Experiment 1: Methods

Participants. Participants for Experiment 1 were 191 USA residents recruited

from Prolific Academic (73 females and 118 males). They were on average aged 31.6 years old. 60.2% of them had a university degree or higher. After completing the experiment, each of them received a flat payment of $1.

Intertemporal choice task. The intertemporal choice task included 81 target

items. Across items, the smaller-sooner (SS) option was held constant, which was always receiving $110 in 2 months. The target items were created by crossing nine different LL outcomes with nine different LL delays. The LL outcome could take nine

18 _{While referring to similar concepts, Stewart et al. (2003) and Vlaev, et al. (2007) use the term} “prospect relativity” and Ebert and Prelec describe this as attention-driven sensitivity to an attribute.

CHAPTER 3 ATTENTION AND INTERTEMPORAL CHOICE

64 values from $120 to $200 by an increment of $10. The LL delay could also take nine different values from 4 months to 20 months by an increment of 2 months. Nine additional screening items with a dominant option, which differed across conditions, were added to the target items to check whether participants were just responding randomly. So, there were totally 90 intertemporal choice items in each condition.

Design. A between-participant design was adopted in Experiment 1. All

participants answered the same target intertemporal choice questions, with only the pattern of presentation being different across conditions. In the outcome-focus

condition, the 81 target items were divided into nine blocks so that only one LL delay, but all LL outcomes, appeared in a block. In each block, an additional screening item was added, whose LL outcome was $100 and LL delay was the same as the LL delay of target items in the same block (e.g., $100 in 6 months), making LL dominated by SS. The screening items were chosen in order not to intrude the one-attribute-varying property within a block. To consolidate the manipulation of attention, the inter-item substitution of LL outcomes was accompanied by a smooth transition animation (1000 ms) with Odometer, a jQuery plugin in HTML (see Figure 3.2 for an illustration).19 The order of blocks and the order of LL outcomes within each block were randomized among participants.

In the delay-focus condition, items were also divided into nine blocks of nine items each. In each block, all LL delays, but only one LL outcome, appeared. An additional screening item was added to each block. The LL outcome of the screening item was the same as the corresponding LL outcome in the block and the LL delay is 2 months (e.g., $150 in 2 months), making LL dominate SS. Within each block, the LL delay varied with the same inter-item transition animation (1000 ms), while others were kept constant across items. The order of blocks and the order of items in each block were randomized across participants.

Figure 3.2. Example screenshots of the experimental procedure in Experiment 1. In the outcome-focus condition, each intertemporal choice (except the first item in each block) was preceded by a transition from a different LL outcome to the current one (1000 ms), while the environment was otherwise kept unchanged. In the delay-focus condition, each intertemporal choice (except the first item in a block) was preceded by a transition from another LL delay to the current one (1000 ms), while the environment was otherwise kept unchanged.

Procedures. Participants were randomly assigned to one of the two conditions.

To further control the effect of the relative position of the two options, the vertical placement of the SS and the LL options were counterbalanced among participants. For half of the participants from each condition, SS was placed above LL; for the other half, SS was placed below LL. After the intertemporal choice task, participants completed the Cognitive Reflection Test (CRT; Frederick, 2005) and demographic questions including gender, age, ethnical identity, annual income, educational qualifications and employment status. Being loosely related to the core focus of the chapter, results related to CRT were reported in Appendix 3C.

CHAPTER 3 ATTENTION AND INTERTEMPORAL CHOICE

66

Data analysis. Data analysis was carried out at the aggregate level. Binary data

points from individual participants were aggregated into binomial data on each item in each condition. Mixed-effect models were applied to the aggregate data with a random intercept for each intertemporal choice item to capture the variation of the relative attractiveness of LL to SS across items.20 _{Statistical model fitting consisted of} two steps. First, the probability of choosing LL was predicted by mixed-effect models with a logit link function. Specifically, two models were used:

(1)pˆ_i logit-1 _j[i] β₁x_iAttention and

(2)pˆ_i logit-1 _j[i] β₁x_iAttention ₂x_iBackgroundContrast ₃x_iAttentionx_iBackgroundContrast , where j[i] (j = 1, …, 81) was the random intercept for each item. The random intercept was used to capture the relative attractiveness between SS and LL for each item. Model (1) had only attention, an effect-coded variable (-0.5 = delay-focus; 0.5 = outcome-focus), as the fixed-effect predictor while Model (2) involved both attention and background contrast. Second, the predicted probability of choosing LL, pˆ_i, was fitted to data with the binomial distribution ki ~Bin(Ni,pˆi), where kiwas frequency of

LL choices and N_i was the corresponding sample size. Results of the two models are

shown in Table 3.1.

Bayesian modelling was used for parameter estimation and model evaluation (see Kruschke, 2010). To be conservative, the prior distribution used for all parameters was the standard normal distribution, rather than the uniform distribution. Estimation of parameters was summarized by the median (Md) and 95% high density interval (95% HDI) of posterior distributions. Model evaluation and comparisons were based on the Deviance Information Criterion (DIC; Gelman, Carlin, Stern, Dunson, Vehtari, & Rubin, 2014a; Spiegelhalter, Best, Carlin, & Van Der Linde, 2002): The lower the DIC value, the better the model. The descriptive accuracy of some models was further checked with posterior predictive check (Gelman, Carlin, Stern, Dunson, Vehtari, & Rubin, 2014b).

Bayesian modelling and Deviance Information Criterion were approximated by Markov chain Monte Carlo (MCMC) in JAGS (Plummer, 2003), which was called

20 _{We did not involve the formal intertemporal choice models (as in Chapter 2) to capture the variation}

across items because it is highly controversial which model is descriptively accurate (see also Cavagnaro et al., 2016a; Dai & Busemeyer, 2014; Read, 2001; Scholten et al., 2014).

by the runjags package (Denwood, in press) in R (R Core Team, 2016). Three independent Markov chains were run for each model. Each chain consisted of 10,000 burn-in samples, which served as the initial adaption period and were discarded from formal approximation, and 50,000 formal samples. Convergence among the three chains was checked with the Gelman-Rubin statistic (Gelman &Rubin, 1992). The three chains summed up to 150,000 samples from the posterior distribution.