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ThestatConfR package provides functions to fit static models ofdecision-making and confidence derived from signal detection theory forbinary discrimination tasks with confidence ratings on the data fromindividual subjects (Rausch et al., 2025). Up to now, the followingmodels have been included:

• signal detection rating model (Green & Swets, 1966),
• Gaussian noise model (Maniscalco & Lau, 2016),
• weighted evidence and visibility model (Rausch et al., 2018),
• post-decisional accumulation model (Rausch et al., 2018),
• independent Gaussian model (Rausch & Zehetleitner, 2017),
• independent truncated Gaussian model (the model underlying themeta-d$^\prime$/d$^\prime$ method, see Rausch et al., 2023),
• lognormal noise model (Shekhar & Rahnev, 2021), and
• lognormal weighted evidence and visibility model (Shekhar & Rahnev,2023).

In addition, thestatConfR package provides functions for estimatingdifferent kinds of measures of metacognition:

• meta-d$^\prime$/d$^\prime$, the most widely-used measure ofmetacognitive efficiency, allowing both Maniscalco and Lau (2012)’sand Fleming (2017)’s model specification. Fitting models of confidenceis a way to test the assumptions underlyingmeta-d$^\prime$/d$^\prime$.

• information-theoretic measures of metacognition (Dayan, 2023),including

  • meta-I, an information-theoretic measures of metacognitivesensitivity,
  • $meta-I_{1}^{r}$ and$meta-I_{2}^{r}$, two measures ofmetacognitive efficiency proposed by Dayan (2023),
  • meta-$I_{1}^{r\prime}$, a novel variant of$meta-I_{1}^{r}$,
  • RMI, a novel measure of metacognitive accuracy, also derived frominformation theory.

The models included in the statConfR package are all based on signaldetection theory (Green & Swets, 1966). It is assumed that participantsselect a binary discrimination response$R$ about a stimulus$S$.Both$S$ and$R$ can be either -1 or 1.$R$ is considered correctif$S=R$. In addition, we assume that in the experiment, there are$K$ different levels of stimulus discriminability, i.e. a physicalvariable that makes the discrimination task easier or harder. For eachlevel of discriminability, the function fits a different discriminationsensitivity parameter$d_k$. If there is more than one sensitivityparameter, we assume that the sensitivity parameters are ordered such as$0 < d_1 < d_2 < ... < d_K$. The models assume that the stimulusgenerates normally distributed sensory evidence$x$ with mean$S\times d_k/2$ and variance of 1. The sensory evidence$x$ iscompared to a decision criterion$c$ to generate a discriminationresponse$R$, which is 1, if$x$ exceeds$c$ and -1 else. Togenerate confidence, it is assumed that the confidence variable$y$ iscompared to another set of criteria$\theta_{R,i}, i=1,2,...,L-1$,depending on the discrimination response$R$ to produce a$L$-stepdiscrete confidence response. The different models vary in how$y$ isgenerated (see below). The following parameters are shared between allmodels:

• sensitivity parameters$d_1, ..., d_K$ ($K$: number of difficultylevels),
• decision criterion$c$,
• confidence criterion$\theta_{-1,1}, ..., \theta_{-1,L-1}, \theta_{1,1}, ,...,\theta_{1,L-1}$ ($L$: number of confidencecategories available for confidence ratings).

According to SDT, the same sample of sensory evidence is used togenerate response and confidence, i.e.,$y=x$. The confidence criteriaassociated with$R=-1$ are more negative than the decision criterion$c$, whereas the confidence criteria associated with$R=1$ are morepositive than$c$.

Conceptually, the Gaussian noise model reflects the idea that confidenceis informed by the same sensory evidence as the task decision, butconfidence is affected by additive Gaussian noise. According to GN,$y$ is subject to additive noise and assumed to be normallydistributed around the decision evidence value$x$ with a standarddeviation$\sigma$, which is an additional free parameter.

