dataesr/scientific-paragraphs-categorization huggingface dataset:
Multiple imputation The principle of multiple imputation (MI) is to generate multiple sets of plausible values for the missing variables by drawing from the posterior predictive distribution of these variables given the observed data. Variables of different types are often included in the PS model. Therefore, we focused on chained equations, in which a specific imputation model, including the outcome, is specified for each partially observed variable, rather than joint modelling to impute the missing data  [38]. M complete datasets are created and analysed independently to produce estimates θk , (k = 1,. , M ) of θ the vector of the parameters of interest (eg. regression coefficients) and estimates W k of their associated variance matrix. Then θk and W k , (k = 1,. , M ) are combined across the M imputed datasets.
Rubin's rules for the mean and variance state that an overall estimate, θMI , of θ and an estimate of the variance of θMI , V ar( θMI ), are  [8] : θMI = 1 M M k=1 θk , V ar( θMI ) = W + 1 + 1 M B, (2) where W is the within-imputation variance-covariance matrix, which reflects the variability of the parameter estimates in each imputed dataset, and B is the between-imputation variance matrix reflecting the variability in the estimates caused by the missing information. These two components are defined as: W = 1 M M k=1 W k , B = 1 M -1 M k=1 θk -θMI 2. ( ) 3 Three variants of MI are MIte, MIps and MIpar. For complete data, the IPTW method involves three steps: i) estimation of the parameters α in the PS model; ii) calculation of individual PS's; and iii) use of equation (1) to calculate μ0 and μ1.
In MIte, all three steps are applied to each of the M imputed datasets separately. Then the M treatment effect estimates obtained are averaged. In MIps, the method recommended by Mitra and Reiter  [9] , the first two steps are applied to each imputed dataset separately. Then the M PS's for each individual are averaged and equation (  1 ) is applied once using these average PS's. In MIpar, the first step is applied separately to each of the imputed datasets. The M values of α and each individual's covariates X i are then averaged over the M datasets. Each individual's PS is then estimated as: êi (x i ) = exp ᾱxi 1 + exp ᾱxi , (4) (i = 1,. , n) with ᾱ the (p + 1) vector of the average PS parameter values (for the p covariates and the intercept) and xi a (p + 1) vector of the average p covariates across imputed datasets for individual i.
Combining the treatment effects after MI (MIte) Let X, the vector of covariates, be split into observed and missing components, X = (X obs , X miss ). X (k) m is the imputed value of X miss in the k th imputed dataset (k = 1,. , M ). We show (see Appendix 2a) that in each imputed dataset: e(X obs , X (k) m ) = E[Z|X obs , X (k) m ]. (5) If X (k) m is imputed from the true model (i. e. correctly specified at the true parameter values), we can also show (Appendix 2b) that a SITA-type assumption holds in each imputed dataset, i. e. : Y z=1 ⊥ Z | X obs , X (k) m and Y z=0 ⊥ Z | X obs , X (k) m These two assumptions are the imputed-data version of what Imbens calls weak unconfoundedness  [39]. Note that we do not have the analogue of the usual, stronger, assumption (Y z=1 , Y z=0 ) ⊥ Z | X obs , X (k) m , which requires the treatment to be independent of the set of potential outcomes.
This is because our imputation model is a model for X miss |Z = z, Y Z=z , X obs. The stronger assumption would require our imputation model to capture X miss |Z = z, Y z=0 , Y z=1 , X obs. However, it is important to note, as Imbens does, that the weak unconfoundedness suffices to obtain unbiased estimates of the causal treatment effect. Balancing properties: We show in Appendix 2 that X obs ⊥ Z | e(X obs , X (k) m ) and X (k) m ⊥ Z | e(X obs , X (k) m ). Thus the true PS in each completed dataset balance both the unobserved and the imputed values of the covariates across treatment groups. This balancing property is what leads to consistency of the MIte estimator. Consistency: Seaman and White  [11]  proved that for an infinite number of imputations, the estimator obtained by combining the treatment effects after MI (MIte) is consistent.
