=0.75, respectively). In other words, the dotted lines capture the shape and the width of the fitted sigmoid. B. Value difference model. Same conventions as in panel A. In this model, the animal would split choices even when one of the two offers is vanishingly small (e.g., in this cartoon, for q_A = 0 and q_B = 1, P_B = 0.75).

Intuitively, the former prediction seems more sensible – it captures the idea that the difficulty of choosing between $9 and $10 is similar to the difficulty of choosing between $0.9 and $1, while choosing between $0 and $1 is much easier. Another advantage of the log value ratio model is that parameters a₀ and a₁ are pure numbers and do not depend on the particular units used to express quantities or values. Finally, using value ratios makes it particularly convenient to write the logistic model when the value function is a product – e.g., when the two determinants of value are probability and quantity, or quantity and time. Conversely, the value difference model is more convenient when the value function is a sum – e.g., when the two determinants are quantity and cost, or when options are bundles. Thus for most of this article we will continue to use the log value ratio model. However, in the discussion of choices between bundles, we will use the value difference model.

Quantifying risk attitudes

In the following sections, we show how logistic analyses may be used to quantify a variety of behavioral phenomena and neural effects. We start by the analysis of choices under risk – i.e., choices between goods that might be obtained with probability <1. The value of a desirable good offered with probability p < 1 naturally depends on p. Moreover, choices often reflect some “risk attitude”. For example, many people would chose $490 for sure to a gamble that pays $1000 with probability p = 0.5 and $0 otherwise. Thus many choices under risk are not simply driven by the expected value of the offers. Risk attitudes can be quantified using logistic analyses and numerous neuroscience studies have taken this approach (Levy et al., 2010; Yamada et al., 2013; Raghuraman and Padoa-Schioppa, 2014; Strait et al., 2014; Ferrari-Toniolo et al., 2021).

Consider the experiment illustrated in Fig.3. In this session, a monkey chose between two juices, labeled A and B, offered in variable quantities and probabilities. Offers were represented by pie-shaped images. For each offer, the color represented the juice type, the radius of the pie represented the quantity, and the angle of the pie represented the probability (Fig.3A). For juice J (= A or B), we indicate the probability and quantity offered on any given trial respectively with p_J and q_J. We define the expected quantity as the product EQ_J = p_J q_J. In the experiment, for each juice J, p_J and q_J varied randomly and independently from each other, subject to the constraint that EQ_J be above a fixed minimum (Fig.3BC). Fig.3D illustrates the choices made by the animal in each trial as a function of the expected quantities EQ_A and EQ_B. The preferred juice was conventionally labeled as juice A.

Figure 3.

Choices under uncertainty: quantifying the risk attitude. A. Experimental design. The trial structure was similar to that of Fig.1. However, in this experiment, offers were represented by incomplete pies. The color indicated the juice type, the radius indicates the juice amount, and the filled angle represented the probability. For example, in the trial shown here, juice A (red) was offered in larger quantity but smaller probability than juice B (blue). B. Example session, offer A distribution. The two axes represent probability (p_A) and quantity (q_A), each symbol represents one trial, and the color represents the chosen juice. Within the quantity range [0 4] and the probability range [0 1], the distribution of offers was uniform subject to the constraint that the expected quantity be not too small (p_A q_A ≥ max(q_A)/10). C. Example session, offer B distribution. Same format as panel B. D. Joint distribution, expected quantities. In this plot, the two axes represent the two expected quantities, namely EQ_A = p_A q_A (y-axis) and EQ_B = p_B q_B (x-axis). E. Joint distribution, expected values. Here the two axes represent the two expected values, namely $E{V}_A=p_A^{\gamma }\rho q_A$ and $E{V}_B=p_B^{\gamma }{q}_B$. On both axes, values are expressed in units of juice B. The black line is the indifference line (P_B=0.5) and indicate the chosen juice. This plot was based on the results of the logistic analysis. Notably, the indifference line divides A choices (red) and B choices (blue) better than any straight line one could draw in panel D (because γ≠1). F. Same data as in panel E, binning trials. Gray shades indicate the fraction of trials in which the animal chose juice B. Bins with <5 trials were excluded. G. Fitted sigmoid surface. Axes represent the log quantity ratio, the log probability ratio and the frequency of B choices, respectively.

We indicate with P_J the probability that on any given trial the animal chose juice J. Again, the goal of the logistic analysis is to provide a function that accurately relates P_A and P_B to p_A, p_B, q_A and q_B. As for the case discussed above, where p_A = p_B = 1, the logistic model assumes that P_J is proportional to the value of juice J raised to some power (Eq.1). In this case, however, the value function also depends on the probability. We assume the following value function:

$$V_J=p_J^{\gamma }{\rho }_J{q}_J$$ (15)

Again, without losing generality, we may set ρ_B = 1 and thus measure values in units of juice B. Thus the resulting logistic model has three free parameters: η is the inverse temperature, ρ = ρ_A/ρ_B is the relative value, and γ captures the risk attitude. Specifically, γ < 1 means that the monkey treats probabilities as higher than they are ($p_J^{\gamma }>p_J$; risk seeking). Conversely, γ > 1 means that the monkey treats probabilities as lower than they are ($p_J^{\gamma }<p_J$; risk aversion). If the monkey makes a very large number of choices, the condition γ = 1 ($p_J^{\gamma }=p_J$; risk indifference) yields the maximum payoff. In this sense, risk indifference is optimal.

An alternative way to model choices under risk is to use the following value function:

$$V_J=p_J{\rho }_J{q}_J^{\kappa }$$ (16)

Eq.15 assumes that risk attitudes reflect a distortion in the computation of subjective probabilities. Conversely, Eq.16 attributes risk attitudes to a distortion of subjective quantities. The latter formulation is referred to as “expected utility theory” (Kreps, 1990). We generally prefer Eq.15 because it makes it transparent that the behavioral phenomenon – risk aversion or risk seeking – is related to the attitude toward probabilities. Furthermore, in this formulation, choices without risk, discussed in the first section, are a specific case. In other words, Eq.15 reduces to Eq.3 when all p_J = 1. Thus we will continue the presentation referring to this model.

