1 The statConfR package for R

The statConfR package provides functions to fit static models of decision-making and confidence derived from signal detection theory for binary discrimination tasks with confidence ratings on the data from individual subjects. Up to now, the following models 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 the meta-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, the statConfR package provides functions for estimating different kinds of measures of metacognition:

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

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

  • meta-I, an information-theoretic measures of metacognitive sensitivity,
  • $meta-I_{1}^{r}$ and $meta-I_{2}^{r}$, two measures of metacognitive 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 from information theory.

2 Mathematical description of implemented generative models of confidence

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

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

2.1 Signal detection rating model (SDT)

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

2.2 Gaussian noise model (GN)

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

2.3 Weighted evidence and visibility model (WEV)

Conceptually, the WEV model reflects the idea that the observer combines evidence about decision-relevant features of the stimulus with the strength of evidence about choice-irrelevant features to generate confidence. For this purpose, WEV assumes that $y$ is normally distributed with a mean of $(1-w)\times x+w \times d_k\times R$ and standard deviation $\sigma$. The standard deviation quantifies the amount of unsystematic variability contributing to confidence judgments but not to the discrimination judgments. The parameter $w$ represents the weight that is put on the choice-irrelevant features in the confidence judgment. The parameters $w$ and $\sigma$ are free parameters in addition to the set of shared parameters.

2.4 Post-decisional accumulation model (PDA)

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

2.5 Independent Gaussian model (IG)

According to IG, the information used for confidence judgments is generated independently from the sensory evidence used for the task decision. For this purpose, it is assumed that $y$ is sampled independently from $x$. The variable $y$ is normally distributed with a mean of $a\times d_k$ and variance of 1. The additional parameter $m$ represents the amount of information available for confidence judgment relative to amount of evidence available for the discrimination decision and can be smaller as well as greater than 1.

2.6 Independent truncated Gaussian model: HMetad-Version (ITGc)

Conceptually, the two ITG models just as IG are based on the idea that the information used for confidence judgments is generated independently from the sensory evidence used for the task decision. However, in contrast to IG, the two ITG models also reflect a form of confirmation bias in so far as it is not possible to collect information that contradicts the original decision. According to the version of ITG consistent with the HMetad-method (Fleming, 2017), $y$ is sampled independently from $x$ from a truncated Gaussian distribution with a location parameter of $S\times d_k \times m/2$ and a scale parameter of 1. The Gaussian distribution of $y$ is truncated in a way that it is 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 $c$. The additional parameter $m$ represents metacognitive efficiency, i.e., the amount of information available for confidence judgments relative to amount of evidence available for discrimination decisions and can be smaller as well as greater than 1.

2.7 Independent truncated Gaussian model: Meta-d’-Version (ITGcm)

According to the version of the ITG consistent with the original meta-d’ method (Maniscalco & Lau, 2012, 2014), $y$ is sampled independently from $x$ from a truncated Gaussian distribution with a location parameter 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 metacognitive efficiency, i.e., the amount of information available for confidence judgments relative to amount of evidence available for the discrimination decision and can be smaller as well as greater than 1.

2.8 Logistic noise model (logN)

According to logN, the same sample of sensory evidence is used to generate response and confidence, i.e., $y=x$ just as in SDT. However, according to logN, the confidence criteria are not assumed to be constant, but instead they are affected by noise drawn from a lognormal distribution. 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 lognormal distribution 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 designed to quantify metacognitive ability. It is assumed that the criterion noise is perfectly correlated across confidence criteria, ensuring that the 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 as free parameters. Instead, we estimate the means of the confidence criteria, i.e., $\mu_{\theta_{-1,1}}, ..., \mu_{\theta_{-1,L-1}}, \mu_{\theta_{1,1}}, ... \mu_{\theta_{1,L-1}}$, as free parameters.

2.9 Logistic weighted evidence and visibility model (logWEV)

The logWEV model is a combination of logN and WEV proposed by . Conceptually, logWEV assumes that the observer combines evidence about decision-relevant features of the stimulus with the strength of evidence about choice-irrelevant features. The model also assumes that noise affecting the confidence decision variable is lognormal. According to logWEV, the confidence decision variable is $y$ is equal to R × y’. The variable y’ is sampled from a lognormal distribution with a location parameter of $(1-w)\times x\times R + w \times d_k$ and a scale parameter of $\sigma$. The parameter $\sigma$ quantifies the amount of unsystematic variability contributing to confidence judgments but not to the discrimination judgments. The parameter $w$ represents the weight that is put on the choice-irrelevant features in the confidence judgment. The parameters $w$ and $\sigma$ are free parameters.

