Data from a study examining factors that influence mock juror confidence in verdicts for criminal trials. The experiment manipulates verdict options (two-option vs. three-option) and presence of conflicting testimonial evidence.
Format
A data frame with 104 observations on 3 variables:
- confidence
numeric. Juror confidence in their verdict, scaled to the open unit interval (0, 1). Original scale was 0-100.
- verdict
factor indicating whether a two-option verdict (guilty vs. acquittal) or three-option verdict (with Scottish 'not proven' alternative) was requested. Sum contrast coding is employed.
- conflict
factor. Is there conflicting testimonial evidence? Values are
nooryes. Sum contrast coding is employed.
Details
The data were collected by Deady (2004) among first-year psychology students at Australian National University. The experiment examined how the availability of a third verdict option ('not proven') and conflicting evidence affect juror confidence.
Smithson and Verkuilen (2006) employed the data, scaling the original confidence
(on a scale 0-100) to the open unit interval using the transformation:
((original_confidence/100) * 103 - 0.5) / 104.
Important note: The original coding of conflict in the data provided from
Smithson's homepage is -1/1 which Smithson and Verkuilen (2006) describe to
mean no/yes. However, all their results (sample statistics, histograms, etc.)
suggest that it actually means yes/no, which was employed in the corrected
MockJurors dataset.
References
Deady, S. (2004). The Psychological Third Verdict: 'Not Proven' or 'Not Willing to Make a Decision'? Unpublished honors thesis, The Australian National University, Canberra.
Smithson, M., and Verkuilen, J. (2006). A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta-Distributed Dependent Variables. Psychological Methods, 11(1), 54–71.
Examples
# \donttest{
require(gkwreg)
require(gkwdist)
data(MockJurors)
# Example 1: Main effects model with heteroscedasticity
# Confidence depends on verdict options and conflicting evidence
# Variability may also depend on these factors
fit_kw <- gkwreg(
confidence ~ verdict + conflict |
verdict * conflict,
data = MockJurors,
family = "kw"
)
summary(fit_kw)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = confidence ~ verdict + conflict | verdict *
#> conflict, data = MockJurors, family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.6484 -0.1143 0.0515 0.0152 0.1827 0.3416
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 0.70572 0.13722 5.143 2.71e-07 ***
#> alpha:verdict -0.31540 0.15835 -1.992 0.04640 *
#> alpha:conflict 0.21806 0.15492 1.408 0.15925
#> beta:(Intercept) -0.14787 0.12674 -1.167 0.24332
#> beta:verdict -0.35402 0.13337 -2.654 0.00794 **
#> beta:conflict 0.10940 0.13351 0.819 0.41258
#> beta:verdict:conflict -0.12932 0.09992 -1.294 0.19558
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 0.4368 0.9747
#> alpha:verdict -0.6258 -0.0050
#> alpha:conflict -0.0856 0.5217
#> beta:(Intercept) -0.3963 0.1005
#> beta:verdict -0.6154 -0.0926
#> beta:conflict -0.1523 0.3711
#> beta:verdict:conflict -0.3252 0.0665
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 2.184
#> beta: 0.9347
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 104
#> Number of parameters: 7
#> Residual degrees of freedom: 97
#> Log-likelihood: 35.56
#> AIC: -57.12
#> BIC: -38.61
#> RMSE: 0.2084
#> Efron's R2: 0.0305
#> Mean Absolute Error: 0.1657
#>
#> Convergence status: Successful
#> Iterations: 18
#>
# Interpretation:
# - Alpha (mean): Additive effects of verdict type and conflict
# Three-option verdicts may reduce confidence
# Conflicting evidence reduces confidence
# - Beta (precision): Interaction suggests confidence variability
# depends on combination of verdict options and evidence type
# Example 2: Full interaction in mean model
fit_kw_interact <- gkwreg(
