dataesr/scientific-paragraphs-categorization huggingface dataset:
ACKNOWLEDGMENTS We thank the  Assistance Publique -Hôpitaux de Paris, Santé publique France ,  Niel Hens ,  Pieter Libin ,  Julia Bielicki ,  Pascal Crepey , and  Raphaëlle Métras  for useful discussions;  Philippe Vanhems ,  Elisabeth Bothello-Nevers ,  Olivier Epaulard ,  Jean Beytout ,  Annabelle Ravni ,  Olivier Dugrip  and the  Academie of the Auvergne-Rhône-Alpes region  for the school screening initiatives; the  Ministry of National Education  for supporting the surveillance activities. This study was partially funded by:  ANR  projects  COSCREEN  ( ANR-21-CO16-0005 ) and  DATAREDUX  ( ANR-19-CE46-0008-03 );  ANRS-MIE  project  EMERGEN  ( ANRS0151 );  EU   H2020 grants MOOD  ( H2020-874850 ) and  RECOVER  ( H2020-101003589 );  EU HORIZON  grant  VERDI (101045989 ); REACTing COVID-19 grant.
DATA SHARING De-identified individual data on contacts of the two schools under study are publicly available at the Sociopatterns project website (http://www. sociopatterns. org/datasets/). De-identified aggregated COVID-19 community surveillance data by age class are publicly available at Sante publique France Data Observatory platform (https://geodes. santepubliquefrance. fr/). De-identified aggregated COVID-19 prevalence data of pilot screenings during the Alpha wave used in this study are available in the tables reported in the Appendix.
ACCEPTED MANUSCRIPT / CLEAN COPY In this study, we define a "sparse" design as any study with only a few sampling points and that challenges the identifiability of the model, which means that the sparse nature of data depends on the complexity of the model of interest. Our work was based on data collected during the development of gantenerumab, a monoclonal antibody for the treatment of Alzheimer's disease. As this drug has a very long half-life, the clinical trials were conducted using a parallel design (more than 13 weeks of follow up), which is not the classical design for PK equivalence studies that are usually conducted using a crossover design. In this real case, we compared the PK data gathered in participants treated with two formulations of gantenerumab. Then, we evaluated the performance of the MB approach on simulations based on data from this study and assessed the impact of study design, model misspecification, and the relevance of a model selection step.
Although this assessment was based on PK data from a monoclonal antibody, our novel method may potentially be used to evaluate BE studies in generic drug development when there is sparse PK sampling. We first present the theoretical background, i. e. , the NCA and MB approach for equivalence TOST tests. We then describe the observed data, the methodology to analyse it and the results of this real case study. We finally present the design, methods and results of the simulation study, and discuss our findings in the last section.
The regulatory guidances for equivalence studies fix the threshold δ = log(1. 25) [3, 4]. By decomposing this null hypothesis in two, we perform Two One-Sided Tests (TOST): H 0,-δ : {θ ≤ -δ} and H 0,δ : {θ ≥ δ} (3) The two t test statistics are rejected at α = 5% if: ACCEPTED MANUSCRIPT / CLEAN COPY Z -δ = θ + δ SE(θ) ≥ q 1-α and Z δ = θ -δ SE(θ) ≤ q α (4) with q α the quantile of order α of a reference distribution. Equivalently, we can reject the null hypothesis if the confidence interval of θ is within [-δ, δ], that is if the confidence interval of the exponential of θ is within [0. 8 ; 1. 25]. The exponential of θ is often shown in the results of the test and is called the geometric mean ratio (GMR).
In this case, there is no period or sequence effect and intra-individual variability cannot be properly evaluated. The models to fit are simply: log(AU C i ) = µ AU C + θ AU C T i + ϵ AU C i (5) log(C max i ) = µ C max + θ C max T i + ϵ C max i (6) with: µ: mean value of variable for the reference treatment; -T i : treatment covariate variable for individual i; θ: coefficient of treatment effect; ϵ i ∼ N (0, σ 2 ): residual error. The treatment effects on the variables of interest and their standard errors are obtained directly from the linear model inference. The geometric mean ratio is, e. g. for AU C: GM R = exp(E(log(AU C T ))) exp(E(log(AU C R ))) = exp(µ AU C + θ AU C ) exp(µ AU C ) = exp(θ AU C ) In non-compartmental PK equivalence analyses (hereafter called NCA-TOST), the standard error is obtained with the Fisher Information Matrix (FIM), which ACCEPTED MANUSCRIPT / CLEAN COPY is asymptotically the inverse of the lower bound of the variance-covariance matrix of regression coefficients.