Conceptually, the WEV model reflects the idea that the observer combinesevidence about decision-relevant features of the stimulus with thestrength of evidence about choice-irrelevant features to generateconfidence. For this purpose, WEV assumes that$y$ is normallydistributed with a mean of$(1-w)\times x+w \times d_k\times R$ andstandard deviation$\sigma$. The standard deviation quantifies theamount of unsystematic variability contributing to confidence judgmentsbut not to the discrimination judgments. The parameter$w$ representsthe weight that is put on the choice-irrelevant features in theconfidence judgment. The parameters$w$ and$\sigma$ are freeparameters in addition to the set of shared parameters.

PDA represents the idea of on-going information accumulation after thediscrimination choice. The parameter$a$ indicates the amount ofadditional accumulation. The confidence variable is normally distributedwith mean$x+S\times d_k\times a$ and variance$a$. The parameter$a$ is fitted in addition to the shared parameters.

According to IG, the information used for confidence judgments isgenerated independently from the sensory evidence used for the taskdecision. For this purpose, it is assumed that$y$ is sampledindependently from$x$. The variable$y$ is normally distributedwith a mean of$a\times d_k$ and variance of 1. The additionalparameter$m$ represents the amount of information available forconfidence judgment relative to amount of evidence available for thediscrimination decision and can be smaller as well as greater than 1.

Conceptually, the two ITG models just as IG are based on the idea thatthe information used for confidence judgments is generated independentlyfrom the sensory evidence used for the task decision. However, incontrast to IG, the two ITG models also reflect a form of confirmationbias in so far as it is not possible to collect information thatcontradicts the original decision. According to the version of ITGconsistent with the HMetad-method (Fleming, 2017),$y$ is sampledindependently from$x$ from a truncated Gaussian distribution with alocation parameter of$S\times d_k \times m/2$ and a scale parameterof 1. The Gaussian distribution of$y$ is truncated in a way that itis impossible to sample evidence that contradicts the original decision:If$R = -1$, the distribution is truncated to the right of$c$. If$R = 1$, the distribution is truncated to the left of$c$. Theadditional parameter$m$ represents metacognitive efficiency, i.e.,the amount of information available for confidence judgments relative toamount of evidence available for discrimination decisions and can besmaller as well as greater than 1.

According to the version of the ITG consistent with the original meta-d’method (Maniscalco & Lau, 2012, 2014),$y$ is sampled independentlyfrom$x$ from a truncated Gaussian distribution with a locationparameter of$S\times d_k \times m/2$ and a scale parameter of 1. If$R = -1$, the distribution is truncated to the right of$m\times c$.If$R = 1$, the distribution is truncated to the left of$m\times c$. The additional parameter$m$ represents metacognitiveefficiency, i.e., the amount of information available for confidencejudgments relative to amount of evidence available for thediscrimination decision and can be smaller as well as greater than 1.

According to logN, the same sample of sensory evidence is used togenerate response and confidence, i.e.,$y=x$ just as in SDT. However,according to logN, the confidence criteria are not assumed to beconstant, but instead they are affected by noise drawn from a lognormaldistribution. In each trial,$\theta_{-1,i}$ is given by$c - \epsilon_i$. Likewise,$\theta_{1,i}$ is given by$c + \epsilon_i$. The noise$\epsilon_i$ is drawn from a lognormaldistribution with the location parameter$\mu_{R,i} = \log(\left| \mu_{\theta_{R,i}} - c\right|)- 0.5 \times \sigma^{2}$,and scale parameter$\sigma$.$\sigma$ is a free parameter designedto quantify metacognitive ability. It is assumed that the criterionnoise is perfectly correlated across confidence criteria, ensuring thatthe confidence criteria are always perfectly ordered. Because$\theta_{-1,1}$, …,$\theta_{-1,L-1}$,$\theta_{1,1}$, …,$\theta_{1,L-1}$ change from trial to trial, they are not estimated asfree parameters. Instead, we estimate the means of the confidencecriteria, i.e.,$\mu_{\theta_{-1,1}}, ..., \mu_{\theta_{-1,L-1}}, \mu_{\theta_{1,1}}, ... \mu_{\theta_{1,L-1}}$,as free parameters.

