dataesr/scientific-paragraphs-categorization huggingface dataset:
As the number of distinct points is S n -invariant, there is a corresponding notion of q-skeleta of samples. The name skeleton is taken from the theory of simplicial complexes and cell complexes. The filtration of sample space into skeleta is rather coarse, and finer stratifications are needed to fully describe the local geometry of sample space. This is done next. Definition 2. 3 (Orbifolds  [52] ). A Hausdorff topological space O is an orbifold, if the following data are given: • An open cover (U i ) of O which is closed under forming finite intersections. • For each i there is an open subset V i ⊂ R N which is invariant under a faithful linear action of a finite group G i on R N and a G i -invariant quotient map π i : V i → V i /G i ∼ = U i. • If U i ⊂ U j then there is an injective group homomorphism ϕ ij : G i → G j and a gluing map ψ ij from V i to an open subset of V j which is G i -equivariant in the sense that ψ i,j (g.
Acknowledgements The authors would like to thank  François-Xavier Vialard  for helpful discussions. P. Harms was funded by the  National Research Foundation Singapore  under the award  NRF-NRFF13-2021-0012  and by  Nanyang Technological University Singapore  under the award NAP-SUG. X. Pennec was funded by the  European Research Council (ERC)  under the European Union's  Horizon 2020 research and innovation program  (grant agreement Nr. 786854  G-Statistics ). He was also supported by the  French government  through the  3IA Côte d'Azur Investments   ANR-19-P3IA-0002  managed by the  French National Research Agency (ANR). S. Sommer is supported by the  Villum Foundation Grants 40582  and the  Novo Nordisk Foundation  grant  NNF18OC0052000.
, 2019; Kallus et al. , 2018; Mayer et al. , 2020). One difficulty in the context of causal inference, is that one has to couple identifiability assumptions for the causal parameter with assumptions on the missing values mechanisms to define an appropriate strategy. According to  Mayer et al. (2020) , identifiability of the causal effect with missing values is ensured by adapting the causal inference assumptions to the missing values setting with an unconfoundedness despite missing values (UDM) assumption  (Rosenbaum and Rubin, 1984). However, these works only consider the case of a single dataset-or potentially multiple datasets with the same data distribution, i. e. , sampled from the same population of interest-and do not treat the case of transporting or generalizing a treatment effect from an RCT to a target distribution defined through an observational dataset; the RCT representing a "distorted" population due to sampling or selection bias (generally defined via eligibility criteria).
Our main contributions are: • we define several multiple imputation strategies adapted for the aforementioned data integration problem in Section 3; • we propose alternative identifiability assumptions which can be seen as an an extension of the unconfoundedness despite missingness (UDM) assumption from the observational data case  (Mayer et al. , 2020)  and suggest adapted estimators in Section 4; • we assess the performance of the proposed estimators and naive complete case estimators in an extensive simulation study in Section 5; • we present the results of the real-world data analysis which has motivated this work in Section 6. For simplicity, we will assume that the relevant covariates that allow for adjustment of the sampling bias are observed both in the RCT and the observational dataset, and we focus on the case of different missing values mechanisms for these covariates.
General concept Multiple imputation (MI) is one of the most powerful approaches to estimate parameters and their variance from incomplete data  (Little and Rubin, 2014; Kim and Shao, 2013; Schafer, 2010). In a nutshell, for a single dataset, it consists in generating M plausible values for each missing entry, which leads to M completed datasets. Then, an analysis is performed on each imputed data set m = 1,. , M , to get an estimate for the parameter of interest, say θ as θ m and an estimate of its variance V m ( θ m ) and the results are combined using  Rubin (1987) 's rules to get correct inference with missing values, namely confidence intervals with the appropriate coverage.
