dataesr/scientific-paragraphs-categorization huggingface dataset:
The multiple imputation multiple factor analysis approach (MI-MFA) To deal with multiple tables with missing rows, we propose the MI-MFA approach, a multiple imputation (MI) adapted to the framework of MFA. The aim of our method is not to get the best possible estimations of the missing values, but to replace them with plausible values in order to provide estimates of the MFA configurations. According to MI methodology, the MI-MFA approach is carried out by performing the following three steps: 1. Imputation: generate M different imputed datasets K (1) ,. , K (m) ,. , K (M) of K. 2. MFA analysis: perform an MFA on each K (m)  imputed dataset leading to M different configurations F 1 ,. , F m ,. , F M. 3. Combination: find a consensus configuration between all F 1 ,. , F M configurations. These steps are outlined in Fig.
Step 2. For each recipient in K j , impute the missing individual by drawing randomly with replacement a donor from the corresponding donor pool. Repeat this procedure until all missing individuals in the J tables have been imputed. Step 3. Repeat Step 2 until M different imputed datasets K (1) ,. , K (M) of K are obtained. By conducting the imputations in this way, it is reasonable to assume that the within-unit between-variables multivariate relationships are preserved.
The third step is to estimate the configuration matrix F m for each imputed dataset K (m) using MFA. The estimated configurations differ from each other because their input data differ. The last step is to combine the M estimated configurations F 1 ,. , F M into a compromise configuration F c using the STATIS method configuration agrees the most with all other configurations. An overview of the STATIS method is presented in Additional file 1: Figure  S2. STATIS analyzes a set of N tables X 1 ,. , X N , where each X n is a table of quantitative variables measured on the same individuals. The first stage of STATIS consists of calculating a matrix of cross-products between individuals for each table W n = X n X T n (A T means the transpose of a vector or a matrix A) reflecting the similarities between individuals within this table.
Implementation of MI-MFA The MI-MFA algorithm can be summarized as follows (see Fig. 2 ): Step 0. Start with an observed, incomplete dataset K. Define the number of imputations M and the dimensionality d of the compromise configuration. Step 1. Perform multiple hot-deck imputation. For m = 1,. , M: • Obtain an imputed version K (m) of K , such that, K (m) = K (m ) for m = m. The imputed datasets are identical for the non-missing data entries, but differ for the imputed values. The imputed version of the data is obtained by using the hot-deck imputation approach. • Perform an MFA using d components on the imputed dataset K (m) to obtain the configuration F m. Step 2. Perform a STATIS on the set of configurations F 1 ,. , F M to obtain F c , the compromise configuration. Note that the number of dimensions d used in the algorithm has to be chosen a priori.
However, the number of dimensions does not affect the estimation of the imputed values and the estimation of the compromise configuration. Moreover, for given K (1) ,. , K (M) imputed datasets, solutions provided by the algorithms are nested (the solution with d dimensions is included in the solution with d+1 dimensions). Since the core of MI-MFA is a weighted PCA, the strategies suggested to choose the number of components in PCA can be adapted to MI-MFA, but work needs to be done to validate the quality of these extensions.
How many imputations? When using MI, one of the uncertainties concerns the number M of imputed datasets needed to obtain satisfactory results. The number of imputed datasets in MI depends to a large extent on the proportion of missing data. The greater the missingness, the larger the number of imputations needed to obtain stable results. However, in multiple hot-deck imputation, the number of imputed datasets is limited by the size of the donor pools. In any case, the total number of possible imputations M total can be calculated before applying the imputation approach (see Additional file 2). If M total is small (M total ≤ 50), then M = M total can be used in MI-MFA. The appropriate number of imputations can be informally determined by carrying out MI-MFA on N replicate sets of M l imputations for l = 0, 1, 2,.