To understand how this consistency relates to the SITA-type assumption above, it is helpful to consider the following expectation: E Y Z e(X obs , X (k) m ) = E E Y Z e(X obs , X (k) m ) X obs , X (k) m = E E[Y z=1 |X obs , X (k) m ] E[Z|X obs , X (k) m ] e(X obs , X (k) m ) (6) = E[E[Y z=1 |X obs , X (k) m ]] (7) = E[Y z=1 ] = θ, Step 6 requires the SITA-type assumption 6. Step 7 relies on PS in the k th imputed dataset being equal to the probability of being treated given the observed and imputed part of the covariates (equation 5).
Combining the PS or the PS parameters after MI (MIps and MIpar) For PS methods, consistency comes from the ability of PS to balance covariates between groups. MIps and MIpar create a single overall PS used to estimate the treatment effect. Thus, consistency for these methods would rely on the ability of these overall PS to balance both the observed and the missing parts of the covariates in the original incomplete dataset. Rosenbaum and Rubin's results show that this can happen only if the single PS (as estimated in MIps or MIpar) is 'finer' than the true (full data) PS (in other words, if the true PS is a function of the single pooled PS)  [3]. However, when combining the PS or the PS parameters, the overall PS used for the analysis is not a function of the observed covariates but rather is a function of the average covariates across imputed datasets, including the imputed values.
Data generation mechanisms We generated datasets of sample size n = 2000, reflecting an observational study comparing a treatment Z = 1 to a control treatment Z = 0 on a fully observed binary outcome Y with 3 measured covariates X = (X 1 , X 2 , X 3 ). X 1 and X 2 were continuous and X 3 was binary. X 2 was fully observed whereas X 1 and X 3 were partially observed. We generated the data as follows: • Covariates: The 3 covariates X = (X 1 , X 2 , X 3 ) are generated from a multivariate normal distribution X ∼ N 3 (0, Σ), with Σ ii = 1 and Σ ij = ρ for i = j. X 3 is then dichotomised according to a threshold of 0 to obtain a prevalence of 0. 5. • Treatment assignment depends on X according to the following model: logit(p(Z = 1|x)) = -1. 15 + 0. 7x 1 + 0. 6x 2 + 0. 6x 3. (8) These coefficients give E(Z) = p Z = 0.
Methods For each studied scenario, 5000 datasets were simulated. We compared 9 methods, all of them using IPTW: treatment effect were estimated on the full dataset (before the imposition of missingness) and using CC, MP, and the 3 MI approaches (MIte, MIps and MIpar). For each of the 3 MI approaches, we considered two imputation models, including or not the outcome and we generated M = 10 imputed datasets. Simulations were performed in R and the mi package was used for multiple imputation  [41] , based on full conditional specification (FCS).
For the MIte approach, standardized differences were calculated using the PS estimated from each imputed dataset, both to assess the balance on the originally simulated complete dataset (before imposing missingness) and on the given imputed dataset. For MIps and MIpar, standardized differences were calculated using the pooled PS to assess balance on (i) the original dataset, (ii) on the average value of the covariates across the imputed datasets. For (ii) we also calculated the standardized differences separately on the observed part of the covariates and the average imputed part. 7 Simulation study results.
Thus, when the missingness rate increases, imbalances (and consequently, bias of the treatment effect estimate) increase. MIte performs well because the PS within each imputed dataset balances the observed and imputed components of the covariates in that imputed dataset (see Table  3 ). Conversely, MIps and MIpar could only be consistent by balancing the observed and missing covariates, which is not the case, as seen in Table  3. Each boxplot represents the coverage distribution for the 8 main scenarios. Because the CC and MP approaches are strongly biased, their coverage rates are not relevant. The coverage rate for the MIte approach is close to the nominal value of 95%, confirming that Rubin's rules perform well in this context provided that the withinimputation variance estimation takes into account the uncertainty in PS estimation.
Application to the motivating example We applied CC analysis, the MP approach and the three MI strategies to estimate the effect of statin treatment on mortality after pneumonia from our motivating example dataset. For simplicity, we analysed the primary outcome, mortality within 6 months, as a binary outcome, and estimated the corresponding relative risk and its 95% confidence interval. For each approach, IPTW was used to account for the confounding. We focused the analysis on the 7158 patients without coronary heart disease. The propensity score was estimated from a logistic regression modeling statin use as a function of the following covariates: age, sex, body mass index (BMI), alcohol consumption, smoking status, diabetes, cardiovascular disease, circulatory disease, heart failure, dementia, cancer, hyperlipidaemia, hypertension and prescription of antipsychotics, hormone replacement therapy, antidepressants, steroids, nitrates, beta-blockers, diuretics, anticoagulants and use of antihypertensive drugs.