Repeating the steps taken in Eq.2, we re-write the probability of choosing juice B as follows:

$$P_B=\frac{1}{1+e^{-X}}$$

$$X=a_0+a_1\text{ log}\left(p_B/p_A\right)+a_2\text{ log}\left(q_B/q_A\right)$$ (17)

where we defined a₀ = −η log(ρ), a₁ = η γ and a₂ = η. Once the logistic regression has been performed, we can compute the parameters that characterize choices, namely the relative value $\rho =e^{-\widehat{a_0}/\widehat{a_2}}$, the inverse temperature $\eta =\widehat{a_2}$, and the risk attitude $\gamma =\widehat{a_1}/\widehat{a_2}$.

For the experimental session in Fig.3, the logistic regression provided the estimates ρ = 2.1, γ = 0.53 and η = 6.2. Notably, γ < 1 indicates that the animal was risk seeking – a common finding in monkeys (Heilbronner and Hayden, 2013). Choices made by the animal in this session may also be plotted as a function of the expected values $E{V}_A=p_A^{\gamma }\rho q_A$ and $E{V}_B=p_B^{\gamma }{q}_B$ (Fig.3EF). In essence, the expected values are similar to the expected quantities (Fig.3D), except that they incorporate the relative value (ρ) and the probability distortion (γ). We observe that the two sets of data points corresponding to choices of juice A and juice B are more clearly separated in Fig.3E than in Fig.3D, confirming the fact that the parameter γ captures an important trait of the animal’s behavior.

Quantifying choice biases

Human and animal choices may be affected by a variety of biases (Kagel et al., 1995). In many cases, such biases have a binary form. For example, in the experiment of Fig.1, other things equal, the animal might be more likely to choose the offer presented on the right (side bias). Similarly, numerous studies reported a history-dependent bias termed choice hysteresis – other things equal, subjects were more likely to choose on any given trial the same good chosen in the previous trial (Padoa-Schioppa, 2013; Alos-Ferrer et al., 2016; Schoemann and Scherbaum, 2019). In other experiments, monkeys chose between two options offered with probability <1. For one of the two options, information about the trial outcome (good or poor luck) would be revealed to the animal before the end of the trial. Even though this information did not affect the reward likelihood, animals’ choices were biased in favor of the informative good (Blanchard et al., 2015). All of these biases can be effectively quantified with a logistic analysis. To illustrate this point, we examine an experiment of choices under sequential offers (Fig.4).

Figure 4.

Choices under sequential offers: quantifying the order bias. A. Experimental design. The experimental design was similar to that in Fig.1. However, in this experiment, offer quantities were quantized and the two offers were presented sequentially. For each offer, the color indicated the juice type and the number of squares indicated the juice amount (in multiples of a fixed quantum). For each pair of offer quantities, the presentation order varied pseudorandomly. The two saccade targets had different colors associated with the two juices, and the left/right position of saccade targets was independent of the order. B. Joint distribution of offers. Dotted lines represent the indifference lines for AB trials (green), BA trials (purple) and all trials pooled (black). C. Choice pattern. An offer type was defined by two offered quantities in a particular order. In the plot, each symbol represents one offer type. Green and purple are associated with AB trials and BA trials, respectively. The sigmoid were obtained from the logistic regression (Eq.21). Other things equal, the animal was more likely to choose juice A in when juice A was offered second (BA trials). In other words, the animal had a bias favoring offer2 (order bias ϵ′ > 0).

In each session, a monkey chose between two juices, labeled A and B, offered in variable amounts. In this case, however, the two juices were presented sequentially. Offers were represented by sets of squares displayed on a computed monitor. For each offer, the color represented the juice type and the number of squares represented the quantity (offers were quantized; Fig5AB). For each pair of quantities [q_A, q_B], the presentation order varied randomly. We refer to trials in which juice A was offered first and trials in which juice B was offered first as “AB trials” and “BA trials”, respectively. We also refer to the first and second offers as “offer1” and “offer2”, respectively. This situation closely resembles that examined in the first section (Fig.1). However, as in other studies where offers were presented in sequence (Krajbich et al., 2010; Ballesta and Padoa-Schioppa, 2019; Rustichini et al., 2021), choices in this session where generally biased in favor of the second offer (Fig.4B). We may model this phenomenon as though the value of any particular good was reduced or enhanced when that good was offered first or second. We can thus write the logistic model as in Eq.1 and write the following value functions:

$$V_A={\rho }_A{q}_A\left(1-ϵ{\delta }_{\mathit{\text{order}},AB}+ϵ{\delta }_{\mathit{\text{order}},BA}\right)$$ (18)

$$V_B={\rho }_B{q}_B\left(1+ϵ{\delta }_{\mathit{\text{order}},AB}-ϵ{\delta }_{\mathit{\text{order}},BA}\right)$$ (19)

where ϵ is a parameter that captures the order bias, δ_order,AB = 1 in AB trials and 0 otherwise, and δ_order, BA = 1 − δ_order,AB. In practice, in each of Eqs.18–19, one δ term always equals one and the other equals zero. Thus each value is multiplied by (1 − ϵ) or (1 + ϵ) depending on the presentation order. Note also that the sign of ϵ captures the direction of the effect. Specifically, if ϵ < 0, the bias favors offer1; if ϵ > 0, the bias favors offer2. As usual, we set ρ_B = 1 and define ρ = ρ_A/ρ_B. We can thus compute the log value ratio:

$$\text{log}\left(V_B/V_A\right)\approx -\text{log}(\rho )+\text{log}\left(q_B/q_A\right)+2ϵ\left({\delta }_{\mathit{\text{order}},AB}-{\delta }_{\mathit{\text{order}},BA}\right)$$ (20)

where we have used Taylor’s expansion log(1 + x) ≈ x. Repeating the steps taken in Eq.2, we write the probability of choosing juice B as follows:

$$P_B=\frac{1}{1+e^{-X}}$$

$$X=a_0+a_1\text{ log}\left(q_B/q_A\right)+a_2\left({\delta }_{\mathit{\text{order}},AB}-{\delta }_{\mathit{\text{order}},BA}\right)$$ (21)

where we defined a₀ = −η log(ρ), a₁ = η, and a₂ = η ϵ. Intuitively, Eq.21 describes two sigmoids, one for AB trials and the other for BA trials. The two sigmoids are parallel to each other (same steepness a₁) but they have different flex points (corresponding to (a₀ ± a₂)/a₁) (Fig.4C).

