Unexpectedly, we did not find evidence for group differences between patients with psychotic disorders and healthy controls with regard to the JTC-bias (draws to decision and the presence of the bias). Moreover, the degree of evidence required before reaching a conclusion (i.e. the number of draws to decision) was neither associated with delusional conviction, positive symptoms in general or persecutory delusions in particular.
Comparison between patients with psychotic disorders and healthy controls in JTC assessed with the fish task
Our results are at odds with meta-analytical findings suggesting a more pronounced JTC and DTD in patients with psychotic disorders when compared to healthy controls . Based on the DTD group differences between patients with psychotic disorders and controls (Hedges g = 0.52) reported in the meta-analysis of Dudley et al. , the present study had a 96% power to detect such a difference. Thus, the possibility that our results are based on random effects is very unlikely.
The fish task was developed to increase the comprehensibility of the beads task paradigm  and has revealed JTC group differences between individuals with lower and higher subclinical paranoia . However, an inspection of the data in Dudley et al.  suggests that group differences between patients with psychotic disorders and nonclinical controls are larger for the traditional beads task as compared with the fish task.
It should be taken into account that although patients were preselected based on PANSS P1 delusion item scores/ PANSS P3 hallucination item scores larger than three, in the sample, the mean PANSS total score was quite low (M = 64.80, SD = 11.38). According to Leucht and colleagues , a PANSS total mean score in a patient sample above 57 should be interpreted as "moderately ill", while a total score of 75 indicates that the patient sample can be viewed as "markedly ill". It could be argued that the relatively moderate mean score of symptom severity in the patient sample could explain the non-significant group differences, as differences between the patient sample and controls might have been reduced. However, this interpretation seems unlikely, as the proportion of patients who presented the JTC-bias in our study with the fish task is comparable with other studies using the beads task (about 50%) [20, 40, 41], while the proportion of controls in our study who presented the JTC-bias (41%) is about three to four times higher compared to other studies with the beads task [10, 23, 42]. However, other studies with the fish task also found that their control sample showed a less cautious behavior with 44%  and 42%  of the control sample displaying a JTC bias, and these studies also found no significant JTC differences between patients with psychotic disorders and controls. Given these results, it seems more likely that the fish task, while formally similar to the beads task, may induce a generally less cautious decision behavior, particularly in controls.
Thus, while our test power was ideal, our control sample seems to present a pronounced tendency to present a JTC-bias, and consequently, we were unable to detect group differences in JTC, in line with the two other studies that comparably used the fish task. Concluding, it is important to take a close look at the task characteristics in the fish task, in contrast to the traditional beads task.
Methodological considerations on the fish task
How can we explain the different findings concerning group differences between patients with psychotic disorders and controls in the beads task vs. the fish task?
One explanation of differences between the beads task and the fish task might be that we used a task variant of the fish task designed to simultaneously measure DTD and the decision threshold of the subjects. In this variant, participants are first asked to decide from which original pond the fish they see on the screen were fished. Next, they are additionally asked to estimate the respective probability for the fish to stem from one of the two ponds (this procedure is then repeated ten times). It is possible that the healthy control sample we assessed in the study might have been triggered by the additional probability estimation task to show a "riskier" response behavior . This explanation is supported by the results of the two other studies that used the same variant of the fish task and compared the JTC-bias between patients with psychotic disorders and healthy controls who also did not find statistically significant group differences between both groups [14, 15].
A second explanation of the conflicting results is based on different experimental procedures. On the one hand, in the classical beads task, participants are asked after each presentation of a bead, if they can decide from which jar the beads are drawn. If the participants decide for a jar, the trial is completed. On the other hand, in the fish task, if participants decide that the fish stem from one pond, the trial is continued until ten fish are drawn. Thus, the traditional beads task might result in a more cautious decision-making behavior in healthy subjects, as the decision is definite and subjects are asked only to decide once they are “completely sure”  and this might explain the more cautious behavior other groups and we detected in the control group.
