The hmetad package is designed to fit the meta-d’ model for confidence ratings (Maniscalco and Lau 2012, 2014). Like the Hmeta-d toolbox (Fleming 2017), the hmetad package uses a Bayesian modeling approach. The hmetad package builds on the Hmeta-d toolbox through 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)
- 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, metacognitive bias)
- 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 0 0 1 3
#> 2 2 0 0 1 4
#> 3 3 0 0 1 2
#> 4 4 1 1 1 1
#> 5 5 1 0 0 1
#> 6 6 0 1 0 2
#> 7 7 1 1 1 2
#> 8 8 0 0 1 2
#> 9 9 0 0 1 2
#> 10 10 0 0 1 1
#> # 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 -0.18 0.16 -0.52 0.11 1.00 3142 3260
#>
#> 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 4527 3273
#> c 0.02 0.04 -0.06 0.10 1.00 3675 3067
#> metac2zero1diff 0.48 0.03 0.41 0.55 1.00 4490 3209
#> metac2zero2diff 0.52 0.04 0.44 0.60 1.00 5076 3242
#> metac2zero3diff 0.55 0.05 0.46 0.65 1.00 5655 2716
#> metac2one1diff 0.46 0.03 0.39 0.52 1.00 4627 2976
#> metac2one2diff 0.43 0.04 0.36 0.50 1.00 4979 2987
#> metac2one3diff 0.52 0.05 0.43 0.61 1.00 5653 2723
#>
#> 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-participants 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 1 1 1 4
#> 2 1 1 2 0 1 0 2
#> 3 1 1 3 0 1 0 1
#> 4 1 1 4 1 1 1 1
#> 5 1 1 5 0 0 1 2
#> 6 1 1 6 1 0 0 1
#> 7 1 1 7 1 1 1 2
#> 8 1 1 8 0 0 1 3
#> 9 1 1 9 1 1 1 4
#> 10 1 1 10 0 0 1 1
#> # 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", file = "vignettes/models/readme2.rds",
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) 0.53 0.16
#> sd(condition2) 0.85 0.25
#> sd(dprime_Intercept) 0.48 0.09
#> sd(dprime_condition2) 0.67 0.14
#> sd(c_Intercept) 0.64 0.10
#> sd(c_condition2) 0.75 0.12
#> sd(metac2zero1diff_Intercept) 0.09 0.06
#> sd(metac2zero1diff_condition2) 0.09 0.07
#> sd(metac2zero2diff_Intercept) 0.15 0.09
#> sd(metac2zero2diff_condition2) 0.17 0.11
#> sd(metac2zero3diff_Intercept) 0.07 0.06
#> sd(metac2zero3diff_condition2) 0.14 0.10
#> sd(metac2one1diff_Intercept) 0.08 0.05
#> sd(metac2one1diff_condition2) 0.12 0.08
#> sd(metac2one2diff_Intercept) 0.12 0.07
#> sd(metac2one2diff_condition2) 0.10 0.07
#> sd(metac2one3diff_Intercept) 0.21 0.12
#> sd(metac2one3diff_condition2) 0.19 0.13
#> cor(Intercept,condition2) -0.84 0.15
#> cor(dprime_Intercept,dprime_condition2) -0.61 0.16
#> cor(c_Intercept,c_condition2) -0.79 0.08
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) -0.10 0.57
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) -0.20 0.56
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) -0.23 0.58
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) -0.07 0.59
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) 0.05 0.56
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) -0.63 0.46
#> l-95% CI u-95% CI
#> sd(Intercept) 0.28 0.89
#> sd(condition2) 0.40 1.39
#> sd(dprime_Intercept) 0.33 0.68
#> sd(dprime_condition2) 0.45 0.98
#> sd(c_Intercept) 0.48 0.86
#> sd(c_condition2) 0.55 1.02
#> sd(metac2zero1diff_Intercept) 0.00 0.22
#> sd(metac2zero1diff_condition2) 0.00 0.25
#> sd(metac2zero2diff_Intercept) 0.01 0.34
#> sd(metac2zero2diff_condition2) 0.01 0.42
#> sd(metac2zero3diff_Intercept) 0.00 0.21
#> sd(metac2zero3diff_condition2) 0.01 0.37
#> sd(metac2one1diff_Intercept) 0.00 0.20
#> sd(metac2one1diff_condition2) 0.01 0.31
#> sd(metac2one2diff_Intercept) 0.01 0.26
#> sd(metac2one2diff_condition2) 0.00 0.27
#> sd(metac2one3diff_Intercept) 0.02 0.46
#> sd(metac2one3diff_condition2) 0.01 0.47
#> cor(Intercept,condition2) -0.99 -0.44
#> cor(dprime_Intercept,dprime_condition2) -0.85 -0.25
#> cor(c_Intercept,c_condition2) -0.91 -0.59
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) -0.96 0.93
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) -0.97 0.93
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) -0.98 0.91
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) -0.96 0.95
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) -0.92 0.95
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) -1.00 0.70
#> Rhat Bulk_ESS
#> sd(Intercept) 1.00 1976
#> sd(condition2) 1.00 1316
#> sd(dprime_Intercept) 1.00 1744
#> sd(dprime_condition2) 1.00 1306
#> sd(c_Intercept) 1.00 940
#> sd(c_condition2) 1.00 908