Figure 5.

Ternary choices. A. Experimental design. In this experiment, the animal chose between three options that varied for the juice type, the quantity and the probability. Each offer was represented by an incomplete pie. The color indicated the juice type, the radius indicated the juice quantity, and the filled angle represented the probability. In each trial, the centers of the three pies were placed on an invisible circle surrounding the fixation point, 120 degrees apart from each other. B. Offer distribution for the three juices. The three panels refer to juices A, B and C, respectively. In each panel, the two axes represent probability and quantity, each symbol represents one trial, and the color represents the chosen juice. The distribution of offers was uniform within a domain and subject to the constraint that the expected quantity be not below a set minimum. C. Joint distribution, expected values. This simplex plot (or ternary plot) illustrates data that varied on three dimensions, namely EV_A, EV_B and EV_C. Each data point represents one trial and, for each trial, data are normalized by the sum of the three expected values. Each corner of the triangle corresponds to one juice type. Consider the top corner (juice C) and the bottom side opposite to it. For each point in the triangle, one may trace the line passing through that point and the top corner. On that line, the distance between the bottom side and the point equals the normalized EV_C. Similarly, the equivalent distances between each side and the opposite corner are the normalized EV_A and EV_B. This session included binary choices (one of the goods = null) and forced choices (two of the goods = null). Data points on the three sides of the triangle represent binary choices, while data on the three corners represent forced choices. Trials in which the normalized EV_C is large are close to the top corner; trials in which the normalized EV_C is small are close to the bottom side (similarly for the other juices). Notably, the three dimensions are identified up to a factor (i.e., if the three EV are doubled, the position does not change). In this plot, the color of each data point also represents the expected values in RBG format, where the red, blue and green components of the color are set by EV_A, EV_B and EV_C, respectively. Thus for any position on the triangle, the point’s brightness captures the value scale. D. Joint distribution, chosen juice. Same plot as in panel C, except that colors indicate the chosen juice. Notably, the three clouds of red, blue and green points are relatively well, but not perfectly, separated.

Once the logistic regression has been performed, we compute the parameters characterizing choices, namely the relative value $\rho =e^{-\widehat{a_0}/\widehat{a_1}}$, the inverse temperature $\eta =\widehat{a_1}$, and the order bias $ϵ=\widehat{a_2}/\widehat{a_1}$. An alternative and valid definition for the order bias is ${ϵ}^{\prime }=2\rho \widehat{a_2}/\widehat{a_1}$. One advantage of this definition is that the order bias quantifies explicitly the distance between the two sigmoids. Indeed, the relative values specific to AB trials and BA trials may be defined, respectively, as ${\rho }_{AB}=e^{-\left(\widehat{a_0}+\widehat{a_2}\right)/\widehat{a_1}}$ and ${\rho }_{BA}=e^{-\left(\widehat{a_0}-\widehat{a_2}\right)/\widehat{a_1}}$. Using Taylor’s expansion, we may then observe that ϵ′ ≈ ρ_BA − ρ_AB. For the experimental session in Fig.4A, the logistic regression provided the estimates ρ = 1.5, η = 3.8 and ϵ′ = 0.55.

The same approach has been used to model choice hysteresis (Padoa-Schioppa, 2013). In this case, the probability of choosing juice B is written as follows:

$$P_B=\frac{1}{1+e^{-X}}$$

$$X=a_0+a_1\text{ log}\left(q_B/q_A\right)+a_2\left({\delta }_{\mathit{\text{prev choice}},B}-{\delta }_{\mathit{\text{prev choice}},A}\right)$$ (22)

where δ_{prev choice,J} = 1 if in the previous trial the animal chose juice J and 0 otherwise. Again, Eq.22 describes two parallel sigmoids, one for trials following choices of juice A and the other for trials following choices of juice B. Following the logistic regression, one can compute the parameters characterizing choices, namely relative value $\rho =e^{-\widehat{a_0}/\widehat{a_1}}$, inverse temperature $\eta =\widehat{a_1}$, and choice hysteresis ${\xi }^{\prime }=2\rho \widehat{a_2}/\widehat{a_1}$.

For any set of choices, one can also examine multiple choice biases, separately or at once (Cai and Padoa-Schioppa, 2019; Constantinople et al., 2019; Kuwabara et al., 2020). In essence, with a single analysis, one can model a family of parallel sigmoids (same inverse temperature). One caveat is that different variables included in the log odds X must be independent of each other. Also, when we fit a family of sigmoids, other things equal, the statistical power of each fitted parameter is reduced. Hence, one typically needs more trials.

Logistic models for ternary choices

All the cases discussed so far involved choices between two options. However, many daily situations involve choices between three or more options. Here we derive the logistic analysis for ternary choices. Consider the experiment illustrated in Fig.5. In this session, a monkey chose between three juices offered in variable quantities and probabilities. We label the three juices A, B and C based on the preference order. Offers were represented by pie-shaped images. For each offer, the color represented the juice type, the pie radius represented the quantity, and the pie angle represented the probability (Fig.5A). For juice J (= A, B or C), we indicate with p_J and q_J the probability and quantity offered on any given trial. We defined the expected quantity as the product EQ_J = p_J q_J. For each juice J, p_J and q_J varied randomly and independently from each other, subject to the constraint that EQ_J be above a fixed minimum (Fig.5B).