(b) Multiple imputation of the observational data: Impute M times the covariates of the observational data using only the covariates (X i ) i: Qi=O. (c) Create M × M complete tables by concatenating all possible combinations of imputed RCT and observational data. Estimate the treatment effect on every combination using any complete case estimator as in Section 2 and aggregate these estimations using Rubin's rules. 2. Ad-hoc joint covariates multiple imputation, ignoring the group variable, see also Figure  3b : (a) Impute M times the joint datasets (the concatenation of the covariates from the RCT and the ones from the observational study) with covariates X and ignore the "source" indicator variable Q during the imputation. (b) Concatenate the outcome Y and treatment A for each imputed RCT.
(c) Compute the M treatment effect estimators using any complete case estimator as in Section 2 and aggregate them using Rubin's rules. 3. Joint covariates multiple imputation, modeling the group variable as a fixed effect, i. e. keep the indicator variable Q indicating the corresponding "group"/"source" during the imputation, see also Figure  3c : (a) Impute M times the joint datasets with covariates X and model the source indicator variable Q as a fixed effect during the imputation. (b) Concatenate the outcome Y and treatment A for each imputed RCT. (c) Compute the M treatment effect estimators using any complete case estimator as in Section 2 and aggregate them using Rubin's rules. The first strategy has the advantage that it takes into account the outcome Y and treatment A which are dependent variables of the covariates X, when imputing the covariates of the RCT as suggested by  Leyrat et al.
(2018) ; van Buuren (2018). All three strategies can be implemented easily using the R package mice (van Buuren, 2018) which uses conditional models such as linear and logistic regressions to perform multiple imputation. The sketched fourth strategy could be implemented using the micemd R package  (Audigier et al. , 2018). Multiple imputation is suited if the missing values are ignorable as described in Section 2. 4 and if the identifiability assumptions of the ATE τ in the full data case are met, namely Assumptions (2) and (3). Note that for the considered imputation strategies, increasing the number of imputations does not improve (significantly) the final result. There exists no clear rule about the number of multiple imputations to achieve good performances, however an accepted rule of thumb is to choose the number of imputations to be similar to the percentage of incomplete cases  (Hippel, 2009)  or to the average percentage of missing data (van Buuren, 2018).
, 2019). Nonparametric estimation. As an example of such a modern nonparametric estimation approach, we propose to estimate the generalized parameters via random forests  (Breiman, 2001; Athey et al. , 2019) , with missing data handled using the missing incorporated in attributes (MIA) method of  Twala et al. (2008). The resulting IPSW, CO and AIPSW estimators will be denoted by τ IP SW M IA , τ CO M IA , and τ AIP SW M IA respectively. In random trees, the MIA approach extends the classical splitting rules such that missing values are incorporated in the splitting criterion. More specifically, consider splitting on the j-th at-tribute and assume that for some individuals, the value of X j is missing, MIA treats the missing values as a separate category or code and considers the following splits: • {i : X ij ≤ t or X ij is missing} vs.
{i : X ij > t} • {i : X ij ≤ t} vs. {i : X ij > t or X ij is missing} • {X ij is missing} vs. {X ij is observed}, for some threshold t. The MIA approach does not seek to model why some features are unobserved; instead, it simply tries to use information about missingness to make the best possible splits for modeling the desired outcome. Thus the MIA strategy works with arbitrary missingness mechanisms and does not require the missing data to follow a specific mechanism. This MIA approach for (generalized) random forests is implemented in the R package grf  (Tibshirani et al. , 2020)  which is also used in the simulation part of this work presented in Section 5. Parametric alternative. Parametric estimation is however possible in the case of logistic and linear regression models. This is based on work by  Jiang et al.