The imbalance between the study groups is illustrated in Figure  5. Complete case (CC) analysis was conducted on the 5168 individuals with complete records. For the missingness pattern approach (MP), 8 patterns were identified. However, some of these patterns were very rare. For instance, only 6 individuals had only the smoking status missing, and only 8 had both smoking status and BMI missing. Thus, we considered only 4 groups: • complete records (n=5168) for which all the covariates listed above are included • individuals with only the alcohol consumption missing (n=455) • individuals with only BMI missing (n=575) • individuals with the smoking status missing (alone or in addition to BMI and alcohol consumption) and individuals with both BMI and alcohol consumption missing (n=960) For multiple imputation (MI), 10 imputed datasets were created.
Second, we showed that combining the treatment effects after MI (MIte approach) is the preferred MI strategy in terms of bias reduction under a MAR mechanism. This estimator is the only estimator of the 3 MI estimators to be consistent and to provide good balancing properties. Even though MIps and MIpar are not consistent estimators of the treatment effect, they can reduce the bias of the CC analysis, if the rate of missing data is not too high. Combining the PS or the PS parameters has no clear advantage for IPTW, but may be useful in the context of PS matching: because it involves only one estimation of the treatment effect, it could provide a computational advantage for large datasets. In addition, MIte for PS matching implies that the M treatment effect estimates are estimated from different matched sets, potentially of different sample sizes, leading to a more complex variance estimation because of these different sample sizes.
Third, as long as the uncertainty in the PS estimation is taken into account in the variance estimation  [34] , Rubin's rules perform well for MIte, even for moderate sample size (n=500). For MIpar, the proposed variance approximation (Appendix 1) showed good performance in our simulation study. The 3 MI approaches differ in terms of their balancing properties. We showed that whereas the PS estimated in each dataset in the MIte approach can balance covariates between groups in each imputed dataset, this is no longer true for MIps and MIpar. However, the best method to assess covariate balance after MI remains unknown. With MIte, the aim being to estimate a treatment effect from each dataset, we require balance between groups within each imputed dataset. In contrast, for MIps and MIpar, further investigation is needed to know if we should assess the balancing properties of the pooled PS on the average covariate values across the imputed dataset or on each dataset.
Our recommendations for the optimal way to implement multiple imputation for PS analysis are in contradiction to those given by Mitra and Reiter, who suggest that pooling propensity scores across imputed datasets can sometimes out-perform our recommended method of pooling treatment effects  [9]. In their simulation studies MIps (or MIpar) led to a bigger reduction in bias than MIte in some scenarios. This was due to a near violation of the positivity assumption in their simulated datasets  [45]  combined with the omission of the outcome from the imputation model. Penning de Vries and Groenwold  [46]  used Mitra and Reiters data-generating mechanism to perform a simulation study but they increased the effective sample size to ensure positivity. Under this condition and when the outcome was included in the imputation model, as we recommend, they showed MIte was superior, in accordance with our findings.
When the assumptions for the MP approach are valid, it avoids the need for an imputation model altogether, and hence avoids the need to use the outcome when calculating the PS. Because the assumptions for MI and MP are different, we believe that MP could be a promising alternative when the assumptions for MI are not valid. Further investigations are needed to understand the usefulness of the MP approach in practice. Moreover, its application to our real-life example dataset was challenging, because the sample size within each missingness pattern was not large enough to estimate the PS. This work has some limitations. We generated only 3 covariates in our simulation, whereas PS are often calculated from a large number of covariates. Moreover, we studied only the common situation where the PS model only includes main effects, and we assumed as well the model including main effects only for the outcome given the covariates.