As for binary choices, the logistic model assumes that the probability P_J of choosing juice J is proportional to the value V_J raised to some power. Furthermore, we can assume the same value function used for binary choices under risk. Thus the logistic model is:

$$P_J=\frac{V_J^{\eta }}{{\sum }_I{V}_I^{\eta }}$$

$$V_J=p_J^{\gamma }{\rho }_J{q}_J$$ (23)

with I and J = A, B or C. Without losing generality, we express values in units of juice C and thus set ρ_C = 1. We thus indicate the relative values ρ_A/ρ_C and ρ_B/ρ_C simply as ρ_A and ρ_B, respectively. We can now write the probability of choosing each juice. For example, the probability of choosing juice A is

$$P_A=\frac{e^{\eta \text{ log}\left(V_A\right)}}{e^{\eta \text{ log}\left(V_A\right)}+e^{\eta \text{ log}\left(V_B\right)}+e^{\eta \text{ log}\left(V_C\right)}}$$

$$P_A=\frac{e^{\eta \text{ log}\left(p_A^{\gamma }{\rho }_A{q}_A\right)}}{e^{\eta \text{ log}\left(p_A^{\gamma }{\rho }_A{q}_A\right)}+e^{\eta \text{ log}\left(p_B^{\gamma }{\rho }_B{q}_B\right)}+e^{\eta \text{ log}\left(p_C^{\gamma }{q}_C\right)}}$$

$$P_A=\frac{1}{1+e^{\eta \text{log}\left({\rho }_B/{\rho }_A\right)+\eta \gamma \text{log}\left(p_B/p_A\right)+\eta \text{log}\left(q_B/q_A\right)}+e^{\eta \text{log}\left(1/{\rho }_A\right)+\eta \gamma \text{log}\left(p_C/p_A\right)+\eta \text{log}\left(q_C/q_A\right)}}$$

$$P_A=\frac{1}{1+e^{-\left(a_0-a_1+a_2\text{log}\left(p_A/p_B\right)+a_3\text{log}\left(q_A/q_B\right)\right)}{+e}^{-\left(a_0+a_2\text{log}\left(p_A/p_C\right)+a_3\text{log}\left(q_A/q_C\right)\right)}}$$ (24)

where we defined a₀ = η log(ρ_A), a₁ = η log(ρ_B), a₂ = η γ, and a₃ = η. Similarly, we can compute:

$$P_B=\frac{1}{1+e^{-\left(-a_0+a_1+a_2\text{log}\left(p_B/p_A\right)+a_3\text{log}\left(q_B/q_A\right)\right)}+e^{-\left(a_1+a_2\text{log}\left(p_B/p_C\right)+a_3\text{log}\left(q_B/q_C\right)\right)}}$$ (25)

$$P_C=\frac{1}{1+e^{-\left(-a_0+a_2\text{log}\left(p_C/p_A\right)+a_3\text{log}\left(q_C/q_A\right)\right)}+e^{-\left(-a_1+a_2\text{log}\left(p_C/p_B\right)+a_3\text{log}\left(q_C/q_B\right)\right)}}$$ (26)

The maximum likelihood estimation is computed as in the binary case (Eq.9) using the objective function

$$\mathcal{F}={\prod }_{i=1\dots N}{P}_{\mathit{\text{chosen juice}}}^i$$ (27)

where N is the number of trials and the probabilities $P_{\mathit{\text{chosen juice}}}^i$ are computed using Eqs.24–26. As a supplement to this primer, we provide a Matlab function that performs logistic analysis for ternary choices.

For the experimental session in Fig.5, the logistic regression provides the estimates ρ_A = 3.6, ρ_B = 2.7, γ = 0.9 and η = 5.3. Based on these fitted parameters, we can compute the three expected values offered in each trial $E{V}_A=p_A^{\gamma }{\rho }_A{q}_A$, $E{V}_B=p_B^{\gamma }{\rho }_B{q}_B$ and $E{V}_C=p_C^{\gamma }{q}_C$, and then construct a simplex plot in which the three expected values determine the position on the triangle (Fig.5CD). Using this format Fig.5D illustrates the choices made by the animal in each trial as a function of the expected values EV_A, EV_B and EV_C.

Choices between bundles

In many situations, options available for choice are not individual goods but rather bundles of goods. For example, while contemplating a restaurant menu, one might choose between a dish of salmon and broccoli and one of tofu and bok choy. Several neuroscience studies have focused on choices between bundles (FitzGerald et al., 2009; Pastor-Bernier et al., 2019; Pastor-Bernier and Schultz, 2021). To discuss logistic analyses of such choices, we refer to the experiment illustrated in Fig.6. In this session, a monkey chose between two bundles, labeled T1 and T2, each constituted by two juices. Different juices constituted each bundle, and the 4 juices were labeled A, B, C and D. Bundle T1 included juices A and B; bundle T2 included juices C and D. Offers were represented by sets of symbols on a computer monitor. The color of the symbols represented the juice type and the number of symbols indicated the quantity. In each bundle, the preferred juice (A for T1; C for T2) was represented by circles and located on the upper half of the screen; the less preferred juice (B for T1; D for T2) was represented by squares and located on the lower half of the screen. Thus in the trial depicted in Fig.6A the animal chose between T1 = [2A+3B] and T2 = [3C+4D]. The monkey revealed its choice with a saccade. After a brief delay, the two juices in the chosen bundle were delivered sequentially, starting from the preferred juice and with a 1 s delay interval between the two juices. (Thus temporal discounting effects remained constant for each juice and can be considered incorporated in the juice’s relative value.)

Figure 6.