, with M 0 < M 1 < M 2 < • • • < M total , until the estimate compromise configurations are stabilized. More precisely, this approach can be carried out by applying the following steps: Step 0. Start with an observed, incomplete dataset K. Define the number of imputations M l with M 0 < M 1 < M 2 < • • • < M total and the number N of replicate sets of M l imputations. Step 1. Create collections I M l n , n = 1,. , N, each one containing M l different imputed datasets of K , such that, I M l n = I M l n , for n = n and I M l-1 n ⊂ I M l n for M 0 < M 1 < M 2 < • • • < M total. Step 2. For n = 1,. , N, perform an MI-MFA using I M 0 n , to obtain N different compromise configurations F c M 0 1 ,. , F c M 0 N. Step 3. Let l = 1. For n = 1,. , N, • perform an MI-MFA using the collection I M l n , to obtain a compromise configuration F c M l n ; • calculate r l n = r(F c M l n , F c M l-1 n ), a measure of the distance or correlation between configurations (for example the RV coefficient  [23] ).
Uncertainty of MI-MFA solutions In an MI-MFA framework, after estimating the configurations from the imputed datasets, a new source of variability due to missing values can be taken into account. Here we describe two approaches to visualize the uncertainty of the estimated MFA configurations attributable to missing row values. First, an individual plot for all estimated MFA configurations is constructed. The individual plot is obtained by projecting each estimated MFA configuration onto the compromise configuration (named the trajectories by Lavit  [22] ). Each individual is represented by M points, each corresponding to one of the M MFA configurations. Confidence ellipses and convex hulls can then be constructed for the M configurations for each individual. The computed convex hull results in a polygon containing all M solutions.
Performance of the method We conducted two case studies to assess the performance of our method. Instead of using theoretical distributions to generate simulated data, our studies were based on two real datasets, denoted as the original datasets. Subsequently, specific patterns of missingness were created in these datasets as illustrated in Fig. 1 , resulting in what we called the incomplete datasets. This approach was used in order to more closely mirror situations that may occur in the omics context. Next, missing row values were estimated and the resulting complete datasets were referred to the imputed datasets. We then compared our MI-MFA method to the RI-MFA approach  [9]  and the mean variable imputation MFA (MVI-MFA) method, in which the missing values are simply replaced by the mean of each variable after which an MFA is performed on the imputed dataset.
This latter approach was considered as the common base for comparing the MI-MFA and RI-MFA methods. For each configuration obtained using MI-, RI-and MVI-MFA, the similarity between the configuration solution and the true configuration (based on an MFA using the original dataset) was assessed from the RV coefficient  [23]. The RV coefficient, which ranges from zero to one, can be interpreted as a correlation coefficient between two matrices, which allows the relative positions of objects to be compared from one configuration to another. We also provide comprehensive graphical comparisons of the true vs. the MI-, RI-or MVI-MFA configurations. The individuals from both configurations are drawn in a same plot and connected by an arrow, the length of which indicates the divergence between the two configurations.
Implementation of the analyses All analyses were performed using the R computing environment  [24]. MFA was performed using the MFA function of the FactoMineR R package  [25]. The statis function of the ade4 R package  [26]  was used to determine the compromise configuration. The RI-MFA method is implemented in the imputeMFA function available in the missMDA R package  [27]. Note that the number of components ncp used to predict the missing entries in the imputeMFA function has to be chosen a priori. This choice is crucial and difficult  [9]. As the true configuration was known in our case, the number of components ncp was chosen to minimize the RV coefficient between the true and the imputeMFA configurations. The appropriateness of the results from MI-, RI-and MVI-MFA was then determined by comparing the configurations resulting from these three strategies with the true MFA configuration.
Liver toxicity The datasets originated from a liver toxicity study  [29]  in which 64 male rats of the inbred Fisher F344/N strain were exposed to toxic doses of acetaminophen (paracetamol) in a controlled experiment. Necropsies were performed 6, 18, 24 and 48 h after exposure and mRNA was extracted. The data consisted of the expression of 3,116 genes and 10 clinical variables considered to be markers of liver injury. The 64 subjects (rats) were cross-classified in eight strata (or treatments) according to two factors: • exposure time: 6, 18, 24 and 48 h; • toxic doses of acetaminophen: high (1500 mg/kg or 2000 mg/kg) or low (50 mg/kg or 150 mg/kg). Eight subjects per stratum were included. These datasets were downloaded from the mixOmics R package  [30].