3 Measures of metacognition

3.1 meta-d$^\prime$/d$^\prime$

The conceptual idea of meta-d$^\prime$ is to quantify metacognition in terms of sensitivity in a hypothetical signal detection rating model describing the primary task, under the assumption that participants had perfect access to the sensory evidence and were perfectly consistent in placing their confidence criteria (Maniscalco & Lau, 2012, 2014). Using a signal detection model describing the primary task to quantify metacognition, it allows a direct comparison between metacognitive accuracy and discrimination performance because both are measured on the same scale. Meta-d$^\prime$ can be compared against the estimate of the distance between the two stimulus distributions estimated from discrimination responses, which is referred to as d$^\prime$: If meta-$^\prime$ equals d$^\prime$, it means that metacognitive accuracy is exactly as good as expected from discrimination performance. If meta-d$^\prime$ is lower than d$^\prime$, it means that metacognitive accuracy is not optimal. It can be shown that the implicit model of confidence underlying the meta-d$^\prime$/d$^\prime$ method is identical to different versions of the independent truncated Gaussian model (Rausch et al., 2023), depending on whether the original model specification by Maniscalco and Lau (2012) or alternatively the specification by Fleming (2017) is used. We strongly recommend to test whether the independent truncated Gaussian models are adequate descriptions of the data before quantifying metacognitive efficiency with meta-d$^\prime$/d$^\prime$ (see Rausch et al., 2023).

3.2 Information-theoretic measures of metacognition

It is assumed that a classifier (possibly a human being performing a discrimination task) or an algorithmic classifier in a classification application, makes a binary prediction $R$ about a true state of the world $S$. Dayan (2023) proposed several measures of metacognition based on quantities of information theory (for an introduction into information theory, see MacKay, 2003; Cover & Thomas, 2006).

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

$$meta-I = I(S; R, C) - I(S; R)$$

It can be shown that this is equivalent to Dayan’s formulation of meta-I as the information that confidence transmits about the correctness of a response:

$$meta-I = I(S = R; C)$$

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

$$meta-I_{1}^{r} = meta-I / meta-I(d')$$

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

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

$$meta-I_{2}^{r} = meta-I / H(S = R)$$

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

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

$$RMI = \frac{meta-I}{\max_{\text{accuracy}}\{meta-I\}}$$

As these measures are prone to estimation bias, the package offers a simple bias reduction mechanism in which the observed frequencies of stimulus-response combinations are taken as the underlying probability distribution. From this, Monte-Carlo simulations are conducted to estimate and subtract the bias from these measures. Note that the bias is only reduced but not removed completely.

4 Installation

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

install.packages("statConfR")

The easiest way to install the development version is using devtools and install from GitHub:

devtools::install_github("ManuelRausch/StatConfR")

5 Usage

5.1 Example data set

The package includes a demo data set from a masked orientation discrimination 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

5.2 Fitting models of confidence and decision making to individual subjects

The function fitConfModels allows the user to fit several confidence models separately to the data of each participant using maximum likelihood estimation. The data should be provided via the argument .data in the form of a data.frame object with the following variables in separate columns:

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

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

The fitting routine first performs a coarse grid search to find promising starting values for the maximum likelihood optimization procedure. Then the best nInits parameter sets found by the grid search are used as the initial values for separate runs of the Nelder-Mead algorithm implemented in optim (default: 5). Each run is restarted nRestart 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 of participant 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.4 for both models, w and sigma for WEV, and m only for ITGcm) as well as for 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

5.3 Visualization of model fits

After obtaining the model fit, it is strongly recommended to visualise the predictions implied by the best-fitting set of parameters and compare these predictions with the actual data (Palminteri et al., 2017). The statConfR package provides the function plotConfModelFit, which creates a ggplot object with empirically observed distribution of responses and confidence ratings as bars on the x-axis as a function of discriminability (in the rows) and stimulus (in the columns). Superimposed on the empirical data, the plot also shows the prediction of one selected model as dots. The parameters of the model are passed to plotConfModelFit by the argument fitted_pars.

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

PlotFitWEV

Observed distribution of accuracy and responses as a function of discriminability and stimulus vs. prediction by the weighted evidence and visibility model

PlotFitITGcm

Observed distribution of accuracy and responses as a function of discriminability and stimulus vs. prediction by the Independent truncated Gaussian model: HMetad-Version (ITGc)

5.4 Estimating measures of metacognition

Assuming that the independent truncated Gaussian model provides a decent account of the data (notably, this is not the case in the demo data set), the function fitMetaDprime can be used to estimate meta-d$^\prime$/d$^\prime$ independently for each subject. The arguments .data and .parallel=TRUE work just in the same way the arguments of fitConfModels. The argument model offers the user the choice between two model specifications, either “ML” to use the original model specification used by Maniscalco and Lau (2012, 2014) or “F” to use the model specification by Fleming (2017)’s Hmetad method. The function fitMetaDprime produces a dataframe with one row for each participant 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, a measure of discrimination performance,
• c: the bias index c from signal detection theory, a measure of discrimination bias,
• Ratio: The meta-d′/d′ index, the most common measure of metacognitive efficiency.