Choices between juice bundles. A. Experimental design. This experiment involved 4 different juices labeled A, B, C and D. Each offer was a bundle of two juices (T1 or T2). Offered quantities were quantized. In the trial shown here, the animal chose between bundle T1, including 2A and 3B, and bundle T2, including 3C and 4D. After the animal indicated its choice with a saccade, the two juices in the chosen bundle were delivered sequentially, starting with the preferred one, and with a 1 s delay between them. Bundles were constructed such that juice A was preferred to juice B and juice C was preferred to juice D. B. B. Joint distribution of quantities in bundle T1. C. Joint distribution of quantities in bundle T2. D. Choice pattern. The x- and y-axis represent the values of bundles T1 and T2, respectively. Each point represents on offer type (defined by 4 juice quantities) and the color represents the fraction of trials in which the animal chose bundle T2 (see color bar). The dotted line is the indifference line.

To model choices between bundles, it is most convenient to use a value difference model. Thus the logistic can be written as follows:

$$P_R=\frac{e^{\eta V_R}}{{\sum }_S{e}^{\eta V_S}}$$ (28)

$$V_{T1}={\rho }_A{q}_A+{\rho }_B{q}_B$$ (29)

$$V_{T2}={\rho }_C{q}_C+{\rho }_D{q}_D$$ (30)

where R and S = T1 or T2. As in other cases discussed above, the model in Eqs.28–29 has one parameter too many (because η multiplies all the ρ_J). We may thus choose to measure values in units of juice D by setting ρ_D = 1. We thus write the probability of choosing bundle T2 as follows:

$$P_{T2}=\frac{1}{1+e^{-X}}$$

$$X=a_1{q}_A+a_2{q}_B+a_3{q}_C+a_4{q}_D$$ (31)

where defined a₁ = −ηρ_A, a₂ = −ηρ_B, a₃ = ηρ_C, and a₄ = η. The relative values ρ_A, ρ_B and ρ_C and the inverse temperature η can be computed after the logistic regression from the fitted parameters. For the session illustrated in Fig.6, this analysis provided the estimates ρ_A = 2.3, ρ_B = 1.4, and ρ_C = 2.0.

Non-linear value functions

Throughout this primer, we always assumed that the value of a quantity q_J of juice J was proportional to q_J (Eq.3; linear value function). In Eqs.29–30 we have also assumed that the value of a bundle equals the sum of the components’ values (additivity). In many situations, however, value functions might be non-linear and/or non-additive. For example, a common intuition is that the subjective value of an additional $100 is higher in times of poverty and lower in times of prosperity (sub-linear value function, or diminishing marginal returns). Along similar lines, the subjective value of bread and cheese together is intuitively different from the value sum of bread alone and cheese alone. These phenomena can be examined using the same logistic model as in Eq.26 while including quadratic and/or interaction terms in the value functions:

$$V_{T1}={\alpha }_0{q}_A+{\alpha }_1{q}_B+{\alpha }_2{q}_A^2+{\alpha }_3{q}_B^2+{\alpha }_4{q}_A{q}_B$$ (32)

$$V_{T2}={\beta }_0{q}_C+{\beta }_1{q}_D+{\beta }_2{q}_C^2+{\beta }_3{q}_D^2+{\beta }_4{q}_C{q}_D$$ (33)

In essence, the quadratic terms α₂, α₃, β₂ and β₃ capture the non-linearities, while the interaction terms α₄ and β₄ capture the non-additivities. The logistic regression then proceeds as for linear value functions. For the session illustrated in Fig.6, this analysis indicated that all quadratic terms were <0 (sub-linearity). Furthermore, both interaction terms were negative (subadditivity). This was generally true across sessions (Xie and Padoa-Schioppa), and other studies yielded similar results (Pastor-Bernier et al., 2019).

The last observation might be viewed as a challenge to the linearity assumption made in the previous sections. To mitigate this concern, referring to the first section, we note that in the log value model non-linear value functions (Eq.3) are naturally expressed by elevating the product ρ_J q_J to some exponent κ_J. If exponents κ_A and κ_B do not differ from each other, both of them are effectively absorbed in the inverse temperature η (Eq.2). Thus, in reality, the value ratio models discussed in previous sections only assume that all value functions have the same nonlinearity (i.e., κ_A = κ_B). This assumption is substantially weaker than that of linear value functions. For the experiment illustrated in Fig.1, it essentially amounts to assuming that the set of quantities [q_A, q_B] that make the animal indifferent between the two juices form a straight line. For this experiment, the assumption seems reasonable (Fig.1). Supporting this point, for the experiment in Fig.6, the quadratic terms α₂, α₃, β₂ and β₃, while all <0, did not differ significantly from each other.

Including neuronal activity in logistic regressions

A number of studies have found that neurons in several brain regions encode the values of offered and/or chosen goods (Wallis and Kennerley, 2010; Schultz, 2015). This fact validates the classic idea that choices rely on the computation and comparison of subjective values. However, values can guide a variety of cognitive processes including perceptual attention, associative learning, emotion and motor control. Hence, that a neuronal population represents values does not necessarily imply that the population participates in a choice process. Furthermore, a long-term goal for decision neuroscience is to shed light on the circuit and neural mechanisms underlying economic decisions. An important step in that direction is to assess whether the activity of a particular neuron (or group of neurons) can contribute to explaining choices above and beyond what is expected based on behavioral variables alone. Consider for example a monkey choosing between juices A and B and a neuron whose activity we denote by r. As discussed above, a logistic analysis provides a function linking, for example, the probability of choosing juice B, P_B, to the offered quantities q_A and q_B. The question is whether a better account of choices may be obtained using some function that also includes r. We write such function as

$$P_B\left(q_A,q_B,r\right)$$ (34)

In setting up this problem, one must consider two issues.