NCI-60 data The NCI-60 dataset contained transcriptomic  [31]  and proteomic  [32]  tables for a collection of 60 cell lines from the National Cancer Institute (NCI-60). The NCI-60 panel included cell lines derived from various cancer types: colon (7 cell lines), renal  (8) , ovarian  (6) , breast (8), prostate (2), lung (  9 ) and central nervous system (6), as well as leukemia (6) and melanoma  (8). The gene expression profiles used here were generated using an Agilent platform  [31]  and downloaded from Cellminer  [33]. Data were log 2 -transformed. To facilitate data interpretation and computation, the transcriptomic data were filtered to exclude probes that did not map to an official HUGO gene symbol and to retain only the probe with the highest average value when multiple probes mapped to the same gene, as previously described in  [34].
Gene invariants across all 60 cell lines, corresponding to genes without any effect between cancer types, were also removed. Filtering produced a dataset of 1,433 genes. The NCI-60 proteome table was also downloaded from Cellminer  [33]. Proteomic data were obtained using high-density reverse-phase lysate microarrays  [32]. Data were log 2transformed and protein abundance levels were available for 162 proteins  [32].
Liver toxicity data analysis A specific pattern of missing values was created as illustrated in Fig. 1. To obtain an incomplete dataset, three individuals per stratum were randomly removed from the transcriptomic table. For this specific pattern, there were 3 × 8 = 24 missing individuals. MI-MFA was then performed on the incomplete dataset using M = 30 imputed datasets. RI-MFA, and MVI-MFA were also performed. Figure  3  shows the divergence of the MI-, RI-and MVI-MFA configurations from the true configuration. As can be seen, the configuration obtained with MI-MFA was very close to the true configuration (Fig. 3 , top right). This result was confirmed by the high RV coefficient (0. 96 for the first two dimensions). The configurations obtained with RI-MFA and MVI-MFA were more distorted and less close to the true configuration with RV coefficients of 0.
77 and 0. 84 respectively (Fig. 3 , bottom). The number M of imputed datasets in MI-MFA for the above incomplete liver toxicity data was determined as described in the How many imputations section. Collections of size N = 30 were generated for each of the following numbers of imputations: M l = 10 l, for l = 1,. , 10. The stability of the estimated MI-MFA configurations was then determined by calculating the RV coefficient between the configurations obtained using M l and M l+1 imputations (see Fig. 4 , left). As the true configuration was known, we also described the stability of the estimated MI-MFA configurations by calculating the RV coefficients between the true configuration and those obtained using M l imputations (see Fig. 4, right ). Although the missing information is substantial, Fig.
4  shows that only a slight increase in precision was obtained by using more than 30 imputations. MI-MFA was also applied to different scenarios of missingness. First, MFA was performed on the original dataset to obtain the true configuration. Secondly, individuals were randomly removed from each stratum in the original transcriptomic table. For this, three scenarios were considered for the number of missing rows (i. e. low, medium and high), in which there were respectively one, two and three missing rows per stratum. The total number of missing rows per scenario was therefore 8, 16 and 24 respectively. Thirdly, for each missingness scenario, 50 incomplete datasets were randomly chosen for analysis. As previously, MI-MFA was performed on each incomplete dataset using M = 30 imputed datasets.
More precisely, missing row values were inserted into each stratum (e. g. high-6h treatment) of the original dataset (including both transcriptomic and clinical tables) according to the scenarios illustrated in Table  1. Twenty incomplete datasets were then selected at random from each stratum (treatment) and each scenario. All analyses (MI-MFA, RI-MFA and MVI-MFA) and the calculation of the RV coefficient were performed in the same way as previously described. Figure  6  (and Additional file 1: Figure  S3 ) shows the means of RV coefficients for the two-dimensional configurations as a function of the scenarios for each method. For almost all the scenarios, the average results obtained with MI-MFA were better than with the other methods. As the number of missing row values increased (Fig. 6  and Additional file 1: Figure  S3 ), the results obtained with the RI-MFA and MVI-MFA algorithms worsened rapidly, especially when the dimension increased, whereas the results with the MI-MFA approach were still satisfactory.
