PhD Véronique Sébille
email: veronique.sebille@univ-nantes.fr

PhD Yseulys Dubuy
email: yseulys.dubuy@univ-nantes.fr

PhD Fanny Feuillet
email: fanny.feuillet@univ-nantes.fr

PhD Myriam Blanchin
email: myriam.blanchin@univ-nantes.fr

PhD Antoine Roquilly
email: antoine.roquilly@chu-nantes.fr

PhD Raphaël Cinotti
email: raphael.cinotti@chu-nantes.fr

Related Quality of Life Assessment Between Patients and Proxies in Patients with Moderate-to-Severe Traumatic Brain Injury?

Keywords: traumatic brain injury, Patient-Reported Outcomes Measures, self-report, proxy-report, Differential Item Functioning

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Due to the tremendous advances in critical care allowing more patients to survive critical illness, a growing interest for longer-term patient-centered outcomes has been observed in Intensive Care Units (ICUs) [START_REF] Needham | Improving long-term outcomes after discharge from intensive care unit: report from a stakeholders' conference[END_REF] . Outcomes such as health-related quality of life (HRQoL), mental health, or cognitive functioning are now recognized as 'patient-important outcomes' [START_REF] Dinglas | Understanding patient-important outcomes after critical illness: a synthesis of recent qualitative, empirical, and consensus-related studies[END_REF] and often monitored during ICU stay or after discharge using Patient-Reported Outcomes Measures (PROM) [START_REF] Hammond | Health-related quality of life in survivors of septic shock: 6month follow-up from the ADRENAL trial[END_REF][START_REF] Sprigg | Tranexamic acid for hyperacute primary IntraCerebral Haemorrhage (TICH-2): an international randomised, placebo-controlled, phase 3 superiority trial[END_REF] . PROM are supposed to be directly reported by the patients without interpretation of their responses by a physician or anyone else. However, there are many instances where patients are unable to self-report due to physical and/or cognitive impairments, especially in vulnerable patient populations, e.g. critically ill 3,5 , stroke (5), geriatric [START_REF] Arons | Quality of life in dementia: a study on proxy bias[END_REF] patients. Thus, proxy-reported measures, e.g. coming from family members, are often needed and used on the patient's behalf. However, studies where responses by patients and their proxies were both examined frequently reported that proxy and patient ratings differ and may not be comparable [START_REF] Lapin | Magnitude and Variability of Stroke Patient-Proxy Disagreement Across Multiple Health Domains[END_REF][START_REF] Hung | Measurement of quality of life using EQ-5D in patients on prolonged mechanical ventilation: comparison of patients, family caregivers, and nurses[END_REF][START_REF] Jones | Assessing agreement between terminally ill cancer patients' reports of their quality of life and family caregiver and palliative care physician proxy ratings[END_REF] . Specifically, while proxies are most often inclined to report worst HRQoL or functioning than patients [START_REF] Lapin | Magnitude and Variability of Stroke Patient-Proxy Disagreement Across Multiple Health Domains[END_REF][START_REF] Hung | Measurement of quality of life using EQ-5D in patients on prolonged mechanical ventilation: comparison of patients, family caregivers, and nurses[END_REF] , discrepancies are sometimes found in the other direction [START_REF] Kroenke | Agreement between older adult patient and caregiver proxy symptom reports[END_REF] . Furthermore, such disparities seem to be more pronounced when the outcome is not directly observable (e.g., emotional functioning) [START_REF] Lapin | Magnitude and Variability of Stroke Patient-Proxy Disagreement Across Multiple Health Domains[END_REF][START_REF] Hung | Measurement of quality of life using EQ-5D in patients on prolonged mechanical ventilation: comparison of patients, family caregivers, and nurses[END_REF][START_REF] Jones | Assessing agreement between terminally ill cancer patients' reports of their quality of life and family caregiver and palliative care physician proxy ratings[END_REF] .

Hence, the substitution of missing patients' self-assessment by proxies' may be questioned, particularly in ICUs where its prevalence may be high as a non-negligible number of patients are often unable to respond for themselves [START_REF] Hammond | Health-related quality of life in survivors of septic shock: 6month follow-up from the ADRENAL trial[END_REF][START_REF] Unroe | One-year trajectories of care and resource utilization for recipients of prolonged mechanical ventilation: a cohort study[END_REF] . Likewise, aggregating patient and proxy responses under these conditions can substantially bias patient-centered outcomes estimates because patient and proxy ratings are not necessarily comparable. Assessing the comparability of patient and proxy responses is therefore critical. However, we should not only consider discrepancies in responses due to respondent type (proxy vs. patient) but also those caused by other potential confounders. For instance, some items (questions) of the PROM, may not perform (or function) similarly in patients and proxies, i.e. patients and proxies could perceive and interpret some items differently and thus report e.g. HRQoL in different ways. In this case, responses to the questionnaire items may not only reflect patient HRQoL but also the respondent's (patient or proxy) own perception of the items. This phenomenon, acknowledged as Differential Item Functioning (DIF) [START_REF] Mellenbergh | Item bias and item response theory[END_REF][START_REF] Rouquette | Differential Item Functioning (DIF) and Subsequent Bias in Group Comparisons using a Composite Measurement Scale: A Simulation Study[END_REF] , can markedly compromise the comparability patient-and proxy-reported measures and lead to biased comparisons [START_REF] Yadegari | Differential item functioning of the SF-12 in a populationbased regional joint replacement registry[END_REF] . Assessing whether DIF is present or not is thus essential. However, as it is likely that patients who are either able or unable to self-report may differ on some important clinical characteristics related to the outcome of interest, e.g. HRQoL, DIF evaluation should also consider these potential confounders.

Our objective was to assess the comparability of moderate-to-severe TBI patient self-reports and proxy reports of HRQoL when the prevalence of proxy reports on patients' behalf is high and thus may compromise the comparability of HRQoL data. Specifically, we aimed to assess the comparability of patients' self-reports and proxies' reports by evaluating the extent to which items perception differs (i.e. presence of DIF) between patients and proxies after controlling for potential confounding factors.

Material and Methods

This is an ancillary study of the COBI 'COntinuous hyperosmolar therapy in traumatic Brain-Injured patients' prospective multicentre randomized-controlled trial which assessed whether a continuous infusion of hypertonic saline solution in addition to standard care would improve neurological status at 6 months in patients with moderate to severe TBI [START_REF] Roquilly | Effect of Continuous Infusion of Hypertonic Saline vs Standard Care on 6-Month Neurological Outcomes in Patients With Traumatic Brain Injury: The COBI Randomized Clinical Trial[END_REF] . The study protocol was approved by the Institutional Review Board of Ile de France VIII (France) on May 8, 2017 and the trial was conducted according to the Declaration of Helsinki. The consent procedure is described in the princeps study [START_REF] Roquilly | Effect of Continuous Infusion of Hypertonic Saline vs Standard Care on 6-Month Neurological Outcomes in Patients With Traumatic Brain Injury: The COBI Randomized Clinical Trial[END_REF] and a specific consent was obtained from the proxies. Eligible patients were aged from 18 to 80 years old and admitted for a moderate to severe TBI (Glasgow Coma Score, GCS ≤12) with abnormal CT scan in one of the 9 participating ICUs.

Outcomes

The primary outcome of the COBI trial was the Extended Glasgow Outcome Scale (GOS-E) score at 6 months. Secondary outcomes included HRQoL, measured with the SF-36 questionnaire [START_REF] Leplège | The French SF-36 Health Survey: translation, cultural adaptation and preliminary psychometric evaluation[END_REF] six months after randomization. The questionnaires were sent by mail and were centralized to be managed by a staff member blinded to the intervention. The SF-36 comprises 36 items distributed within eight domains (Physical Functioning, Mental Health, General Health, Role Physical, Role Emotional, Bodily Pain, Vitality, and Social Functioning). The scores range from 0 to 100 in each domain, a higher score representing better HRQoL. For this study, the two domains Role Physical (RP) and Role Emotional (RE), only containing binary items, were used. HRQoL was self-reported by patients who were able to complete the questionnaire, patients' HRQoL was reported by proxies otherwise. Proxies included family members or health-care professionals (e.g. physicians).

Main objective

We assessed the comparability of HRQoL assessment at 6 months after trauma between patients with moderate-to-severe TBI and proxies by evaluating the extent to which items perception differs (i.e. DIF) between respondents (patients or proxies).

Statistical analysis

Continuous and categorical data are expressed as mean (SD) and frequencies (%) and were compared according to the two groups (patients or proxy respondents) using t-tests and chi-square tests, respectively. DIF between patients who could self-report and proxies who responded on patients' behalf was investigated using logistic regression [START_REF] Swaminathan | Detecting Differential Item Functioning Using Logistic Regression Procedures[END_REF] . For each item of the RP and RE domain, the dependent variable is the logit of the probability of giving a favorable response to the item (e.g. answering 'no' to the item 'accomplished less than you would like?') and the main independent variables are the group (respondent type), the domain rest-score (score calculated without the item under examination) and their interaction. If the group or/and interaction effect is/are significant, DIF is assumed for the item.

To account for several potential confounders simultaneously in the DIF analyses, multivariable logistic models adjusted on the potential confounders were used. Specifically, for each HRQoL domain (RP and RE), confounders associated with being a proxy respondent and HRQoL were considered in the logistic models.

We drew on Crane et al. [START_REF] Crane | Differential item functioning analysis with ordinal logistic regression techniques. DIFdetect and difwithpar[END_REF] and Maldonado & Greenland [START_REF] Maldonado | Simulation study of confounder-selection strategies[END_REF] to assess whether the highlighted DIF could be considered meaningful. DIF was regarded as meaningful if the relative change in the regression coefficient associated with the rest-score was at least 10% between the model without DIF and the model with DIF (significant at the 0.05 level), adjusted on the potential confounders.

Details on DIF analyses are provided in the Electronic Supplementary Material ESM1. Statistical analyses were performed using R version 4.1.0. P-values < 0.05 are considered statistically significant.

Results

The trial failed to demonstrate an improved neurological status at 6 months after continuous infusion of hypertonic saline compared with standard care [START_REF] Roquilly | Effect of Continuous Infusion of Hypertonic Saline vs Standard Care on 6-Month Neurological Outcomes in Patients With Traumatic Brain Injury: The COBI Randomized Clinical Trial[END_REF] . Of the 370 patients included in the COBI trial, SF-36 data corresponding to 240 (65%) patients were collected, due to deaths or lost to follow-up. In total, the SF-36 was completed by 110 (46%) patients and 130 (54%) proxies. Overall, 40% of the questionnaires were completed and returned by mail and 60% were obtained by telephone interview.

Among the 130 proxies, 75% (97/130) were family members, and 25% (33/130) were healthcare professionals, the majority of whom were physicians (88%). The distribution of the proportion of patient or proxy respondents was similar between the two randomization treatment groups at 6 months.