1 Sample N >> n observations X 1 ,. , X N from the target population P(X); 2 Sample S according to the logistic model (15) on X; 3 Keep the {S = 1} indexed observations X R ← X {i: Si=1} as the RCT; 4 Sample A according to a Bernoulli distribution B(0. 5) (coin flip); 5 Sample Y according to the linear model (  16 ) on X R ; 6 Sample m observations X O from the target population P(X) as the observational data; 7 Concatenate the datasets: X ← [X T R , X T O ] T and append the indicator Q to the data (X ← [X , Q]); 8 Sample missing values for the n + m observations according to either (17), (  18 ) or (19); For ease of comparison with the alternative simulation scenario, we provide the expression of the joint distribution over (X, R, A, Y, S, Q) factorized according to the underlying generative model: p(X, R, A, Y, S, Q) ∝ p(R|S, Q, X, A, Y )p(S, Q, Y |X, A)p(A|X)p(X) ∝ p(R|S, Q, X, A, Y )p(Q|S)p(S|X, A)p(Y |X, A)p(A|X)p(X) (20) In the MCAR case (8), this factorization simplifies as follows: p(X, R, A, Y, S, Q) ∝ p(R)p(Q|S)p(S, Y |X, A)p(A|X)p(X).
, 2020). • MIA (see Section 4. 1): we use generalized random forests with MIA splitting criterion to estimate the generalized models α * and µ * a,1 on the incomplete data, using the R package grf  (Athey et al. , 2019). • Multiple imputation (MI, Section 3): we apply the standard full data estimators from Section 2 on the imputed data (5-10 imputations obtained using the R package mice) where we use either within-study multiple imputation (WI-MI), or ad-hoc multiple imputation (AH-MI), or fixed effect multiple imputation (FE-MI). We do not assess the random effect multiple imputation (i. e. , the joint covariates multiple imputation with a 2-level model accounting for the multiple data sources, sketched Strategy 4 from Section 3) in this simulation study because this does not correspond to our motivating data example from the introduction.
6 n view of the results, if one has to recommend a method, it is preferable, from the present empirical evidence, to choose the MIA-AIPSW estimator or the joint multiple imputation coupled with the CW estimator. The MIA approach is simple to use in R thanks to the grf package which directly handles incomplete variables with the MIA splitting criterion for random forests. However, in terms of computational costs, this approach can be more expensive due to (automatic) parameter tuning. The joint multiple imputation approaches are easy to implement (with the mice package) but the relative computational running time is of similar magnitude as the MIA approach.
Application on critical care data In this part, we come back to the medical question introduced in the beginning of this work about the potential effect of tranexamic acid (TXA) on mortality in patients with intracranial bleeding. We recall that, in order to answer this question, we have at disposal two data sources: (1) CRASH-2, a multi-center international RCT, (2) Traumabase, an observational national registry. A detailed data analysis of the observational registry to address the above medical question has been conducted by  Mayer et al. (2020). We thus refer to this previous analysis for a detailed description of the observational registry as well as their findings. In a nutshell, leveraging only the observational registry does not provide evidence towards a beneficial (or detrimental) effect of TXA on trauma patients with TBI in terms of head-injury-related mortality.
The necessary variables are also available in the Traumabase, either exactly or in form of proxies, which allows the estimation of the trial inclusion model on the combined data. Additional covariates. Note that other covariates are (partially) available in both data sets, while not responsible of trial inclusion according to CRASH-2 investigators. But as this could still be covariates moderating the outcome and treatment effect, we include them in the outcome models used in the CO and AIPSW estimators (  5 ) and (  6 ). According to the two data sets, we could add three of them: sex (binary), type of injury (categorical, 1 =blunt, 2 = penetrating, 3 = blunt and penetrating), and initial Glasgow coma scale (numeric, integers from 3 to 15). Note that this three covariates are all mentioned in the baseline of CRASH-2 results  (Shakur-Still et al.
, 2009) , arguing that they should impact the outcome. The variables central capillary refill time and respiratory rate are also mentioned but are not available in the Traumabase, we thus omit them from this joint study. Missing values. First, note that the RCT contain almost no missing values, whereas the variables for determining eligibility in the observational data contains important fractions of missing values, as shown in Table  8 , while the sample sizes of the data sets are similar, see Table  3. While the MCAR assumption is admitted for the RCT  (Shakur-Still et al. , 2009) , the missing values in the observational Traumabase are more complex and, according to the medical experts monitoring the collection process, partly non-ignorable. For example, the pre-hospital systolic blood pressure (SBP) is likely to be missing for patients with severe ongoing bleeding.