Lack of associations with psychopathology
While the task format may have been responsible for a riskier decision behavior, subjects still differ regarding the evidence required before they reach a decision. We also set out to investigate whether DTD relates to delusional conviction as well as positive symptomatology in general and persecutory delusions in particular. Surprisingly, we did not find a correlation between DTD and delusional conviction in the patient sample. This is not in line with past research, as several studies that used the beads task suggest that a low DTD score is related especially to a more pronounced delusional conviction in patients with psychotic disorders [20, 21] and delusion-prone individuals [20–22]. Nevertheless, our results are in line with the two recent meta-analyses that did not find significant associations between DTD and delusion severity in currently deluded patients with psychotic disorders and delusion-prone individuals [8, 9], also their results are primarily or solely based on studies with the beads task.
Interestingly, studies that used the fish task also did not find significant correlations between DTD and measures of delusion severity in controls  and patients with psychotic disorders . Thus, as discussed above, it is important to pay close attention to our task variant, the fish task, to explain the different results.
With regard to associations between the JTC-bias and delusions, results point generally more in the direction that associations are less pronounced than previously assumed and that the importance of the JTC-bias for delusion severity has been overestimated to some degree in the past in theoretical models. Although the present study is not a direct replication of previous studies regarding the measurement of the JTC-bias, it is important to take our results into careful account, especially in light of a large scale study that also found that delusional ideation did not predict DTD (measured with the beads task) . Additionally, meta-analytical results primarily obtained with the beads task [8, 9] based on a large number of studies also point in the same direction.
Interestingly, in the fish task variant other groups and we used, participants first view one (and then two) orange fish and are repeatedly asked whether they can decide if the fish stem from the orange pond. Mathematical analysis reveals probabilities for a correct decision of 80% after the first draw and 94,1% after the second draw for the fish task (using a ratio of 80:20), providing the participants with Bayesian factors of four (first draw) respectively 32 (second draw). Bayesian factors of this size are considered “positive” respectively “strong” evidence in the context of data analysis in research [46, 47], deciding at that stage rather rational than biased.
Since not all decisions after one or two fish should be nominated as “jumping to conclusions”, we stress the importance of an analysis of the probabilities. This could explain the rather “equivocal” findings on associations between JTC and delusions (review of Garety and Freeman ) that might depend on the probabilities of a correct decision based on the specific fish/bead sequence and ratio framing the JTC-bias (decision after 1-2 fish/beads) as rather more or less risky/rational.
Furthermore, as misunderstandings of the instructions of the beads task occur more often in patients with psychosis than in healthy controls , they could also contribute and enlarge the frequently found group differences between patients and healthy controls. This interpretation is supported by the findings of a large study by Tripoli et al. , which found that the association between JTC and psychotic symptoms in patients with psychosis seems to be mediated to a large extent by the IQ of the subjects. Therefore, JTC may be a consequence of more basal cognitive deficits, which may contribute to a misunderstanding of the task. If the fish task is indeed easier to understand in comparison to other task variants, this could be accompanied by a more "rational" response behavior on the part of the patients, which is more in line with the response behavior of non-clinical control subjects. Consequently, the fish task could be a valid measurement method for the JTC bias, which, however, occurs much less frequently in patients with psychosis, if their correct understanding of the task is ensured, as compared to studies using the traditional beads task paradigm that is harder to correctly understand.
It is possible that the decision behavior we measure in the beads task/fish task (whether it is risky or rational) might be still an interesting causal factor involved in the formation and maintenance of symptoms of psychotic disorders. Nevertheless, it might not be causally related to delusions in particular, but rather associated with other symptoms of psychotic disorders such as negative symptoms [49, 50], insight  or neurocognitive problems  (Interestingly, in an exploratory analysis, we found negative symptoms to be associated with JTC, r = -.166, p < .05).