The characteristics of the patients who could self-report (patient group) and those who couldn't and needed a proxy report (proxy group) appear in Table 1. Patients in the proxy group were significantly older, had lower baseline Glasgow Coma Scale scores (meaning higher trauma severity), and were more likely to have hypoxia at baseline. Besides, they more frequently had an unfavorable neurological recovery (GOS-E ≤5) and were less likely to live at home at 6 months.

Proxies reported significantly lower HRQoL scores than patients self-reported for the RP (absolute difference, -15.0 [95% CI, -25.1 to -4.9]) and RE (absolute difference, -19.6 [95% CI, -30.2 to -9.0]) domains (Table 2).

Differential Item Functioning

To investigate DIF, multivariable logistic regression controlling for confounders were used.

For the RP domain containing 4 items, the general instruction for answering each item is: 'During the past 4 weeks, have you had any of the following problems with your work or regular daily activities as a result of your physical health?'. Three out of the four items of this domain were evidenced to function differently between patients and proxies (i.e. meaningful DIF), the interaction term 'Restscore*Type of respondent' being systematically significant at 0.05 level and the relative change in the regression coefficient associated with the rest-score higher than 10% (range: 20%-25%, data not shown) between the model with and without DIF (Table 3):

-items 4a 'cut down the amount of time you spent on work or other activities' -item 4c 'were limited in the kind of work or other activities' -item 4d 'had difficulty performing the work or other activities (for example it took extra effort)'.

For instance, for item 4c (Figure 1), it can be seen, on the left hand-side of the figure, that proxies have a lower probability (close to 0) of responding favorably to this item (i.e., answering 'no') compared to patients, despite reporting a same rest-score equal to 0 (substantial role limitations due to physical health problems). This trend was reversed when the rest-score was above 60, with proxies tending to respond more favorably than patients, despite reporting comparable rest-scores. The same general response pattern was observed for the other RP domain items (see ESM2 Supplementary Figure 1, Figure 2).

For the RE domain, the general instruction for answering the items is: 'During the past 4 weeks, have you had any of the following problems with your work or regular daily activities as a result of any emotional problems (such as feeling depressed or anxious)?'. One out of the three items of this domain was evidenced to function differently between patients and proxies (i.e. meaningful DIF), the interaction term being significant at 0.05 level and the relative change in the regression coefficient associated with the rest-score higher than 10% (i.e. 20%, data not shown) between the model with and without DIF (Table 4): item 5a 'cut down the amount of time you spent on work or other activities'. The same response pattern was observed for this item as for the RP domain items, i.e. proxies tended to give a more pessimistic response than patients in case of substantial role limitations due to personal or emotional problems (i.e. rest-score close to 0), and a more optimistic one in case of no limitations (i.e. rest-score of 100) (ESM2 Supplementary Figure 3).

Discussion

Differential item functioning, considered meaningful, was evidenced in three out of the four items of the RP domain measuring role limitations due to physical health problems and in one out of the three items of the RE domain measuring role limitations due to personal or emotional problems. As stated in Wu et al. [START_REF] Wu | Is Difference in Measurement Outcome between Groups Differential Responding, Bias or Disparity? A Methodology for Detecting Bias and Impact from an Attributional Stance[END_REF] , p.4 'DIF is manifest when an item response differs among groups who possess an equal level of the construct that an item intends to measure'. Overall, despite an expected similar level of role limitations between patients who were able to respond and those for whom proxies responded, the responses provided by patients and proxies differed. Specifically, proxies tend to give more pessimistic responses than patients in the case of major limitations and more optimistic responses than patients in the case of minor limitations.

DIF may have different clinical implications [START_REF] Kwon | Patient-Reported Outcomes Measurement in Radiation Oncology: Interpretation of Individual Scores and Change over Time in Clinical Practice[END_REF] , according to the level of limitations. In case of major limitations, proxies tend to report that patients are a little worse off than they are. This could lead to more intense care than needed for some patients. Such care could of course be beneficial but also potentially detrimental in terms of patients' comfort and HRQOL. In case of minor limitations, proxies tend to report that patients are a little better than they are. While one may fear that patients may then not benefit from optimal care, one would hope that the interpretation would be relatively similar in terms of limitations, and hence, healthcare decision-making. Overall, the global impact on the domain scores (RP and RE) if patients respond versus proxies was quite high as a difference in score of at least 15 points was observed. Using proxies' responses could therefore lead to underestimate patients' role limitations due to physical or emotional problems which may in turn result in inappropriate healthcare management. However, of note, the global impact of the type of respondent on the domain scores includes both DIF and confounding effects.

By controlling for confounders using multivariable models to study DIF, thus making the two patient populations, able or unable to respond, more comparable, we approximate 'causal DIF' [START_REF] Liu | Investigating Causal DIF via Propensity Score Methods[END_REF] , stating that DIF is caused by group composition. Of note, other approaches have been suggested to approach 'causal DIF', including propensity score (PS) matching and conditional logistic regression [START_REF] Liu | Investigating Causal DIF via Propensity Score Methods[END_REF] . These had been initially planned for the analysis. However, this strategy was subsequently abandoned for the following reasons: i) the small number of confounders (only three), ii) persistent between-group imbalance in confounding factors despite PS matching (standardized mean difference greater than 0.10), iii) convergence problems encountered when estimating conditional logistic regression models. As these methods can also be used and might be of interest to some researchers, they are also described in ESM1.

When DIF is present, the comparability of HRQoL scores of patients and proxies is compromised as these scores may not only reflect HRQoL levels but also the respondents' differential interpretation of the items. Moreover, e.g. as 3 out of the 4 items of the RP domain were flagged with DIF, one could suspect that the concept measured by the items (role limitations due to physical health problems) is not the same for patients and proxies respondents [START_REF] Demars | Differential Item Functioning Detection With Latent Classes: How Accurately Can We Detect Who Is Responding Differentially?[END_REF] . Hence, the scores of the RP domain coming from patients or proxies reports should probably not be 'simply' pooled and more methodological research is now needed to account for this 'causal DIF'.

Our results are in line with some previous studies where DIF has been investigated between proxies and TBI [START_REF] Chan | Analysis of patient and proxy ratings on the Dysexecutive Questionnaire: an application of Rasch analysis[END_REF] or stroke [START_REF] Lapin | Patient versus proxy response on global health scales: no meaningful DIFference[END_REF][START_REF] Whynes | Testing for differential item functioning within the EQ-5D[END_REF] patients. Chan & Bode [START_REF] Chan | Analysis of patient and proxy ratings on the Dysexecutive Questionnaire: an application of Rasch analysis[END_REF] have shown that TBI patients and proxy data were not interchangeable due to their different perspectives when reporting the occurrence of dysexecutive behavior, as revealed by DIF in some of the items of the Dysexecutive Questionnaire [START_REF] Wilson | Behavioural assessment of the dysexecutive syndrome[END_REF] .

Results were mixed in stroke patients where DIF was either evidenced [START_REF] Whynes | Testing for differential item functioning within the EQ-5D[END_REF] or not [START_REF] Lapin | Patient versus proxy response on global health scales: no meaningful DIFference[END_REF] between patients and proxies when reporting HRQoL. While providing interesting information regarding the differences in interpretation that patients and proxies may have of the items of a questionnaire, those works did not address confounding due to differing populations between patients who selfreported and patients who needed a proxy report. Hence, we do not know whether DIF, or its absence, is due to the composition of the group of respondents or to confounding factors.

By considering confounding in DIF analyses, we could examine the extent to which the differential interpretation of the items by patients and proxies were attributable to the type of respondent based on unbiased comparisons. Provided that the majority of available confounders were accounted for in the identification of DIF, the difference in HRQoL reported by patients and proxies would then not only reflect true HRQoL difference between patients who are able or unable to respond but differences attributable to the people responding. This statement should nevertheless be taken with caution as other potential confounders (e.g. attitude, proxy disposition, personality, coping style, ethnicity) were not captured in this study.

As is often the case in ICUs, our study reflects situations in which a large majority of patients are unable to respond. While the impact of DIF could then be major, it also exemplifies circumstances where proxy reports could be very useful in allowing the 'missing voice of the critically ill' 28 to be heard. However, the usefulness of proxy reports in place of patients' is highly dependent on their ability to provide a reliable assessment of the patient-centered outcomes of interest (e.g. HRQoL).

In the ICU setting, proxy responses appear to be often substituted for those of the patients for whom they respond [START_REF] Hammond | Health-related quality of life in survivors of septic shock: 6month follow-up from the ADRENAL trial[END_REF][START_REF] Crescioli | Long-term mortality and health-related quality of life of lower versus higher oxygenation targets in ICU patients with severe hypoxaemia[END_REF][START_REF] Unroe | One-year trajectories of care and resource utilization for recipients of prolonged mechanical ventilation: a cohort study[END_REF][START_REF] Geense | New Physical, Mental, and Cognitive Problems 1 Year after ICU Admission: A Prospective Multicenter Study[END_REF] and patients and proxies PROM data are usually combined, as was done in the princeps publication of the COBI trial [START_REF] Roquilly | Effect of Continuous Infusion of Hypertonic Saline vs Standard Care on 6-Month Neurological Outcomes in Patients With Traumatic Brain Injury: The COBI Randomized Clinical Trial[END_REF] . One of the reasons given for using proxy data as substitutes for the patients' is to minimize selection bias by considering data from patients who are unable to respond [START_REF] Jette | Evaluation of patient and proxy responses on the activity measure for postacute care[END_REF] . Unfortunately, it is likely that not only is selection bias not resolved, but a possible bias due to the non-comparability of patient and proxy reports may also be present.

This leads to several considerations. First, patient and proxy reports may not be comparable and substituting proxy data for patient data will most likely bias PROM estimates. Second, we can hypothesize that the effect of an intervention could also be biased would there be an imbalance of patients and proxies between the treatment groups being compared. This should not be the case in the COBI trial where the proportion of proxy respondents was balanced; although not resolved, bias was therefore a similar issue in both groups. Third, while the notion of reliability of proxy's assessments was used, it may be a too pejorative term. Proxies' perceptions are of course very valuable and acknowledged as important, as witnessed by family engagement initiatives in ICUs and for TBI patients [START_REF] Kleinpell | Promoting Family Engagement in the ICU: Experience From a National Collaborative of 63 ICUs[END_REF][START_REF] Richard-Lalonde | Family members' perceptions of pain behaviors and pain management of adult patients unable to self-report in the intensive care unit: A qualitative descriptive study[END_REF][START_REF] Foster | Encouraging family engagement in the rehabilitation process: a rehabilitation provider's development of support strategies for family members of people with traumatic brain injury[END_REF][START_REF] Fisher | Family-directed approach to brain injury (FAB) model: a preliminary framework to guide family-directed intervention for individuals with brain injury[END_REF] . The next step would be to understand why patient and proxy perceptions differ. For instance, would this reveal some form of psychological distress reported in family members of ICU and TBI patients [START_REF] Haines | Psychosocial outcomes in informal caregivers of the critically ill: a systematic review[END_REF][START_REF] Grayson | The impact of cognitive-communication difficulties following traumatic brain injury on the family; a qualitative, focus group study[END_REF] that may need attention? It should be noted that, in our study, proxy respondents were either relatives (75%) or health professionals (25%). No differences were noted in responses among these two proxy groups (ESM2 Supplementary Table 1) but sample size did not allow to explore whether DIF was present by proxy type. Higher levels of agreement between ICU patients' ratings of their symptoms and those made by their family members compared with assessments made by nurses and physicians have been reported previously but without any investigation of DIF [START_REF] Puntillo | Challenge of assessing symptoms in seriously ill intensive care unit patients: can proxy reporters help?[END_REF] . Although a difference in perception (i.e. DIF) according to proxy type is likely, this would warrant further investigation.