NCI-60 data analysis To confirm the performance of our method, MI-MFA was also performed on the NCI-60 dataset. A pattern of missing values was created as illustrated in Table  2. One or two individuals were removed per table for all types of cancer cell lines, except for the prostate cancer line (which only had two individuals). This specific pattern would reflect a study in which a lot of rows were missing. To compare the MFA configurations, one incomplete dataset was chosen from a large range of possibilities of incomplete datasets (6 × 10 14 ) according to the scenario of missingness illustrated in Table  2. We then computed our MI-MFA method on this incomplete dataset by using M = 50 imputed datasets. As with the liver toxicity data, RI-MFA and MVI-MFA were also performed. We chose M = 50 in order to achieve stable results.
Results are given for all of the first three two-dimensional possible configurations as a function of the scenarios presented in Table  1. The discrete RV values are joined by lines for ease of understanding. The performance of the different methods on the liver toxicity data with missing individuals in the low-dose treatment are presented in Additional file 1: Figure S3 For each incomplete dataset, MI-MFA was performed using M = 50 imputed datasets. RI-MFA and MVI-MFA were also computed. Figure  8  shows the RV coefficient for the two-dimensional configurations as a function of each case. The results obtained with MI-MFA and RI-MFA were similar, with RV coefficients of approx. 0. 97, whereas the results obtained with MVI-MFA were much further from the true configuration. Thus, even with complex patterns of missingness, the MI-MFA approach still provided satisfactory results, as did RI-MFA in this case.
Why is it essential to evaluate uncertainty? This question was addressed through an example using the NCI-60 dataset. A specific pattern of missing values was created. The missing rows were randomly introduced for four melanoma and two leukemia cancer lines in Consequently, since the configurations changed, the positions of all the individuals also changed and thus all the individuals had confidence areas, even individuals not being imputed (Fig. 9 , individuals represented by filled circles). However, the greatest uncertainty occurred around the imputed individuals (Fig. 9 , empty circles). In order to highlight the importance of the uncertainty of MI-MFA configurations induced by missing rows, additional rows were removed from the transcriptomic table resulting in 30 % missing rows. We then carried out MI-MFA on the incomplete dataset using M = 50 imputed datasets.
Discussion In the present paper, we propose a MI approach to handle missing row values in an MFA framework and therefore resolve one of the major issues associated with multiple omics data tables. The proposed method, which we called MI-MFA, provides point estimates of MFA configurations with a notion of uncertainty due to the missing values. The aim of our method was not to obtain the best possible estimates for the missing values, but to replace them so as to be able to estimate MFA configurations. The MI-MFA method generates M imputed datasets from an MFA model, where multiple hot-deck imputation is used to fill in the missing values. The hot-deck approach resolves the most important limitation of other model-based techniques (such as JM and FCS) in that it can be applied to large datasets.
Likewise, if the overall sample size is very small, then typically there are also too few potential donors. However, all imputation techniques are challenged by small sample sizes since these reduce the availability of information required to create suitable conditional statements  [19]. An important aspect of our strategy is the choice of the number M of imputed datasets. As shown in the two case studies, we think that this number should be a good compromise between the need to obtain stable estimates and to avoid computation bottlenecks. The STATIS method was proposed to combine the results of MI-MFA. As the core of MFA is a PCA, combining the results from MFA is the same as combining the results from PCA. Several procedures have previously been proposed to combine results from PCA such as the Mean Varimax Method (MVM) or Mean Correlation Matrix (MCM) approaches (as discussed in  [35] ).