This is reminiscent of another cognitive bias, a deficit in theory of mind, that first seemed like a vital factor in the formation of delusions but later proved to be rather associated with negative and disorganization symptoms . Moreover, as more and more studies raise doubts about the importance of the JTC-bias for delusions, future studies could also focus on other possible factors in the formation and maintenance of delusions, like emotional processes such as worrying, low self-esteem, or negative affect .
In future studies that aim to analyze group differences between patients with psychotic disorders and controls, first, it is important to use a more reliable assessment of JTC, possibly by combining different fish ratios (80:20, 60:40) and different sequences of fish/beads (see Speechley et al.  for an analysis of probability estimations for different fish sequences and ratios) and pay close attention to the probability for a correct decision after one or two fish in the task that is used, as it could vary largely and result in either a statistically more premature or a more risky decision. Interestingly, neither the traditional beads task nor the fish task paradigm provides a definite correct decision regarding the required number of beads. It would be interesting to address the question of the optimal decision threshold further in future research, possibly by providing incentives for more cautious decisions and by analyzing ROC-curves of decisions. Future research could also build on variants of the fish task that incentivize optimal levels of data gathering , which in turn would allow a clear identification of "premature" decisions. Also, the measurement of the JTC-bias could be performed with different task materials (e.g., emotional salient material) , additional measures (e.g. the Cognitive Biases Questionnaire for Psychosis)  or new task variants (e.g. the box-task  and the “what is this” task [43, 57]).
Furthermore, future studies are well-advised to preregister their hypotheses and analyses, as studies that are not preregistered tend to overestimate effects . Finally, in future meta-analyses on JTC, it would also be interesting to take the probability of a correct decision after one or two fish into account, as it might not always be an “extreme” decision , but in the case of the fish task, often a rational decision.
Strength and limitations
The present study has some notable features. We set out to investigate the JTC-bias in a large, patient sample to fill a gap that became apparent in the meta-analyses mentioned above . Due to the large sample size, the broad variance in positive symptoms, and the measurement of delusions using two interviews, the non-significance of our results should encourage a discussion of the measurement of the JTC-bias in general, which goes beyond the mere criticism of the fish task.
However, the study has some important limitations. First, only hypothesis 2 (association of JTC with delusional symptoms) was preregistered in the trial proposal. The control group was only included in the study design during the study and group comparisons between healthy control subjects and patients were not the main objective of the study, which primarily aimed to compare the effectiveness of CBTp and supportive therapy in treating patients with psychotic symptoms and to analyze putative mediators (e.g. cognitive biases such as the JTC-bias) of change in CBTp. Thus, our results regarding group differences between patients with psychosis and controls are exploratory and should be investigated again in a preregistered design. Furthermore, only a limited budget was available for the assessment of a non-clinical control group, which is why a further criticism of the study might relate to the unequal group size regarding the patient sample and the healthy control subjects. Although unbalanced group sizes may be associated with statistical problems, we would have achieved a power of approximately beta = .99 for absolute differences in JTC rates (based on a JTC rate of 50% in patients and 20% in healthy controls [10, 20, 23, 40, 41]), and .96 for group differences in DTD at the effect sizes reported in the literature . As none of the assumptions for the statistical tests carried out are violated, the results can be considered relatively robust despite the different group sizes . Nevertheless, we have re-examined the results using the non-parametric Mann-Whitney-U test, which also does not show significant group differences in DTD. Furthermore, as discussed before, the task variant with the simultaneous measurement of the decision threshold might be responsible for the non-significance of our results. Additionally, as we did not measure the delusion severity of the healthy subjects, we cannot rule out the possibility that our healthy control sample consisted of delusion-prone individuals who might also display a higher level of JTC-bias, in line with other studies in delusion-prone subjects [59, 60]. While we excluded controls who had a mental disorder in their past/ present, we did not assess delusional ideation in the control group (e.g. with questionnaires like the Peters et al. Delusions Inventory) . Thus, our control sample might be more prone to delusional ideation and consequently, we did not find group differences in JTC.