dataesr/scientific-paragraphs-categorization huggingface dataset:
LFF assessment using the on-field device This study was performed using the on-field Myocene ® device (Figure  2 ). Participants sat on the seating of the device with their leg in contact with the "Myo-sensor," that is, a dedicated force sensor recording evoked forces at a rate of 4 kHz. Muscle electrical stimulations (biphasic square wave with a pulse width of 400 µs) were applied using three electrodes (MyoPro-1-electrode, Myocene, Liège, Belgium). The cathode (5 × 10 cm) was placed transversely across the width of the proximal portion of the quadriceps femoris, and the anodes (5 × 5 cm) were, respectively, placed over the vastus lateralis and the vastus medialis muscles. Pre-programmed electrical stimuli trains were directly sent by the device that was driven by the Myocene ® software. The series of stimuli consisted of sets including 1) a single pulse, 2) a train of  5 stimuli at low frequency (20 Hz), and 3) a train of 18 stimuli at high frequency (120 Hz).
One second separated each stimulation (Figure  1 ). In total, 16 sets were performed, with 5 s in-between, and the intensity of stimulation was progressively increased by steps of 1 mA at each set (from 25 mA to 40 mA). The total duration of LFF assessment using the device was 2 min. The Myocene ® software integrates a specific algorithm allowing the instantaneous and automatic measurement of LFF. Within each set, the ratio of low-to high-frequency evoked forces was calculated, and the median value of all ratios was provided as an outcome, that is, the so-called Powerdex. This Powerdex value was therefore retained for analysis in the present study.
Quadriceps neuromuscular function evaluation Participants sat on a custom-built chair with knees and hip at 90 °of flexion. Force was recorded using a calibrated load cell (Meiri F2732 200 daN, Celians, Montauban, France) fixed to the chair and attached above the ankle malleoli of participants via a non-compliant strap. Movements of the upper body were minimized using two belts across the thorax. The force signal was acquired using a PowerLab data acquisition system (16/30-ML880/P, ADInstruments, Bella Vista, Australia) at a sampling rate of 2 kHz. During all measurements, participants were provided with real-time feedback of their force traces on a screen. Strong verbal encouragements were given during MVCs. Neuromuscular function evaluation (Figure  1 ) began with a free warm-up (only in PRE) consisting of submaximal contractions of the quadriceps.
Experimental Results Implementation. We implemented our automatic invariants generation algorithm in Python 3. 8 and used Sage  [61]  for matrix manipulation. All results were obtained on an Intel Core i7(2. 00 GHz) machine with 64 GB memory, running Ubuntu 18. 04. All µ values were obtained by explicitly solving the roots of characteristic polynomial of the transition matrix and the polynomial  (8). Benchmarks. Our benchmarks are affine while loops chosen from some original linear transition systems benchmarks (which we rewrite as programs) from the StInG invariant generator  [59]  that implements the most related approach in  [56] , some linear dynamical systems in  [41] , some loop programs in  [60]  and some other linear dynamical systems resulting from well-known linear recurrences such as Fibonacci numbers, Lucas numbers, etc.
Results. The experimental results are presented in Table  1. In the table, the column 'Loop' specifies the name of the benchmark, 'Dim(ension)' specifies the number of program variables in the benchmark, 'µ' specifies the values through eigenvalues of the transition matrix or boundary points of the intervals in the feasible domain, 'Invariants' lists the generated affine invariant from our approach, 'C(om)p(a)r(ison)' compares our approach with the existing cited work where '=' means the generated invariants are identical, '>' means our generated invariants are tighter, '/' means the invariants can only be generated in this work, and 'Time' records the amount of runtime for our approach measured in seconds. Analysis. The results in Table  1  show that our approach can derive affine invariants with irrational coefficients which result from the calculated irrational µ's from eigenvalues of the transition matrix and boundary points of the feasible domain, which cannot be achieved by the invariant generator StInG.