However, Van Ginkel and Kroonenberg  [10, 35]  demonstrated that Generalized Procrustes Analysis (GPA) was more suited to this purpose. GPA fits the PCA configurations obtained from the imputed datasets Fig. 9  Visualization of the uncertainty induced by the missing individuals. Projection of the 50 obtained configurations on the compromise configuration from MI-MFA on the incomplete data with 10 % missing rows (top) and 30 % missing rows (bottom). The 95 % confidence ellipses (left) and convex hulls (right) show the uncertainty for each individual (dots). Empty circles represent imputed individuals from the transcriptomic data. For ease of understanding, not all individuals for the 50 configurations obtained were plotted to a single fixed reference configuration to produce the final solution, the centroid configuration (the mean of all transformed solutions).
Indeed, ellipses and convex hull areas offer great assistance by either supporting the MFA results if they are small or suggesting that caution be exercised otherwise. It should be noted that RI-MFA (or MVI-MFA) also provides a configuration of individuals whatever the missingness pattern; however, there is no way of knowing if the results obtained are plausible and if the user can interpret the results without making any mistakes. We have illustrated our approach by applying it to two real case studies using the liver toxicity and NCI-60 datasets. Incomplete artificial datasets with different patterns of missingness were created within these datasets. The configurations resulting from MI-, RI-and MVI-MFA were compared with the MFA configurations of the corresponding original population (the true configuration).
Performance of the methods was assessed by considering the RV coefficient with respect to the true configuration. In the liver toxicity study, the performances of the methods were compared in two different missingness settings. First the number of missing rows in each stratum of the transcriptomic table was chosen to be low (1 row), medium  (2)  or high  (3). Secondly, seven scenarios were created by inserting missing rows in the original dataset which included both transcriptomic and clinical tables. The overall results showed that MI-MFA clearly outperformed the RI-MFA and MVI-MFA approaches in nearly all settings. In the NCI-60 study, we illustrated the performance of our method on complex patterns of missingness where substantial numbers of rows were missing from both tables of the NCI-60 dataset.
Conclusion We propose here a new method, MI-MFA, an extension of MFA, designed to deal with multiple tables with missing row values. MI-MFA is a useful and attractive method to estimate the coordinates of individuals for MFA configurations despite the missing rows. The study cases showed that the other proposed methods either encountered serious problems or were unable to adequately assess the accuracy due to missing data. The configurations obtained with our method were closer to the true configuration even when a significant number of individuals were missing, and thus provided better results. Moreover, the uncertainty due to the missing rows could be visualized on the compromise configuration. The software for our MI-MFA method is available in an easy-to-use code for the R statistical environment.
Acknowledgements VV is a PhD fellow supported by the  INRA GA (Génétique Animale) , the  INRA PHASE (Physiologie Animale et Systèmes d'Élevage)  and the  région Languedoc-Roussillon Midi-Pyrénées. The authors would like to thank  Helen Mundutéguy-Hutchings  and  Diana Goodfellow  for the English revision. The authors thank the reviewers and the editors for their useful comments and suggestions that helped improve the quality of this paper.
Availability of data and materials The liver toxicity data can be accessed from the mixOmics R package  [30]  (https://cran. r-project. org/web/packages/mixOmics). The NCI-60 data can be downloaded from the Cellminer  [33]  at https://discover. nci. nih. gov/cellminer/ loadDownload. do. The methods introduced here are implemented using the R computing environment  [24]. The code, reference manual and examples are freely available as Additional file 3.
Genomes from two reference strains (BL23 and ATCC334, respectively, called S27 and S13) were collected from public database. Strains diversity was also assessed by phylogenetic distances calculated using distribution of orthologous groups representative genes in each genome (Supplementary Fig. S1B ). Orthologous groups definition and distance calculation were obtained using hierarchical clustering as described previously  [21]  (Supplementary Fig. S1B ). Strains originate from fermented dairy products, human gut, animal gut, or plants. Bacteria were cultured in Man-Rogosa-Sharpe broth at 378C for 16 h, washed and suspended in phosphate buffer saline (PBS) at concentration 1Â and pH 7. 4 before being incubated with cells. When indicated, antibiotics (penicillin-streptomycin; Life Technologies, Villebon sur Yvette, France) were added to the cell culture media.