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hmetad

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 R formulas
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 rows

You can fit an intercepts-only meta-d’ model using fit_metad:

library(hmetad)

m <- fit_metad(N ~ 1, data = d)

#>  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 rows

To 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.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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