Risky Choice and CPT - Statistics Assignment Help

Assignment Task

Assignment: Risky Choice and CPT

The Data
You will be working with a dataset (DATA_Study2_Rieskamp_2008.xls) from a paper by Rieskamp (2008). The dataset contains choices between pairs of gambles. The sheet gambles summarises all unique choice pairs. The key is provided but most gamble pairs consists of lotteries with two outcomes.

The Modelling
Your objective will be to fit three different versions of Cumulative Prospect Theory (see Stott, 2006; Tversky & Kahneman, 1992) to the data. The stochastic choice rule should be the logit rule. All fitting must be done on the individual level. There are 30 participants in total (columns in sheet choices). You may still aggregate your model fitting results (e.g., model parameters, parameter recovery) to answer some of the questions below.
Version 1: Fit the model with the α parameter for the curvature of the value function for both gains and losses. In addition include parameter λ for loss aversion:

Version 2: Include an additional parameter β to capture the curvature of the value function for losses (in addition to α for gains). This model should also include λ for loss aversion:

Version 3: In addition to the value function in Equation 2, you should also include the CPT probability weighting function as described in Tversky and Kahneman (1992). This function should not be used for cumulative probabilities/decision weights (i.e., you should apply it to the single event probabilities):


Finally, the subjective valuation of a gamble A with outcomes x_m > ... > x₁ ≥ 0 > y₁ > ... > y_n with corresponding probabilities p_m...p₁ and q₁...q_n is:

The logit function defining the probability of a gamble A being selected (compared to a gamble B) is:

The Task
Across your modelling attempts, you must illustrate the following tasks:
1. Estimate the parameter values for each model (i.e., model fitting). Summarise the results and comment on the findings. Do you find parameter values according to the literature (e.g., λ > 2)? What do the results suggest about participants’ risk preferences? Make sure that the model fits are not affected by local minima.
2. Model Comparison: Compare the 3 different models using appropriate methods and comment on the results. You should be able to answer questions such as “Is the subjective transformation of probabilities via the CPT probability weighting function a necessary component to explain risky choice?” and “Do we need separate parameters (i.e., α and β) to capture the subjective transformation of gains and losses?". You should compare the model using the AIC and the likelihood ratio test (LHR), if models are nested. Do the two methods give you the same results?
3. Make scatter plots for each pairing of parameters, with dots representing each participant. You will see that some parameters are correlated over participants. For example, participants with low α have high bias/temperature parameter (in the logit rule), and vice versa. Why is this? What does it tell us about the psychological processes underlying risky choice?
4. Simulate data based on the initial model fits for each model (from a single set of starting parameter values, i.e., the best-fitting parameters for each model). Then, re-fit all three models to the simulated/generated datasets to assess whether the generating parameters can be recovered. Describe parameter recoverability across all three models. If recoverability is poor, explain why this might be the case.

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