Limitations are worth mentioning. First, hidden bias, due to unmeasured confounders may still be present. Second, DIF analyses only focused on the two domains containing binary items and used logistic regression. Examining other domains including items with more than two response categories using logistic regression for ordinal data usually requires the proportional odds assumption which might be too restrictive and may not hold [START_REF] Crane | Differential item functioning analysis with ordinal logistic regression techniques. DIFdetect and difwithpar[END_REF] . In addition, the next stage would be to adjust for the identified DIF in the analyses and see if the difference between patient-and proxy-report persists to assess proxy-response bias. However, to date, more methodological developments are needed, such as those based on latent variable models [START_REF] Rouquette | Differential Item Functioning (DIF) and Subsequent Bias in Group Comparisons using a Composite Measurement Scale: A Simulation Study[END_REF] . Third, DIF may also be present between TBI patients with differing severity levels, as was reported by Chang et al. [START_REF] Chang | Psychometric Comparisons of the Quality of Life after Brain Injury between Individuals with Mild and Those with Moderate/Severe Traumatic Brain Injuries[END_REF] with the QOLIBRI questionnaire, and would require further investigation. Fourth, HRQoL data was not simultaneously collected from patients and their proxies which would have allowed for a case-matched approach. However, while matched analyses may have indeed strengthened our analysis, it may also have limited the generalizability of the results as it might not have reflected some clinical settings where a large proportion of patients are unable to respond for themselves, as was the case for this study. Fifth, responses to the questionnaire represented 65% of the 370 patients of the COBI trial, hence, possible selection bias cannot be ignored.

Conclusion

Patients with moderate-to-severe TBI and proxies were shown to have different perceptions of the items measuring role limitations due to physical health or emotional problems. Our results infer that patient and proxy ratings of HRQOL do not seem to be comparable. Reporting the two separately (patient and proxy data) is suggested until more studies can be conducted to evaluate the impact of DIF on interpreting results. Substituting proxy data for patients unable to respond may indeed bias HRQoL estimates and t could therefore impact patient-centered outcome assessment and medical decision making based on these outcomes.

The COBI 'COntinuous hyperosmolar therapy in traumatic Brain-Injured patients' study was supported by a grant from the French Ministry of Health Programme Hospitalier de Recherche Clinique Inter-regional 2016 (PHRCI 2016, RC16_0474). The Nantes University Hospital acted as the sponsor of the study. where the dependent variable is the logit of the probability of giving a favorable response ¥ to each item of the Role Physical (RP) domain and the independent variables are the type of respondent (patients or proxy), the RP domain rest-score € , and their interaction. Differential Item Functioning (DIF) is assumed for the item if the group or/and interaction effect is significant. DIF is considered meaningful if the relative change in the regression coefficient associated with the rest-score is at least 10% § between the model without DIF and the model with DIF. €: score computed without the item under examination §: relative change in the regression coefficient associated with the rest-score is at least 10% between the model without DIF and the model with DIF Table 4. Multivariable logistic regression adjusted on the covariates identified as confounders £ where the dependent variable is the logit of the probability of giving a favorable response ¥ to each item of the Role Emotional (RE) domain and the independent variables are the type of respondent (patients or proxy), the RE domain rest-score € , and their interaction. Differential Item Functioning (DIF) is assumed for the item if the group or/and interaction effect is significant. DIF is considered meaningful if the relative change in the regression coefficient associated with the rest-score is at least 10% § between the model without DIF and the model with DIF. ☐ ☐

Variables

Variables

The answer "No" indicates an absence of limitations and is therefore favorable for patients' HRQol. On the contrary, the answer "Yes" is unfavorable.

Hence, giving a favorable response to the item = Responding "No" to the item Domain scores are computed as follows:

Domain scores range from 0 (i.e., responding "Yes" to all items, indicating limitations due to the physical or emotional health) to 100 (i.e., responding "No" to all items, indicating no limitations due to the physical or emotional health).

b) Searching for DIF with the logistic regression procedure (raw data)

The logistic regression procedure for DIF analysis that we used in our manuscript relies on the estimation of the following logistic regression model for each item of a given domain:

=

Or equivalently,

In this model, the probability of giving a favorable response to a given item (i.e. responding "No" to the item) is predicted by:

-The rest-score of the considered domain (obtained by recomputing the score without the item under examination1 ). Of note, the rest-score serves as an approximation for the patients' level/severity of limitations due to either the physical or the emotional health in our study. -The grouping variable (a dummy variable indicating the group: 1 = Proxy respondent and 0 = Patient respondent) -The interaction between and DIF is evidenced for the item being studied if at least one of the regression coefficients or is significant. Indeed, in this case, the probability of succeeding the item does not only depends on the level/severity of limitations (approximated by the rest-score) but also on the group membership. To search for DIF taking into account confounders, two main strategies can be undertaken:

1) The first one has been proposed by Liu et al. [Liu 2016] and Wu et al. [Wu 2017]. It consists in performing the DIF analysis on data matched using propensity scores. Although not retained for the manuscript, the principle of this strategy is explained in the section below with an illustration based on the COBI trial data.

2) The second one consists in introducing the confounding covariates directly into the logistic regression model as independent variables. This is the strategy that was used in this study. The general model formulation is given by: Where:

is the dummy variable indicating the group: 1 = Proxy respondent and 0 = Patient respondent designate the covariates considered as confounders (see below the Supplementary Table 1 for the lists of the covariates considered as confounders for each domain)

These models are referred to as propensity score (PS) models. Propensity scores (PS) can be estimated for each individual from these logistic regression models.

Supplementary

More precisely, PS correspond to the estimated probabilities of being a proxy respondent (i.e. = 1)

given the observed patients' characteristics suspected to be potential confounders:

) b) Propensity score matching Individuals from the patient group (patients who could self-report) can then be matched with individuals from the proxy group (patients who needed a proxy report) who had the same (or very similar) propensity scores. The aim is to balance the characteristics of the two groups ( = 1: "Proxy respondent" and = 0: "Patient respondent") and thus minimize the confounding. To do so, a method called Optimal Full Matching (with a combination of matching one-to-multiple and multiple-to-one) can be used [Rosenbaum 1991]. This type of matching is implemented in the MatchIt R package [sStuart, King, Imai & Ho, 2011].

This matching has been realized twice on our data: once for the analysis of the RP domain and once for the analysis of the RE domain:

-For the RP domain we obtained 59 clusters composed on average of 4 individuals (range = [2-6]). Among the clusters, there were on average 1.8 patients who could self-report and 2.2 patients who needed a proxy report.

-For the RE domain we obtained 65 clusters composed on average of 3.7 individuals (range = [2-6]). Among the clusters, there were on average 1.7 patients who could self-report and 2.0 patients who needed a proxy report.

For each analysis, the covariate balance between the groups being compared (patient respondent and proxy respondent) can be examined using:

 The percentage of bias reduction (PBR):

and refer to the mean difference between groups computed before (pre) or after (post) matching, respectively.

For quantitative covariates , mean differences were computed as:

Notations:

: estimated mean of among patient respondents (group ) : estimated mean of among proxy respondents for patients (group ) For dummy covariates [0 (= reference) versus 1], mean differences were computed as:

Notations: : estimated proportion of category 1 among patient respondents (group ) : estimated proportion of category 1 among proxy respondents for patients (group )

A percentage of bias reduction greater than 70% is considered as large. Between 40% and 70%, the level of bias reduction is considered as medium [Liu 2016]. Of note, to obtain means ( and ) and proportions ( and ) after matching, one's need to use the matching weights.

 The standardized mean differences (SDiff) For quantitative covariates , standardized mean difference can be computed as:

Notations : estimated mean of among patient respondents (group ) : estimated mean of among proxy respondents for patients (group ) : estimated variance of among patient respondents (group ) : estimated variance of among proxy respondents for patients (group )

For dummy covariates [0 (= reference) versus 1], standardized mean difference can be computed as:

Notations: : estimated proportion of category 1 among patient respondents (group ) : estimated proportion of category 1 among proxy respondents for patients (group )

Standardized mean differences are generally compared to the threshold 0.1 [Stuart, Lee & Leacy 2013] (a Standardized mean difference below this threshold being recommended for declaring balance but this threshold is somewhat arbitrary). Of note, here again, one's needs to use the matching weights to obtain means ( and ) and proportions ( and ) after matching.

The R code for calculating these indices is available in section 3.

c) Searching for DIF with matched data (data matched on PS)

Regular logistic regression analysis is not appropriate for matched data. Instead, one should rely on conditional logistic regression which is more appropriate as it takes into account the dependence structure in the data due to the matched clusters. Of note, the formulation of the logistic regression model to search for DIF is the same, but the estimation for conditional logistic regression is based on the likelihood function considering only the discordant clusters (i.e. the clusters among which at least two individuals choose different response categories for the item under examination for DIF). The concordant clusters (i.e. the clusters among which all individuals choose the same response category for the item under examination) are disregarded because they do not provide any information for likelihood estimation. DIF is evidenced for the item being studied if at least one of the regression coefficients or is significant: in this case, the probability of a favorable response to the item does not only depends on the level/severity of limitations (approximated by the rest-score) but also on the group membership.

d) DIF detection results

As the proportion of concordant clusters was non-negligible, we chose not to perform conditional logistic regression for investigating DIF. Instead, as mentioned in section 1.b., we chose to perform a conventional logistic regression controlling for all confounders included in the PS models. All DIF detection results are given in Tables 3 and4 of the manuscript. Besides, significant DIF effects are graphically represented in Figure 1 plot(Y0~X,ylim=c(0,1),ylab="Probability of answering NO to item 4A",type="l",lwd=2,xlab="Rest-score") points(Y0~X,pch=16,cex=1) lines(Y1~X, col ="red",lwd=2) points(Y1~X,pch=16,cex=1,col="red") # DIF analysis with matching (Full optimal matching, combination of matching one-to-multiple and multiple-to-one)

Figure 1 .