Model-based approach Regulatory requirements may not be met in studies with sparse sampling design, and NCA-TOST may then become less accurate. Indeed, it can be hard to compute individual AU C and C max if we only have a few points per subject. In an effort to leverage population data over time to inform predictions for individuals, a model-based alternative has been proposed  [8, 10] , in which we build a structural PK model and use a non-linear mixed effect model (NLMEM) to estimate the treatment effect. The corresponding statistical model can be written as follows in the case of parallel studies: y ij = f (t ij , ϕ i ) + g(t ij , ϕ i )ϵ ij (7) log(ϕ il ) = log(µ l ) + θ l T i + η il (8) with: t ij : time j for individual i; y ij : concentration for individual i at time t ij ; ϕ i : vector of parameters for individual i (typically of size 3 to 10); f (t ij , ϕ i ): non-linear structural PK model depending on ϕ i ; g(t ij , ϕ i ): error model; ϵ ij ∼ N (0, 1): residual error; µ l : fixed effect for parameter l; -T i : treatment covariate variable; θ l : coefficient of treatment effect for parameter l; η il ∼ N (0, ω l ): between subject random effect for parameter l; ω l : standard deviation of the inter-individual random effect for parameter l.
Data In our analysis, PK data was collected from two phase I randomised clinical trials on healthy male or female subjects between 40-70 years of age. These trials investigated the relative bioavailability, tolerability, and dose-exposure relationship of a high concentration liquid formulation (HCLF G3) versus a lyophilised formulation (LyoF G2) of gantenerumab, a monoclonal antibody used for the treatment of Alzheimer's disease. Hereafter we considered the high concentration liquid formulation as the reference formulation. Both formulations were administered by subcutaneous injection. The first study (NCT01636531, here called S1) was composed of five parallel arms with 24 participants each: three reference arms at different dose levels (105, 225 and 300 mg) and two test arms (105 and 225 mg).
In the second study (NCT02133937, here called S2), composed of one reference arm of 25 participants and one test arm of 23 participants, the dose tested was 225 mg. PK sampling was performed in participants for up to 13 weeks using the following scheme: 0. 25, 1, 2, 3, 4, 7, 13, 20, 42, 63, and 84 days post dose. There was one additional sampling time in S2, one hour post dose (0. 04 days). We evaluated PK equivalence of the two formulations in terms of C max and AU C ∞.
Implementation Analyses were run on R version 4. 0. 2. Parameters of the PK models were estimated by maximising the likelihood using the Stochastic Approximation of Expectation Maximisation algorithm (SAEM)  [16] , in the saemix R package  [17]  (development version: https://github. com/saemixdevelopment/saemixextension). For NCA-TOST, AU C ∞ was computed by extrapolation with the PKNCA R package  [18]  version 0. 9. 4, using the observed concentration at t last. Sampling points for the sparse designs were chosen with the PFIM  [15]  R package version 4. 0 which enables to optimise population design using the Fedorov-Wynn algorithm. The same model, a two-compartment model (V 1 /F : apparent volume of the principal compartment, V 2 /F : apparent volume of the peripheral compartment, Q/F : apparent inter-compartmental clearance) with linear absorption (ka: absorption constant) and elimination (CL/F : apparent clearance constant) with an absorption delay (T lag ), was selected to be the best (among the considered candidates) at describing the drug PK across studies/arms (taken as three separate datasets).
ACCEPTED MANUSCRIPT / CLEAN COPY T lag , θ ka , θ CL/F , θ V 1/F , θ Q/F , and θ V 2/F ). On all datasets, based on BIC, the inter-individual random effect on V 2 /F was withdrawn, and a correlation between the inter-individual random effects of CL/F and V 1 /F was estimated. On S1-105 and S1-225, the error model was multiplicative. On S1-225, no inter-individual random effect was kept on Q/F. On S2-225, the error model was combined. The models selected were therefore very similar. Table  A1  in Appendix 3 gives the parameter estimates obtained across datasets. As shown in Fig. 2 , illustrating the GMR and their confidence intervals in the different datasets investigated, the different methods gave consistent results: for S1-105, with both NCA-TOST and MB-TOST Asympt, the 90% confidence interval of the GMR of AU C and C max fell within [0.
