Theories of visual confidence have largely been grounded in the gaussian signal detection framework. This framework is so dominant that idiosyncratic consequences from this distributional assumption have remained unappreciated. This article reports systematic comparisons of the gaussian signal detection framework to its logistic counterpart in the measurement of metacognitive accuracy. Because of the difference in their distribution kurtosis, these frameworks are found to provide different perspectives regarding the efficiency of confidence rating relative to objective decision (the logistic model intrinsically gives greater meta-dʹ/dʹ ratio than the gaussian model). These frameworks can also provide opposing conclusions regarding the metacognitive inefficiency along the internal evidence continuum (whether meta-dʹ is larger or smaller for higher levels of confidence). Previous theories developed on these lines of analysis may need to be revisited as the gaussian and logistic metacognitive models received somewhat equivalent support in our quantitative model comparisons. Despite these discrepancies, however, we found that across-condition or across-participant comparisons of metacognitive measures are relatively robust against the distributional assumptions, which provides much assurance to conventional research practice. We hope this article promotes the awareness for the significance of hidden modeling assumptions, contributing to the cumulative development of the relevant field.

^{1}The logistic SDT has been actively in use for psychophysics (e.g., DeCarlo, 1998; Kornbrot, 2006; Pleskac, 2015), and there has been an emerging interest in its application to type 2 ROC analyses (Kristensen, Sandberg, & Bibby, 2020). Earlier literature stated that behaviors of the gaussian and logistic SDTs are practically indistinguishable (Luce, 1959), and they are unlikely to support discrepant conclusions in empirical analyses (Macmillan & Creelman, 2005). However, as we shall demonstrate later, the difference in their distributional forms has a considerable impact on the relative magnitude of meta-dʹ against dʹ. In an extreme case scenario, the gaussian and logistic SDTs even provide qualitatively opposite conclusions regarding the observer's metacognitive efficiency (i.e., m-ratio > 1 or < 1), which would have significant implications for the theories of metacognition.

^{2}

^{3}Here, the genuine 2AFC refers to the task design which offers two explicit stimulus intervals on every trial and requires participants to identify which of the two includes the target stimulus (Macmillan & Creelman, 2005, chap. 7). In order to evaluate metacognitive accuracy's criterion-dependency, we have only targeted experiments that employed four or more levels of confidence rating. The following analyses were conducted on the free statistical language R (Version 4.0.5).

_{S1}) and the S1-response-specific type 2 false alarm rate (far2

_{S1}) were defined by the following equations, where τ

_{gaussian_S1}is the confidence criterion on the S1 response side, meta-dʹ

_{gaussian}is the gaussian meta-dʹ, Φ

_{0}is the cumulative standard gaussian distribution function, and Φ

_{meta-}

_{dʹ}

_{_gaussian}is the cumulative gaussian distribution function with the standard deviation of 1 and the mean value that is equal to the gaussian meta-dʹ. In the fitting of multilevel confidence rating data, τ

_{gaussian_S1}becomes a vector containing a series of confidence criteria, and hr2

_{S1}and far2

_{S1}are defined at each location of those.

_{S2}) and the S2-response-specific type 2 false alarm rate (far2

_{S2}) were defined as follows, where τ

_{gaussian_S2}is the confidence criterion (or criteria) on the S2 response side.

_{logistic}is the logistic meta-dʹ, Λ

_{0}is the cumulative logistic distribution function with the scale parameter of 1 and the location parameter of 0, while Λ

_{meta-}

_{dʹ}

_{_logistic}is the cumulative logistic distribution function with the scale parameter of 1 and the location parameter that is equal to the logistic meta-dʹ. Also, τ

_{logistic_S1}and τ

_{logistic_S2}are the confidence criterion (or criteria) on each response side. As in the case of the gaussian fitting, the parameters were estimated by the maximization of the log-likelihood, which is again given by the sum of the products of the log-estimated probability of each response category and the observed response frequency for the corresponding category.

*t*-tests showed that mean meta-dʹ was significantly smaller than mean dʹ under the gaussian meta-SDT (

*t*(3817) = −16.99,

*p*< 0.001, Figure 5A) and the logistic meta-SDT (

*t*(4126) = −4.03,

*p*< 0.001, Figure 5B), indicating metacognitive inefficiency on average basis. However, caution would be advised because the logistic meta-SDT indicated greater mean m-ratio than the gaussian meta-SDT (

*t*= 28.09,

*p*< 0.001, Figure 5C), and there are even cases where the gaussian meta-SDT showed m-ratio < 1 whereas the logistic meta-SDT demonstrated m-ratio > 1 (399 of 3552 cases in Figure 5C). These results exemplify the important consequences from the distributional assumptions. In an extreme scenario, it is even possible that the models advocate for the qualitatively opposite theoretical operations (i.e., contamination of metacognitive noise vs. acquisition of metacognitive evidence).

*p*< 0.001). Despite these differences, however, m-ratios estimated by these models were highly consistent across the individual cases (Pearson's

*r*= 0.942), ensuring the models’ reliability in the assessment of metacognitive accuracy (Figure 5C).

^{4}

*t*= −3921.54,

*p*< 0.001) and gaussian m-ratio (

*t*= −16.52,

*p*< 0.001), which is consistent with the previous report of greater metacognitive inefficiency at higher confidence criteria (Rahnev, 2021; Shekhar & Rahnev, 2021). Importantly, however, the logistic meta-SDT showed positive criterion-dependency for meta-dʹ (

*t*= 11.27,

*p*< 0.001) and m-ratio (

*t*= 38.20,

*p*< 0.001), indicating that metacognition is more efficient at higher confidence criteria. The reversal of the criterion dependency again showcases the serious consequence of the auxiliary modeling assumptions.

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