Attributing ourselves agency, or causation of our actions and their outcomes, is central to our experience of moving intentionally. Previous research has suggested that agency depends on a comparison between the predicted and observed consequences of our actions, resulting in a prediction error signal if they do not match. Under this widely accepted comparator model of agency, it is the prediction error signal that leads to our feeling of agency (FoA), which we assess in order to make judgments of agency (JoAs) (Carruthers, 2012; Frith et al., 2000; Haggard, 2017). There are several sources of noise to these signals, and agency processing, much like other perceptual processing, occurs under varying degrees of uncertainty. Noisy agency signals are suggested to play a role in clinical cases involving striking disruptions to agency, such as in schizophrenia (Corlett et al., 2010; Fletcher and Frith, 2009; Moore and Fletcher, 2012; Robinson et al., 2016), so it is critical to understand precisely how agency signals change with uncertainty, in both healthy and clinical populations.

Some existing models of agency have suggested that noise affects the comparator signal because the latter is the result of a cue integration process, in which the motor and sensory information are integrated and weighted by their reliability (Moore and Fletcher, 2012; Moore and Haggard, 2008). In line with these models, participants’ FoA, as measured by an implicit temporal binding effect, has been found to rely less on cues that are noisy or imprecise, and more on more informative cues (Moore et al., 2009; Wolpe et al., 2013). Further, one recent study put forward a possible computational model in which FoAs are affected by sensory noise (Legaspi and Toyoizumi, 2019). However, work on uncertainty in lower level agency processing has not addressed the role of noise in the formation of JoAs. It is not clear that explicit JoAs incorporate noise in the same way as lower level FoAs, as JoAs are generally considered to be at a higher level of the processing hierarchy (Gallagher, 2007; Haggard and Tsakiris, 2009; Sato, 2009; Synofzik et al., 2008).

Explicit JoAs could incorporate noise in two main ways. One alternative is that JoAs are simply reports of FoAs, that do not involve secondary assessments of their reliability. A second alternative is that JoAs monitor the uncertainty of the first-order signal through computations that include a secondary assessment of the sensory uncertainty. The crucial difference between these two alternatives is whether JoAs require an estimate of the precision of the first-order signal, or can be determined without any information about it. This kind of second-order representation of the precision of a first-order signal is a defining feature of what has been called a metacognitive process (Shea, 2012). Accordingly, metacognitive judgments are often operationalized as confidence ratings following discrimination responses (Fleming and Lau, 2014), and these second-order metacognitive ratings have been shown to monitor the uncertainty of the perceived internal signal (de Gardelle et al., 2015; Navajas et al., 2017; Rausch et al., 2017; Sanders et al., 2016; Spence et al., 2016).

In the literature, JoAs have sometimes been considered to be metacognitive reports about otherwise first-order agency signals (Carruthers, 2015; Metcalfe et al., 2012; Metcalfe and Greene, 2007; Miele et al., 2011; Potts and Carlson, 2019; Terhune and Hedman, 2017; Wenke et al., 2010; Zalla et al., 2015). However, it is unclear if the link to metacognition is merely conceptual, or whether JoAs result from second-order uncertainty representations, and are computationally comparable to other metacognitive judgments. Here, we commit to a definition of metacognition as a process that involves second-order uncertainty monitoring computations, to test the link between JoAs and metacognition in this narrow but computationally tractable sense.

We propose that for agency judgments to be metacognitive in a computational sense, they should monitor the noise in the perceived prediction error signal, incorporating a second-order judgment of the uncertainty of one’s agency processing. We examined this by setting up a two-criterion test. The two criteria refer to behavioral and computational measures, respectively. The first criterion for JoAs to be computationally metacognitive was for sensory noise to influence JoAs in a secondary way, beyond just altering their variance across trials. We expected the effect of the delay to become smaller in high perceptual uncertainty trials, leading JoAs to be less extreme, as compared to trials with lower uncertainty. This would suggest that the reliability of the signal was estimated and factored in for each rating, against the simpler alternative that the rating was made on the basis of a linear readout of the less reliable signal. We examined this by assessing the effects of noise and delay on explicit agency ratings, using a sensory noise manipulation orthogonal to the delay.

