Imprecise Probabilities for Sunday Weather and Boeing Stock Task
Source:R/gkwreg-datasets.R
ImpreciseTask.RdData from a cognitive psychology experiment where participants estimated upper and lower probabilities for events to occur and not to occur. The study examines judgment under uncertainty with imprecise probability assessments.
Format
A data frame with 242 observations on 3 variables:
- task
factor with levels
Boeing stockandSunday weather. Indicates which task the participant performed.- location
numeric. Average of the lower estimate for the event not to occur and the upper estimate for the event to occur (proportion).
- difference
numeric. Difference between upper and lower probability estimates, measuring imprecision or uncertainty.
Details
All participants in the study were either first- or second-year undergraduate students in psychology at Australian universities, none of whom had a strong background in probability theory or were familiar with imprecise probability theories.
For the Sunday weather task, participants were asked to estimate the probability that the temperature at Canberra airport on Sunday would be higher than a specified value.
For the Boeing stock task, participants were asked to estimate the probability that Boeing's stock would rise more than those in a list of 30 companies.
For each task, participants were asked to provide lower and upper estimates for the event to occur and not to occur.
References
Smithson, M., Merkle, E.C., and Verkuilen, J. (2011). Beta Regression Finite Mixture Models of Polarization and Priming. Journal of Educational and Behavioral Statistics, 36(6), 804–831. doi:10.3102/1076998610396893
Smithson, M., and Segale, C. (2009). Partition Priming in Judgments of Imprecise Probabilities. Journal of Statistical Theory and Practice, 3(1), 169–181.
Examples
# \donttest{
require(gkwreg)
require(gkwdist)
data(ImpreciseTask)
# Example 1: Basic model with task effects
# Probability location varies by task type and uncertainty level
fit_kw <- gkwreg(location ~ task * difference,
data = ImpreciseTask,
family = "kw"
)
summary(fit_kw)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = location ~ task * difference, data = ImpreciseTask,
#> family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.3556 -0.0788 0.0274 0.0019 0.0944 0.3499
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 1.25134 0.11593 10.794 <2e-16 ***
#> alpha:taskSunday weather -0.09543 0.11108 -0.859 0.3903
#> alpha:difference 0.28623 0.21709 1.318 0.1874
#> alpha:taskSunday weather:difference 0.43791 0.25964 1.687 0.0917 .
#> beta:(Intercept) 2.74022 0.15340 17.864 <2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 1.0241 1.4786
#> alpha:taskSunday weather -0.3131 0.1223
#> alpha:difference -0.1393 0.7117
#> alpha:taskSunday weather:difference -0.0710 0.9468
#> beta:(Intercept) 2.4396 3.0409
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 3.805
#> beta: 15.47
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 242
#> Number of parameters: 5
#> Residual degrees of freedom: 237
#> Log-likelihood: 162.2
#> AIC: -314.3
#> BIC: -296.9
#> RMSE: 0.1198
#> Efron's R2: 0.101
#> Mean Absolute Error: 0.0981
#>
#> Convergence status: Successful
#> Iterations: 21
#>
# Interpretation:
# - Alpha: Task type and uncertainty (difference) interact to affect
# probability estimates
# - Different tasks may have different baseline probability assessments
# Example 2: Heteroscedastic model
# Precision of estimates may vary by task and uncertainty
fit_kw_hetero <- gkwreg(
location ~ task * difference |
task + difference,
data = ImpreciseTask,
family = "kw"
)
summary(fit_kw_hetero)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = location ~ task * difference | task + difference,