 1 Figure 1. Probability of answering 'no' to item 4c of the Role Physical (RP) domain as a function of the rest-score (score calculated without this item) according to whether the respondent is a patient or a proxy

  DIF was regarded as meaningful if the relative change in the regression coefficient associated with the rest-score was at least 10% between the model without DIF and the model with DIF (significant at the 0.05 level).

  and ESM2, Figures1, 2, and 3.

3 . 1 #

 31 R code for the analysis of the RP domain # R version: R version 4.1.0 (2021-05-18) --"Camp Pontanezen" # Packages: library(MatchIt) # Matching (version ) library(optmatch) # Matching (version ) library(Epi) # Conditional logistic regression (version ) library(naniar) # Missing data imputation (version ) # Data import data <-read.csv(……) # To be completed # Data dictionary# Dataset name: data # Variables: # SF36_Q4A, SF36_Q4B, SF36_Q4C, SF36_Q4D: Items from the RP domain of SF-36 reported et M6 # Response categories and codes: 1 = Yes / 2 = No # SF36_Q5A, SF36_Q5B, SF36_Q5C: Items from the RE domain of SF-36 reported at M6 # Response categories and codes: 1 = Yes, 2 = No # PROXY: Dummy variable indicating who reported the SF-36 at M6 # Categories: 0 = patient who could self-report, 1 = proxy who reported in place of the patient # AGE: Patient age at baseline # GCS: Glasgow Coma Scale score at baseline # RETURN_HOME: Dummy variable indicating whether the patient has returned home at M6 # Categories: 0 = No, 1 = Yes # GOSE: Glasgow Outcome Scale Extended: Dummy variable indicating whether the outcome at M6 is favorable/unfavorable according to the Glasgow Outcome Scale Extended # Categories: 0= Unfavorable outcome (GOS-E <= 5), 1 = Favorable outcome (GOS-E > 5) # Domain RP # Missing data visualization data$NB_MISS_RP <-is.na(data$SF36_Q4A) + is.na(data$SF36_Q4B) + is.na(data$SF36_Q4C) + is.na(data$SF36_Q4D) # Contains the number of missing data for the RP domain table(data$NB_MISS_RP) id_miss <-which(is.na(data$SF36_Q4A)| is.na(data$SF36_Q4B)| is.na(data$SF36_Q4C)| is.na(data$PR_SF36_Q4D))vis_miss(data[id_miss ,c("PR_SF36_Q4A" ,"PR_SF36_Q4B","PR_SF36_Q4C" , "PR_SF36_Q4D" ) ])# Missing data imputation # Impute missing data of individuals having at least 3 non-missing items by the personal mean score # For instance, the code below can be used to impute item 4B id_miss_4B <-which(data$NB_MISS_RP == 1 & is.na(data$SF36_Q4B)) items_RP <-c("SF36_Q4A","SF36_Q4B","SF36_Q4C","SF36_Q4D") data$SF36_Q4B[id_miss_4B] <-round(rowMeans(data[,items_RP], na.rm = T)[ id_miss_4B]) # Select individuals with complete data after imputation data_RP <-data[data$NB_MISS_RP <= 1,] # Computation of the score to the domain + the rest-scores # Total score data_RP$SCORE_RP <-(data_RP$SF36_Q4A + data_RP$SF36_Q4B + data_RP$SF36_Q4C + data_RP$SF36_Q4D-4)/4*100 # Rest-score without item 4A data_RP$RESTSCORE_RP_4A <-(data_RP$SF36_Q4B + data_RP$SF36_Q4C + data_RP$SF36_Q4D-3)/3*100 # Rest-score without item 4B data_RP$RESTSCORE_RP_4B <-(data_RP$SF36_Q4A + data_RP$SF36_Q4C + data_RP$SF36_Q4D-3)/3*100 # Rest-score without item 4C data_RP$RESTSCORE_RP_4C <-(data_RP$SF36_Q4A + data_RP$SF36_Q4B + data_RP$SF36_Q4D-3)/3*100 # Rest-score without item 4D data_RP$RESTSCORE_RP_4D <-(data_RP$SF36_Q4A + data_RP$SF36_Q4B + data_RP$SF36_Q4C-3)/3*100 # Recoding items for logistic regression to search for DIF # New codes: Yes = 0, No = 1 (No being the favorable response category) data_RP$SF36_4A_RECOD = data_RP$SF36_Q4A -1 data_RP$SF36_4B_RECOD = data_RP$SF36_Q4B -1 data_RP$SF36_4C_RECOD = data_RP$SF36_Q4C -1 data_RP$SF36_4D_RECOD = data_RP$SF36_Q4D -Logistic regression procedure to search for DIF on raw data (unadjusted on confounders) #Item 4A Glm4A_RAW <-glm(SF36_4A_RECOD~ PROXY* RESTSCORE_RP_4A, family = 'binomial', data = data_RP) summary(Glm4A_RAW) #Item 4B Glm4B_RAW <-glm(SF36_4B_RECOD~ PROXY* RESTSCORE_RP_4B, family = 'binomial', data = data_RP) summary(Glm4B_RAW) #Item 4C Glm4C_RAW <-glm(SF36_4C_RECOD~ PROXY* RESTSCORE_RP_4C, family = 'binomial', data = data_RP) summary(Glm4C_RAW) #Item 4D Glm4D_RAW <-glm(SF36_4D_RECOD~ PROXY* RESTSCORE_RP_4D, family = 'binomial', data = data_RP) summary(Glm4D_RAW) # Graphical representation of the results (example for item 4A, must be adapted for the other items) # The curve related to patients who could self-report (respectively patients who needed a proxy report) is in black (respectively in red) X <-c(0,33,66,100) # Contains the possible rest-scores # Model predictions for each rest-score among patients (G=0) and proxy (G=1) linear_pred0 <-Glm4A_RAW$coefficients[1]+ Glm4A_RAW$coefficients[3]*X Y0 <-exp(linear_pred0)/(1+ exp(linear_pred0)) linear_pred1 <-Glm4A_RAW$coefficients[1]+ Glm4A_RAW$coefficients[2]+ Glm4A_RAW$coefficients[3]*X+ Glm4A_RAW$coefficients[4]*X Y1 <-exp(linear_pred1)/(1+ exp(linear_pred1))

  Describe any methods used to examine subgroups and interactions NA (c) Explain how missing data were addressed NA (d) Cohort study-If applicable, explain how loss to followup was addressed Case-control study-If applicable, explain how matching of cases and controls was addressed Cross-sectional study-If applicable, describe analytical methods taking account of sampling strategy NA (e) Describe any sensitivity analyses NA Results Participants 13* (a) Report numbers of individuals at each stage of studyeg numbers potentially eligible, examined for eligibility, confirmed eligible, included in the study, completing follow-up, and analysed 5 (b) Give reasons for non-participation at each stage 5 (c) Consider use of a flow diagram Descriptive data 14* (a) Give characteristics of study participants (eg demographic, clinical, social) and information on exposures and potential confounders

Table 1 .

 1 Characteristic of traumatic brain injury patients according to whether they were able (Patient group) or unable (Proxy group) to complete the SF-36 questionnaire. Glasgow Coma Scale; Hypoxia was defined as oxygen saturation less than 92%for more than 5 minutes or PaO2 less than 10 kPa. *: One missing data

	Patient group	Proxy group	P-value
	N=110	N=130

Table 2 .

 2 SF-36 scores at 6 months for patient-and proxy-reported outcomes

	Patient report	Proxy report	P-value
	N=110	N=130

Table 3 .

 3 Multivariable logistic regression adjusted on the covariates identified as confounders £

  : score computed without the item under examination §: relative change in the regression coefficient associated with the rest-score is at least 10% between the model without DIF and the model with DIF Title: Does Differential Item Functioning jeopardizes the comparability of health-related quality of life assessment between patients and proxies in moderate to severe traumatic brain injury patients?

	Electronic Supplementary Materials Supplementary 1
	Estimate (SE) 1. Searching for DIF within the domains RP and RE of the SF-36 with P-value
	Item 5a: 'Cut down the amount of time you spent on work or other activities?' logistic regression
	Intercept	-0.57 (0.83)			0.494
	Type of respondent, proxy Rest-score § a) RP and RE domains: composition and scoring -1.44 (0.67) 0.03 (0.01)		0.032 <0.001
	Rest-score*Type of respondent, proxy	0.04 (0.01)		0.005 Answer
	Item 5b: 'Accomplished less than you would like?' -1.86 (1.02) Type of respondent, proxy Intercept 0.06 (0.91) Role physical (4 items) Rest-score 0.05 (0.01) During the past 4 weeks, have you had any of the following problems with Rest-score*Type of respondent, proxy 0.00 (0.01) your work or regular daily activities as a result of your physical health? 4a. Cut down the amount of time you spent on work or other activities	Yes 1 ☐	0.067 0.951 <0.01 0.826	No 2 ☐
	4b. Accomplished less than you would like Item 5c: 'Didn't do work or other activities as careful as usual?' Intercept -0.84 (0.71) 4c. Were limited in the kind of work or other activities Type of respondent, proxy -0.51 (0.50) Rest-score 0.02 (0.01) took extra effort) 4d. Had difficulty performing the work or other activities (for example, it	☐ ☐ ☐	0.238 <0.001 0.310	☐ ☐ ☐
	Rest-score*Type of respondent, proxy Role emotional (3 items) The general instruction for answering each item is: 'During the past 4 weeks, have you had any of the 0.01 (0.01) 0.175 During the past 4 weeks, have you had any of the following problems with following problems with your work or regular daily activities as a result of any emotional problems your work or other regular daily activities as a result of any emotional (such as feeling depressed or anxious)?'. problems (such as feeling depressed or anxious)?
	5a. Cut down the amount of time you spent on work or other activities £: covariates identified as confounders, i.e. associated with being a proxy respondent and HRQoL : ☐ ☐ 5b. Accomplished less than you would like ☐ ☐ return at home at 6 months [Yes/No], Glasgow Outcome Scale Extended [Favorable/Unfavorable outcome] at 6 months, age at baseline, 5c. Didn't do work or other activities as carefully as usual
	¥: e.g., answering 'no' to the item 'Accomplished less than you would like'

€

  The first stage of this strategy consists in establishing a propensity score model for each studied HRQoL domain (i.e. RP: Role Physical and RE: Role emotional from the SF-36). For a given HRQoL domain, the propensity score model is a logistic regression model predicting the probability of being a proxy respondent as a function of the covariates related to both the group variable (being a proxy respondent) and the outcome (score on the HRQoL domain RP or RE) 2 .

