The hmetad package is designed to fit the meta-d’ model for confidence ratings (Maniscalco & Lau, 2012, 2014). The hmetad package uses a Bayesian modeling approach, building on and superseding previous development of the Hmeta-d toolbox (Fleming, 2017). A key advance is implementation as a custom family in the brms package, which itself provides a friendly interface to the probabilistic programming language Stan.
This provides major benefits:
- Model designs can be specified as simple
Rformulas - Support for complex model designs (e.g., multilevel models, distributional models, multivariate models)
- Interfaces to other packages surrounding
brms(e.g.,tidybayes,ggdist,bayesplot,loo,posterior,bridgesampling,bayestestR) - Computation of model-implied quantities (e.g., mean confidence, type 1 and type 2 receiver operating characteristic curves)
- Derivation of novel metacognitive bias metrics, and well as established metrics of metacognitive efficiency
- Increased sampling efficiency and better convergence diagnostics
Installation
hmetad is currently under submission to CRAN, which means soon you will be able to install it using:
install.packages("hmetad")For now, you can install the development version of hmetad from GitHub with:
# install.packages("pak")
pak::pak("metacoglab/hmetad")Quick setup
Let’s say you have some data from a binary decision task with ordinal confidence ratings:
#> # A tibble: 1,000 × 5
#> trial stimulus response correct confidence
#> <int> <int> <int> <int> <int>
#> 1 1 1 1 1 1
#> 2 2 1 1 1 4
#> 3 3 0 1 0 2
#> 4 4 1 0 0 3
#> 5 5 1 1 1 3
#> 6 6 0 1 0 1
#> 7 7 0 0 1 1
#> 8 8 1 0 0 2
#> 9 9 0 1 0 1
#> 10 10 1 1 1 2
#> # ℹ 990 more rowsYou can fit an intercepts-only meta-d’ model using fit_metad:
#> Family: metad__4__normal__absolute__multinomial
#> Links: mu = log
#> Formula: N ~ 1
#> Data: data.aggregated (Number of observations: 1)
#> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup draws = 4000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept -0.06 0.15 -0.37 0.22 1.00 3499 3160
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> dprime 1.08 0.08 0.92 1.25 1.00 4034 3225
#> c -0.05 0.04 -0.13 0.03 1.00 4026 3038
#> metac2zero1diff 0.50 0.04 0.44 0.57 1.00 5219 3390
#> metac2zero2diff 0.48 0.04 0.41 0.56 1.00 5492 3136
#> metac2zero3diff 0.46 0.05 0.37 0.55 1.00 5509 3055
#> metac2one1diff 0.54 0.04 0.46 0.61 1.00 4859 3329
#> metac2one2diff 0.54 0.04 0.47 0.62 1.00 5115 2993
#> metac2one3diff 0.49 0.05 0.40 0.58 1.00 6067 3040
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).Now let’s say you have a more complicated design, such as a within-participant manipulation:
#> # A tibble: 5,000 × 7
#> # Groups: participant, condition [50]
#> participant condition trial stimulus response correct confidence
#> <int> <int> <int> <int> <int> <int> <int>
#> 1 1 1 1 0 0 1 2
#> 2 1 1 2 0 0 1 4
#> 3 1 1 3 0 1 0 4
#> 4 1 1 4 0 1 0 1
#> 5 1 1 5 0 1 0 3
#> 6 1 1 6 1 1 1 4
#> 7 1 1 7 0 0 1 2
#> 8 1 1 8 1 0 0 1
#> 9 1 1 9 0 0 1 1
#> 10 1 1 10 1 0 0 2
#> # ℹ 4,990 more rowsTo account for the repeated measures in this design, you can simply adjust the formula to include participant-level effects:
m <- fit_metad(
bf(
N ~ condition + (condition | participant),
dprime + c +
metac2zero1diff + metac2zero2diff + metac2zero3diff +
metac2one1diff + metac2one2diff + metac2one3diff ~
condition + (condition | participant)
),
data = d, init = "0",
prior = prior(normal(0, 1)) +
prior(normal(0, 1), dpar = dprime) +
prior(normal(0, 1), dpar = c) +
prior(normal(0, 1), dpar = metac2zero1diff) +
prior(normal(0, 1), dpar = metac2zero2diff) +
prior(normal(0, 1), dpar = metac2zero3diff) +
prior(normal(0, 1), dpar = metac2one1diff) +
prior(normal(0, 1), dpar = metac2one2diff) +
prior(normal(0, 1), dpar = metac2one3diff)
)#> Family: metad__4__normal__absolute__multinomial
#> Links: mu = log; dprime = identity; c = identity; metac2zero1diff = log; metac2zero2diff = log; metac2zero3diff = log; metac2one1diff = log; metac2one2diff = log; metac2one3diff = log