The logWEV model is a combination of logN and WEV proposed by .Conceptually, logWEV assumes that the observer combines evidence aboutdecision-relevant features of the stimulus with the strength of evidenceabout choice-irrelevant features. The model also assumes that noiseaffecting the confidence decision variable is lognormal. According tologWEV, the confidence decision variable is$y$ is equal to R × y’.The variable y’ is sampled from a lognormal distribution with a locationparameter of$(1-w)\times x\times R + w \times d_k$ and a scaleparameter of$\sigma$. The parameter$\sigma$ quantifies the amountof unsystematic variability contributing to confidence judgments but notto the discrimination judgments. The parameter$w$ represents theweight that is put on the choice-irrelevant features in the confidencejudgment. The parameters$w$ and$\sigma$ are free parameters.

The conceptual idea of meta-d$^\prime$ is to quantify metacognition interms of sensitivity in a hypothetical signal detection rating modeldescribing the primary task, under the assumption that participants hadperfect access to the sensory evidence and were perfectly consistent inplacing their confidence criteria (Maniscalco & Lau, 2012, 2014). Usinga signal detection model describing the primary task to quantifymetacognition, it allows a direct comparison between metacognitiveaccuracy and discrimination performance because both are measured on thesame scale. Meta-d$^\prime$ can be compared against the estimate ofthe distance between the two stimulus distributions estimated fromdiscrimination responses, which is referred to as d$^\prime$: Ifmeta-$^\prime$ equals d$^\prime$, it means that metacognitiveaccuracy is exactly as good as expected from discrimination performance.If meta-d$^\prime$ is lower than d$^\prime$, it means thatmetacognitive accuracy is not optimal. It can be shown that the implicitmodel of confidence underlying the meta-d$^\prime$/d$^\prime$ methodis identical to different versions of the independent truncated Gaussianmodel (Rausch et al., 2023), depending on whether the original modelspecification by Maniscalco and Lau (2012) or alternatively thespecification by Fleming (2017) is used. We strongly recommend to testwhether the independent truncated Gaussian models are adequatedescriptions of the data before quantifying metacognitive efficiencywith meta-d$^\prime$/d$^\prime$ (see Rausch et al., 2023).

It is assumed that a classifier (possibly a human being performing adiscrimination task) or an algorithmic classifier in a classificationapplication, makes a binary prediction$R$ about a true state of theworld$S$. Dayan (2023) proposed several measures of metacognitionbased on quantities of information theory (for an introduction intoinformation theory, see MacKay, 2003; Cover & Thomas, 2006).

• Meta-I is a measure of metacognitive sensitivity defined as the mutualinformation between the confidence and accuracy and is calculated asthe transmitted information minus the minimal information given theaccuracy of the classification response:

It can be shown that this is equivalent to Dayan’s formulation of meta-Ias the information that confidence transmits about the correctness of aresponse:

• Meta-$I_{1}^{r}$ is meta-I normalized by the value of meta-Iexpected assuming a signal detection model (Green & Swets, 1966) withGaussian noise, based on calculating the sensitivity index d’:

• Meta-$I_{1}^{r\prime}$ is a variant of meta-$I_{1}^{r}$, whichnormalizes by the meta-I that would be expected from an underlyingnormal distribution with the same accuracy (this is similar to thesensitivity approach but without considering variable thresholds).

• Meta-$I_{2}^{r}$ is meta-I normalized by its theoretical upperbound, which is the information entropy of accuracy,$H(S = R)$:

Notably, Dayan (2023) pointed out that a liberal or conservative use ofthe confidence levels will affected the mutual information and thus allinformation-theoretic measures of metacognition.

In addition to Dayan’s measures, Meyen et al. (submitted) suggested anadditional measure that normalizes meta-I by the range of possiblevalues it can take. Normalizing meta-I by the range of possible valuesrequires deriving lower and upper bounds of the transmitted informationgiven a participant’s accuracy.

