Weather Task with Priming and Precise and Imprecise Probabilities
Source:R/gkwreg-datasets.R
WeatherTask.RdData from a cognitive psychology experiment on probabilistic learning and probability judgments. Participants estimated probabilities for weather events under different priming and precision conditions.
Format
A data frame with 345 observations on 4 variables:
- agreement
numeric. Probability indicated by participants, or the average between minimum and maximum estimates in the imprecise condition. Response variable scaled to (0, 1).
- priming
factor with levels
two-fold(case prime) andseven-fold(class prime). Indicates the partition priming condition.- eliciting
factor with levels
preciseandimprecise(lower and upper limit). Indicates whether participants gave point estimates or interval estimates.
Details
All participants in the study were either first- or second-year undergraduate students in psychology, none of whom had a strong background in probability or were familiar with imprecise probability theories.
Task description: Participants were asked: "What is the probability that the temperature at Canberra airport on Sunday will be higher than 'specified temperature'?"
Experimental manipulations:
Priming: Two-fold (simple binary: above/below) vs. seven-fold (multiple temperature categories)
Eliciting: Precise (single probability estimate) vs. imprecise (lower and upper bounds)
The study examines how partition priming (number of response categories) and elicitation format affect probability judgments. Classical findings suggest that more categories (seven-fold) lead to different probability assessments than binary categories (two-fold).
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(WeatherTask)
# Example 1: Main effects model
# Probability judgments affected by priming and elicitation format
fit_kw <- gkwreg(
agreement ~ priming + eliciting,
data = WeatherTask,
family = "kw"
)
summary(fit_kw)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = agreement ~ priming + eliciting, data = WeatherTask,
#> family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.2941 -0.1053 -0.0494 -0.0063 0.0899 0.7053
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 0.39534 0.06067 6.516 7.24e-11 ***
#> alpha:primingseven-fold -0.18903 0.05241 -3.606 0.00031 ***
#> alpha:elicitingimprecise 0.21471 0.05233 4.103 4.07e-05 ***
#> beta:(Intercept) 1.81969 0.09799 18.571 < 2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 0.2764 0.5143
#> alpha:primingseven-fold -0.2918 -0.0863
#> alpha:elicitingimprecise 0.1122 0.3173
#> beta:(Intercept) 1.6276 2.0117
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 1.524
#> beta: 6.164
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 345
#> Number of parameters: 4
#> Residual degrees of freedom: 341
#> Log-likelihood: 201.6
#> AIC: -395.2
#> BIC: -379.9
#> RMSE: 0.1541
#> Efron's R2: 0.08752
#> Mean Absolute Error: 0.1253
#>
#> Convergence status: Successful
#> Iterations: 25
#>
# Interpretation:
# - Alpha: Seven-fold priming may shift probability estimates
# Imprecise elicitation may produce different mean estimates
# Example 2: Interaction model with heteroscedasticity
# Priming effects may differ by elicitation format
# Variability may also depend on conditions
fit_kw_interact <- gkwreg(
agreement ~ priming * eliciting |
priming + eliciting,
data = WeatherTask,
family = "kw"
)
summary(fit_kw_interact)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = agreement ~ priming * eliciting | priming +
#> eliciting, data = WeatherTask, family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.2930 -0.1159 -0.0320 -0.0052 0.0910 0.6891
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 0.26198 0.08991 2.914 0.00357
#> alpha:primingseven-fold 0.10566 0.10814 0.977 0.32850
#> alpha:elicitingimprecise 0.21199 0.11980 1.769 0.07681
#> alpha:primingseven-fold:elicitingimprecise 0.08181 0.10438 0.784 0.43317
#> beta:(Intercept) 1.46021 0.15558 9.385 < 2e-16
#> beta:primingseven-fold 0.85229 0.20297 4.199 2.68e-05
#> beta:elicitingimprecise 0.09174 0.19953 0.460 0.64566
#>
#> alpha:(Intercept) **
#> alpha:primingseven-fold
#> alpha:elicitingimprecise .