8; 1. 25], but for S1-225, equivalence could not be shown on C max with either of the two methods. On S2-225, equivalence could not be shown on C max with both methods. For AU C, equivalence was shown using MB-TOST but not using NCA-TOST, although the estimates were close (MB-TOST Asympt: 90% CI=[0. 801;1. 218], p-value=0. 049; NCA-TOST: 90% CI=[0. 782;1. 205], p-value=0. 070). The data used to produce Fig. 2  are provided in Table  A2  in Appendix 3. The sparse design optimised using PFIM led to the following sampling scheme: 0. 25, 3, 7, 20, 84 days post dose for S1-105, 0. 25, 4, 20, 42, 84 days for S1-225, and 0. 04, 4, 13, 42, 84 days post dose for S2-225. The selected PK model was a one compartment model with linear absorption and an absorption delay on the two S1 datasets, and a one compartment model with zero order absorption and no absorption delay on S2.
Again, a treatment effect was estimated on all apparent parameters in each case. On all datasets, a correlation between the interindividual random effects of CL/F and V /F was selected. On S1-105 and S1-225, the error model selected was multiplicative. On S2-225, the error model selected was combined. On S1-225 and S2-225, no inter-individual random effect was kept on T lag. Table  A1  in Appendix 3 gives the parameters estimated on all these subsets. Although the PK models selected on the sparse data were different from the ones selected on the observed data, the results of the equivalence study using MB-TOST were consistent, across all computation methods of SE, and comparable to those obtained on rich design (Fig. 2 ). Fig. A1  shows the VPC and Fig. A2  reports the normality of residuals for S1-225 original and sparse design.
25)) and then the treatment effect on V 1/F was varied to obtain the desired treatment effect on C max without impacting the treatment effect on AU C. Table  3  gives the values of the different levels of treatment effects simulated. For each of the 5 treatment effects, 1000 datasets were simulated. On rich design simulations, we compared the performances of NCA-TOST and MB-TOST Asympt in terms of type I error and study power. We first fitted the simulated structural PK model, estimating treatment effects on all 5 apparent parameters (referred to as model 2cpt par). We also explored the performance of MB-TOST Asympt when modeling the treatment effects differently, i. e. , twocompartment model with treatment effects estimated on the absorption parameter only, i. e. , ka, and an additional scale/bioavailability parameter defined by F , with µ F fixed to 1, and ω F estimated (called hereafter 2cpt F).
ACCEPTED MANUSCRIPT / CLEAN COPY We also explored the relevance of a PK model selection step, on the reference arm, on the BIC, prior to the equivalence test, on rich and sparse design simulations (two models to compare in each case). We observed the impact of this approach in terms of type I error. Estimation Errors (EE) and Standard Errors (SE) of treatment effects were computed to evaluate the agreement between the estimations of NLMEM and the real values under which we simulated the data. Empirical SE were computed as the standard deviation on the 1000 estimates of each parameter in each scenario.
In our evaluation of these PK BE studies, we assume that the PK characteristics of the reference drug are well-known ACCEPTED MANUSCRIPT / CLEAN COPY and the change of treatment does not affect the underlying PK structural model. In our simulations, we assumed the residuals are independent of the treatment covariate in the model. Also, we assumed that the study population would be adequately randomised to avoid imbalance between the treatment arms, so we did not evaluate the impact of covariates in our MB approaches. However, it is important to acknowledge that covariates would likely have a greater impact on a PK BE study with a parallel study design as compared to a crossover design. Moreover, adding covariates that affect the PK would decrease the between subject variability. Thus, future research may be warranted in this area.
In the simulation study, when using the simulated model, MB-TOST Asympt achieved controlled type I errors that were similar to those obtained with NCA-TOST on rich designs. These results complement previous studies showing the efficiency of MB approaches for equivalence tests  [8]. In general, regulatory authorities recommend that PK sampling includes 12 to 18 samples with at least three sampling points after the peak  [3, 4]. These recommendations present unique challenges for PK studies with sparse designs. Indeed, the sparse design we extracted from the full design did not comply with those requirements, consequently we did not apply NCA-TOST to the datasets simulated with the sparse design. In this setting, we used MB-TOST as it relies on NLMEM which demonstrated improved accuracy of the estimates in particular when dealing with sparse designs  [22].