Motivation: Imputing missing values is common practice in label-free quantitative proteomics. Imputation aims at replacing a missing value with a userdefined one. However, the imputation itself may not be optimally considered downstream of the imputation process, as imputed datasets are often considered as if they had always been complete. Hence, the uncertainty due to the imputation is not adequately taken into account. We provide a rigorous multiple imputation strategy, leading to a less biased estimation of the parameters' variability thanks to Rubin's rules. The imputation-based peptide's intensities' variance estimator is then moderated using Bayesian hierarchical models. This estimator is finally included in moderated t-test statistics to provide differential analyses results. This workflow can be used both at peptide and protein-level in quantification datasets.
Introduction Dealing with incomplete data is one of the main challenges as far as statistical analysis is concerned. Different strategies can be used to tackle this issue. The simplest way consists of deleting from the dataset the observations for which there are too many missing values, leading to a complete-case dataset. However, it causes information loss, might create bias and could ultimately result in poorly informative datasets. Another way to cope with missing data is to use methods that account for the missing information. For the last decades, researchers advocated the use of a single technique called imputation. Imputing missing values consists of replacing a missing value with a value derived using a user-defined formula (such as the mean, the median or a value provided by an expert, thus considering the user's knowledge).
Hence it makes it possible to perform the analysis as if the data were complete. More particularly, the vector of parameters of interest can be then estimated. Single imputation means completing the dataset once and considering the imputed dataset as if it was never incomplete, see Figure  1. However, single imputation has the major disadvantage of discarding the variability from the missing data and the imputation process. It may also lead to a biased estimator of the vector of parameters of interest. Multiple imputation described by  Little and Rubin (2019)  closes this loophole by generating several imputed datasets. These datasets are then used to build a combined estimator of the vector of parameters of interest, by usually using the mean of the estimates among all the imputed datasets, see Figure  2.
This combined estimator is known as the first Rubin's rule. The second Rubin's rule states a formula to estimate the variance of the combined estimator, decomposing it as the sum of the intra-imputation variance component and the between-imputation component. The rule of thumb suggested by  White et al. (2011)  takes the number of imputed datasets as the percentage of missing values in the original dataset. Recent work focused on better estimating the Fraction of Missing Information  (Pan, Q. et al. )  or improving that rule (von Hippel, P. T. ). Note that Rubin's rules cannot be used in order to get a combined imputed dataset but instead provide an estimator of the vector of parameters of interest and an estimator of its covariance matrix both based on multiple imputation, see Figure  2. (2) Multiple imputation provides D estimators for the vector of parameters of interest.
In the MSstats R package (available on Bioconductor),  Choi et al. (2014)  distinguish missing completely at random values and missing values due to low intensities. The user can then choose to impute the censored value using a threshold value or an Accelerated Failure Time model. The Perseus software by  Tyanova et al. (2016)  offers three methods for single imputation: either imputing by "NaN", impute by a user-defined constant or impute according to a Gaussian distribution in order to simulate intensities, which are lower than the limit of detection. Recently,  Goeminne et al. (2020)  implemented a single imputation method based on a hurdle model in their MSqRob R package  (Goeminne et al. , 2018). As far as machine learning is concerned,  Song and Yu (2021)  suggested a method for imputing missing values in label-free mass spectrometry-based proteomics datasets, called XGboost.
The ProStaR software based on the DAPAR R package and developed by  Wiec-zorek et al. (2017)  splits missing values into two categories, whether they are Missing in an Entire Condition (MEC) or Partially Observed Values (POV)  (Lazar et al. , 2016). The software allows single imputation, using either a small quantile from the distribution of the considered biological sample, the k-Nearest Neighbours (kNN) algorithm or the Structured Least Squares Adaptative algorithm or by choosing a fixed value. The PANDA-view software developed by  Chang et al. (2018)  also enables the use of the kNN algorithm or a fixed value. Moreover, both software programs give the possibility to impute the dataset several times before combining the imputed datasets in order to get a final dataset without any missing values.