confidence ~ verdict * conflict |
verdict * conflict,
data = MockJurors,
family = "kw"
)
summary(fit_kw_interact)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: kw
#>
#> Call:
#> gkwreg(formula = confidence ~ verdict * conflict | verdict *
#> conflict, data = MockJurors, family = "kw")
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.6078 -0.1026 0.0131 0.0076 0.1653 0.3822
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 0.78602 0.13396 5.867 4.43e-09 ***
#> alpha:verdict -0.31171 0.13396 -2.327 0.01997 *
#> alpha:conflict 0.30523 0.13396 2.278 0.02270 *
#> alpha:verdict:conflict 0.41032 0.13396 3.063 0.00219 **
#> beta:(Intercept) -0.09854 0.12599 -0.782 0.43415
#> beta:verdict -0.34600 0.12599 -2.746 0.00603 **
#> beta:conflict 0.11003 0.12599 0.873 0.38250
#> beta:verdict:conflict 0.10701 0.12599 0.849 0.39569
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) 0.5235 1.0486
#> alpha:verdict -0.5743 -0.0491
#> alpha:conflict 0.0427 0.5678
#> alpha:verdict:conflict 0.1478 0.6729
#> beta:(Intercept) -0.3455 0.1484
#> beta:verdict -0.5929 -0.0991
#> beta:conflict -0.1369 0.3570
#> beta:verdict:conflict -0.1399 0.3539
#>
#> Link functions:
#> alpha: log
#> beta: log
#>
#> Fitted parameter means:
#> alpha: 2.549
#> beta: 0.9711
#> gamma: 1
#> delta: 0
#> lambda: 1
#>
#> Model fit statistics:
#> Number of observations: 104
#> Number of parameters: 8
#> Residual degrees of freedom: 96
#> Log-likelihood: 40.12
#> AIC: -64.24
#> BIC: -43.09
#> RMSE: 0.2078
#> Efron's R2: 0.03573
#> Mean Absolute Error: 0.1631
#>
#> Convergence status: Successful
#> Iterations: 18
#>
# Interpretation:
# - Full interaction: Third verdict option may have different effects
# depending on whether evidence is conflicting
# Test interaction significance
anova(fit_kw, fit_kw_interact)
#> Analysis of Deviance Table
#>
#> Model 1: confidence ~ verdict + conflict | verdict * conflict
#> Model 2: confidence ~ verdict * conflict | verdict * conflict
#>
#> Resid. Df Resid. Dev Df Deviance Pr(>Chi)
#> fit_kw 97.00000 -71.12225
#> fit_kw_interact 96.00000 -80.24264 1 9.12039 0.0025278 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Example 3: McDonald distribution for extreme confidence patterns
# Jurors may show very high confidence (ceiling effects) or very low
# confidence depending on conditions
fit_mc <- gkwreg(
confidence ~ verdict * conflict | # gamma: full interaction
verdict * conflict | # delta: full interaction
verdict + conflict, # lambda: additive extremity effects
data = MockJurors,
family = "mc",
control = gkw_control(
method = "BFGS",
maxit = 1500,
reltol = 1e-8
)
)
summary(fit_mc)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: mc
#>
#> Call:
#> gkwreg(formula = confidence ~ verdict * conflict | verdict *
#> conflict | verdict + conflict, data = MockJurors, family = "mc",
#> control = gkw_control(method = "BFGS", maxit = 1500, reltol = 1e-08))
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.4395 -0.0471 0.0520 0.1064 0.2500 0.7032
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> gamma:(Intercept) 3.2186 5.7452 0.560 0.5753
#> gamma:verdict 2.0199 5.5102 0.367 0.7139
#> gamma:conflict -0.7418 0.6489 -1.143 0.2530
#> gamma:verdict:conflict 0.9145 0.1266 7.224 5.04e-13 ***
#> delta:(Intercept) -3.6223 0.3943 -9.186 < 2e-16 ***
#> delta:verdict -0.8930 0.3948 -2.262 0.0237 *
#> delta:conflict 0.2426 0.3944 0.615 0.5386
#> delta:verdict:conflict 0.7301 0.3960 1.843 0.0653 .
#> lambda:(Intercept) -2.2131 5.7326 -0.386 0.6995
#> lambda:verdict -2.3000 5.5063 -0.418 0.6762
#> lambda:conflict 1.0677 0.6352 1.681 0.0928 .