#> Formula: N ~ condition + (condition | participant)
#> dprime ~ condition + (condition | participant)
#> c ~ condition + (condition | participant)
#> metac2zero1diff ~ condition + (condition | participant)
#> metac2zero2diff ~ condition + (condition | participant)
#> metac2zero3diff ~ condition + (condition | participant)
#> metac2one1diff ~ condition + (condition | participant)
#> metac2one2diff ~ condition + (condition | participant)
#> metac2one3diff ~ condition + (condition | participant)
#> Data: data.aggregated (Number of observations: 50)
#> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup draws = 4000
#>
#> Multilevel Hyperparameters:
#> ~participant (Number of levels: 25)
#> Estimate Est.Error
#> sd(Intercept) 0.56 0.17
#> sd(condition2) 0.66 0.22
#> sd(dprime_Intercept) 0.36 0.08
#> sd(dprime_condition2) 0.62 0.12
#> sd(c_Intercept) 0.53 0.09
#> sd(c_condition2) 0.58 0.10
#> sd(metac2zero1diff_Intercept) 0.12 0.07
#> sd(metac2zero1diff_condition2) 0.11 0.08
#> sd(metac2zero2diff_Intercept) 0.12 0.08
#> sd(metac2zero2diff_condition2) 0.40 0.13
#> sd(metac2zero3diff_Intercept) 0.12 0.08
#> sd(metac2zero3diff_condition2) 0.15 0.10
#> sd(metac2one1diff_Intercept) 0.07 0.05
#> sd(metac2one1diff_condition2) 0.09 0.07
#> sd(metac2one2diff_Intercept) 0.10 0.07
#> sd(metac2one2diff_condition2) 0.12 0.09
#> sd(metac2one3diff_Intercept) 0.12 0.08
#> sd(metac2one3diff_condition2) 0.15 0.11
#> cor(Intercept,condition2) -0.16 0.37
#> cor(dprime_Intercept,dprime_condition2) -0.37 0.24
#> cor(c_Intercept,c_condition2) -0.50 0.16
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) -0.18 0.56
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) -0.31 0.50
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) -0.05 0.57
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) -0.03 0.57
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) -0.26 0.57
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) -0.28 0.57
#> l-95% CI u-95% CI
#> sd(Intercept) 0.27 0.95
#> sd(condition2) 0.28 1.14
#> sd(dprime_Intercept) 0.21 0.54
#> sd(dprime_condition2) 0.40 0.89
#> sd(c_Intercept) 0.39 0.73
#> sd(c_condition2) 0.42 0.81
#> sd(metac2zero1diff_Intercept) 0.01 0.27
#> sd(metac2zero1diff_condition2) 0.01 0.30
#> sd(metac2zero2diff_Intercept) 0.01 0.30
#> sd(metac2zero2diff_condition2) 0.14 0.68
#> sd(metac2zero3diff_Intercept) 0.01 0.31
#> sd(metac2zero3diff_condition2) 0.01 0.38
#> sd(metac2one1diff_Intercept) 0.00 0.20
#> sd(metac2one1diff_condition2) 0.00 0.26
#> sd(metac2one2diff_Intercept) 0.01 0.26
#> sd(metac2one2diff_condition2) 0.00 0.33
#> sd(metac2one3diff_Intercept) 0.01 0.31
#> sd(metac2one3diff_condition2) 0.01 0.41
#> cor(Intercept,condition2) -0.74 0.71
#> cor(dprime_Intercept,dprime_condition2) -0.75 0.16
#> cor(c_Intercept,c_condition2) -0.77 -0.14
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) -0.96 0.92
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) -0.96 0.86
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) -0.95 0.95
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) -0.94 0.94
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) -0.98 0.93
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) -0.98 0.90
#> Rhat Bulk_ESS
#> sd(Intercept) 1.00 1615
#> sd(condition2) 1.00 1167
#> sd(dprime_Intercept) 1.00 1769
#> sd(dprime_condition2) 1.01 1194
#> sd(c_Intercept) 1.00 1025
#> sd(c_condition2) 1.00 1036
#> sd(metac2zero1diff_Intercept) 1.00 1467
#> sd(metac2zero1diff_condition2) 1.00 1954
#> sd(metac2zero2diff_Intercept) 1.00 1642
#> sd(metac2zero2diff_condition2) 1.01 1010
#> sd(metac2zero3diff_Intercept) 1.00 1229
#> sd(metac2zero3diff_condition2) 1.00 1477
#> sd(metac2one1diff_Intercept) 1.00 1844
#> sd(metac2one1diff_condition2) 1.00 1926
#> sd(metac2one2diff_Intercept) 1.00 1966
#> sd(metac2one2diff_condition2) 1.00 1637
#> sd(metac2one3diff_Intercept) 1.00 1478