PANDA-view relies on the mice R package by  Van Buuren and Groothuis-Oudshoorn, (2011) , whereas ProStaR accounts for the nature of missing values and imputes them with the imp4p R package implemented by Giai Gianetto et al. (2020). However, both software programs consider the final dataset as if it had always been complete. The uncertainty due to multiple imputation is not properly taken into account downstream of the imputation step. In the following, we will conduct the multiple imputation process to its end, as described by  Little and Rubin (2019)  and use the imputed datasets to provide a combined estimator of the vector of parameters of interest as well as a combined estimator of its variance-covariance matrix estimator. We will then project this matrix to get a unidimensional variance estimator before moderating it using the empirical Bayes procedure defined in the seminal paper of  Smyth et al.
Simulated datasets We evaluated our methodology on three types of simulated datasets. First, we considered an experimental design where the distributions of the two groups to be compared scarcely overlap. This design led to a fixed effect one-way analysis of variance model (ANOVA), which can be written as: y ij = µ + δ ij + ij (1) with µ = 100, δ ij = 100 if 1 ≤ i ≤ 10 and j = 2 and δ ij = 0 otherwise and ijk ∼ N (0, 1). Here, y ij represents the log-transformed abundance of peptide i in the j-th sample. Thus, we generated 100 datasets by considering 200 individuals and 10 variables, divided into 2 groups of 5 variables, using the following steps: 1. For the first 10 rows of the data frame, set as differentially expressed, draw the first 5 observations (first group) from a Gaussian distribution with a mean of 100 and a standard deviation of 1.
We generated 100 datasets by considering 1000 individuals and 20 variables, divided into 2 groups of 10 variables, using the following steps: 1. Generate the peptide-wise effect P i by drawing 1000 observations from a Gaussian distribution with a mean of 1. 5 and a standard deviation of 0. 5. 2. Generate the group effect G ik by drawing 200 observations (for the 200 individuals set as differentially expressed) from a Gaussian distribution with a mean of 1. 5 and a standard deviation of 0. 5 and 800 observations fixed to 0. 3. Build the first group dataset by replicating 10 times the sum of P i and the random error term, drawn from a Gaussian distribution of mean 0 and standard deviation 0. 5. 4. Build the second group dataset by replicating 10 times the sum of P i , G ik and the random error term drawn from a Gaussian distribution of mean 0 and standard deviation 0.
5. 5. Bind both datasets to get the complete dataset. Finally, we considered an experimental design similar to the second one, but with random effects P i and G ik. The 100 datasets were generated as follows. 1. For the first group, replicate 10 times (for the 10 variables in this group) a draw from a mixture of 2 Gaussian distributions. The first one has the following parameters: a mean of 1. 5 and a standard deviation of 0. 5 (corresponds to P i ). The second one has the following parameters: a mean of 0 and a standard deviation of 0. 5 (corresponds to ij ). 2. For the second group replicate 10 times (for the 10 variables in this group) a draw from a mixture of the following 3 distributions. (a) The first one is a Gaussian distribution with the following parameters: a mean of 1. 5 and a standard deviation of 0.
5 (corresponds to P i ). (b) The second one is the mixture of a Gaussian distribution with a mean of 1. 5 and a standard deviation of 0. 5 for the 200 first rows (set as differentially expressed) and a zero vector for the remaining 800 rows (set as not differentially expressed). This mixture illustrates the G ik term in the previous model. (c) The third distribution has the following parameters: a mean of 0 and a standard deviation of 0. 5 (corresponds to ij ). All simulated datasets were then amputed to produce MCAR missing values in the following proportions: 1%, 5%, 10%, 15%, 20% and 25%.
Real datasets We challenged our methodology on several real datasets coming from two different experiments described hereafter. We consider a first real dataset from  Muller et al. (2016). The experiment involved six peptide mixtures, composed of a constant yeast (Saccharomyces cerevisiae) background, into which increasing amounts of UPS1 standard proteins mixtures (Sigma) were spiked at 0. 5, 1, 2. 5, 5, 10 and 25 fmol, respectively. In a second well-calibrated dataset, yeast was replaced by a more complex total lysate of Arabidopsis thaliana in which UPS1 was spiked in 7 different amounts, namely 0. 05, 0. 25, 0. 5, 1. 25, 2. 5, 5 and 10 fmol. For each mixture, technical triplicates were constituted. The Saccharomyces cerevisiae dataset was acquired on a nanoLC-MS/MS coupling composed of nanoAcquity UPLC device (Waters) coupled to a Q-Exactive Plus mass spectrometer (Thermo Scientific, Bremen, Germany) as extensively described in  Muller et al.