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Confidence intervals (95%):
#> 3% 98%
#> gamma:(Intercept) -8.0418 14.4790
#> gamma:verdict -8.7798 12.8196
#> gamma:conflict -2.0136 0.5300
#> gamma:verdict:conflict 0.6664 1.1627
#> delta:(Intercept) -4.3952 -2.8494
#> delta:verdict -1.6667 -0.1193
#> delta:conflict -0.5305 1.0156
#> delta:verdict:conflict -0.0461 1.5063
#> lambda:(Intercept) -13.4488 9.0225
#> lambda:verdict -13.0922 8.4921
#> lambda:conflict -0.1773 2.3127
#>
#> Link functions:
#> gamma: log
#> delta: logit
#> lambda: log
#>
#> Fitted parameter means:
#> alpha: 1
#> beta: 1
#> gamma: 100.7
#> delta: 0.4194
#> lambda: 0.8955
#>
#> Model fit statistics:
#> Number of observations: 104
#> Number of parameters: 11
#> Residual degrees of freedom: 93
#> Log-likelihood: 19.98
#> AIC: -17.96
#> BIC: 11.13
#> RMSE: 0.2857
#> Efron's R2: -0.8222
#> Mean Absolute Error: 0.2204
#>
#> Convergence status: Successful
#> Iterations: 38
#>
# Interpretation:
# - Lambda: Models asymmetry and extreme confidence
# Some conditions produce more polarized confidence (very high or very low)
# Example 4: Exponentiated Kumaraswamy alternative
fit_ekw <- gkwreg(
confidence ~ verdict * conflict | # alpha
verdict + conflict | # beta
conflict, # lambda: conflict affects extremity
data = MockJurors,
family = "ekw",
control = gkw_control(
method = "BFGS",
maxit = 1500
)
)
#> Warning: NaNs produced
summary(fit_ekw)
#>
#> Generalized Kumaraswamy Regression Model Summary
#>
#> Family: ekw
#>
#> Call:
#> gkwreg(formula = confidence ~ verdict * conflict | verdict +
#> conflict | conflict, data = MockJurors, family = "ekw", control = gkw_control(method = "BFGS",
#> maxit = 1500))
#>
#> Residuals:
#> Min Q1.25% Median Mean Q3.75% Max
#> -0.7253 -0.2066 -0.0576 -0.0722 0.0744 0.2647
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> alpha:(Intercept) 0.45685 NaN NaN NaN
#> alpha:verdict -0.05041 0.12348 -0.408 0.683
#> alpha:conflict 0.09766 NaN NaN NaN
#> alpha:verdict:conflict 0.13324 0.10132 1.315 0.189
#> beta:(Intercept) -0.64393 NaN NaN NaN
#> beta:verdict -0.17246 0.12703 -1.358 0.175
#> beta:conflict -0.07422 NaN NaN NaN
#> lambda:(Intercept) 0.45685 NaN NaN NaN
#> lambda:conflict 0.09766 NaN NaN NaN
#>
#> Confidence intervals (95%):
#> 3% 98%
#> alpha:(Intercept) NaN NaN
#> alpha:verdict -0.2924 0.1916
#> alpha:conflict NaN NaN
#> alpha:verdict:conflict -0.0654 0.3318
#> beta:(Intercept) NaN NaN
#> beta:verdict -0.4214 0.0765
#> beta:conflict NaN NaN
#> lambda:(Intercept) NaN NaN
#> lambda:conflict NaN NaN
#>
#> Link functions:
#> alpha: log
#> beta: log
#> lambda: log
#>
#> Fitted parameter means:
#> alpha: 1.608
#> beta: 0.5339
#> gamma: 1
#> delta: 0
#> lambda: 1.589
#>
#> Model fit statistics:
#> Number of observations: 104
#> Number of parameters: 9
#> Residual degrees of freedom: 95
#> Log-likelihood: 9.75
#> AIC: -1.499
#> BIC: 22.3
#> RMSE: 0.2213
#> Efron's R2: -0.09353
#> Mean Absolute Error: 0.1677
#>
#> Convergence status: Successful
#> Iterations: 5
#>
# Compare 3-parameter models
AIC(fit_ekw, fit_mc)
#> df AIC
#> fit_ekw 9 -1.499307
#> fit_mc 11 -17.957086
# }