Nevertheless, because the PS estimated in either MIps or MIpar is not a balancing score, PS matching (like IPTW -and like also PS stratification and PS covariate adjustment) would not be expected to perform well when used in combination with MIPs or MIPar. Conversely, because in MIte, the estimated PS can balance both the observed and imputed component of the covariates in the imputed dataset, we would expect all the PS-based methods (PS matching, stratification, covariate adjustment and IPTW) to perform well to estimate the treatment effect within each imputed dataset as long as the underlying assumptions for these methods are fulfilled. In conclusion, for IPTW, multiple imputation followed by pooling of treatment effect estimates is the preferred approach amongst those studied, when data are missing at random, and the outcome must be included in the imputation model.
After dichotomization, corr(X 1 ; X 3 ) = corr(X 2 , X 3 ) = 0. 24 or 0. 46 RR 1 or 2 Relative risk In model  (9) , θ c = 0 or 1. 221 when ρ = 0. 3, and 0 or 1. 289 when ρ = 0. 6. γ Y 0 or -0. 4 Association between the outcome and the probability of missingness When γ Y = 0, γ 0 = -1. 5 and when γ Y = -0. 4, γ 0 = -1. 3 in model (  10 ) and  (11)  For each scenario, 5000 datasets of sample size 2000 were generated. The prevalence of the treatment was 0. 3 and each partially observed covariate had 30% of data missing. 10 imputed datasets were created. For each scenario of the sensitivity analysis, 5000 datasets were generated. The other parameter values were: RR=2, ρ = 0. 6 and γ Y = -0. 4. Table  4 : Bias of the log(RR), its estimated variance and coverage rate for the 3 MI approaches according the sample size n for one scenario (RR=2, ρ = 0.
Finally, the quantities K 0 and K 1 depend on the measure of interest for the treatment effect. For a difference in mean or a risk difference, K 0 = K 1 = 1. For a relative risk, K 0 = μ0 -1 and K 1 = μ1 -1 , and for an odds ratio, K 0 = { μ0 (1 -μ0 )} -1 and K 1 = { μ1 (1 -μ1 )} -1. C α is n 2 ×the variance/covariance matrix of the estimated PS parameters α in the full data, which we estimate by the within-imputation matrix W. The additional part of the variance is = 2vC α * , where * = 1 nE[Z/e(x; α F )] E x (Y -µ F 1 )Z (e(x; α F ) -e(x; α F ) e(x; α F ) + 1 nE[(1 -Z)/(1 -e(x; α F ))] E x (Y -µ F 0 )(1 -Z) (1 -e(x; α F )) -(1 -e(x; α F ))) 1 -e(x; α F ) This is the only part of the variance which involves the full (unobserved) data. The magnitude of this term is driven partly by the difference between the propensity score evaluated at the average covariates across imputed datasets and the true (possibly unobserved) covariate values.
For complete cases, therefore, this term is 0. We could attempt to estimate this component using the imputed datasets. However, we take a pragmatic approach and assume that pooling of the propensity scores is undertaken due to a desire to avoid the need to retain all K imputed datasets. Therefore, we ignore this term in the simulation study and evaluate the performance of the estimator without this term. Thus, we assume that approximately V W E I|W [ θMI ] = Vun -vT Wv (14) This is our estimator for the second term in 13. For the first term, we begin by noting that, conditional on the observed data (Y, Z, X obs ), we have approximately, V.
I|W ( θMIpar ) = ∂ θMIpar ∂α T Cov I|W (α) ∂ θMIpar ∂αNoting that the average of the propensity score parameters across imputed datasets is also the standard MI estimate of these parameters,Cov(α) = 1 + 1 M Bwhere B is the between-imputation covariance matrix of the propensity score parameters. Differentiating the estimate above gives∂ θMI ∂α = -i (Y -μ1)Zxe(x;α)(1-e(x;α)) (e(x;α)) 2 i Z e(x;α) -i (Y -μ0)(1-Z)xe(x;α)(1-e(x;α))(1-e(x;α)) 2 i(1-Z) (1-e(x;α)).
