Propranolol

We modeled participants’ behavior using maximum likelihood estimation as implemented in MATLAB (Mathworks, Natick, MA) to fit a prospect-theory-inspired model (Tversky & Kahneman, 1992 (link)) to choices, identical to that used in previous studies (Sokol-Hessner, et al., 2013 (link); Sokol-Hessner, et al., 2014 (link); Sokol-Hessner, et al., 2009 (link)).
$$u\left(x^{+}\right)=p\times {\left(x^{+}\right)}^{\rho }$$
$$u\left(x^{-}\right)=-\lambda \times p\times {(-x^{-})}^{\rho }$$
$$p(\mathit{\text{gamble}})=\frac{1}{1+e^{-\mu \times (u(\mathit{\text{gamble}})-u(\mathit{guaranteed}))}}$$
Equations 1 and 2 calculate the utility of gains and losses respectively. These are used to compute the utility of the gamble and the guaranteed alternative, which are then converted into a probability of choosing the gamble using the softmax in Equation 3. The model’s parameters quantify loss aversion (λ, the relative multiplicative weight placed on losses compared to gains), risk attitudes (ρ, feelings about chance/diminishing marginal sensitivity to value), and choice consistency (μ, noisiness in choices, also called the softmax temperature). All analyses of loss aversion used log(λ); the logarithm is commonly used since λ is positively skewed.
This task has the ability to separate changes in loss aversion from those in risk attitudes by including both gain-loss and gain-only trial types. Gain-loss trials consist of a gamble with positive and negative possible outcomes, and a guaranteed alternative of zero. Choices in these trials reflect both risk attitudes (because the gamble is risky) and loss aversion (because losses are being evaluated). Gain-only trials consist of a gamble with possible outcomes of a large positive amount or zero, and a guaranteed alternative of a small positive amount. In these trials, risk attitudes affect behavior (because the gamble is still risky), but loss aversion does not (because there are no losses). When gain-loss and gain-only trials are simultaneously fit, risk attitudes are estimated that account for behavior across all trials, while loss aversion accounts for the remainder of gain-loss gambling behavior not explained by risk attitudes. In studies that only include gain-loss trial types, it is impossible to identify to what extent loss aversion or risk aversion is driving behavior because both processes are present and have similar gross effects on behavior (increasing or decreasing gambling). Both gain-loss and gain-only trial types are necessary to separately identify risk attitudes and loss aversion.
To analyze changes in choice behavior, we first regressed the change in log(λ) across days on Day (the constant), Medication (−1 placebo to propranolol; +1 propranolol to placebo), BMI group (+1 Low BMI group; −1 High BMI group), and the interaction between Medication and BMI group. We also did this exact regression using risk attitudes (ρ) and choice consistency (μ) instead of log(λ). To clarify the effect of the interaction, we subtracted the strong Day effect from the change in log(λ) leaving the residual change in loss aversion (Δλ_R) due to propranolol. The resulting values were still in “Day” space (i.e. reflecting changes from Day 1 to Day 2), so we flipped the sign of the values corresponding to individuals who received placebo on Day 1 so that all values were in a “Medication” space (reflecting propranolol-to-placebo changes). We called this Δλ_R as it reflected the residual change in loss aversion due to propranolol, and performed subsequent analyses on these residuals.
We think it likely that there is some more continuous relationship between BMI and a constant 80mg dose of propranolol. However, this relationship is certainly non-linear, characterized by both ceiling effects (in which 80mg of propranolol has a constant, maximal effect below some BMI) and floor effects (in which 80 mg of propranolol has no observable effect above a particular BMI). To avoid arbitrary assumptions of functional forms, and allow our data to shape the monotonic transformation of BMI, we used nonlinear curve fitting procedures in MATLAB (NonLinearModel.fit) to fit a model based on the following two equations:
$$\mathit{tBMI}=z\left(\frac{1}{1+e^{\alpha +\gamma \times z\left(\mathit{BMI}\right)}}\right)$$
$$\log \left({\lambda }_{\mathit{\text{Day}}2}\right)-\log \left({\lambda }_{\mathit{\text{Day}}1}\right)={\beta }_1\times \mathit{\text{Day}}+{\beta }_2\times \mathit{\text{Medication}}+{\beta }_2\times \mathit{tBMI}+{\beta }_4\times \mathit{\text{Medication}}\times \mathit{tBMI}$$ in which z() indicates the use of z-scoring. This regression is identical to that described earlier, except instead of a median split on BMI, we allowed the modified softmax function (Equation 4, parameterized by α and γ) to transform BMI into tBMI as best fit the change in loss aversion. This transformation function avoids arbitrary assumptions as it can approximate a wide variety of possible monotonic relationships including linear, sigmoidal, curvilinear, or step functions, and is thus capable of modeling both ceiling and floor effects. For analyses using BMI as a strictly linear covariate, see Supplementary Materials, though note non-linearity caveats above.