#> data = ImpreciseTask, family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.3607 -0.0787 0.0333 0.0010 0.0907 0.3574
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 1.7403 0.1329 13.092 < 2e-16 ***
#> alpha:taskSunday weather -0.4913 0.1365 -3.598 0.000320 ***
#> alpha:difference -0.1068 0.2145 -0.498 0.618539
#> alpha:taskSunday weather:difference 0.3001 0.2410 1.245 0.213134
#> beta:(Intercept) 4.4688 0.4707 9.493 < 2e-16 ***
#> beta:taskSunday weather -1.4832 0.4354 -3.407 0.000658 ***
#> beta:difference -1.3788 0.5970 -2.309 0.020925 *
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 1.4798 2.0009
#> alpha:taskSunday weather -0.7589 -0.2237
#> alpha:difference -0.5272 0.3136
#> alpha:taskSunday weather:difference -0.1723 0.7725
#> beta:(Intercept) 3.5462 5.3914
#> beta:taskSunday weather -2.3366 -0.6298
#> beta:difference -2.5489 -0.2086
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 4.164
#> beta: 25.37
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 242
#> Number of parameters: 7
#> Residual degrees of freedom: 235
#> Log-likelihood: 170.6
#> AIC: -327.3
#> BIC: -302.9
#> RMSE: 0.1197
#> Efron's R2: 0.102
#> Mean Absolute Error: 0.09725
#>
#> Convergence status: Successful
#> Iterations: 26
#>
# Interpretation:
# - Beta: Variability in estimates differs between tasks
# Higher uncertainty (difference) may lead to less precise estimates
# Example 3: McDonald distribution for extreme uncertainty
# Some participants may show very extreme probability assessments
fit_mc <- gkwreg(
location ~ task * difference | # gamma: full interaction
task * difference | # delta: full interaction
task, # lambda: task affects extremity
data = ImpreciseTask,
family = "mc",
control = gkw_control(
method = "BFGS",
maxit = 1500
)
)
#> using C++ compiler: ‘g++ (Ubuntu 13.3.0-6ubuntu2~24.04) 13.3.0’
#> Warning: NaNs produced
summary(fit_mc)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: mc
#>
#> Call:
#> gkwreg(formula = location ~ task * difference | task * difference |
#> task, data = ImpreciseTask, family = "mc", control = gkw_control(method = "BFGS",
#> maxit = 1500))
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.2631 0.1058 0.2147 0.1669 0.2255 0.6379
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> gamma:(Intercept) 0.12000 NaN NaN NaN
#> gamma:taskSunday weather 0.08321 0.37727 0.221 0.825
#> gamma:difference 0.03513 NaN NaN NaN
#> gamma:taskSunday weather:difference 0.01876 NaN NaN NaN
#> delta:(Intercept) -0.08309 NaN NaN NaN
#> delta:taskSunday weather -0.05747 0.58059 -0.099 0.921
#> delta:difference -0.02637 NaN NaN NaN
#> delta:taskSunday weather:difference -0.01482 1.06142 -0.014 0.989
#> lambda:(Intercept) 0.25123 NaN NaN NaN
#> lambda:taskSunday weather 0.17376 NaN NaN NaN
#>
#> Confidence intervals (95%):
#> 3% 98%
#> gamma:(Intercept) NaN NaN
#> gamma:taskSunday weather -0.6562 0.8226
#> gamma:difference NaN NaN
#> gamma:taskSunday weather:difference NaN NaN
#> delta:(Intercept) NaN NaN
#> delta:taskSunday weather -1.1954 1.0805
#> delta:difference NaN NaN
#> delta:taskSunday weather:difference -2.0952 2.0655
#> lambda:(Intercept) NaN NaN
#> lambda:taskSunday weather NaN NaN
#>
#> Link functions:
#> gamma: log
#> delta: logit
#> lambda: log
#>
#> Fitted parameter means:
#> alpha: 1
#> beta: 1
#> gamma: 1.212
#> delta: 4.668
#> lambda: 1.457
#>
#> Model fit statistics:
#> Number of observations: 242
#> Number of parameters: 10
#> Residual degrees of freedom: 232
#> Log-likelihood: 20.49
#> AIC: -20.97
#> BIC: 13.92
#> RMSE: 0.2123
#> Efron's R2: -1.824
#> Mean Absolute Error: 0.1885
#>
#> Convergence status: Successful
#> Iterations: 7
#>
# Interpretation:
# - Lambda varies by task: Weather vs. stock may produce
# different patterns of extreme probability assessments
# }