This first criterion formed our pre-registered hypothesis and was necessary, but not sufficient, for JoAs to be considered metacognitive. It therefore served as a prerequisite for the second criterion: Agency and confidence judgments should show similar sensitivity to internal estimates of the sensory noise, suggesting the involvement of second-order uncertainty monitoring following the same computational principles. We assessed this by contrasting two computational models against distributions of JoA data, one that would reflect metacognitive monitoring of noise and an alternative model that would not. We found that JoAs satisfied the first, but not the second criterion: Sensory noise did indeed influence JoAs, but this influence did not reflect any second-order noise monitoring, suggesting that JoAs may not be metacognitive in the computational sense.

Each participant completed two tasks: A confidence-rating task, consisting of a two-interval forced choice (2IFC) followed by a confidence rating on a scale from 1 to 6; and an agency-rating task consisting of a JoA on an equivalent scale. Both tasks used the same basic stimuli, namely, a movement of participants’ index finger tracked by a LEAP Motion infrared tracker and displayed on the screen as a virtual hand movement either in synchrony or with small temporal delays. In both tasks, we manipulated sensory noise in the same way, by changing the illumination of the scene. We created two conditions (low and high sensory noise) by displaying the virtual hand under bright, high-contrast illumination or under dim, low-contrast illumination respectively (Figure 1A). This change in illumination, a manipulation that was orthogonal to the stimulus intensity, effectively changed the reliability of the mapping between the external and internal signal without affecting the stimulus.

Figure 1

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Confirmatory analyses

Confidence-rating task

On each trial of the confidence task, participants were cued to make two consecutive movements (which constituted the two intervals of the 2IFC) of their right index finger, with their hand out of sight. For only one of the two movements, we added a temporal delay to the virtual hand shown on the screen (in the other interval, the virtual hand was displayed to match the participant’s hand movement in real time). Participants then discriminated in which interval they felt more agency over the movement of the virtual hand, and rated confidence in their response (Figure 1B). We assumed that participants compared their degree of control over the virtual hand in the two movements to solve the task, and rated confidence in this comparison. This paradigm brings agency into a standard framework for studying metacognition (Wang et al., 2020). Importantly, under this operationalization, we can define correct responses to the 2IFC discrimination task as those where participants report that they felt more agency for the stimulus without any added delay, allowing us to quantify discrimination accuracy. If the illumination manipulation served to increase sensory noise in the intended way, we expected lower discrimination accuracy under high sensory noise compared to low noise (Macmillan and Creelman, 1991). Further, based on previous work using similar confidence paradigms (Bang and Fleming, 2018; de Gardelle et al., 2015; Spence et al., 2016), and a normative model of confidence (Sanders et al., 2016), we predicted an interaction between sensory noise and accuracy on confidence, in particular with confidence decreasing in high noise following correct trials and increasing in high noise following incorrect trials. To test the effect of the illumination manipulation, we first built a logistic regression model on response accuracy, including sensory noise as a fixed effect, and by-participant random intercepts (see Table 1 for the explicit model syntax). As expected, we found a main effect of Noise, revealing significantly lower accuracy in the high-noise compared to the low-noise condition (Mdiff = 10%, SE = 1.4%, χ²(1) = 97.60, p < 0.001, BF₁₀ = 1.78 ✕ 10²⁰, OR = 1.55, 95% CI [1.42, 1.70]; Figure 1C).

Table 1

Task	Hypothesis	Model formula
Confidence Task	Sensory noise influences response accuracy	logit(Response Accuracy)~ Noise + (1 | Participant)
Confidence Task	Sensory noise influences confidence following correct decisions	Confidence~ Response + Response Accuracy*Noise + (1 | Participant)
Agency Rating Task	Sensory noise influences the effect of delay on JoA	JoA ~ Delay*Noise + (Delay:Noise | Participant)