	2. Searching for DIF with matched data (based on propensity score
	matching)
	a) Specification of propensity score models

Table 1 .

 1 Confounders included in each propensity score (PS) model

	PS model for RP domain	PS model for RE domain
	Patients' characteristics at baseline:	Patients' characteristics at baseline:
	Age, GCS score	Age
	Patients' characteristics at M6:	Patients' characteristics at M6:
	Return at home, GOS-E	Return at home, GOS-E
	RP: Role physical domain, RE: Role emotional, GCS: Glasgow Coma Scale score, GOS-E: Glasgow
	Outcome Scale Extended (dummy variable: favorable/unfavorable outcome)

Does Differential Item Functioning jeopardize the comparability of health-related quality of life assessment between patients and proxies in moderate to severe traumatic brain injury patients? Electronic Supplementary Material 2 Figure legends Figure 1.

  Probability of answering 'no' to item 4c of the Role Physical (RP) domain as a function of the rest-score (score calculated without this item) according to whether the respondent is a patient or a proxy

	# Matching with Matchit package m.out <-matchit(PROXY~ AGE+GCS + RETURN_HOME + GOSE, data = data_RP, method="full",distance="logit",min.controls = 1/5,max.controls=5) sdiff = (m0-m1)/sqrt((sd0^2+sd1^2)/2) Electronic Supplementary Materials Supplementary 1 summary(m.out,improvement=T) # Default package output m0 = w.m(x=match.data$AGE[match.data$ PROXY==0],w=w[match.data$PROXY==0]) sd0 = w.sd(x=match.data$ AGE[match.data$ PROXY==0], w=w[match.data$ PROXY==0]) m0 ; sd0 m1 = w.m(x=match.data$AGE[match.data$ PROXY==1],w=w[match.data$PROXY==1]) sd1 = w.sd(x=match.data$ AGE[match.data$ PROXY==1], w=w[match.data$ PROXY==1]) m1 ; sd1 mean_diff = m0-m1 # Logistic regression procedure to search for DIF on raw data, adjusted on confounders included in the PS #Item 4A glm4A = glm(F36_4A_RECOD~PROXY*RESTSCORE_RP_4A + RETURN_HOME + GOSE+ AGE+GCS , family = "binomial", data = match.data) # Same for the other items Quantitative variables 11 Explain how quantitative variables were handled in the analyses. If applicable, describe which groupings were Title: Supplementary Material chosen and why
	match.data = match.data(m.out) # Save the data with the weights # The results can be graphically represented by retrieving the predictive margins Electronic Supplementary Materials Supplementary 2
	# Example with a binary variable: RETURN AT HOME
	# Clusters description # BEFORE MATCHING
	length(unique(match.data$subclass)) # Number of clusters p0 = round(mean(match.data$RETURN_HOME[match.data$REP_PROXY==0]),3)
	mean(table(match.data$subclass)) # Average number of individuals per cluster p1 = round(mean(match.data$RETURN_HOME[match.data$REP_PROXY==1]),3)
	min(table(match.data$subclass)); max(table(match.data$subclass)) # Range mean_diff = p0-p1
	colMeans(table(match.data$subclass,match.data$REP_PROXY)) #Average number of patients sdiff = (p0-p1)/sqrt((p0*(1-p0) + p1*(1-p1))/2)
	and proxy respondents per cluster
	# AFTER MATCHING
	# Description Before / After matching p0 = w.m(x=match.data$RETURN_HOME[match.data$PROXY==0],
	w=w[match.data$PROXY==0]) # Example with a continuous variable: AGE round(p0,3)
	# BEFORE MATCHING p1 = w.m(x=match.data$RETURN_HOME[match.data$PROXY==1],
	m0 = round(mean(match.data$AGE[match.data$PROXY==0]),1) w=w[match.data$PROXY==1])
	sd0 = round(sd(match.data$AGE[match.data$PROXY==0]),1)
	m1 = round(mean(match.data$AGE[match.data$PROXY==1]),1) round(p1,3)
	sd1 = round(sd(match.data$AGE[match.data$PROXY==1]),1) mean_diff = p0-p1
	m0; sd0 sdiff = (p0-p1)/sqrt((p0*(1-p0) + p1*(1-p1))/2)
	m1 ; sd1
	mean_diff = m0-m1 sdiff = (m0-m1)/sqrt((sd0^2+sd1^2)/2) # Conditional logistic regression procedure to search for DIF on matched data
	#Item 4A
	CLogistic4A <-clogistic(SF36_4A_RECOD~ PROXY* RESTSCORE_RP_4A,
	# AFTER MATCHING strata =subclass, data=match.data)
	# Function to compute weighted mean
	w.m <-function(x,w){ sum(x*w)/sum(w)} # Same for the other items
	# Function to compute weighted SD
	w.sd <-function(x, w){sqrt(sum(((x -w.m(x,w))^2)*w)/(sum(w)-1))}
	# Get weights from matching
	w=m.out$weights

  Give unadjusted estimates and, if applicable, confounder-adjusted estimates and their precision (eg, 95% confidence interval). Make clear which confounders were adjusted for and why they were included 5, 6 and Table 3, Table 4 (b) Report category boundaries when continuous variables were categorized NA (c) If relevant, consider translating estimates of relative risk into absolute risk for a meaningful time period NA Continued on next page Other analyses 17 Report other analyses done-eg analyses of subgroups and interactions, and sensitivity analyses NA Other information Funding 22 Give the source of funding and the role of the funders for the present study and, if applicable, for the original study on which the present article is based 8, 9 *Give information separately for cases and controls in case-control studies and, if applicable, for exposed and unexposed groups in cohort and cross-sectional studies. An Explanation and Elaboration article discusses each checklist item and gives methodological background and published examples of transparent reporting. The STROBE checklist is best used in conjunction with this article (freely available on the Web sites of PLoS Medicine at http://www.plosmedicine.org/, Annals of Internal Medicine at http://www.annals.org/, and Epidemiology at http://www.epidem.com/). Information on the STROBE Initiative is available at www.strobe-statement.org.

	(b) Indicate number of participants with missing data for each variable of interest (c) Cohort study-Summarise follow-up time (eg, average and total amount) Cohort study-Report numbers of outcome events or summary measures over time Case-control study-Report numbers in each exposure category, or summary measures of exposure Cross-sectional study-Report numbers of outcome events or summary measures 18 Summarise key results with reference to study objectives Outcome data 15* Main results 16 Key results Limitations 19 Discuss limitations of the study, taking into account sources of potential bias or imprecision. Discuss both direction and magnitude of any potential bias Interpretation 20 Give a cautious overall interpretation of results considering objectives, limitations, multiplicity of analyses, results from similar studies, and other relevant evidence (a) Discussion Generalisability 21 Discuss the generalisability (external validity) of the study results	5 and Table 1 5 and Table 1, Table 2, ESM2 Supplementary Table 1 NA NA NA 5 6 8 8 7, 8

Note:

For instance, to determine whether item 4a (domain RP) is affected by DIF using the logistic regression, the rest-score was computed as:

i.e. the covariates considered as confounders in the manuscript.

Funding

Authors' Contributions

All authors contributed to the development of the ideas, writing, editing, and final review of the submitted article. All authors have read and approved the full version of this article.

Author Disclosure Statement

No competing financial interests exist. Supplementary Table 1. SF-36 scores and items responses at 6 months for proxy report according to whether they were family members or healthcare professionals.

Family member N=97


Thierry Paul

AND Mario Pulvirenti

Sergio Simonella

MEAN FIELD LIMIT FOR THE KAC MODEL AND GRAND CANONICAL FORMALISM

 where a large class of models, including quantum systems, are considered. Here we outline the main ideas in the context of grand canonical measures, for which both the evolution equations and the proof simplify.

 explained the long time behaviour of the homogeneous Boltzmann equation without using the H-theorem nor global properties, but exploiting a Dyson expansion over trajectories. Our contribution is, in spirit, close to this approach.

Introduction

The kinetic description of particle systems is strongly related to propagation of chaos. This property allows one to substitute the complex dynamics of a huge number of particles by a single nonlinear partial differential equation for the probability density of a given particle. More precisely, one adopts a statistical description. At time zero, the N -particle system is assumed to be "chaotic" in the sense that each particle is distributed identically and independently from the others, at least up to an error which is vanishing when N diverges. The dynamics creates correlations and the independence property is lost at any positive time. However, under suitable scaling limits, the statistical independence of any finite group of particles can be recovered, after averaging, in the limit N → ∞. As a consequence, a given particle evolves according to an effective kinetic equation. The nature of this dynamics is determined by the microscopic details of the system and by the regime of physical parameters. Such a mechanism works in the formal (in a few cases, rigorous) derivation of the most common kinetic equations.

If the system is described by a symmetric probability measure W N (Z N , t) at time t, where Z N = (z 1 , ⋯, z N ) is a configuration of the system, being z i = (x i , v i ) position and velocity of the particle i, the probability density of finding the first j particles in the microscopic state Z j is given by the marginal f N j (Z j , t) ∶= ∫ dz j+1 ⋯dz N W N (Z N , t) . In case of a two-body interaction, we have that f N 1 depends on f N 2 , which depends on f N 3 and so on, and these marginals are not tensorized. Indeed, the dynamics creates correlations even though at time zero it is assumed that

W N = (f 0 ) ⊗N .
In many cases, the physical regime of interest can be expressed in terms of suitable scaling limits (N → ∞, together with additional prescriptions), which one can hope to use to prove that [START_REF] Bogolyubov | Problems of a dynamical theory in Statistical Physics[END_REF] f N j (t) → f (t) ⊗j , where f (t) is a solution of the aforementioned kinetic equation.

In three dimensions, fundamental scaling limits for Hamiltonian systems are the following.

• Mean-field limit for classical systems: N → ∞ and the coupling constant of the two-body interaction is scaled by 1 N . Namely, a system of point particles with the mean-field Hamiltonian

H N = 1 2 N ∑ i=1 v 2 i + 1 2N ∑ i≠j φ(x i -x j ) ,
where φ is a two-body potential. The corresponding expected kinetic equation is, in this case, the Vlasov equation.

• Low-density or Boltzmann-Grad limit: N → ∞ and the diameter of hard-sphere like particles is ∼ N -1 2 . The corresponding kinetic equation is the Boltzmann equation.

• Weak-coupling limit: N → ∞ and ε ∼ N -1 3 is a scale parameter coding the interaction length. The radial interaction potential -say, smooth and decaying -is scaled as φ( r ε ), so that the Hamiltonian of the system is

H N = 1 2 N ∑ i=1 v 2 i + √ ε 2 ∑ i≠j φ ( |x i -x j | ε ) .
The corresponding kinetic equation is the Landau equation.