(2016). The Arabidopsis thaliana dataset was acquired on a nanoLC-MS/MS coupling composed of nanoAcquity UPLC device (Waters) coupled to a Q-Exactive HF-X mass spectrometer (Thermo Scientific, Bremen, Germany) as described in Supplementary data, Section S7. 3. For the Saccharomyces cerevisiae and Arabidopsis thaliana datasets, Maxquant software was used to identify peptides and derive extracted ion chromatograms. Peaks were assigned with the Andromeda search engine with full trypsin specificity. The database used for the searches was concatenated in-house with the Saccharomyces cerevisiae entries extracted from the UniProtKB-SwissProt database (16 April 2015, 7806 entries) or the Arabidopsis thaliana entries (09 April 2019, 15 818 entries) and those of the UPS1 proteins (48 entries). The maximum false discovery rate was 1% at peptide and protein levels using a target-decoy strategy.
For the Arabidopsis thaliana + UPS1 experiment, data were extracted both with and without Match Between Runs and 2 pre-filtering criteria were applied before statistical analysis: only peptides with, on the one hand, at least 1 out of 3 quantified values in each condition and, on the other hand at least 2 out of 3, were kept. Thus, 4 datasets derived from the Arabidopsis thaliana + UPS1 were considered. The same filtering criteria were applied for the Saccharomyces cerevisiae + UPS1 experiment, but only on data extracted with Match Between Runs, leading to 2 datasets being considered.
Multiple imputation methods Several methods for imputing missing values in mass spectrometry-based proteomics datasets were developed in the last decade. However, the recent benchmarks of imputation algorithms do not reach a consensus (as shown in Supplementary data, Table  S1. 1). This is mainly due to the complex nature of the underlying missing values mechanism. In this work, we chose to focus on some of the most commonly used methods 1. The k-Nearest Neighbours method imputes missing values by averaging the k-nearest observations of the given missing value in terms of Euclidean distance. This method was described by  Hastie et al. (1999)  and  Troyanskaya et al (2001)  and implemented in  Hastie et al. (2021). The Maximum Likelihood Estimation method imputed missing values using the EM algorithm proposed by  Schafer (1997) and implemented by Giai Gianetto et al.
Estimation The objective of multiple imputation is to estimate from D drawn datasets the vector of parameters of interest and its variance-covariance matrix. Notably, accounting for multiple-imputation-based variability is possible thanks to Rubin's rules, which provide an accurate estimation of these parameters. Let us consider a D-time imputed dataset with P individuals (corresponding to P peptides or proteins) and N observations (corresponding to N biological samples), divided between I groups (corresponding to I conditions to be compared). Let β P be the vector of parameters of interest, such as : β P = (β P ,1 ,. , β P ,I ) (3) The first Rubin's rule provides the combined estimator of β P : βp = 1 D D d βp,d (4) where βp,d is the estimator of β P in the d-imputed dataset. The second Rubin's rule gives the combined estimator of the variance-covariance matrix for each estimated vector of parameters of interest for peptide p through the D imputed datasets such as: Σp = 1 D D d=1 W d + D + 1 D(D -1) D d=1 ( βp,d -βp ) T ( βp,d -βp ) (5) where W d denotes the variance-covariance matrix of βp,d , i.
Therefore, the results of this testing procedure account both for the specific structure of the data and the uncertainty caused by the multiple imputation step. T pi[mod] = βpi ŝ2 p[mod] (X T X) -1 k,k (9) with (X T Ω p X) -1 k,k the k-th diagonal element in the matrix (X T Ω p X) -1. Under the H 0 hypothesis, T pi[mod] follows a Student distribution with d p + d 0 degrees of freedom. As many tests as the number of peptides considered are performed. Hence, the proportion of falsely rejected hypotheses has to be controlled. Here, the False Discovery Rate control procedure from  Benjamini and Hochberg (1995)  was performed using the cp4p R package by Giai Gianetto et al. (2016).