Then, we built a linear mixed-effects model on confidence to test the second prediction (an interaction effect between sensory noise and response accuracy). The model included the interaction between response accuracy and noise level and each factor as fixed effects, as well as by-participant random intercepts. We also included response identity (first or second interval) as a fixed effect, as presentation order could have biased confidence ratings (Jamieson and Petrusic, 1975; Yeshurun et al., 2008). In line with our predictions, we found a significant interaction between Noise and Response Accuracy on confidence, F(1,8858) = 14.43, p < 0.001, BF₁₀ = 2.19, η²_p = 0.0016 (Figure 1D), with a stronger difference in confidence between correct and incorrect trials under low noise (Mdiff_{Correct-Incorrect} = 0.58, SE = 0.044) compared to high noise (Mdiff_{Correct-Incorrect} = 0.34, SE = 0.042). In addition to the interaction effect, we found that confidence following incorrect decisions was lower in the high-noise compared to the low-noise condition. Although we expected confidence following incorrect decisions to increase under high noise, the ‘double-increase’ confidence pattern seen here has also been shown in the literature (Adler and Ma, 2018; Rausch et al., 2017; Rausch et al., 2019). Finally, we found a significant main effect of Response, F(1,8872) = 82.64, p < 0.001, BF₁₀ = 5.71 ✕ 10¹⁴, η²_p = 0.0092, with pairwise comparisons revealing significantly higher confidence ratings when participants reported feeling more agency over the stimulus in the second interval, compared to the first (Mdiff = –0.27, SE = 0.030), t(8872) = –9.09, p < 0.001. These results were also all confirmed by repeating this analysis with ordinal models (Appendix 2).

In order to further ensure that participants’ confidence ratings reflected metacognitive processing, especially given relatively low accuracy under high noise, we checked that their metacognitive efficiency (M-Ratio) was above-chance in both conditions (Figure 1E; see Appendix 3 for full analysis). Together, these results suggest that the illumination manipulation affected sensory noise as we intended, and influenced metacognitive confidence judgments as previous studies have shown.

Agency-rating task

On each trial of the agency-rating task, participants made a single movement of their index finger and watched the virtual hand model either move synchronously with their movement (25% of trials) or with a delay of either 70, 100, or 200 ms. After every movement, participants rated their agency on a scale from 1 to 6 (Figure 2A). By adding sensory noise to the perceived sensory outcome of the movement (namely, the virtual hand movement), we added noise to the comparator signal that participants assessed with their JoAs (Figure 2B). We then formulated a two-criterion test to assess whether JoAs are computationally metacognitive, and hence involve second-order estimates of the noise. The first — behavioral — criterion was that mean JoAs should depend on both delay and sensory noise. This would suggest that sensory noise influences agency ratings beyond just increasing their variability, with agency ratings instead including an indication of the signal’s precision (Figure 2B). Alternatively, if the mean JoA per delay does not depend on the sensory noise level, this would indicate that JoAs are simply a first-order report of the perceived comparator signal, with mean JoA reflecting the mean of the comparator signal distribution (Figure 2B). The second criterion of our test, if the first criterion was met, was at the computational level, for agency ratings to involve underlying metacognitive computations such as those involved in confidence, in particular, the second-order monitoring of sensory noise. To test this, we compared two computational models built based on the results of the first criterion, a Second-order agency model that involves metacognitive processing, and an alternative First-order model that does not.

Figure 2

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Behavioral results

While the results of the confidence task confirmed that the illumination manipulation affected sensory noise overall as intended, we were interested in examining precisely how JoAs responded to sensory noise, and hence required that the high-noise condition actually increased sensory noise for all participants included in the analysis of the agency rating task. We therefore excluded from the following analyses any participants for whom discrimination accuracy in the high-noise condition was not lower than in the low-noise condition in the confidence task. We note that all the results described below remain largely the same, and the conclusions unchanged, when we include the data from all participants in the analyses (see Appendix 1). We predicted that if sensory noise affected JoAs similarly to metacognitive processes, we would observe a significant interaction between Noise and Delay on JoA (Figure 2B). This would be the first of our two criteria. In particular, if uncertainty led to a metacognitive scaling of JoAs, similarly to confidence-weighted agency ratings, we would expect to see less extreme JoAs under high noise, with low certainty pulling ratings closer to the decision criterion. We investigated this using a linear mixed-effects model on JoAs that included the interaction between noise level and delay as fixed effects, and allowed for by-participant random effects of the interaction, and random intercepts (Table 1). We found a significant interaction effect between Noise and Delay, F(1,52) = 61.16, p < 0.001, BF₁₀ = 3.78 ✕ 10⁶, η²_p = 0.54, 95% CI [0.35, 0.67], with a less extreme negative slope across delay values in the high-noise condition (β_High = -5.93, SE = 0.69), compared to low-noise (β_Low = -9.84, SE = 0.77) (Figure 2C), suggesting that JoAs met our first criterion. We also found a significant main effect of Delay, F(1,39) = 132.05, p < 0.001, BF₁₀ = 14486.52, η²_p = 0.77, 95% CI [0.64, 0.85], replicating previous findings that showed increasing delays of the virtual hand movement to lead to lower JoAs (Krugwasser et al., 2019; Stern et al., 2020). We found this effect of JoAs decreasing with delay for the majority of participants (37 out of 40) included in this analysis in both conditions, indicating that participants were able to make meaningful ratings, even in the high-noise condition. We also repeated these analyses with ordinal models, which confirmed our results (Appendix 2).