First, for any choice task, some population of neurons encodes the actual choice outcome. For example, if the task requires the monkey to indicate its choice with a leftward or rightward saccade, neurons encoding the direction of impending saccades will be excellent predictors of choice. However, calculating P_B from their activity would be trivial and not provide any insight into the decision process. Thus one should ensure that the activity r included in Eq.34 precedes the decision, either because it takes place before a decision can be made or because it is from neurons that encode some pre-decision variable.

Second, for any choice task, there are typically neurons encoding the values of individual offers. In particular, when monkeys choose between juices A and B (Fig.7A), two groups of cells in the orbitofrontal cortex (OFC) encode the two offer values. Their activity is linearly related to quantities q_A and q_B. Thus on any given trial, the activity of a cell encoding the offer value B may be written as

$$r_B=c_0+c_1{q}_B+\delta r_B$$ (35)

where c₀ and c₁ are the parameters of the linear relation and δr_B is the trial-by-trial stochastic variability in the cell activity, also referred to as neuronal noise. An equivalent expression may be written for cells encoding the offer value A. An interesting exercise consists in predicting choices based only on firing rates r_A and r_B. If Eq.35 is accurate and δr_B is not too large, one can calculate P_B with relatively good accuracy (Knutson et al., 2007; Webb et al., 2019). However, the goal pursued in this section is more ambitious, as we want to assess whether measures of neuronal activity can contribute to predicting the choice made on any given trial above and beyond what would be predicted based on the offers alone. Thus for the neuron in Eq.35, the question effectively is whether δr_B can contribute to predicting choices.

Figure 7.

Including neural activity in logistic analyses. A. Experimental design. B. Offer value cells, activity profiles. The analysis focused on offer types where choices were split between the two juices. For each offer type, trials were divided depending on the animal’s choice (E chosen; O chosen) and the activity was averaged separately for the two groups of trials. The resulting traces were averaged across offer types for each cell and then across cells. Neurons with positive and negative encoding were pooled, inverting the E and O for cells with negative encoding. The time window of interest (500 ms following the offer) is highlighted. C. ROC analysis. For each cell, the difference between the light blue and light red traces was quantified with an ROC analysis, which provided a choice probability (CP). The histogram illustrates the distribution of CPs obtained across the population. D. Logistic analysis of residuals. For each cell, we conducted a logistic analysis of the residuals (Eq.36). For each neuron, the neuronal activity explained choices with a marginal power (or neuronal bias) equal to $\mu =\widehat{a_2}/\widehat{a_1}$. The histogram illustrates the distribution of μ measured across the population. E. Chosen juice cells, activity profiles. The analysis included all trials. Offer types were divided depending on the animal’s choice (E chosen; O chosen) and on whether the animal consistently chose the same juice (easy) or split choices between juices (split). For each group of trials, the activity profiles were averaged across trials for each cell and then across cells. The time window of interest (500 ms preceding the offer) is highlighted. F. ROC analysis. For each cell, the difference between the light blue (E chosen, split) and light red (O chosen, split) traces was quantified with an ROC analysis. The histogram illustrates the distribution of CPs obtained across the population. G. logistic analysis of residuals. For each cell, we conducted a logistic analysis of the residuals (Eq.37). The neuronal bias was equal to $v=\widehat{a_2}/\widehat{a_1}$. The histogram illustrates the distribution of ν measured across the population. Panels B-E were adapted from Padoa-Schioppa (2013).

When values are close to indifference, choices are split between the two offers (choice variability). The standard way to quantify the relation between neuronal variability and choice variability is based on an ROC analysis, which provides a choice probability (Britten et al., 1992). However, the question of interest here may be addressed more precisely using logistic regressions. This approach was developed to analyze neuronal activity in the lateral intraparietal area during a probabilistic inference task (Yang and Shadlen, 2007). More recently, it was used to examine neuronal activity in OFC during economic choices (Padoa-Schioppa, 2013; Masset et al., 2020). We will discuss two example analyses.

Consider the experiment in Fig.7. Neuronal recordings in monkeys performing this task have identified in the OFC different groups of neurons encoding individual offer values (offer value A or offer value B), the binary choice outcome (chosen juice) and the chosen value (Padoa-Schioppa and Assad, 2006). The first question focuses on offer value cells. Their activity recorded in a 0.5 s time window starting with the offer onset is described by Eq.35. Each neuron encodes the value of one juice; we refer to the encoded juice as juice E and to the other juice as juice O. To assess whether the activity of these cells can significantly contribute to predicting choices, we construct the following logistic model:

$$P_E=\frac{1}{1+e^{-X}}$$

$$X=a_0+a_1\text{ log}\left(q_E/q_O\right)+a_2\delta r_E$$ (36)

where δr_E is the residual activity remaining after the firing rate is regressed against q_E. After the logistic regression, one can compute $\mu =\widehat{a_2}/\widehat{a_1}$, which we might term neuronal bias. This analysis can be repeated for each offer value cell in the data set, and one can compute the distribution for the neuronal bias μ across the population. For the experiment in Fig.7, an analysis of 229 offer value cells indicated mean(μ) = 0.001 (p=0.12, t-test; Fig.7C). Notably, this measure is in the expected direction, but rather small. In this respect, it is worth noting that, similar to choice probabilities obtained from ROC analyses, neuronal biases are closely related to noise correlations. That is, if the relevant population is sufficient large, one should expect neuronal biases to be close to zero unless trial-to-trial stochastic fluctuations of different cells are correlated (Haefner et al., 2013). Interestingly, noise correlations in OFC are substantially smaller than in many other brain regions, which explains small neuronal biases (Conen and Padoa-Schioppa, 2015).