#> sd(metac2zero1diff_Intercept) 1.00 1721
#> sd(metac2zero1diff_condition2) 1.00 1858
#> sd(metac2zero2diff_Intercept) 1.00 1372
#> sd(metac2zero2diff_condition2) 1.00 1294
#> sd(metac2zero3diff_Intercept) 1.00 1676
#> sd(metac2zero3diff_condition2) 1.00 1669
#> sd(metac2one1diff_Intercept) 1.00 2318
#> sd(metac2one1diff_condition2) 1.00 1592
#> sd(metac2one2diff_Intercept) 1.00 1303
#> sd(metac2one2diff_condition2) 1.00 1848
#> sd(metac2one3diff_Intercept) 1.00 863
#> sd(metac2one3diff_condition2) 1.00 1038
#> cor(Intercept,condition2) 1.00 1379
#> cor(dprime_Intercept,dprime_condition2) 1.00 1635
#> cor(c_Intercept,c_condition2) 1.00 1090
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) 1.00 4554
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) 1.00 2823
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) 1.00 2675
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) 1.00 2354
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) 1.00 4178
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) 1.00 1520
#> Tail_ESS
#> sd(Intercept) 2725
#> sd(condition2) 1774
#> sd(dprime_Intercept) 2325
#> sd(dprime_condition2) 2546
#> sd(c_Intercept) 1722
#> sd(c_condition2) 1190
#> sd(metac2zero1diff_Intercept) 2037
#> sd(metac2zero1diff_condition2) 1691
#> sd(metac2zero2diff_Intercept) 1808
#> sd(metac2zero2diff_condition2) 1908
#> sd(metac2zero3diff_Intercept) 2164
#> sd(metac2zero3diff_condition2) 2429
#> sd(metac2one1diff_Intercept) 2459
#> sd(metac2one1diff_condition2) 1920
#> sd(metac2one2diff_Intercept) 1510
#> sd(metac2one2diff_condition2) 2032
#> sd(metac2one3diff_Intercept) 1528
#> sd(metac2one3diff_condition2) 1590
#> cor(Intercept,condition2) 2113
#> cor(dprime_Intercept,dprime_condition2) 2358
#> cor(c_Intercept,c_condition2) 1538
#> cor(metac2zero1diff_Intercept,metac2zero1diff_condition2) 2956
#> cor(metac2zero2diff_Intercept,metac2zero2diff_condition2) 2275
#> cor(metac2zero3diff_Intercept,metac2zero3diff_condition2) 2654
#> cor(metac2one1diff_Intercept,metac2one1diff_condition2) 2669
#> cor(metac2one2diff_Intercept,metac2one2diff_condition2) 3249
#> cor(metac2one3diff_Intercept,metac2one3diff_condition2) 2395
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> Intercept -0.10 0.16 -0.44 0.19 1.00 3095
#> dprime_Intercept 0.90 0.11 0.67 1.12 1.00 1523
#> c_Intercept -0.02 0.13 -0.27 0.24 1.01 474
#> metac2zero1diff_Intercept -0.95 0.06 -1.07 -0.84 1.00 5998
#> metac2zero2diff_Intercept -1.02 0.07 -1.16 -0.89 1.00 4764
#> metac2zero3diff_Intercept -1.05 0.07 -1.18 -0.92 1.00 6306
#> metac2one1diff_Intercept -0.96 0.06 -1.09 -0.85 1.00 5967
#> metac2one2diff_Intercept -0.97 0.06 -1.10 -0.84 1.00 5015
#> metac2one3diff_Intercept -1.09 0.08 -1.26 -0.93 1.00 4329
#> condition2 -0.05 0.24 -0.53 0.43 1.00 2754
#> dprime_condition2 0.03 0.15 -0.27 0.33 1.00 2083
#> c_condition2 -0.19 0.15 -0.48 0.11 1.00 809
#> metac2zero1diff_condition2 -0.09 0.08 -0.25 0.07 1.00 5915
#> metac2zero2diff_condition2 -0.14 0.10 -0.32 0.05 1.00 4944
#> metac2zero3diff_condition2 0.01 0.10 -0.17 0.20 1.00 5620
#> metac2one1diff_condition2 -0.12 0.08 -0.28 0.04 1.00 5205
#> metac2one2diff_condition2 -0.01 0.08 -0.17 0.15 1.00 5177
#> metac2one3diff_condition2 0.10 0.10 -0.09 0.31 1.00 5508
#> Tail_ESS
#> Intercept 2616
#> dprime_Intercept 2172
#> c_Intercept 1239
#> metac2zero1diff_Intercept 2829
#> metac2zero2diff_Intercept 3016
#> metac2zero3diff_Intercept 3005
#> metac2one1diff_Intercept 2897
#> metac2one2diff_Intercept 2952
#> metac2one3diff_Intercept 2879
#> condition2 2771
#> dprime_condition2 2679
#> c_condition2 1446
#> metac2zero1diff_condition2 2905
#> metac2zero2diff_condition2 3129
#> metac2zero3diff_condition2 2461
#> metac2one1diff_condition2 2420
#> metac2one2diff_condition2 2707
#> metac2one3diff_condition2 3413
#>
#> 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, Stephen M. 2017. “HMeta-d: Hierarchical Bayesian Estimation of Metacognitive Efficiency from Confidence Ratings.” Neuroscience of Consciousness 2017 (1): nix007.
Maniscalco, Brian, and Hakwan Lau. 2012. “A Signal Detection Theoretic Approach for Estimating Metacognitive Sensitivity from Confidence Ratings.” Consciousness and Cognition 21 (1): 422–30.
———. 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, 25–66. Springer.