#> alpha:primingseven-fold:elicitingimprecise
#> beta:(Intercept) ***
#> beta:primingseven-fold ***
#> beta:elicitingimprecise
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 0.0858 0.4382
#> alpha:primingseven-fold -0.1063 0.3176
#> alpha:elicitingimprecise -0.0228 0.4468
#> alpha:primingseven-fold:elicitingimprecise -0.1228 0.2864
#> beta:(Intercept) 1.1553 1.7651
#> beta:primingseven-fold 0.4545 1.2501
#> beta:elicitingimprecise -0.2993 0.4828
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 1.566
#> beta: 7.428
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 345
#> Number of parameters: 7
#> Residual degrees of freedom: 338
#> Log-likelihood: 210.9
#> AIC: -407.8
#> BIC: -380.9
#> RMSE: 0.1537
#> Efron's R2: 0.09164
#> Mean Absolute Error: 0.124
#>
#> Convergence status: Successful
#> Iterations: 31
#>
# Interpretation:
# - Alpha: Interaction tests if partition priming works differently
# for precise vs. imprecise probability judgments
# - Beta: Precision varies by experimental condition
# Test interaction
anova(fit_kw, fit_kw_interact)
#> Analysis of Deviance Table
#>
#> Model 1: agreement ~ priming + eliciting
#> Model 2: agreement ~ priming * eliciting | priming + eliciting
#>
#> Resid. Df Resid. Dev Df Deviance Pr(>Chi)
#> fit_kw 341.00000 -403.23406
#> fit_kw_interact 338.00000 -421.81480 3 18.58074 0.00033376 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Example 3: McDonald distribution for polarized responses
# Probability judgments often show polarization (clustering at extremes)
# particularly under certain priming conditions
fit_mc <- gkwreg(
agreement ~ priming * eliciting | # gamma
priming * eliciting | # delta
priming, # lambda: priming affects polarization
data = WeatherTask,
family = "mc",
control = gkw_control(method = "BFGS", maxit = 1500)
)
summary(fit_mc)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: mc
#>
#> Call:
#> gkwreg(formula = agreement ~ priming * eliciting | priming *
#> eliciting | priming, data = WeatherTask, family = "mc", control = gkw_control(method = "BFGS",
#> maxit = 1500))
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.2027 -0.0727 -0.0416 0.0351 0.1084 0.7564
#>
#> Coefficients:
#> Estimate Std. Error z value
#> gamma:(Intercept) 0.090319 0.222111 0.407
#> gamma:primingseven-fold 0.036067 0.426063 0.085
#> gamma:elicitingimprecise 0.054185 0.522357 0.104
#> gamma:primingseven-fold:elicitingimprecise 0.023760 0.566762 0.042
#> delta:(Intercept) -0.039825 0.534925 -0.074
#> delta:primingseven-fold -0.010793 0.644386 -0.017
#> delta:elicitingimprecise -0.025605 1.096937 -0.023
#> delta:primingseven-fold:elicitingimprecise -0.008725 1.274826 -0.007
#> lambda:(Intercept) 0.165563 0.026202 6.319
#> lambda:primingseven-fold 0.062143 0.216997 0.286
#> Pr(>|z|)
#> gamma:(Intercept) 0.684
#> gamma:primingseven-fold 0.933
#> gamma:elicitingimprecise 0.917
#> gamma:primingseven-fold:elicitingimprecise 0.967
#> delta:(Intercept) 0.941
#> delta:primingseven-fold 0.987
#> delta:elicitingimprecise 0.981
#> delta:primingseven-fold:elicitingimprecise 0.995
#> lambda:(Intercept) 2.64e-10 ***
#> lambda:primingseven-fold 0.775
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> gamma:(Intercept) -0.3450 0.5256
#> gamma:primingseven-fold -0.7990 0.8711
#> gamma:elicitingimprecise -0.9696 1.0780
#> gamma:primingseven-fold:elicitingimprecise -1.0871 1.1346
#> delta:(Intercept) -1.0883 1.0086
#> delta:primingseven-fold -1.2738 1.2522
#> delta:elicitingimprecise -2.1756 2.1244
#> delta:primingseven-fold:elicitingimprecise -2.5073 2.4899
#> lambda:(Intercept) 0.1142 0.2169
#> lambda:primingseven-fold -0.3632 0.4874
#>
#> Link functions:
#> gamma: log
#> delta: logit
#> lambda: log
#>
#> Fitted parameter means:
#> alpha: 1
#> beta: 1
#> gamma: 1.151
#> delta: 4.851
#> lambda: 1.216
#>
#> Model fit statistics:
#> Number of observations: 345
#> Number of parameters: 10
#> Residual degrees of freedom: 335
#> Log-likelihood: 175.9
#> AIC: -331.8
#> BIC: -293.4
#> RMSE: 0.1669
#> Efron's R2: -0.07105
#> Mean Absolute Error: 0.1242
#>
#> Convergence status: Successful
#> Iterations: 7
#>
# Interpretation:
# - Lambda varies by priming: Seven-fold priming may produce more
# extreme/polarized probability judgments
# }