The second analysis focuses on chosen juice cells. Several sources of evidence suggest that different groups of neurons identified in OFC constitute the building blocks of a neural circuit in which decisions are formed. Moreover, this circuit may be modeled as a neural network and described by a dynamic system (Rustichini and Padoa-Schioppa, 2015). In this view, chosen juice cells represent the output layer of the network. The activity of these neurons at the end of the decision process is binary – high when the animal chooses one juice and low when it chooses the other juice. We refer to the encoded juice (eliciting high activity) as juice E and to the other juice as juice O. The analysis focuses on the activity of these neurons recorded in a 0.5 s time window before the offers appear on the monitor. This pre-offer activity captures the initial conditions of the dynamic system, which can substantially influence the network dynamics. Preliminary observations indicated that this pre-offer activity correlates with the eventual choice outcome. To quantify this effect we construct the following logistic model:

$$P_E=\frac{1}{1+e^{-X}}$$

$$X=a_0+a_1\text{log}\left(q_E/q_0\right)+a_2{r}_E$$ (37)

where r_E is the pre-offer activity of chosen juice cells. After the logistic regression, one can compute $v=\widehat{a_2}/\widehat{a_1}$, which we might term initial bias. This analysis may be repeated for each chosen juice cell in the data set, and one can compute distribution for the initial bias ν. For the experiment in Fig.7, an analysis of 257 chosen juice cells indicated that mean(ν) was significantly above zero (Fig.7C) (Padoa-Schioppa, 2013).

Probit model and the interpretation of choice variability

We now return to the case of simple choices between goods that vary in quantity (Fig.1). The analysis discussed in the first section and summarized in Eqs.4–5 is referred to as logistic or “logit”. An alternative approach is to model the sigmoid with a different function, specifically with the cumulative function of the standard normal distribution. To do so, we replace Eqs.4–5 with:

$$P_B=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^X{e}^{-t^2/2}dt$$ (38)

$$X=a_0+a_1\text{ log}\left(q_B/q_A\right)$$ (39)

Eqs.38–39 are referred to as a “probit” model. As for the logit model, parameters a₀ and a₁ are fitted using maximum likelihood (Eq.9). Several neuroscience studies have used probit models (Padoa-Schioppa and Assad, 2006; Ballesta et al., 2020). In practice, the numerical values obtained for $\widehat{a_0}$ and $\widehat{a_1}$ with the logit and with the probit models are very similar. For example, for the session in Fig.1, the two analyses yielded ρ_logit = 2.42 and ρ_probit = 2.40. Thus the two models can often be used interchangeably.

The main advantage of the probit model is that it provides an alternative interpretation for choice variability. Referring to Fig.1, the phrase “choice variability” essentially captures the fact that the sigmoid is not infinitely steep – when two offers have unequal but similar value, monkeys split their choices. In the probit model, the sigmoid is a cumulative function, and the underlying normal distribution may be interpreted as a probability distribution for the log relative value. Thus the probit model affords two interpretations for choice variability:

1. The relative value of the two juices is fixed. Choice variability reflects noise in the process of value comparison

2. The relative value of the two juices is stochastic. Choice variability reflects trial-to-trial fluctuations in the relative value

Interpretation (a) is that implicitly discussed so far, and it is the most natural for the logit model. Using the probit, the definition of relative value remains unchanged $\left(\rho =e^{-\widehat{a_0}/\widehat{a_1}}\right)$ and the fact that ρ equals the quantity ratio q_B/q_A that makes the animal indifferent between the two juices remains true. Similarly, the definition of sigmoid steepness $\eta =\widehat{a_1}$ is unchanged, and the fact that η is proportional to the slope of the sigmoid at the indifference point remains true.

Interpretation (b) is conceptually quite different. In this view, the process of value comparison (i.e., the decision per se) is noiseless and may be described by a step function. In other words, whenever the quantity ratio q_B/q_A is higher than the relative value ρ, the animal chooses juice B faultlessly. However, the relative value ρ is not fixed – it varies stochastically from trial to trial (Fig.8A). More precisely, according to (b), the log relative value is a stochastic variable with a normal distribution that has $\text{mean}(\text{log }\rho )=-\widehat{a_0}/\widehat{a_1}$ and $\text{std}(\text{log }\rho )=1/\widehat{a_1}$. In formulas:

$$D(\text{log }\rho )=N\left(-\widehat{a_0}/\widehat{a_1},1/\widehat{a_1}\right)$$ (40)

A mathematical proof of Eq.40 is provided in the Supplementary Material.

Figure 8.

Hypothetical choice patterns. A. Probit analysis and stochastic values. The black sigmoid is the choice pattern fitted with a probit function – that is, the probability that the animal choose juice B as a function of the log quantity ratio. The underlying Gaussian, shown in light red, can be interpreted as a probability distribution for the log relative value. B. Choice pattern with a double step. The choice pattern (black) has a double step. The underlying probability distribution function (light red) is bimodal. Such choice pattern suggests the presence of a systematic choice bias, similar to that revealed in Fig.4. B. Choice pattern with lapses. This choice pattern has a sigmoidal shape but it does not saturate. That is, choices are split between the two options even when one of the offers is strongly dominant. In the cartoon, we set λ = 0.3. The dotted line indicates P_B = 1. D. Non-monotonic choice pattern. This pattern cannot be interpreted as the cumulative function of a probability distribution.

Intuitively, both interpretations seem to capture something real. On the one hand, the subjective value of any good may well vary on a short time scale due to noise in value computation, recent history, or some other reason (b). On the other hand, like any mental function, the comparison of values is presumably a noisy process (a). In other words, multiple factors likely contribute to choice variability. That said, the economics literature provides an important lesson: teasing apart different contributions to choice variability is very difficult based on choices alone. In other words, both (a) and (b) are valid interpretations and one cannot adjudicate between them based on choices. However, one might be able to disambiguate between (a) and (b) – and, more generally, to identify different sources of choice variability – based on neuronal measures. In fact, doing so is arguably one of the ultimate goals of research in decision neuroscience.

Limits and failures of logistic analysis

As described in previous sections, logistic analysis can effectively model choices under a broad range of experimental protocols, and quantify a variety of behavioral effects. However, not all choices can be described using a logistic analysis. This section discusses a few examples.