Exploratory analyses

Computational modeling

We found in our behavioral analysis that the mean JoA depended on both delay and noise, meeting our first criterion for JoAs being metacognitive. This allowed us to move on to our second test-criterion and investigate whether there are strictly metacognitive computations underlying agency judgments. In order to test our second criterion, namely, whether JoAs can be explained by the same computations as confidence, we compared two possible models of agency ratings (the Second-order model and the First-order model, Figure 3A), that differed in their predicted distributions of JoAs across noise conditions and delays (Figure 3B). Both models could in principle account for the observed interaction effect between noise and delay on JoA, satisfying the first criterion. However, only the Second-order model required participants to be able to reflect on the noise of their own sensory processing. The Second-order model assumed that agency ratings behave like confidence as described by Bayesian-confidence models, namely as the posterior probability that a decision is correct, given the strength of the internal evidence and the decision (Navajas et al., 2017; Sanders et al., 2016). The computation of this probability requires the observer to have second-order access to estimate their own sensory noise (Figure 3A), beyond just adopting precision-weighted cues from the lower level. So, in this model, JoAs are metacognitive because they require a representation of the precision of the first-order agency signal.

Figure 3

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As an alternative, we considered the First-order model, which parallels the Second-order one except that ratings are based on first-order point-estimates of evidence and therefore do not factor in metacognitive estimates of sensory noise (Figure 3A). This First-order model accounts for the observed relationship between noise and JoAs by considering that participants might simply treat the noise conditions independently. In practical terms, this implies that participants judged agency in low-noise trials relative to one another, and high-noise trials relative to one another. Although our design aimed to prevent this by interleaving the conditions, it is still possible that participants did this to some extent, as our noise manipulation was visually very apparent. Critically, this condition-dependent scaling is achieved without making any estimates of the sensory noise of the conditions. The only role noise plays in this model is to make the conditions distinguishable from one another such that participants might treat them independently and use the rating scale accordingly. This means that in each condition, the maximum of the scale, or a ‘6’, refers to the maximum agency experienced only within trials of that condition. Therefore, in the First-order model, a ‘6’ may differ between conditions in terms of the actual strength of the agency experience. In contrast, in the Second-order model, a JoA of ‘6’ would always reflect the same perceived level of certainty about the agency decision and hence the same agency experience, since JoAs combine the evidence strength and sensory noise.

We compared the two models in their ability to account for the distributions of JoAs per delay and noise level (Figure 4A) at the group level (with pooled data) and at the single-participant level. We also measured protected exceedance probabilities (PEPs) to assess the probability that each model was the most frequently occurring one in the population, adjusted for the ability to reject the null hypothesis (that the models are equally likely in the population). To compare the models, we first found the best fitting parameter values for each model — low noise σ_L, high noise σ_H, decision criterion (c), and the mapping parameter, or the number of ratings to be considered as ‘Yes’ responses (NRYes), for both models, as well as the scale range parameter (s) for the First-order model — using maximum likelihood estimation (MLE), and then performed a Bayesian-information criterion (BIC) comparison. Following standard recommendations (Raftery, 1995), we required a minimum BIC difference of 2 in order to consider one model a better explanation of the data than the other. To perform the Bayesian model selection and calculate the PEPs, we used the ‘bmsR’ package (Lisi, 2021) with model evidence computed from the Akaike weights (Penny, 2012; Rigoux et al., 2014; Stephan et al., 2009; Wagenmakers and Farrell, 2004). We also performed a model recovery analysis across the set of winning parameter values of each model across participants, confirming that the models were distinguishable from one another over the entire relevant parameter space (see Appendix 4).