It has already been used in other studies as the simulated model  [11]. Here, it cannot properly fit the data as θ F reflects a treatment effect on all distribution and elimination parameters, which does not agree with the way we simulated the data (i. e. , without an effect on the peripheral clearance and volume). This explains why the effect on AU C is underestimated. With biosimilars, differences in the PK characteristics of a drug may be due to factors other than differences in the absorption phase. In contrast, the misspecified one-compartment model with treatment effects estimated on all ACCEPTED MANUSCRIPT / CLEAN COPY apparent parameters, fitted on sparse designs, is a different PK structural model than the simulated one. The choice of the number of compartments is an essential step in structural PK model building.
One compartment model with first-order absorption, linear elimination The model 1cpt par represented in Table  1  corresponds to the equation: C(t) = D V 1 ka Cl V 1 -ka exp(-kat) -exp(- Cl V 1 t) (10) with ka the absorption constant rate. Here, there are l = 3 parameters: µ = c(ka, V 1 , Cl). Two compartment model with zero-order absorption, linear elimination C(t) =            D T k 0 A α (1 -exp(-αt)) + B β (1 -exp(-βt)) if t ≤ T k 0 D T k 0 A α (1 -exp(-αT k 0 )) exp(-α(t -T k 0 )) if t > T k 0 + B β (1 -exp(-βT k 0 )) exp(-β(t -T k 0 )) with: ACCEPTED MANUSCRIPT / CLEAN COPY -A = 1 V 1 k 21 -α β -α the first macro-constant; -B = 1 V 1 k 21 -β α -β the second macro-constant; -α = k 21 k β the first rate constant; -β = 1 2 k 12 + k 21 + k -(k 12 + k 21 + k) 2 -4k 21 k the second rate constant; -k 12 = Q V 1 the distribution rate constant between the principal and the peripheral compartment; -k 21 = Q V 2 the distribution rate constant between the peripheral and principal compartment; -Q the inter-compartmental clearance; -V 1 the volume of distribution of the principal compartment; -V 2 the volume of distribution of peripheral compartment.
ACCEPTED MANUSCRIPT / CLEAN COPY This leads to an alternative way of parameterising the model, by including F in the model. Because F cannot be identified without intravenous data, we fix the population value at F =1 and only allow for some inter-individual variability. We put a treatment effect only on the absorption parameters and F. Also, no correlation is allowed between the random effects of volumes and clearances, as these correlations are assumed to be carried by F. This parameterisation allows to compute fewer treatment effect coefficients. 2 Appendix 2: Method to compute the treatment effect on C max and its SE As part of a PK equivalence analysis, after fitting a NLMEM, we want to compute treatment effects on the PK parameters of interest and their SE when there is no explicit relationship with the direct parameters of the model.
For instance, with C R max,k the maximum over the vector C R k and C T max,k the maximum over the vector C T k : θ C max,k = log C T max,k C R max,k We obtain a vector of K estimated treatment effects. We estimate the global treatment effect as the mean of this vector mean(θ Cmax,k ), and its standard error as the standard deviation of this vector sd(θ Cmax,k ). Consequently, the geometric mean ratio is computed as the exponential of the point estimate of θ computed, GM R Cmax = exp(mean(θ Cmax,k )) for instance. We evaluate the performance of this method on the estimated standard error of the treatment effect on AU C, because it has an explicit formulation with direct parameters: AU C = D CL AU C R k = D CL k AU C T k = D CL k e θ CL,k ACCEPTED MANUSCRIPT / CLEAN COPY θ AU C k = log AU C T k AU C R k = -θ CL,k E( sd(θ AU C )) = E[sd((θ CL,k ))] = sd(θ CL ) That shows that the method would give an estimate of the standard error of θ AU C consistent with the method based on the explicit link with the direct parameters.
ACCEPTED MANUSCRIPT / CLEAN COPY Empirical percentiles: percentiles of the observed data, calculated either for each unique value of time, or pooled by adjacent time intervals (bins). Theoretical percentiles: percentiles of simulated data are computed from multiple Monte Carlo simulations with the model of interest and the design structure of the original dataset (i. e. , dosing, timing, and number of samples). For each simulation, the same percentiles are computed across the same bins as for empirical percentiles. Prediction intervals for each percentile are then estimated across all simulated data and displayed as colored areas. If the model is correct, the observed percentiles should be close to the predicted percentiles and remain within the corresponding prediction intervals. The cumulative distribution function (cdf) of the predictive distribution of the concentrations observed can be computed using Monte-Carlo simulations.