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 the function estimateMetaI. It expects the same kind of data.frame as fitMetaDprime and fitConfModels, returning a dataframe with one row for each participant and the following columns:

• participant: the participant id,
• meta_I is the estimated meta-I value (expressed in bits, i.e. log base 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-reduced variant for which the observed frequencies are taken as underlying probability distribution to estimate the sampling bias. The estimated bias is then subtracted from the initial measures. This approach uses Monte-Carlo simulations and is therefore not deterministic. This is the preferred way to estimate the information-theoretic measures, but it may take ~ 6 s for each subject. To invoke bias reduction, the argument bias_reduction needs to be set to TRUE:

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

6 Documentation

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

7 Contributing to the package

The package is under active development. We are planning to implement new models of decision confidence when they are published. Please feel free to contact us to suggest new models to implement in the package, or to volunteer adding additional models.

Implementing custom models of decision confidence is only recommended for users with experience in cognitive modelling! For readers who want to use our open code to implement models of confidence themselves, the following steps need to be taken:

• Derive the likelihood of a binary response ($R=-1, 1$) and a specific level of confidence ($C=1,...K$) according to the custom model 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 sources and adapt the likelihood function according to your model. According to our convention, name the new file a ‘int_llyourmodelname.R’. Note that all parameters are fitted on the reals, i.e. positive parameters should be transformed outside the log-likelihood function (e.g. using the 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 and adapt the fitting function to reflect the new model.
  • The initial grid used during the grid search should include a plausible range of all parameters of your model.
  • If applicable, the parameters of the initial grid needs be transformed so the parameter vector for optimization is real-valued).
  • The optimization routine should call the new log-likelihood function.
  • If applicable, the parameter vector i obtained during optimization needs to back-transformation for the the output object res.
  • Name the new file according to the convention ‘int_fityourmodelname.R’.
• Add your model and fitting-functions to the high-level functions fitConf and fitConfModels.
• Add a simulation function in the file ‘int_simulateConf.R’ which uses the same structure as the other functions but adapt the likelihood of the responses.

8 Contact

For comments, bug reports, and feature suggestions please feel free to write to either [email protected] or [email protected] or submit an issue.

9 References

• Cover, T. M., & Thomas, J. A. (2006). Elements of information theory. 2nd edition. Wiley.
• Dayan, P. (2023). Metacognitive Information Theory. Open Mind, 7, 392–411. https://doi.org/10.1162/opmi_a_00091
• Fleming, S. M. (2017). HMeta-d: Hierarchical Bayesian estimation of metacognitive efficiency from confidence ratings. Neuroscience of Consciousness, 1, 1–14. https://doi.org/10.1093/nc/nix007
• Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Wiley.
• Hellmann, S., Zehetleitner, M., & Rausch, M. (2023). Simultaneous modeling of choice, confidence, and response time in visual perception. Psychological Review, 130(6), 1521–1543. https://doi.org/10.1037/rev0000411
• Maniscalco, B., & Lau, H. (2012). A signal detection theoretic method for estimating metacognitive sensitivity from confidence ratings. Consciousness and Cognition, 21(1), 422–430. https://doi.org/10.1016/j.concog.2011.09.021
• MacKay, D. J. (2003). Information theory, inference and learning algorithms. Cambridge University Press.
• Maniscalco, B., & Lau, H. (2016). The signal processing architecture underlying subjective reports of sensory awareness. Neuroscience of Consciousness, 1, 1–17. https://doi.org/10.1093/nc/niw002
• Maniscalco, B., & Lau, H. C. (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 S. M. Fleming & C. D. Frith (Eds.), The Cognitive Neuroscience of Metacognition (pp. 25–66). Springer. https://doi.org/10.1007/978-3-642-45190-4_3
• Palminteri, S., Wyart, V., & Koechlin, E. (2017). The importance of falsification in computational cognitive modeling. Trends in Cognitive Sciences, 21(6), 425–433. https://doi.org/10.1016/j.tics.2017.03.011
• Rausch, M., Hellmann, S., & Zehetleitner, M. (2018). Confidence in masked orientation judgments is informed by both evidence and visibility. Attention, Perception, and Psychophysics, 80(1), 134–154. https://doi.org/10.3758/s13414-017-1431-5
• Rausch, M., & Zehetleitner, M. (2017). Should metacognition be measured by logistic regression? Consciousness and Cognition, 49, 291–312. https://doi.org/10.1016/j.concog.2017.02.007
• Shekhar, M., & Rahnev, D. (2021). The Nature of Metacognitive Inefficiency in Perceptual Decision Making. Psychological Review, 128(1), 45–70. https://doi.org/10.1037/rev0000249
• Shekhar, M., & Rahnev, D. (2024). How Do Humans Give Confidence? A Comprehensive Comparison of Process Models of Perceptual Metacognition. Journal of Experimental Psychology: General, 153(3), 656–688. https://doi.org/10.1037/xge0001524