As these measures are prone to estimation bias, the package offers asimple bias reduction mechanism in which the observed frequencies ofstimulus-response combinations are taken as the underlying probabilitydistribution. From this, Monte-Carlo simulations are conducted toestimate and subtract the bias from these measures. Note that the biasis only reduced but not removed completely.

The latest released version of the package is available on CRAN via

install.packages("statConfR")

The easiest way to install the development version is usingdevtoolsand install from GitHub:

devtools::install_github("ManuelRausch/StatConfR")

The package includes a demo data set from a masked orientationdiscrimination task with confidence judgments (Hellmann et al., 2023,Exp. 1).

library(statConfR)data("MaskOri")head(MaskOri)

##   participant stimulus correct rating diffCond trialNo## 1           1        0       1      0      8.3       1## 2           1       90       0      4    133.3       2## 3           1        0       1      0     33.3       3## 4           1       90       0      0     16.7       4## 5           1        0       1      3    133.3       5## 6           1        0       1      0     16.7       6

The functionfitConfModels allows the user to fit several confidencemodels separately to the data of each participant using maximumlikelihood estimation. The data should be provided via the argument.data in the form of a data.frame object with the following variablesin separate columns:

• stimulus (factor with 2 levels): The property of the stimulus whichdefines which response is correct
• diffCond (factor): The experimental manipulation that is expected toaffect discrimination sensitivity
• correct (0-1): Indicating whether the choice was correct (1) orincorrect(0).
• rating (factor): A discrete variable encoding the decision confidence(high: very confident; low: less confident)
• participant (integer): giving the subject ID. The argumentmodel isused to specify which model should be fitted, with ‘WEV’, ‘SDT’, ‘GN’,‘PDA’, ‘IG’, ‘ITGc’, ‘ITGcm’, ‘logN’, and ‘logWEV’ as availableoptions. If model=“all” (default), all implemented models will be fit,although this may take a while.

Setting the optional argument.parallel=TRUE parallizes model fittingover all but 1 available core.Note that the fitting procedure takesmay take a considerable amount of time, especially when there aremultiple models, several difficulty conditions, and/or multipleconfidence categories. For example, if there are five difficultyconditions and five confidence levels, fitting the WEV model to onesingle participant may take 20-30 minutes on a 2.8GHz CPU. We recommendparallelization to keep the required time tolerable.

The fitting routine first performs a coarse grid search to findpromising starting values for the maximum likelihood optimizationprocedure. Then the bestnInits parameter sets found by the gridsearch are used as the initial values for separate runs of theNelder-Mead algorithm implemented in optim (default: 5). Each run isrestartednRestart times (default: 4).

fitted_pars<- fitConfModels(MaskOri,models=c("ITGcm","WEV"),.parallel=TRUE)

The output is then a data frame with one row for each combination ofparticipant and model and separate columns for each estimated parameter(d_1, d_2, d_3, d_4, c, theta_minus.4 theta_minus.3, theta_minus.2,theta_minus.1, theta_plus.1, theta_plus.2, theta_plus.3, theta_plus.4for both models, w and sigma for WEV, and m only for ITGcm) as well asfor different measures for goodness-of-fit (negative log-likelihood,BIC, AIC and AICc).

head(fitted_pars)