Aggregation The methodology implemented in the mi4p R package can be applied to peptidelevel quantification data as well as protein-level quantification data. However, we were interested in evaluating our method on a peptide-level dataset and inferring results at a protein level, as it is common practice in proteomics. Therefore, for intensity aggregation, we chose to sum all unique peptides' intensities for each protein. The detailed pipeline for intensity aggregation is described in Supplementary data in Section S2.
Measures of performance We compared our methodology to the limma testing pipeline implemented in the state-of-the-art ProStaR software, through the DAPAR R package. To assess the performances of both methods, we used the following measures: sensitivity (also known as true positive rate or recall), specificity (also known as true negative rate), precision (also known as positive predictive value), F -score and Matthews correlation coefficient. In our work, we define a true positive (respectively negative) as a peptide/protein that is correctly considered as (not) differentially expressed by the testing procedure. Similarly, we define a false positive (respectively negative) as a peptide/protein that is falsely considered as (not) differentially expressed by the testing procedure. The expressions of the previously mentioned performance indicators are given in Supplementary data in Section S3.
Simulated datasets Figure  3  describes the evolution of the distribution of differences in sensitivity and specificity between mi4p and DAPAR depending on the proportion of missing values in the first set of simulations. For a small proportion of missing values (1%), where the imputation process induces little variability, performances in terms of sensitivity, specificity and F -Score are equivalent between both methods. No improvement nor deterioration was observed for sensitivity, as it remains at 100% regardless of the missing value proportion. Specificity and F -Score are improved with the mi4p workflow above 5% missing values. The same observations can be drawn for precision and Matthews coefficient correlation (see Figure  S4. 1 in Supplementary data). Detailed results can be found in Table  S4.
2. Figure  4  describes the evolution of the distribution of differences in sensitivity and specificity between mi4p and DAPAR depending on the proportion of  missing values in the second set of simulations. For all proportions of missing values, we observe a trade-off between sensitivity and specificity. A slight loss in specificity (remaining above 99%) provides a greater gain in terms of sensitivity. The mean F -score across the 100 datasets is also increased with the mi4p workflow than with the DAPAR one. The Matthews correlation coefficient highlights the gain in performances (as illustrated in Supplementary data, Figure  S5. 2). Extending the simulation model from fixed effects to random effects using the last set of simulations provides similar results, as shown in Supplementary data (Figure  S6.
Real datasets The trade-off suggested by the simulation study is confirmed by the results obtained on the real datasets. In the Saccharomyces cerevisiae + UPS1 experiment, a decrease of 70% in the number of false positives is observed, improving the specificity and precision (Table  S8. 23 in Supplementary data). However, this costs in the number of true positives (see Table  2 ), decreasing of sensitivity. The same trend is observed in the Arabidopsis thaliana + UPS1 experiment; the number of false positives is decreased by 50% (see Table  3  and Table S8. 17), thus improving specificity and precision at the cost of sensitivity. The loss in sensitivity is bigger in the highest points of the range in both experiments. The structure of the calibrated datasets used here can explain these observations.
Indeed, the quantitative dataset considered takes into account all samples from all conditions, while the testing procedure focuses on one-vs-one comparisons. Two issues can be raised: • The data preprocessing step can lead to more data filtering than necessary. For instance, we chose to use the filtering criterion such that rows with at least one quantified value in each condition were kept. The more conditions are considered, the more stringent the rule is, possibly leading to a poorer dataset (with fewer observations) for the conditions of interest. • The imputation process is done on the whole dataset, as well as the estimation step. Then, while projecting the variance-covariance matrix, the estimated variance (later used in the test statistic) is the same for all comparisons. Thus, if one is interested in comparing conditions with fewer missing values, the variance estimator will be penalised by the presence of conditions with more missing values in the initial dataset.