• High-density limit:

N → ∞ and ε ∼ N -1 4
, where the Hamiltonian is

H N = 1 2 N ∑ i=1 v 2 i + ε 2 ∑ i≠j φ ( |x i -x j | ε ) .
The corresponding kinetic equation is the Lenard-Balescu equation.

For a recent analysis of these scalings we refer to [START_REF] Nota | Interacting particle systems with long-range interactions: scaling limits and kinetic equations[END_REF][START_REF] Nota | Interacting particle systems with long-range interactions: Approximation by tagged particles in random fields[END_REF], where allowed classes of interaction potentials are also discussed. Scaling limits and propagation of chaos have a long history starting from the pioneering work of Bogoliubov [START_REF] Bogolyubov | Problems of a dynamical theory in Statistical Physics[END_REF], see e.g. [START_REF] Spohn | Large Scale Dynamics of Interacting Particles[END_REF][START_REF] Pulvirenti | A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory[END_REF] for surveys.

The mechanism for which we expect that propagation of chaos actually holds is deeply different in the above cases. For the mean-field limit, the force exerted by particle j on particle i is O( 1 N ) so that the effect of one particle on the other is triviallynegligible. Hence z i and z j can be considered as independent random variables in the limit N → ∞, if they were so at time zero. At variance, in the case of the Boltzmann-Grad limit, the forces are strong; but due to the low density regime, the probability of two given particles colliding in (0, t) is small, although any given particle is subject to a collision with strictly positive rate. Finally in the weak-coupling limit, the forces are of order 1

√ ε ∼ N 2 3
, but the potential range is small and the interaction takes place in a time interval of order ε. Overall the variation of momentum of particle i due to the interaction with particle j is of order √ ε, thus negligible. A similar argument holds in the high-density case. Going further in this analysis, one may ask about quantitative estimates on the defect of chaos. In recent times, the authors have approached this problem in [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF] by the method of "correlation errors" (also called "v-functions"), which goes back to previous work in the context of kinetic limits [START_REF] Boldrighini | Pellegrinotti : Non equilibrium fluctuations in particle systems modelling Reaction-Diffusion equations[END_REF]- [START_REF] De Masi | Truncated correlations in the stirring process with births and deaths[END_REF], [START_REF] Pulvirenti | The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error[END_REF]. Correlation errors are a tool to measure the tendency of f N j to factorize and converge in the mean field limit. We will give a precise definition in the next section (Definition 1 below; see also Def. 2.1 in [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF]). In [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF], they have been used to prove a result on the Kac's model, which we recall next.

We consider N particles following a stochastic dynamics: to each particle, say particle i, we associate a velocity v i ∈ R 3 and the vector

V N = {v 1 , ⋯, v N }
changes by means of two-body collisions at random times, with random scattering angle. The probability density W N (V N , t) evolves according to the master equation (forward Kolmogorov equation)

(2)

∂ t W N = 1 N T N W N where T N W N = ∑ i<j T i,j W N , (3)
T i,j W N (V N ) = ∫ dωB(ω; v i -v j ){W N (V i,j N ) -W N (V N )} ,
and where

V i,j N = {v 1 , ⋯, v i-1 , v ′ i , v i+1 , ⋯, v j-1 , v ′ j , v j+1 , ⋯, v N } and the pair v ′
i , v ′ j gives the outgoing velocities after a collision with scattering (unit) vector ω and incoming velocities v i , v j .

B(ω;v i -v j ) |v i -v j |
is the differential crosssection of the two-body process and we shall assume here, for simplicity, that B is bounded. The resulting kinetic equation is the homogeneous Boltzmann equation

∂ t f (v) = ∫ dv 1 ∫ dωB(ω; v -v 1 ){f (v ′ )f (v ′ 1 ) -f (v)f (v 1 )} =∶ Q(f, f )(v) (4)
Such a model has been introduced and studied by Kac in [START_REF] Kac | Foundations of kinetic theory[END_REF][START_REF] Kac | Probability and related topics in physical sciences[END_REF] and has given rise to a wide literature later on (see e.g. [START_REF] Grünbaum | Propagation of chaos for the Boltzmann equation[END_REF]26,[START_REF] Graham | Stochastic particle approximations for generalized Boltzmann models and convergence estimates[END_REF][START_REF] Hauray | On Ka's chaos and related problems[END_REF][START_REF] Mischler | Kac's program in kinetic theory[END_REF][START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF]). For instance in [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF], we prove the following.

Theorem 1.1 ([22]). For all t > 0 and all j = 1, ..., N , the marginals satisfy

(5) ∥f N j (t) -f (t) ⊗j ∥ L 1 ≤ C 2 e C 1 t j 2 N
where C 1 , C 2 > 0 are explicitly computable constants.

In the present paper, we are going to recover a similar result, albeit in a different setting which we refer to as grand canonical ensemble. This is a standard formalism in statistical mechanics, adapted here to the kinetic problem. With respect to the model introduced above, the novelty of the grand canonical framework is twofold: (i) we allow the total number of particles N to be a (Poisson) random variable with intensity µ → ∞, and consequently (ii) we do not label the particles from 1 to N (as done in the definition of marginal), but compute only averages of symmetrized quantities. This leads to the notion of correlation functions, denoted below {f µ j } ∞ j=1 , which play the role of the marginals {f N j } ∞ j=1 in this context. The advantage of the grand-canonical formalism is that the correlations due to conservation of energy and mass are zero by assumption, and one can focus more easily on the correlations having a purely dynamical origin. In this spirit we will reformulate the method of [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF] leading to a result analogous to [START_REF] Carlen | Central limit theorem for Maxwellian molecules and truncation of the wild expansion[END_REF], in terms of correlation functions (Theorem 3.2 below). We shall see that this leads to simplifications of the corresponding proof in the canonical ensemble. Indeed, even if f N j and f µ j are asymptotically equivalent, the latter functions encode fewer correlations. This is reflected eventually in a smaller number of terms in the basic evolution equations for the error (see Remark 3.1 below).

The paper is organized as follows. In the next section, we introduce the Kac's model in the grand canonical ensemble, and define the correlation errors. In Section 3 we discuss the evolution equations, whose derivation is postponed to the appendix. In Section 3 we establish the main result, Theorem 3.2, and its proof is presented in the last section.

Kac model and grand canonical ensemble

We start by recalling the main ensembles which can be adopted to describe a collisional dynamics, in the spirit of the abovementioned Kac's model.

• Microcanonical. The binary collisions preserve energy, and the stochastic dynamics lives in the energy surface

E = 1 N ∑ i v 2 i .
• Canonical. E is random and only N is fixed, as in the model discussed in the Introduction. • Grand canonical. N is random. We introduce a sequence of initial density distributions {W n 0 } n≥0 on the phase space ⋃ n≥0 R 3n . By definition, 1 n! W n 0 (V n ) is the symmetric probability density of the configuration V n , being n the total number of particles. The average number of particles is then

µ ∶= ∑ n≥0 n ∫ 1 n! W n 0 (V n )dV n .
Here µ > 0 is a free parameter (eventually µ → ∞) and the distributions do depend on µ, although this dependence is dropped in our notation (W n 0 = W n,µ 0 ). At time t > 0, the state of the system is described by the time-evolved measure {W n (t)} n≥0 solving (cf. ( 2)) ( 6)

∂ t W n = 1 µ T n W n .
Focusing now on the grand canonical case, let us introduce a natural description of correlations and propagation of chaos. In order to examine correlations due exclusively to the dynamics, we choose a perfectly tensorized and still symmetric initial state, namely [START_REF] Caprino | A particle systems approximating stationary solutions to the Boltzmann equation[END_REF] W n 0 = f ⊗n 0 e -µ µ n (however most of our discussion could be extended to a larger class of initial measures). Instead of computing probability densities of particles with labels {1, ⋯, j}, we define rescaled correlation functions by

f µ j (V j , t) = µ -j ∑ k≥0 1 k! ∫ dv j+1 ⋯dv j+k W j+k (V j+k , t) .
Loosely speaking, this corresponds to computing the amount of j-tuples of particles in the configuration V j at time t. For a purely factorized state the average total number of j-tuples is µ j , which explains the rescaling factor.

In particular with the choice [START_REF] Caprino | A particle systems approximating stationary solutions to the Boltzmann equation[END_REF], one checks that ∥f µ j ∥ L 1 = 1 (initially and for all times).

Suppose that (8) lim

N →∞ f µ j = f (t) ⊗j ,
say in L 1 -norm. We may ask about convergence rates. We observe that, in general (e.g. for j ∼ µ) f µ j (t) is not expected to be asymptotically equivalent to (f µ 1 (t)) ⊗j , even if it is assumed to be so at time zero. One can then pose the question: how large can j = j(µ) be in such a way that (8) holds true?

Having at our disposal the estimate

(9) ∥f µ j -f (t) ⊗j ∥ L 1 ≤ C j ( 1 µ ) α
for some α ∈ (0, 1) and C > 0, we cannot make better predictions than j(µ) ∼ log µ. On the other hand if for a moderately large j, say j ≈ µ γ , γ ∈ (0, 1),

f µ j ≈ (f µ 1 (t)) ⊗j , then it is natural to consider (10) (f µ 1 (t) -f (t)) ⊗j =∶ E j ;
namely the product of the differences rather than the difference of the products. Expanding [START_REF] De Masi | Glauber evolution with Kac potentials[END_REF] one finds that

(11) E j (t) ∶= ∑ K⊂J (-1) |K| f µ J/K (t)f (t) ⊗K
where J = {1, 2, ⋯, j} is the set of the first j indices, K is any subset of J, J/K is the relative complement of K in J and |K| is the cardinality of K. f µ A (t) stands for the |A|-marginal f µ |A| (t) computed in the configuration (z i ) i∈A . Similarly f (t) ⊗K = f (t) ⊗|K| evaluated in (z i ) i∈K . The formula [START_REF] De Masi | Truncated correlations in the stirring process with births and deaths[END_REF] has an inverse formula, (see [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF]) that is

(12) f µ j (t) = ∑ K⊂J E J/K (t)f (t) ⊗K
where the notation E J/K is the same one as for F µ J/K , and where we set

E 0 = E ∅ = 1.
This motivates the following Definition 1 (Correlation error (grand-canonical)). For any j ≥ 1, setting J = {1, 2, ⋯, j}, we define the "correlation error" of order j by

(13) E j (t) ∶= ∑ K⊂J (-1) |K| f µ J/K (t)f (t) ⊗K ,
where the terms K = ∅ and K = J have to be interpreted as f µ J = f µ j and (-1) j f ⊗J = (-1) j f ⊗j , respectively.