#> sd(metac2one3diff_condition2) 1.00 1347
#> cor(Intercept,condition2) 1.00 1144
#> cor(dprime_Intercept,dprime_condition2) 1.00 1085
#> cor(c_Intercept,c_condition2) 1.00 1223
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) 1.00 3940
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) 1.01 393
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) 1.00 2608
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) 1.00 3472
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) 1.00 2784
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) 1.00 2769
#> Tail_ESS
#> sd(Intercept) 2278
#> sd(condition2) 1380
#> sd(dprime_Intercept) 2801
#> sd(dprime_condition2) 2036
#> sd(c_Intercept) 1600
#> sd(c_condition2) 1772
#> sd(metac2zero1diff_Intercept) 1698
#> sd(metac2zero1diff_condition2) 2374
#> sd(metac2zero2diff_Intercept) 1932
#> sd(metac2zero2diff_condition2) 1397
#> sd(metac2zero3diff_Intercept) 2100
#> sd(metac2zero3diff_condition2) 1730
#> sd(metac2one1diff_Intercept) 2428
#> sd(metac2one1diff_condition2) 1522
#> sd(metac2one2diff_Intercept) 2451
#> sd(metac2one2diff_condition2) 2009
#> sd(metac2one3diff_Intercept) 1322
#> sd(metac2one3diff_condition2) 1660
#> cor(Intercept,condition2) 1059
#> cor(dprime_Intercept,dprime_condition2) 1750
#> cor(c_Intercept,c_condition2) 1782
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) 2843
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) 999
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) 2383
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) 2659
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) 2700
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) 3032
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> Intercept 0.12 0.17 -0.25 0.42 1.00 2319
#> dprime_Intercept 0.88 0.09 0.70 1.06 1.00 2760
#> c_Intercept -0.12 0.11 -0.33 0.11 1.00 577
#> metac2zero1diff_Intercept -1.04 0.06 -1.16 -0.92 1.00 5908
#> metac2zero2diff_Intercept -0.99 0.07 -1.13 -0.86 1.00 5749
#> metac2zero3diff_Intercept -1.01 0.08 -1.17 -0.87 1.00 5687
#> metac2one1diff_Intercept -1.01 0.06 -1.13 -0.89 1.00 6546
#> metac2one2diff_Intercept -0.93 0.06 -1.05 -0.81 1.00 6844
#> metac2one3diff_Intercept -1.10 0.07 -1.24 -0.96 1.00 5729
#> condition2 -0.02 0.22 -0.48 0.39 1.00 2216
#> dprime_condition2 0.17 0.14 -0.11 0.45 1.00 2108
#> c_condition2 -0.10 0.12 -0.34 0.14 1.00 675
#> metac2zero1diff_condition2 0.05 0.09 -0.12 0.22 1.00 5745
#> metac2zero2diff_condition2 0.09 0.12 -0.15 0.33 1.00 3428
#> metac2zero3diff_condition2 0.10 0.11 -0.12 0.31 1.00 5774
#> metac2one1diff_condition2 0.07 0.08 -0.09 0.23 1.00 6787
#> metac2one2diff_condition2 -0.13 0.09 -0.30 0.04 1.00 6040
#> metac2one3diff_condition2 0.03 0.10 -0.16 0.23 1.00 6259
#> Tail_ESS
#> Intercept 2856
#> dprime_Intercept 2966
#> c_Intercept 1166
#> metac2zero1diff_Intercept 3189
#> metac2zero2diff_Intercept 2959
#> metac2zero3diff_Intercept 3199
#> metac2one1diff_Intercept 2716
#> metac2one2diff_Intercept 2893
#> metac2one3diff_Intercept 3484
#> condition2 2325
#> dprime_condition2 2670
#> c_condition2 1246
#> metac2zero1diff_condition2 2892
#> metac2zero2diff_condition2 3034
#> metac2zero3diff_condition2 2857
#> metac2one1diff_condition2 2743
#> metac2one2diff_condition2 3031
#> metac2one3diff_condition2 3001
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).References
Fleming, S. M. (2017). HMeta-d: Hierarchical bayesian estimation of metacognitive efficiency from confidence ratings. Neuroscience of Consciousness, 2017(1), nix007.
Maniscalco, B., & Lau, H. (2012). A signal detection theoretic approach for estimating metacognitive sensitivity from confidence ratings. Consciousness and Cognition, 21(1), 422–430.
Maniscalco, B., & Lau, H. (2014). Signal detection theory analysis of type 1 and type 2 data: Meta-d′, response-specific meta-d′, and the unequal variance SDT model. In The cognitive neuroscience of metacognition (pp. 25–66). Springer.