To provide an example, the average patient age in the RCT is 7-9 years below the average age in the observational study; and there are only 16% of female patients in the RCT while there are over 20% in the observational study. Figure  9 : Distributional shift and difference in terms of univariate means of the trial inclusion criteria (red: group mean greater than overall mean, blue: group mean less than overall mean, white: no significant difference with overall mean, numeric values: group mean (resp. proportion for binary variables). Graph obtained with the catdes function of the FactoMineR package  (Lê et al. , 2008). Ignorability. Due to the design of the CRASH-2 study, namely the eligibility criteria which all need to be informed to decide upon trial eligibility, the modified ignorability assumption CIS (11) is less plausible to hold in this case and we rather consider the standard assumptions (2) and (3) to be satisfied by the CRASH-2 and Traumabase studies.
(2020)  when analyzing observational data, these are the estimators kept in this analysis. When summarizing the results from the separate analyses on the RCT and the observational data respectively and the results from the joint analysis of both studies, we observe on Figure  11  a discrepancy between the different results. The only approach with consistent estimations throughout all estimators is the EM approach that points towards a beneficial effect of TXA on all-cause mortality. The joint fixed effect multiple imputation IPSW and AIPSW estimators (MI IPSW and MI AIPSW) also conclude on a beneficial effect of the treatment. However, the calibration weighting estimator (MI CW) does not find a significant effect. The large confidence intervals could be partly explained by the measurement noise in the administration delay variable in the Traumabase 9  : contrary to the RCT, the Traumabase does not encode the exact delay of treatment administration, but is defined by a noisy proxy (delay between accident and admission to the resuscitation bay).
This illustrates again the importance of choosing adequate assumptions for identifiability with missing covariate values and corresponding estimation strategies. On the methodological side, for all considered approaches and simulation settings, the question of varying missing values mechanisms across data sources remains to be addressed in more detail. Note as well that we have focused on missing values in both data sets, but the recommendations extend to the case where there are only missing values in the observational data and not the RCT due to different levels of systematic data collection. Finally, the problem of incomplete observations addressed here is different from the problem of inconsistent variable sets between an RCT and an observational dataset, e. g. , one variable is completely missing in one set, which is a challenging problem of a different kind that is left for future work.
Acknowledgements We would like to thank  Shu Yang  for fruitful discussions and her valuable feedback on our work. We thank  Tobias Gauss ,  Jean-Denis Moyer  and  François-Xavier Ageron  for their medical insights and interpretation of our data analysis; finally we thank the CRASH-2 trial investigators for sharing the trial data with us. Funding IM was supported by a  EHESS   PhD fellowship  and a  Google PhD fellowship. The funding institutions had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.
Data and methods Panel data and sample issues. We conducted an original empirical work based on a 96 countries dataset between the first of January and the 15th of July 2020, covering the entire "first wave" of the pandemic. As mentioned earlier, we only focused on the first wave to minimize the biases, which might be raised from updating beliefs from human behavior. In addition, we can disentangle the interference with other government policies, vaccination, and non-pharmaceutical policies since these regulations did not happen intensively and interactively. For the first section, we obtain a panel with 96 countries and around 7359 observations. We collected (1)  the number of confirmed COVID-19 cases and deaths for the countries in our sample from the European Center for Disease, Control and Prevention between 1st January 2020 and July, 15th 2020, (2) the estimated population in 2019 from the World Bank's World Development Indicators database, and (3) the mask-wearing variable from the Kreuter et al.