For instance

E 1 (v 1 ) = f µ 1 (v 1 ) -f (v 1 ) E 2 (v 1 , v 2 ) = f µ 2 (v 1 , v 2 ) -f µ 1 (v 1 )f 1 (v 2 ) -f (v 1 )f µ 1 (v 1 ) + f (v 1 )f (v 2 ) E 3 (v 1 , v 2 , v 3 ) = f µ 3 (v 1 , v 2 , v 3 ) -f µ 2 (v 1 , v 2 )f (v 3 )⋯ + f µ 1 (v 1 )f (v 2 )f (v 3 )⋯ -f (v 1 )f (v 2 )f (v 3 ) .

Dynamical equations

It is well known, and can be easily checked from [START_REF] Carlen | On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation[END_REF], that the sequence {f µ j } satisfies the following hierarchy of equations (called BBKGY in analogy with the Hamiltonian case):

(14) ∂ t f µ j = T j µ f µ j + C j+1 f µ j+1 for j = 1, ⋯, ∞ , where (15)
C j+1 f µ j+1 (V j ) = j ∑ i=1 C i,j+1 f µ j+1 (V j ) and (16) C i,j+1 f µ j+1 (V j ) = ∫ dv j+1 ∫ dωB(ω; v i -v j ){f µ j+1 (V i,j+1 j+1 ) -f µ j+1 (V j+1
)} with the same notations as in [START_REF] Caprino | A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result[END_REF].

This hierarchy must be compared with the limiting hierarchy (in the formal limit µ → ∞)

(17) ∂ t f j = C j+1 f j+1
which is satisfied by products f j (t) = f (t) ⊗j , being f (t) solution of the homogeneous Boltzmann equation (4). Introducing correlation errors given by ( 13), we deduce from ( 14) the corresponding evolution equations. We find (see the Appendix)

∂ t E j = T j µ E j + D j E j + D 1 j (E j+1 ) + D -1 j (E j-1 ) + D -2 j (E j-2 ) (18)
where D 1 j = C j+1 and the operators D j , D -1 j , D -2 j are defined by:

D j E j = ∑ i∈J C i,j+1 [f ⊗{i} E J i ∪{j+1} + f ⊗{j+1} E J ] D -1 j E j-1 = 1 µ ∑ i,s∈J T i,s f ⊗{i} E J i D -2 j E j-2 = 1 2µ ∑ i,s∈J T i,s f ⊗{i,s} E J i,s , ( 19
)
where J is the set of the first j indices, J i = J/{i}, J i,s = J/{i, s} and

f ⊗{i} = f (v i ), f ⊗{i,s} = f (v i )f (v s ) .
Here we use the convention the D -2 1 = 0. Moreover these operators may depend on time, although we will drop out this dependence from the notation.

Note that the first line contains operators which do not change the particle number, C j+1 is an operator decreasing by one the number of particles, while D -1 j and D -2 j are operators increasing the number of particles by one and two respectively.

Remark 3.1. The marginals in the canonical setting satisfy the same hierarchy of equations as in (14) (with µ replaced by N ), up to a combinatorial factor (N -j)/N in front of the collision operator C j+1 . This slight difference produces several spurious terms in the definitions of the operators D j , D -1 j , D -2 j (see Eq.s (42)-( 44) in [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF]), which are absent in the definition [START_REF] Nota | Interacting particle systems with long-range interactions: scaling limits and kinetic equations[END_REF]. Therefore the grand-canonical equations single out the contributions having a purely dynamical interpretation, in terms of variation of clusters (groups of particles) with j mutually correlated particles. Since the interactions are binary, such a cluster can be created in five ways only, corresponding respectively to the five terms in [START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF], and depending on how many independent particles enter the game through the described collision.

We conclude this section by establishing our main result. Theorem 3.2. Let us assume f µ j (0) = f ⊗j 0 being f 0 the initial datum for the Boltzmann equation. Then for all t > 0, there exists G > 1 such that, for j √ µ sufficiently small,

(20) ∥f µ j (t) -f ⊗j (t)∥ L 1 ≤ Gj 2 µ .
Here we are assuming a full factorization at time zero. Looking at the proof it will be clear that this condition can be relaxed. Here we prefer to work in the simplest context to outline the main ideas.

Let us remark that, as it can be shown from of the proof of Theorem 3.2 the condition " j √ µ sufficiently small" can be quantified as " j √ µ ≤ αe -βt for some contants α, β > 0".

Towards the Boltzmann equation

Starting from [START_REF] Hauray | On Ka's chaos and related problems[END_REF], by the Dyson (Duhamel) expansion we have:

f µ j (t) = ∞ ∑ n=0 ∫ t 0 dt 1 ∫ t 1 0 dt 2 ⋯ ∫ t n-1 0 dt n (21) U µ j (t -t 1 )C j+1 . . . U µ j+n-1 (t n-1 -t n )C j+n U µ j+n (t n )f 0,n+j , where f 0,j = f ⊗j 0 and U µ j (t) = e T j µ t .
On the other hand we also have, for f j (t) = f (t) ⊗j , being f (t) the solution to the Boltzmann equation

f j (t) = ∞ ∑ n=0 ∫ t 0 dt 1 ∫ t 1 0 dt 2 ⋯ ∫ t n-1 0 dt n C j+1 . . . C j+n f 0,n+j (22) = ∞ ∑ n=0 t n n! C j+1 . . . C j+n f 0,n+j .
Assuming that B is bounded then we have ( 23)

∥T j ∥ ≤ C 1 j(j -1) 2 , ∥C j+1 ∥ ≤ C 1 j.
From now on we denote by ∥ ⋅ ∥ the L 1 -norm and by C i , i = 1, 2⋯ different numerical (positive) constants.

Then since ∥U µ j (t)∥ = 1 both the right hand side of ( 21) and ( 22) are bounded by

∑ n≥0 C n 2 t n n! j(j + 1)⋯(j + n -1) ≤ 2 j ∑ n≥0 (2C 2 t) n ,
which is converging for t small. Therefore the rest of the two series is negligible, uniformly in µ, and this allows us to perform the limit µ → ∞. Indeed realizing that, for all j,

lim µ→∞ U µ j (t) = I
strongly, we have also the term by term convergence of ( 21) to ( 22) in L 1 . Once having the convergence for t < 1 2C 2 , we can iterate the procedure since the new initial conditions, f µ j (t) satisfy ∥f µ j (t)∥ = 1. Then we have convergence for any arbitrary time.

What is the size of chaos in this case? By size of chaos we mean the maximal α for which we have convergence for all j such that j µ α → 0 as µ → ∞. Our main Theorem 3.2 shows that the size of chaos is 1 2 . On the other hand, as we shall see in a moment, we do not expect a size of chaos strictly greater than 1 2 so we believe that our result can be considered as optimal as regards this feature.

We observe that lim µ,j→∞ U µ j (t) = I only if j 2 µ → 0. Indeed expressing U µ (t) in terms of the series expansion

U µ (t)f 0,j = ∑ n t n n! 1 µ n ∑ i 1 <k 1 ⋯ ∑ i n <k n T i 1 ,k 1 ⋯T i n ,k n f 0,j
we realize that the L 1 norm of the right hand side can be bounded by

∑ n (Ct) n n! ( j(j -1) 2µ ) n .
Therefore we expect that an estimate of the size of chaos cannot be better than 1 2 and this can be considered as optimal. Coming back to the dynamical description given by ( 18), we estimate the size of the L 1 norms of the operators appearing in [START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF] by

∥D j ∥ = O(j), ∥D 1 j ∥ = O(j), ∥D -1 j ∥ = O ( j 2 µ ) , ∥D -2 j ∥ = O ( j 2 µ
) .

Note that [START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF] is inhomogeneous so that it has nontrivial solutions even for initial data E j (0) = 0, j > 0 (namely when the initial state is tensorized) which is the case we consider here.

The control of E passes trough a Dyson expansion around the two parameters semigroup Vj generated by

T j µ E j + D j E j , namely the solution of ∂ t Vj (t, s) = ( T j µ + D j ) Vj (t, s) Vj (s, s) = I.
We easily derive the following bound [START_REF] Pulvirenti | A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory[END_REF] ∥ Vj (t, s)∥ ≤ e

C( j 2 µ +j)(t-s) .
The action of the other operators C j+1 , D -1 j , D -2 j represents a positive, a negative and a negative double jump respectively in the space of indices.

We recall that initially the distribution factorizes, namely

E j (0) = δ j,0 , then E j (t) = ∞ ∑ n=0 ∑ k 1 ⋯k n ∫ t 0 dt 1 ∫ t 1 0 dt 2 ⋯ ∫ t n-1 0 dt n (25) Vs 1 (t, t 1 )D k 1 . . . Vs n (t n-1 , t n )D k n Vs n+1 (t n )E j 0 (0).
where k i ∈ {-1, -2, 1}. The sequence {s 1 ⋯s n } is defined recursively by

s r+1 = s r + k r , s 1 = j.
and

j 0 = j + n ∑ i=1 k i the initial index. Here D k = D k
s k and we drop the indices for notational simplicity.

We have to consider the case j 0 = 0 but in the following it will be convenient to consider general j 0 ≥ 0.

Note also that E 0 = 1 and the only possible term involving E 0 is

D -2 2 E 0 = 1 µ T 2 f ⊗2
which is perfectly defined in L 1 (R 2d ) being f = f (t) the solution of the homogeneous Boltzmann equation. For simplicity and without loss of generality we choose the parameters in such a way that

∥T j ∥ ≤ j 2 2 , ∥D j ∥ ≤ j 2 , ∥D 1 j ∥ ≤ j, ∥D -1 j ∥ ≤ j 2 µ , ∥D -2 j ∥ ≤ j 2 µ and we can prove Proposition 4.1. Suppose that, (26) ∥E j (0)∥ ≤ ( j 2 µ ) j/2 C j 0 , for all j ≥ 1,
for some C 0 ≥ 1. Then, there exists t 0 sufficiently small and A > 0 (both explicitily computable) such that for any t ≤ t 0 ,

(27) ∥E j (t)∥ ≤ ( j 2 µ ) j/2
(AC 0 ) j , for all j ≥ 1.

We first observe that we can assume j ≤ 2

C 0 √ µ because otherwise ∥E j (t)∥ ≤ 2 j ≤ 2 j ( jC 0 2 √ µ ) j = ( jC 0 √ µ ) j .
Next from [START_REF] Spohn | Large Scale Dynamics of Interacting Particles[END_REF] we have

E j (t) = M -1 ∑ n=0 ∑ k 1 ⋯k n ∫ t 0 dt 1 ∫ t 1 0 dt 2 ⋯ ∫ t n-1 0 dt n Vs 1 (t, t 1 )D k 1 . . . Vs n (t n-1 , t n )D k n Vs n+1 (t n )E j 0 (0) + ∑ k 1 ⋯k M ∫ t 0 dt 1 ⋯ ∫ t M -1 0 dt M Vs 1 (t, t 1 )D k 1 . . . Vs M (t M -1 -t M )D k M E j 0 (t M )
and denote by T 1 j (t) and T 2 j (t) respectively the two terms in the right hand side of the above expression.