##   model participant negLogLik    N  k      BIC     AICc      AIC        d_1## 1 ITGcm           1  2719.492 1620 15 5549.837 5469.247 5468.985 0.02791587## 2   WEV           1  2621.110 1620 16 5360.464 5274.520 5274.221 0.20268438## 3 ITGcm           2  1926.296 1620 15 3963.445 3882.854 3882.592 0.01889636## 4   WEV           2  1827.221 1620 16 3772.684 3686.741 3686.441 0.05119639## 5 ITGcm           3  1695.957 1620 15 3502.766 3422.176 3421.914 0.32340627## 6   WEV           3  1661.617 1620 16 3441.476 3355.533 3355.233 0.41460563##          d_2       d_3      d_4      d_5          c theta_minus.4 theta_minus.3## 1 0.43212223 1.0210704 3.472310 4.395496 -0.2499098     -1.584000     -1.055322## 2 0.61422596 1.0796567 3.474608 4.079890 -0.2957338     -2.066516     -1.248524## 3 0.06496444 0.6926183 4.209053 5.463259 -0.1068211     -2.109575     -2.009674## 4 0.19195858 1.0412267 4.142295 5.288622 -0.1474590     -2.044069     -1.950015## 5 0.60550967 2.3776478 7.924170 9.428593 -1.2804566     -1.793311     -1.448681## 6 0.85608686 2.7115290 6.916448 7.986348 -1.3742943     -2.762529     -1.919228##   theta_minus.2 theta_minus.1 theta_plus.1 theta_plus.2 theta_plus.3## 1    -0.6463512    -0.4645142  -0.09770594    0.2168548    1.0019751## 2    -0.4151617     0.1296425  -0.61959026    0.1544368    1.3976350## 3    -1.4620933    -0.9950160   0.78839560    1.4081014    2.1950659## 4    -1.3982493    -0.9030114   0.82007352    1.4484447    2.2446957## 5    -1.0652684    -0.9656961  -0.92027462   -0.6053266    0.3337906## 6    -0.3723945     0.9327974  -2.76951959   -1.1312635    0.7714093##   theta_plus.4         m     sigma         w## 1    1.6044716 1.1177354        NA        NA## 2    2.1879187        NA 1.0104584 0.5361153## 3    2.3601086 1.5701944        NA        NA## 4    2.4029896        NA 0.6390763 0.5019978## 5    0.9382662 0.7404757        NA        NA## 6    1.7520050        NA 1.3288815 0.3817864

After obtaining the model fit, it is strongly recommended to visualisethe predictions implied by the best-fitting set of parameters andcompare these predictions with the actual data (Palminteri et al.,2017). ThestatConfR package provides the functionplotConfModelFit,which creates aggplot object with empirically observed distributionof responses and confidence ratings as bars on the x-axis as a functionof discriminability (in the rows) and stimulus (in the columns).Superimposed on the empirical data, the plot also shows the predictionof one selected model as dots. The parameters of the model are passed toplotConfModelFit by the argumentfitted_pars.

PlotFitWEV<- plotConfModelFit(MaskOri,fitted_pars,model="WEV")PlotFitITGcm<- plotConfModelFit(MaskOri,fitted_pars,model="ITGcm")

PlotFitWEV

PlotFitITGcm

Assuming that the independent truncated Gaussian model provides a decentaccount of the data (notably, this is not the case in the demo dataset), the functionfitMetaDprime can be used to estimatemeta-d$^\prime$/d$^\prime$ independently for each subject. Thearguments.data and.parallel=TRUE work just in the same way thearguments offitConfModels. The argumentmodel offers the user thechoice between two model specifications, either “ML” to use the originalmodel specification used by Maniscalco and Lau (2012, 2014) or “F” touse the model specification by Fleming (2017)’s Hmetad method. ThefunctionfitMetaDprime produces a dataframe with one row for eachparticipant and the following columns:

• participant: the participant id,
• model: indicating which model specification has been used,
• dprime: the sensitivity index d′ from signal detection theory, ameasure of discrimination performance,
• c: the bias index c from signal detection theory, a measure ofdiscrimination bias,
• Ratio: The meta-d′/d′ index, the most common measure of metacognitiveefficiency.