study from University of Maryland using the Facebook platform  50. Our mask wearing variable is based on the Covid-19 World Survey Data API  29  and  50. The surveys ask respondents how many people in their household are experiencing Covid-19-like symptoms, among other questions. These surveys are voluntary, and individual survey responses are held by University of Maryland and are shareable with other health researchers under a data use agreement. No individual survey responses are shared back to Facebook. Using this survey response data, we estimate the percentage of people in a given geographic region that use face mask cover. We use the smoothed weighted (two sets of sample respondents separately are used (CMU US and UMD global surveys)) percentage of survey respondents across an one week window that have reported use mask cover (see also https://covidmap.
umd. edu/document/css_methods_brief. pdf for more details). The descriptive statistics (average mask-wearing proportion, in SI Appendix) seem coherent with other sources from the literature. For example, data based on Health Metrics and Evaluation at the University of Washington in Seattle reported in the USA. While Fetzer et al. 30  and Van Bavel et al. 86  constructed weights that account for the differential sample size across countries, the mask-wearing variable approach is also consistent with the previous data processing. By doing this, the dataset can be eliminated the sampling bias from country differences. The dimension of our panel (96 countries) has been constrained by data availability about our main interest 'mask' variable and a significant number of respondents to the mask adoption survey.
In addition, we have to take into account the fact that we have a panel of 96 countries with an important heterogeneity concerning the take-off Covid-19 periods (time with the first infected people) and so different Covid-19 dynamics over time: the first wave of Covid-19 epidemic has started later in Brazil than in Italy. Considering only homogeneous European countries is therefore a mean to test the presence of sample bias. We will show that results reported on European countries in our database are completely in accordance with the global results conducted on the global 96 countries sample.
The lower the human mobility, the lower the COVID-19 cases. These effects can be explained by the fewer social interactions  14, 68, 89. To sum up, based on the previous literature, our estimations are designed rigorously and consider the previously updated literature to maintain reliable findings. When it comes to the cross-sectional data, we constructed our data at the country-level to cover the socioeconomic determinants. To be more precise, we capture the population features (Population density -Density; the aging population -Age65), environmental degradation (CO2), chronic diseases (Diabetic and Overweight), economic and educational status (economic status as GDP, school enrollment as School or reading ability PISA), differences in cultural and behavioral dimensions (Altruism index, Tolerance index, Risk_aversion index, and political trust Government_confidence ).
). Finally, persistence and multicollinearity are other usual issues in panel studies. We have controlled for both by computing autocorrelations LM (Lagrange Multiplier) tests and VIF/Tolerance ratios after each estimated regression. In Appendix, we also consider other robustness tests related to the specification of our econometric model, the choice of an alternative estimator, and several changes in the sample composition. The Cross-section model to investigate face mask adoption. We collect data from different sources (SI Appendix) to estimate a very simple cross-section model of the following form at a country level: where y i captures proportion of individuals declaring wearing a face mask on 15th of July, 2020 and X i is a set of control variables including population density, CO2 emissions (and squared CO2 emissions), GDP level (in logarithms), government effectiveness index, altruism index, individualism index, tolerance index, risk aversion index, trust in government index, Covid-19 policy stringency index, urban population percentage, education level, diabetic population percentage and overweight population percentage have been considered.
The mask adoption is effective on a global scale We first conducted an econometric exercise for 96 countries for the period April, 23 to July, 15, 2020. To account for lags in epidemic dynamics, we collected data on infected cases and fatalities for a longer period from January, 1, 2020. Our global panel data covers all parts of the globe, but we narrow this down to European countries for robustness checks. Our dynamic panel data estimates (Mean Group here) reveal that masks wearing has a clear ( P < 0. 01 ) statis- tical negative impact on infected cases with a 7-to 28-day lag but that this negative effect disappears after 42 days (Table  1 ). The coefficients of the masks variable have been standardized (multiplied by 100,000) in Table  1  and Tables 3 , 4 , 5 6 to facilitate the reading.