For choices between two goods, logistic analysis is valid only when the choice pattern is reasonably well captured by some sigmoidal function. Fig.8B illustrates with a cartoon a situation where this is not the case. In this hypothetical session, a monkey chooses between two juices offered in variable amounts (Fig.1A). The choice pattern has a double step – for a sizeable range of quantity ratios, the animal splits its choices evenly between the two juices. If the distance between the two steps is large compared to the width of each step, any sigmoidal fit will miss an important aspect of this choice pattern. In practice, Fig.8B suggests the presence of some bias that could potentially be modeled with a logit. Indeed, the double step function may be thought of as the average of two independent sigmoid functions. If one can identify some factor that determines whether a particular trial contributes to one sigmoid function or the other, one may add that factor to the logistic regression and thus account for the whole data set. We have already encountered such a situation in Fig.4, where choices were described by two sigmoids. In that case, the factor separating trials was the order in which the two juices were offered, and the distance between the two sigmoids was thus interpreted as an order bias. In other cases, the relevant factor may be the spatial configuration (spatial bias), the outcome of the previous trial (choice hysteresis), etc.

Another interesting case is depicted in Fig.8C. Here the choice pattern is well described by a sigmoid, but choices do not saturate. In other words, even when the two values are very different, the animal chooses the lesser option in some fraction of trials. In perceptual decisions, such choices are termed “lapses” (Busse et al., 2011; Carandini and Churchland, 2013; Pisupati et al., 2021). In general, lapses can be considered errors, but they may also reflect a tendency towards exploration (Pisupati et al., 2021; Wilson et al., 2021). If lapses are present and symmetric (i.e., if they occur equally on the two ends of the sigmoid), choices may be modeled using the following sigmoid function in lieu of the logit:

$$P_B=\frac{1}{2}+\frac{1-a_2}{\pi }\text{ atan}\left(a_0+a_1\text{ log}\left(q_B/q_A\right)\right)$$ (41)

where atan indicates the inverse tangent and π = 3.141… Notably, while log(q_B/q_A) varies in the range (−∞, ∞), P_B varies in the range (a₂/2, 1 − a₂/2). Parameters a₀, a₁ and a₂ can be fitted using maximum likelihood (Eq.9). One can thus compute the parameters characterizing choices, namely the relative value $\rho =e^{-\widehat{a_0}/\widehat{a_1}}$, the inverse temperature $\eta =\widehat{a_1}$, and the lapse rate $\lambda =\widehat{a_2}$. In this formulation, the lapse rate can be interpreted as the fraction of trials in which the animal chooses randomly between the two offers. As usual, ρ is the quantity ratio q_B/q_A that makes the animal indifferent between the two offers. Importantly, here both the inverse temperature and the (negative) lapse rate contribute to choice accuracy. This point is intuitive (if there are fewer lapses, choices are more accurate); it may also be confirmed by computing the sigmoid slope at the indifference point:

$${\frac{d{P}_B}{d\left(\text{log}\left(q_B/q_A\right)\right)}|}_{X=0}=\frac{1}{\pi }(1-\lambda )\eta$$ (42)

Fig.8C illustrates a more dramatic situation, where the choice pattern is non monotonic. To picture how such a choice pattern might occur, imagine a judicious child who is given a $1 coin by her parents to go and buy gummy bears. Once at the candy store, the child talks with the vendor. In scenario (1), the vendor says that for $1 he can offer only 1 gummy bear; recognizing the bad deal, the child keeps her money and walks away. In scenario (2), the vendor offers 5 gummy bears; after some hesitation, the child decides to flip her coin and to choose accordingly – in other words, she will take the gummy bears with probability p=0.5. In scenario (3), the vendor offers 10 gummy bears; in this case, the child takes the bears without hesitation. Finally, in scenario (4), the vendor offers 100 gummy bears; that offer seems overwhelming and not quite right, and the child chooses to keep the money and walks back to her parents. Those of us with kids may find the story somewhat unlikely, but not completely implausible. Most relevant here, it would be impossible to model the child’s choices with a logistic analysis.

Lastly, logistic analysis can sometimes fail when subjects choose between multiple options. For example, consider the case of trinary choices. The logistic model of Eq.23 assumes that choices are “independent of irrelevant alternatives” (IIA) (Luce, 1959; McFadden, 1974). In other words, given two offers A and B, the probability ratio P_A/P_B should not depend on the presence or value of a third option C. If the value of C is low, the sum of P_A + P_B will be close to 1. If the value of C is larger, both P_A and P_B will be smaller, but the ratio P_A/P_B should not change. To confirm that IIA holds in Eq.23, note that the ratio P_A/P_B depends on V_A and V_B, but not on V_C. Conversely, if IIA is violated, it would be impossible to write a logistic model such as that in Eq.23, where each V_J only depends on the properties of offer J. Intuitively, assuming that choices are IIA seems to make sense. However, a number of studies have shown that human choices violate IIA in various ways (Tversky and Simonson, 1993; Soltani et al., 2012; Spektor et al., 2021). These phenomena, often referred to as “decoy effects”, cannot be easily modeled with logistic analysis.

Conclusions

Goods available for choice can vary on a number of dimensions, including quality (e.g., the juice type), quantity, probability, delivery time, cost, etc. Furthermore, choices may involve three or more options, offers may be presented simultaneously or in sequence, and options may be formed by bundles of different goods. Each of these dimensions affects choices in ways that can be quantified through logistic analysis, provided that certain conditions are met. Logistic regressions can also quantify correlations between neuronal activity and choices, providing a valid alternative to ROC analyses. Importantly, different logistic models make different assumptions and provide alternative interpretations for the origins of choice variability. For all these reasons, logistic analysis has become and will continue to be a fundamental tool in decision neuroscience.

Logistic regressions are the primary method to analyze choice behavior in decision neuroscience studies. In this primer, Padoa-Schioppa describes how logistic analysis can quantify a variety of behavioral traits. He also discusses the assumptions and interpretations of different logistic models.