MetaDs<- fitMetaDprime(data=MaskOri,model="ML",.parallel=TRUE)head(MetaDs)

##   model participant   dprime          c    metaD     Ratio## 1    ML           1 1.441199 -0.2597310 1.423263 0.9875551## 2    ML           2 1.253587 -0.1175263 2.074045 1.6544885## 3    ML           3 2.253395 -1.0013475 1.508996 0.6696544## 4    ML           4 1.515356  0.1231483 3.192407 2.1067045## 5    ML           5 1.314925 -0.1047285 2.740354 2.0840380## 6    ML           6 1.260150 -0.1400093 1.872001 1.4855389

Information-theoretic measures of metacognition can be obtained by thefunctionestimateMetaI. It expects the same kind of data.frame asfitMetaDprime andfitConfModels, returning a dataframe with one rowfor each participant and the following columns:

• participant: the participant id,
• meta_I is the estimated meta-I value (expressed in bits, i.e. logbase is 2),
• meta_Ir1 is meta-$I_{1}^{r}$,
• meta_Ir1_acc is meta-$I_{1}^{r\prime}$,
• meta_Ir2 is meta-$I_{2}^{r}$, and
• RMI is RMI.

metaIMeasures<- estimateMetaI(data=MaskOri,bias_reduction=FALSE)head(metaIMeasures)

##   participant    meta_I meta_Ir1 meta_Ir1_acc  meta_Ir2       RMI## 1           1 0.1154252 1.300914     1.384554 0.1434999 0.3687714## 2           2 0.2034822 2.781828     2.815966 0.2432133 0.6708776## 3           3 0.1921722 1.549884     2.038785 0.2529526 0.6001439## 4           4 0.2223333 2.403126     2.429517 0.2884294 0.6969924## 5           5 0.2277498 2.922673     2.945799 0.2774337 0.7380805## 6           6 0.1648054 2.232405     2.276843 0.1969847 0.5433609

All information-theoretic measures can be calculated with a bias-reducedvariant for which the observed frequencies are taken as underlyingprobability distribution to estimate the sampling bias. The estimatedbias is then subtracted from the initial measures. This approach usesMonte-Carlo simulations and is therefore not deterministic. This is thepreferred way to estimate the information-theoretic measures, but it maytake ~ 6 s for each subject. To invoke bias reduction, the argumentbias_reduction needs to be set to TRUE:

metaIMeasures <- estimateMetaI(data = MaskOri, bias_reduction = TRUE)

After installation, the documentation of each function ofstatConfRcan be accessed by typing?functionname into the console.

The package is under active development. We are planning to implementnew models of decision confidence when they are published. Please feelfree tocontact us tosuggest new models to implement in the package, or to volunteer addingadditional models.

Implementing custom models of decision confidence is only recommendedfor users with experience in cognitive modelling! For readers who wantto use our open code to implement models of confidence themselves, thefollowing steps need to be taken:

• Derive the likelihood of a binary response ($R=-1, 1$) and aspecific level of confidence ($C=1,...K$) according to the custommodel and a set of parameters, given the binary stimulus($S=-1, 1$), i.e. $P(R, C | S)$.
• Use one of the files named ‘int_llmodel.R’ from the package sourcesand adapt the likelihood function according to your model. Accordingto our convention, name the new file a ‘int_llyourmodelname.R’. Notethat all parameters are fitted on the reals, i.e. positive parametersshould be transformed outside the log-likelihood function (e.g. usingthe logarithm) and back-transformed within the log-likelihood function(e.g. using the exponential).
• Use one of the files ‘int_fitmodel.R’ from the package sources andadapt the fitting function to reflect the new model.
  • The initial grid used during the grid search should include aplausible range of all parameters of your model.
  • If applicable, the parameters of the initial grid needs betransformed so the parameter vector for optimization isreal-valued).
  • The optimization routine should call the new log-likelihoodfunction.
  • If applicable, the parameter vector i obtained during optimizationneeds to back-transformation for the the output objectres.
  • Name the new file according to the convention‘int_fityourmodelname.R’.
• Add your model and fitting-functions to the high-level functionsfitConf andfitConfModels.
• Add a simulation function in the file ‘int_simulateConf.R’ which usesthe same structure as the other functions but adapt the likelihood ofthe responses.

For comments, bug reports, and feature suggestions please feel free towrite to eithermanuel.rausch@hochschule-rhein-waal.de orsebastian.hellmann@tum.de orsubmit anissue.

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