Moreover we fix M = 2

C 0 √ µ. With this choice j + s m ≤ j + M ≤ 4 C 0 √ µ
so that all the operators D -1 j and D -2 j appearing in the definition of T 1 j and T 2 j satisfy the bound

∥T j ∥ ≤ j, ∥D -1 j ∥ ≤ j 2 µ ≤ j, ∥D -2 j ∥ ≤ j 2 µ ≤ j
provided that µ is large enough. As a consequence, by [START_REF] Pulvirenti | A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory[END_REF] (28) ∥ Vj (t)∥ ≤ e jt .

We first estimate T 2 j (t). Observe that, for t 0 ≤ 1 for which e jt 0 ≤ e j ∥ Vs 1 (t, t 1 )D k 1 . . . Vs M (t M -1 -t M )D k M E j 0 (t M )∥ ≤ e j e M t (j + M ) M 2 j+M as follows by the estimates e s 1 (t-t 1 ) e s 2 (t 1 -t 2 ) ⋯e s M (t M -1 -t M ) ≤ e (j+M )t ≤ e j e M t

(j + k 1 )(j + k 1 + k 2 )⋯(j + k 1 + k 2 + ⋅k M ) ≤ (j + M ) M ≤ (2M ) M ∥E j 0 (t M )∥ ≤ 2 j+M .
Therefore, setting τ = te t , using the bound M M ≤ C M 1 M ! and choosing τ so small that τ 8C 1 ≤ 1 2 , we have

∥T 2 j (t)∥ ≤ 4 M τ M M ! M M e j 2 j+M ≤ (2e) j ( 1 2 ) M . Finally since sup M M j ( 1 2 ) M ≤ (C 2 j) j we conclude that (29) ∥T 2 j (t)∥ ≤ ( C 3 j √ µ ) j .
To estimate T 1 j (t) we split it into two contributions T 1,< j (t) and T 1,> j (t) relative to the events j 0 ≤ j and j 0 > j respectively. Then, proceeding as above

∥T 1,< j (t)∥ ≤ ∑ n≤M -1 ∑ k 1 ⋯k n e j τ n n! (j + n) n ( j 2 µ ) j-j 0 2 ∥E j 0 ∥.
The contribution ( j 2 µ ) j-j 0 2

arises from the fact that to reach j 0 from j we need at least j-j 0 2 negative jumps, each of them produces a factor bounded by j 2 µ . Finally using (26) and the usual arguments we arrive at

∥T 1,< j (t)∥ ≤ ∑ n≥0 (C 4 τ ) n e j ( j 2 µ ) j-j 0 2 ( C 2 0 j 2 0 µ ) j 0 2 ≤ ( C 5 C 2 0 j 2 µ ) j 2
.

Finally

∥T 1,> j (t)∥ ≤ ∑ n≤M -1 ∑ k 1 ⋯k n e j τ n n! (j + n) n ( j 2 0 µ ) j 0 2 C j 0 0 . Setting now ℓ = ℓ(k 1 , k 2 ⋯k n ) = j 0 -j we have ( (j + ℓ) 2 µ ) j+ℓ 2 C ℓ 0 = ( j 2 µ ) j/2 ( (j + ℓ) 2 µ ) ℓ 2 ( j + ℓ j ) j C ℓ 0 ≤ ( j 2 µ ) j/2 ( 2 C 0 ) ℓ e ℓ C ℓ 0 ≤ ( j 2 µ ) j/2 (2e) n . (30)
where, in the last step we used that (remind that both j and M are bounded by

2 √ µ C 0 ) (j + ℓ) √ µ ≤ j + M √ µ ≤ 4 C 0 , ( j + ℓ j ) j ≤ e ℓ , ℓ ≤ n.
Thus, for τ small enough:

∥T 1,> j (t)∥ ≤ e 2j C j 0 ∑ n≥0 C n 5 τ n ( j 2 µ ) j 2 ≤ ( C 6 C 2 0 j 2 µ ) j 2
and the proof is achieved.

As a Corollary we can easily prove Proposition 4.2. Let us suppose that E j (0) = 0 for j ≥ 1 (E ∅ (0) = 1) corresponding to a fully factorized initial state. Then for all t > 0 and all together with the following notation. For any set of particle indices K = {i 1 , ⋯, i k } of cardinality k = |K|, we write

f µ K = f µ k (z i 1 , ⋯, z i k ) , E K = E k (z i 1 , ⋯, z i k ) , f ⊗K = f (z i 1 )⋯f (z i k ) with the conventions f µ ∅ = E ∅ = f ⊗∅ = 1. We also write Q(f, f ) i = Q(f, f )(v i )
for the Boltzmann operator (4) evaluated in particle i.

We start by the simple computation for j = 1:

∂ t E {1} = ∂ t (f µ {1} -f ⊗{1} ) = C 2 f µ {1,2} -Q(f, f ) 1 = C 2 (f ⊗{1,2} + f ⊗{1} E {2} + E {1} f ⊗{2} + E {1,2} ) -Q(f, f ) 1
where in the last step we used the inverse formula (cf. ( 12)) for j = 2. Noticing that, by ( 15)-( 16

), C 2 (f ⊗{1,2} ) = Q(f, f ) 1 , we are left with ∂ t E {1} = C 2 (f ⊗{1} E {2} + E {1} f ⊗{2} ) + C 2 (E {1,2} )
and the two terms on the right hand side correspond to D 1 E {1} and D 1 1 (E {1,2} ) respectively. Therefore ( 18) is verified for j = 1.

We assume now that {E k } k≤j-1 satisfy (18), and prove the same thing for k = j. Using definition (32) and the evolution equations for f µ J and f , we find that

∂ t E J = ∂ t ⎛ ⎜ ⎝ f µ J -∑ K⊂J K≠∅ f ⊗K E J/K ⎞ ⎟ ⎠ = T j µ f µ J + ∑ i∈J C i,j+1 f µ J∪{j+1} -∑ K⊂J K≠∅ f ⊗K ∂ t E J/K -∑ K⊂J K≠∅ ∑ i∈K f ⊗K/{i} Q(f, f ) i E J/K . (33)
We compute the three terms in the second line, one by one. First, using the inverse formula and denoting J i = J/{i}, J i,s = J/{i, s} etc.,

T j µ f µ J = T j µ E J + ∑ i,s∈J i≠s T i,s µ {f ⊗{i} E J i + 1 2 f ⊗{i,s} E J i,s } + ∑ K⊂J K≠∅ f ⊗K T j-k µ E J/K + ∑ K⊂J |K|≠0,1 ∑ i∈K s∈J/K T i,s µ f ⊗{i} f ⊗K i E J/K + ∑ K⊂J |K|≠0,1,2 ∑ i,s∈K i≠s T i,s 2µ
f ⊗{i,s} f ⊗K i,s E J/K (34) and, similarly,

∑ i∈J C i,j+1 f µ J∪{j+1} = ∑ i∈J C i,j+1 {E J f ⊗{j+1} + f ⊗{i} E J i ∪{j+1} + E J∪{j+1} } + ∑ K⊂J |K|≠0 ∑ i∈J/K C i,j+1 f ⊗K f ⊗{j+1} E J/K + ∑ K⊂J K≠∅ ∑ i∈J/K C i,j+1 f ⊗K {f ⊗{j+1} E J/K + f ⊗{i} E (J∪{j+1}/K) i + E J∪{j+1}/K } . (35)
Secondly, using the inductive hypothesis and the explicit definition of the operators D j-k , D 1 j-k , D -1 j-k , D -2 j-k , we obtain that

-∑ K⊂J K≠∅ f ⊗K ∂ t E J/K = -∑ K⊂J K≠∅ f ⊗K T j-k µ E J/K -∑ K⊂J K≠∅ ∑ i,s∈J/K i≠s T i,s µ f ⊗K {f ⊗{i} E (J/K) i + 1 2 f ⊗{i,s} E (J/K) i,s } -∑ K⊂J K≠∅ ∑ i∈J/K C i,j+1 f ⊗K {f ⊗{j+1} E J/K + f ⊗{i} E (J∪{j+1}/K) i + E J∪{j+1}/K } . ( 36
)
We note that the first line in (34) and the first line in (35) reproduce the right hand side of the desired equation [START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF]. Therefore we need to check that the remaining terms cancel out. This is straightforward:

• the second line in (34) cancels with the first term on the right hand side in (36); • last line in (34) cancels with the second line in (36) (change variables K → K ∪ {i} and K → K ∪ {i, s}); • second line in (35) cancels with last line in (33); • last line in (35) cancels with last line in (36). Roma -Italia Email address: sergio.simonella@uniroma1.it

[

  26] A.-S. Sznitman: Topics in propagation of chaos. In: École d'été de Probabilités de Saint-Flour XIX 1989, Lect. Notes in Math. 1464, Springer, Berlin (1991) 165-251. (T.P) CNRS Laboratoire Ypatia des Sciences Mathématiques (LYSM), Roma, Italia & Laboratoire Jacques-Louis Lions (LJLL), Sorbonne Université, Paris, France Email address: thierry.paul@sorbonne-universite.fr (M.P.) Sapienza Università di Roma, Dipartimento di Matematica G. Castelnuovo, Piazzale A. Moro 5, 00185 Roma -Italia and International Research Center on the Mathematics and Mechanics of Complex Systems, ï¿oeMeMoCS, University of L'Aquila, Italy Email address: pulviren@mat.uniroma1.it (S.S.) Sapienza Università di Roma, Dipartimento di Matematica G. Castelnuovo, Piazzale A. Moro 5, 00185

Acknowledgment

We thank the "Istituto Nazionale di Alta Matematica"(INdAM) for partial support.

j > 1, one has (31) ∥E j (t)∥ ≤ (Ae Bt ) j ( j √ µ ) j where B ≥ 0 and A ≥ 1.

Note that from the evolution equation

and Proposition 4.2 we also obtain the estimate ∥E 1 (t)∥ ≤ tA 2 e 2Bt 4 µ which will be useful in a moment.

Proof of Theorem 3.2 We find that, for some time-dependent constants C 7 , C 8 , G, using Proposition 4.2 and the estimate for E 1 (t) above,

Appendix. Derivation of [START_REF] Mischler | A new approach to quantitative propagation of chaos for drift, diffusion and jump processes[END_REF] In this appendix, we deduce the correlation error equations starting from the BBGKY hierarchy, ( 14), which we rewrite for the reader's convenience:

We shall see that the computation is drastically simpler with respect to the analogous derivation in the canonical ensemble reported in the Appendix of [START_REF] Paul | On the Size of Chaos in the Mean Field Dynamics[END_REF] (where additional terms appear due to the canonical constraint). We recall the definition of correlation error (32)

This concludes the derivation of ( 18).