Study1_Testing_the_Optimal_Policy_by_Preference.R

############################################
# Testing the Optimal Policy by Preference #
############################################

# Import Libraries
library("tidyverse")
library('lmerTest')

# Import data
Testing_the_Optimal_Policy_by_Preference <- as_tibble(read.csv("Study1_Testing_the_Optimal_Policy_by_Preference.csv"))

# Preview data
head(Testing_the_Optimal_Policy_by_Preference)

# data dictionary:
## "Participant" = participant number
## "Function" = Function number (range 1-4; refer to payoff function figure; function 5/6 omitted for this analysis (see details in paper)) 
## "Trial" = trial number (range: 11-150; only includes trials which participant had control of policies decisions)
## "Optimal_Policy_Selected" = Notes whether optimal policy (i.e., the policy choice that yields maximum output over time; 1=yes, 0=no) 
## "Ambiguity" = Notes whether function ambiguity type (low or high)
## "Congruence" = Notes whether optimal policy choice was also participants preferred policy (1=yes; 0=no)


# Transform Predictor variables to use 'effects coding'
Testing_the_Optimal_Policy_by_Preference$Ambiguity_Coded <- ifelse(Testing_the_Optimal_Policy_by_Preference$Ambiguity=="Low", .5, -.5)
Testing_the_Optimal_Policy_by_Preference$Congruence_Coded <- ifelse(Testing_the_Optimal_Policy_by_Preference$Congruence=="1", .5, -.5)

# aggregate data by ambiguity and congruence 
Testing_the_Optimal_Policy_by_Preference_2 <- Testing_the_Optimal_Policy_by_Preference %>%
  group_by(Participant, Ambiguity_Coded, Congruence_Coded) %>%
  summarise(Optimal_Policy_Selected_Percent=mean(Optimal_Policy_Selected), 
            n=n())


# maximal model does not converge: 
# model_1 <- lmer(mean_OP_Coded ~ Congruence2*Ambiguity2 +
#                  (1+Congruence2*Ambiguity2|User),
#                data=OP_dfz_simple_Study1)
# summary(model_1)
# Error: number of observations (=125) <= number of random effects (=164) for term (1 + Congruence2 * Ambiguity2 | User); the random-effects parameters and the residual variance (or scale parameter) are probably unidentifiable

# model 
model_2 <- lmer(Optimal_Policy_Selected_Percent ~ Congruence_Coded*Ambiguity_Coded +
                 (1+Congruence_Coded*Ambiguity_Coded||Participant),
               data=Testing_the_Optimal_Policy_by_Preference_2)
summary(model_2)

# output:
# Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
# Formula: Optimal_Policy_Selected_Percent ~ Congruence_Coded * Ambiguity_Coded +  
#     (1 + Congruence_Coded * Ambiguity_Coded || Participant)
#    Data: Testing_the_Optimal_Policy_by_Preference_2
# 
# REML criterion at convergence: 67.6
# 
# Scaled residuals: 
#      Min       1Q   Median       3Q      Max 
# -1.64612 -0.48544  0.07728  0.58592  2.31899 
# 
# Random effects:
#  Groups        Name                             Variance Std.Dev.
#  Participant   (Intercept)                      0.001066 0.03264 
#  Participant.1 Congruence_Coded                 0.058547 0.24196 
#  Participant.2 Ambiguity_Coded                  0.077886 0.27908 
#  Participant.3 Congruence_Coded:Ambiguity_Coded 0.049408 0.22228 
#  Residual                                       0.056910 0.23856 
# Number of obs: 125, groups:  Participant, 41
# 
# Fixed effects:
#                                  Estimate Std. Error       df t value Pr(>|t|)    
# (Intercept)                       0.52922    0.02330 41.28756  22.709  < 2e-16 ***
# Congruence_Coded                  0.37737    0.05910 38.17932   6.386 1.65e-07 ***
# Ambiguity_Coded                   0.26471    0.06303 38.13140   4.200 0.000155 ***
# Congruence_Coded:Ambiguity_Coded -0.06686    0.09749 34.85804  -0.686 0.497404    
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Correlation of Fixed Effects:
#             (Intr) Cngr_C Ambg_C
# Congrnc_Cdd  0.055              
# Ambigty_Cdd  0.052  0.001       
# Cngrn_C:A_C -0.006  0.053  0.050