Figure 4

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Against the notion that agency ratings and confidence arise from analogous uncertainty monitoring computations, the group-level analysis revealed that the First-order model could better explain the JoA data (ΔBIC_Second-First = 2728, Figure 4B). The best fitting parameters for this First-order model were σ_L = 0.16, σ_H = 0.16, (c) = 0.16, NRYes = 3, and (s) = 1.11. In comparison, the best fitting parameters for the Second-order model were σ_L = 0.19, σ_H = 0.24, (c) = 0.16, and NRYes = 3. The predicted densities of each rating per delay and noise level for each model’s best fitting parameters can be seen in Figure 4A. Fitting the two models to each participant revealed results consistent with the group-level analysis: The First-order model could better explain the data for 38 out of 40 participants, whereas the Second-order model provided a better fit for only 2 (Figure 4C). Also in line with this, the PEPs indicated that the First-order model occurs most frequently in the population (Figure 4D), with the predicted frequencies shown in Figure 4E. This was despite the First-order model including one additional free parameter, compared to the Second-order model.

We then performed the same model comparison on confidence ratings from the confidence task in order to confirm metacognitive computations underlying confidence using this modeling approach, for comparison to our agency-rating results. As expected, the Second-order model could better explain confidence ratings (ΔBIC_First-Second = 1121, Figure 5A), suggesting confidence to involve metacognitive computations, in contrast to JoAs. The PEPs also suggested the Second-order model to be the most frequently occurring model in the population (Figure 5C), with estimated frequencies shown in Figure 5D. The subject-wise BIC comparison revealed the Second-order model to provide a better fit in 24 out of 40 participants, the First-order model to provide a better fit in 13, and a BIC difference of less than 2 (suggesting neither model being a conclusively better fit) in 3 (Figure 5B).

Figure 5

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Previous research has investigated the effect of noise on implicit proxies of agency such as temporal binding effects, but it has remained unclear how explicit agency judgments incorporate uncertainty. It is possible that JoAs adopt the effect of uncertainty at the first-order level, by taking a direct readout of the variable lower level signal. However, it is also possible that JoA computations involve a second-order estimate of the noise of first-order signals, which we would take to be a metacognitive computation. Here, we tested whether JoAs are metacognitive in a computational sense by examining whether they involve the same second-order uncertainty monitoring computations as metacognitive confidence ratings, or whether they can only be said to be metacognitive at a broad conceptual level. We found that uncertainty did not affect agency ratings in the same way as it does confidence ratings, suggesting that any effect of noise is better considered as an influence on first-order signals, without involving metacognitive noise estimates.

By combining two tasks, we brought JoAs into a standard metacognitive framework and compared them to confidence ratings following a 2IFC decision. We examined how discrimination accuracy and confidence changed with sensory noise in the confidence-rating task, to first confirm that the sensory noise manipulation had the intended effect. Observing the expected effects of sensory noise on confidence allowed us to consider whether JoAs in the agency-rating task responded to sensory noise in a computationally analogous way. We reasoned that, if this were the case, JoAs would satisfy two criteria: First, they would depend on the precision of the comparator information, reflecting more than just a readout of the perceived signal. Second, this dependence on noise would reflect underlying metacognitive computations such as those involved in confidence, in particular, second-order estimates of one’s own sensory noise. The JoAs satisfied the first of the test-criteria. We found that noise did indeed influence mean JoA across delays, indicating that the noise condition is factored into JoAs. However, because the noise manipulation changed the display in a visually obvious way, this information about the condition could have influenced judgments in a way that did not reflect participants making metacognitive estimates of the noise of their own processing. The second test-criterion investigated precisely this possibility.

