Dear Editors, I am very happy to endorse the publication of this manuscript and have no further comments as the authors have done a fabulous job at revising. I look forward to learning about the results of the study in due course and wish the authors the best of luck!

I first like to thank the authros again for a very through and thoughtful work and response to our comments. I also apology for the delay in revieiwng the paper.

While the work is excellant and through it is also very complex. I fear this will be a future barrier for readers to appraciate the quality, especially if you want to communicate the results to medical and clinicians. am not sure what the authors can do to simplfy it, but maybe it so something to be minded in the report of the results. 

Two final thoughts:

Control group:

1) I apperciate that an MCI control group is likely to be hetrogenous and may include some ovrelapping PD like eatiology. Please make sure you bring it up in study limitation - that difference between PD-MCI vs. PD-NC may reflect comorbidity with other degenration symptoms rather than relates to PD severity per se. 

2) Given the potential difference in socio-demographics between PD-NC and PD-MCI, maybe include to healthy control groups with a match to each group. or based on the adapt analysis approch using two t-tests: PD-NC vs HC, PD-MCI vs PD-NC, than make sure the HC matches the PD-NC better.

Thank you

PS, Zoltan I do not need to review it again, I am happay with whatever decision you will make regarding the authors responses.  

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Loenneker and colleagues proposed to register a study that examine simple arithmetic abilities in two group of patients suffering from Parkinson, those who show no cognitive deficits (PD-NC)  and those who show mild cognitive impairment (PD-MCI) in comparison to healthy controls (HC). The study is paired with an additional registered study that examine non-symbolic quantity abilities in the same sample.
Participants will be assess on addition, subtraction, multiplication and division. Accuracy, reaction time and some derivatives measures (differences between trials: borrow and carry over effects) will be the dependent measures. The authors will also consider confounding factors related to socio-demographics and clinical symptoms.

The authors aim to answer three questions: Is PD relates to calculus deficits?; Can tobserved calculus deficits be accounted by core cognitive abilties (e.g. attention, executive function,   working memory)? Can calculation problem can be used to detect early PD (PD-NC vs. HC), or emerge at later PD stages (PD-NC vs PD-MCI).
 
Overall, this is a very large and impressive project. It is very comprehensive and it is clear that the authors have considered the study design very carefully and thoughtfully. The project is exploratory by nature and given it extent and number of measurements it is quite complex.  
 
Some points to consider
-       The main point I want to raise regards the nature of the healthy control group. This is a general comment that probably relates to most studies that uses neuropsychological measures to characterise a specific patient group. the quetsion is who is an appropriate control group that differ only on the factor of interest, here PD.

o   The authors should be commented that they aim to recruit carer of the PD patients, which is likely to account for some of the socio-demographic confounds.

o   I suggest the recruitment of the control group be more purposeful with the aim of matching them as much as possible with all aspect of the two PD groups.
§  Matching the three groups on age, gender, education, SES and at least childhood testified confidence in math.
§  Including two HC groups, one with no mild cognitive impairment and one with MCI is important for arguing that it is the PD that lead to acalculia rather than MCI.

-   If I understand correctly the upper limit for participants number is 120.

o Will these be equally divided across all three groups?

o Could you provide a power analysis for 120 participants for a medium effect size, as this is the maximum power you can be achieved.

o It can be that when considering all the additional covariates 120 is too small group.

-       The analysis suggests to exclude participants who got the first 10 question wrong in one of the tasks. This may dilute your effect. Maybe in the practice session include some very simple arithmetic to ensure that they understand the task instruction. Then if they understand the task but nevertheless fail the first 10 questions you can just award them a zero score.  
o   Will the first 10 be the easier examples, less complex?
o   Will participants be asked if they want to stop or continue?
o   Will participants received feedback on their answers?

-       The number of planed tests is huge. It is good you opted for Bayes factor as you will not need to compare for multiple comparison. However, if only a single of these test will results in significant results – what would you conclude on the relation of PD and calculus processing?

-       When computing the 'borrow effect' could you please provide a clear definition of problem size.
-       Could you please provide a clear definition of the task complexity factor that will be used to answer Q2.
-       Given the number of potential covariates it maybe more useful to remove variability of socio-demographic and clinical factors from the tasks performances before moving into Q2 (impact of core functions). Then removing the additional variability of cognitive covariate from task performances before using the data to answer Q3 (ability of calculus to discriminate between the three groups).
-       Could you please provide a measure that will ensure that the quality of the data is high (e.g. expecting to replicate basic accuracy and RT effect on task complexity, independent of PD, or across all groups)

- dividing the text by sub-headings, will aid the reading and comrehension of all section. using some additional tables to summarsie the information (description of tasks, measures collected) can also help.   
 

Thank you for asking me to review this detailed, well written and interesting RR. The research question has good validity and the rational of study as well as hypothesis make sense. The sample size and inclusion/exclusion criteria are well defined and justified. Overall, the analysis is feasible and appropriate to answer the questions. However, I have identified the following concerns that warrant expanding/clarification:

-          I understand the RR will be ran alongside the other similar study already accepted-in-principle. Can you give us details on how task order and administration of both studies will be managed and lay out clearly what data overlaps between the two studies?

-          The introduction is quite long and needs to be more succinct focusing on the question at hand and leading to research question and hypothesis. Its missing a good overview of PD and its epidemiology and symptoms. Its not common to have hypothesis in supplemental so I would incorporate this within main text.

-          Within the arithmetic tasks however you propose to analyse RT. However, the response is entered by experimenter. Could you use a microphone instead and record verbal response? Otherwise, your RT measure will not be very reliable.

-          The authors should a section on limitations of the study and how these will be dealt with regarding design, analysis, and previous findings. Issues covered could include for example issues such as heterogenous clinical sample, fatigue, diagnosis errors. Finally a section on potential unexcepted outcomes should also be included including analysis or other steps taken to overcome these.

Minor issues

‘the magnetic resonance imaging volume of the angular gyri’ – this is unusual language here and instead I would remove MRI and just say cortical volume?

A few typos noted (e.g., page 5 end paragraph, line 7).

Spell out acronyms first time within text (e.g., ADL; MOCA)

Some of the paragraphs are long and should be split into separate paragraphs (e.g., page 3).