ACCEPTED MANUSCRIPT / CLEAN COPY We define the prediction discrepancies (pd) as the value of this cdf at each observation. pd are computed as the percentiles of each observation in the empirical distribution of the simulations. By construction, pd are expected to follow U(0, 1), but only in the case of one observation per subject; within-subject correlations introduced when multiple observations are available for each subject induce an increase in the type I error of the test. To correct for this correlation, we compute the empirical mean and empirical variance-covariance matrix over the simulations. Decorrelation is performed simultaneously for simulated data and for observed data. Decorrelated pd are then obtained using the same formula but with the decorrelated data, and we call the resulting variables prediction distribution errors (pde).
Parameters estimates and treatment effect coefficients (relative standard errors), given by saemix on all separate studies, with the original and sparse designACCEPTED MANUSCRIPT / CLEAN COPYTable A3 Type I errors for AU C and C max , under H 0:0. 8 and H 0:1. 25 , on rich (R) design simulations with NCA-TOST and MB-TOST Asympt, and on sparse (S) design simulations with MB-TOST Asympt, Gallant and Post Significant p-values are highlighted in bold. Table A4 Study power to detect a treatment effect on AU C and C max , under H 1:0. 9 , H 1:1 and H 1:1. 11 , on rich (R) design simulations with NCA-TOST and MB-TOST Asympt, and on sparse (S) design simulations with MB-TOST Asympt, Gallant and Post.
Acknowledgments The authors would like to thank all their collaborators in the project " Evaluation of model-based bioequivalence (MBBE) statistical approaches for sparse design PK  studies" under the FDA contract  75F40119C10111. They also thank  Hervé Le Nagard  and  Lionel de la Tribouille  for the use of the CATIBioMed calculus facility. The illustrative example data were obtained from studies sponsored by  F Hoffmann-La Roche. We thank the participants and investigators who participated in these studies. We are grateful to the gantenerumab project team, which allowed us to use these data and provided comments to improve the quality of this manuscript.
265] 0. 437 0. 439 0. 899 [0. 720 ; 1. 123] 0. 450 0. 467 0. 070 0. 314 0. 049 0. 284 0. 044 0. 171 0. 059 0. 190 MB-TOST Post AU C Cmax 1. 003 0. 899 [0. 807 ; 1. 247] [0. 728 ; 1. 111] 0. 048 0. 182 Table A2Gantenerumab analysis -Geometric mean ratios (GMR), their 90% confidence interval and the p -value of the test, for AU C and C max , with NCA-TOST and MB-TOST Asympt on original data and with MB-TOST Asympt, Gallant and Post on sparse data Significant p-values are highlighted in bold. ACCEPTED MANUSCRIPT / CLEAN COPY ACCEPTED MANUSCRIPT / CLEAN COPY ACCEPTED MANUSCRIPT / CLEAN COPY.
Acknowledgments This work partakes of the research program  PEGASE "Percolation et Graphes Aléatoires pour les Systèmes Ecologiques " that aims a better understanding of the role of ecological corridors in biodiversity. PEGASE is supported by  Région Hauts-de-France  and  FEDER  funding. We also acknowledge the support of  CNRS  throught  MONACAL Prime80 's grant. O. G. is also supported by  Labex CEMPI  ( ANR-11-LABX-0007-01 ). The authors thank the Referees for careful reading and useful comments.
The secondauthors acknowledge that the research leading to these results has received funding from the  European Research Council  under the European Union's  Seventh Framework Programme  ( FP7/2007-2013) /  ERC  agreement no. 615112  HAPDEGMT. The third author acknowledges financial support from the  Spanish Ministry of Science and Innovation , through the " Severo Ochoa Programme for Centres of Excellence in R&D"  ( CEX2019-000904-S ). The third and last authors were partially supported by the  Spanish Ministry of Science and Innovation ,  MTM PID2019-107914GB-I00. 1 2 PASCAL AUSCHER, LI CHEN,  JOS É MAR ÍA MARTELL ,  AND CRUZ PRISUELOS-ARRIBAS 2.
Such a case of so-called functional statistics is actually in continuous progress. At this stage, the financial area constitutes an attractive field for functional data modeling. The necessity to provide a mathematical models allowing the real-time control of volatility and exploring the microstructure of the financial data are the principal motivations of the use of functional statistics for financial time series data. However, most works use the parametric ideas to fit the financial data as functional observations. For previous studies on this topic, we cite Muller et al. [18] , Kokoszka et al. [19]  or  Shang et al. [20]. Additional tools for financial risk management based on functional data analysis can be found in Cai  [21]. He provided a sophistical algorithm to analyze the risk measurement of China's financial market.