To assess the second criterion, we compared two computational models. As a prerequisite, both models satisfied the first test-criterion. We contrasted a Second-order agency model, which included metacognitive noise estimates, with a First-order model that did not imply metacognitive processing. We tested these models in their ability to fit participants’ JoAs as well as confidence judgments, to understand the computations underlying both. In the case of confidence, the model comparison revealed that participants' judgments were best explained by the Second-order model, confirming the metacognitive computations underlying them (Kepecs and Mainen, 2012; Meyniel et al., 2015; Pouget et al., 2016; Sanders et al., 2016). In striking contrast, this model comparison against participants’ agency ratings revealed that JoAs were better explained by the First-order model as compared to the Second-order agency-model. The First-order model accounted for the observed behavioral relationship between JoA and noise by assuming that participants compared trials only to other trials within the same condition, and set condition-specific maxima of their rating scales accordingly. It would be an interesting direction for future work to test how JoAs depend on sensory noise under a noise manipulation that is not easily detectable, to investigate if the behavioral relationship we observe between noise and JoAs is limited to cases in which participants can treat the manipulation conditions as independent contexts, as our models suggest.

Taken together, our results suggest that while JoAs can be influenced by sensory noise, this influence is not indicative of metacognitive processing, and JoAs may better reflect first-order assessments of agency signals. We therefore argue that greater care should be taken when discussing agency within a metacognitive context, as the assumptions made about agency judgments being metacognitive do not hold on a computational level. Although this work used confidence as a benchmark for metacognitive processing, the computation of interest is second-order monitoring of the precision of one’s processing, which has become the narrower focus of recent metacognition work (Fleming and Lau, 2014). While JoAs may still satisfy broad definitions of metacognition, our results suggest that they may not satisfy this narrower definition that is associated with a concrete computational view. In this sense, these results may help specify and clarify the assumed relationships between explicit agency judgments and metacognition. At the same time, they may add to our understanding of how JoAs respond to uncertainty — similarly to first-order perceptual judgments — which is critical for interpreting agency reports, especially in certain clinical cases involving agency disruptions.

Importantly, we also found that participants can make metacognitive confidence judgments about agency decisions, but that subjective agency ratings do not share these computations, despite the same basic agency task and noise manipulation. The use of 2IFC agency tasks with confidence has recently been proposed as a promising step toward a more reliable and complete investigation of agency processing, both in healthy and clinical populations (Wang et al., 2020). Here, using a virtual hand, we extended this approach into a more proximal form of embodied agency, closer to agency over the body itself (Christensen and Grünbaum, 2018; Dogge et al., 2019; Stern et al., 2020; Wen, 2019), and provided an initial step in demonstrating that participants can meaningfully monitor the accuracy of these agency decisions. We suggest that confidence judgments about agency should be considered as the metacognitive level of an agency processing hierarchy, with agency judgments as explicit first-order judgments. This also brings agency in line with recent motor metacognition research that considers agency-like judgments such as decisions of which trajectory was caused by one’s movement to be the first-order motor judgments, followed by metacognitive confidence ratings (Arbuzova et al., 2021).

Although we suggest that JoAs do not imply metacognitive uncertainty monitoring, the dependence of agency ratings on the noise condition should not be overlooked and may be highly relevant both for future experimental design and in the interpretation of explicit agency reports. This finding is in line with multifactorial accounts of JoAs as involving a variety of both internal and external cues (Synofzik et al., 2008), and with the expanding empirical work investigating a range of contextual effects on agency (Minohara et al., 2016; Wen, 2019). Our work also fits within cue integration theories of agency (Moore and Fletcher, 2012; Synofzik et al., 2013), with the delay information being weighted less heavily when the signal is made less precise. Further, these results are relevant to empirical work examining cue integration in agency, as they suggest that having a perceivable manipulation such as reduced visibility in order to add noise to feedback cues may itself act as an additional factor influencing agency judgments, which should be accounted for in design and analysis.

These findings also complement recent work that has aimed to find computational models of agency, but has focussed on low-level FoA and implicit measures such as temporal binding effects (Legaspi and Toyoizumi, 2019). Here, we bring explicit agency judgments into a Bayesian and SDT framework, implementing formal computational models that could be used to further assess computations underlying different JoAs. Our findings support the suggestions of previous work that, while Bayesian confidence computations may underlie pre-reflective FoA, explicit JoAs reflect a different computational mechanism, and factor in different contextual information (Legaspi and Toyoizumi, 2019; Wen, 2019).

Taken into the context of two-step models of agency (Synofzik et al., 2008), our results suggest that sensory noise information may influence lower level, perceptual agency signals by making them more variable, and that these more variable signals then feed into higher level JoAs, but that estimates of the noise do not contribute as an additional cue to inform the JoAs. Hence, higher order agency states remain naive with regard to how noisy the lower level signals are. This could illuminate agency processing deficits in clinical cases which might involve perceptual agency cues becoming very noisy due to low level processing deficits. If explicit JoAs do not consider this uncertainty (as our findings indicate), this could lead to extreme agency reports despite unreliable evidence and inaccurate agency inferences. This is in line with work on agency misattributions in schizophrenia suggesting that they may be due to particularly noisy low level agency information, possibly due to dysregulated neurotransmitter activity (Corlett et al., 2010; Fletcher and Frith, 2009; Moore and Fletcher, 2012; Robinson et al., 2016).

By combining an agency rating task with an agency discrimination and confidence task, we were able to consider JoAs against the benchmark of metacognitive confidence judgments, while keeping the basic stimuli and noise manipulations the same in both cases. However, the tasks still differed on the nature of the ratings: Ratings in the agency task were about the feeling of agency over a single movement, whereas ratings in the confidence task were about the accuracy of the decision in the 2IFC task. Importantly, we note that the findings that we describe here do not rely on a comparison between tasks. The confidence task merely served as a positive control task to ensure that our noise manipulation had the expected effect on a metacognitive judgment and that the second-order model we built best predicted the results of computations we knew to be higher-order.

The experiment was pre-registered (osf.io/pyjhm), and we describe deviations from the pre-registered plan.

Agency ratings in the Second-order model

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The Second-order model assumed that agency ratings scale as a function of internal evidence in the same way as confidence, namely scaling with the posterior probability of being correct, given a choice and the internal signal (Sanders et al., 2016). Therefore, in this model, the agency rating is computed by estimating the probability that the agency detection was correct, given the perceived signal and detection decision. Because this probability computation depends on the level of sensory noise, the Second-order model predicts that noise will be factored into participants’ JoAs.

In both models, we obtained the criterion values that split the continuous range of possible f(x) values into equidistant bins. For the Second-order model, because confidence reflects a probability, it is naturally bounded to 1. So for the Second-order model (with a 3:3 mapping, see below) this amounted to finding the criterion values that would lead to confidence levels of ⅓, ⅔, and 1. To estimate the positions of the criteria on the internal signal axis, we followed an analytical solution that defines confidence as

(1) $\{\mathrm{\Phi }\left(\frac{c-x}{\sigma }\right)ifx\ge c$

$\{1-\mathrm{\Phi }\left(\frac{c-x}{\sigma }\right)ifx<c$

which we implemented, for convenience, as in a previous study (Navajas et al., 2017):

(2) $\Phi \left(\frac{\left|x-c\right|}{\sigma }\right)$

This confidence measure can be interpreted as the perceived probability that the true delay signal was on the same side of the decision criterion as the internal signal, hence making the decision correct (Figure 6).

Figure 6

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Raw data is publicly available under https://gitlab.com/MarikaConstant/metaAgency (copy archived at swh:1:rev:10a9c40ac4a8b45c81c5966297ef9911b2d33043).

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Author details

1. Marika Constant

   1. Faculty of Life Sciences, Department of Psychology, Humboldt-Universität zu Berlin, Berlin, Germany
   2. Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
   3. Berlin School of Mind and Brain, Humboldt-Universität zu Berlin, Berlin, Germany

   Contribution

   Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review and editing

   For correspondence

   marika.constant@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3756-0362

2. Roy Salomon

   Gonda Multidisciplinary Brain Research Center, Bar-Ilan University, Ramat-Gan, Israel

   Contribution

   Resources, Software, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-6688-617X

3. Elisa Filevich

   1. Faculty of Life Sciences, Department of Psychology, Humboldt-Universität zu Berlin, Berlin, Germany
   2. Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
   3. Berlin School of Mind and Brain, Humboldt-Universität zu Berlin, Berlin, Germany

   Contribution

   Conceptualization, Funding acquisition, Supervision, Writing – original draft, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1158-8220

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