The framework additionally allows the addition of mass and mixing corrections induced at higher orders for the involved external states. We explain our procedure to cancel infra-red divergences for such cases, which is achieved through an infra-red counter-term taking into account corrected Goldstone boson vertices. We compare our results for sfermion, gluino and Higgs decays in the minimal supersymmetric standard model (MSSM) against the public codes SFOLD, FVSFOLD and HFOLD and explain observed differences. Radiative induced gluino and neutralino decays are compared against the original implementation in SPheno in the MSSM. We exactly reproduce the results of the code CNNDecays for decays of neutralinos and charginos in R-parity violating models. The new version SARAH 4. 11. 0 by default includes the calculation of two-body decay widths at the full one-loop level.
With the increasing limits on the SUSY masses within the MSSM, other ideas for new physics are seeing more and more attention. In order to be able to also give more accurate predictions for the decays in non-minimal SUSY models or also in non-supersymmetric extensions of SM, a high-level of automatisation is needed. A very powerful ansatz to obtain robust results for BSM models has been established with the Mathematica package SARAH  [81] [82] [83] [84] [85] [86] : SARAH derives from a short model file all analytical properties of a given model. This information together with generic expressions for various observables is then used to generate Fortran code for SPheno  [87, 88]  which can be used to obtain numerical results. Up to now, one-and two-loop masses  [89] [90] [91] , one-loop flavour and precision observables  [92] , as well as two-and three-body tree-level decays could be obtained via this setup.
SARAH-SPheno interface The possibility to calculate one-loop decay widths is available from SARAH 4. 11. 0. This is a new feature of the SARAH interface to SPheno which was established with SARAH 3. 0. 0: SARAH generates Fortran code which can be compiled together with the standard SPheno package to obtain a spectrum generator for a given model. The main features of a spectrum generator obtained in that way are a precise mass spectrum calculation including two-loop corrections to real scalars  [89] [90] [91] , a prediction for many precision and flavour observables  [92]  and up to now the calculation of two-and three-body decays mainly at tree-level. The general procedure to obtain the SPheno code for a given model starts with the download of the most recent SARAH version from HepForge:. / b i n /SPheno$MODEL $MODEL/ LesHouches.
in. $MODEL h The output is written to SPheno. s p c. $MODEL and contains all running parameters at the renormalisation scale, the loop-corrected mass spectrum, as well as all other observables calculated for the given model and parameter point. The time for generating the Fortran code for the one-loop two-body decays as well as the compilation time of SPheno are extended by these new routines. Therefore, in case that the user in not interested in the loop-corrected two-body decays, they can be turned off via: MakeSPheno [ IncludeLoopDecays -> False ]; They can be permanently turned off for a given model by adding 1 SA ' AddOneLoopDecay = False ; to SPheno. m. Usually, the calculation of the one-loop decays triggers also the calculation of the RGEs even when using the option "OnlyLowEnergySPheno = True;" to generate the SPheno code.
After these examples we continue with a detailed comparison of our implementation with existing codes, among them SFOLD, HFOLD and CNNDecays. Whereas SFOLD and HFOLD are also based on a DR renormalisation of the parameters of the MSSM, the code CNNDecays calculates neutralino and chargino decays in the MSSM, NMSSM and in models with R-parity violation again renormalising the electric charge in the Thomson limit. Thereafter, we compare loop-induced decays with the original implementation of SPheno and lastly show the effect of U -factors in the calculation of two-body decay widths. A more thorough comparison for Higgs boson decays is left for future work.
Sfermion decays in the MSSM: SARAH vs. SFOLD Next we turn to the comparison of the two-body decays of sfermions in the MSSM. For this purpose, we compared our results against the public code SFOLD 1. 2. We have applied several modifications to the code SFOLD: • The variable to use loop-masses either in the loops or in the kinematics are set to 0 in SFOLD. F: 1 osextmassesOn = 0 2 osloopmassesOn = 0 This is done to ensure that both codes use exactly the same masses everywhere. • We find a disagreement for the bremsstrahlung routines for S → SV decays. Therefore, we add to line 440 in Bremsstrahlung. F of SFOLD the terms: • We find for S → SS and S → SV decays huge numerical loop-corrections that could even cause a negative width. We could trace back the problem to diagrams with two massive vector bosons in the loop  5.
In that case, i. e. when using SFOLD out of the box, the corrections for the decays with a Z or W boson in the final state can be a factor of 10 larger. For the decays into a pair of fermions we also find very good agreement with only very small differences for small values of m. Similarly we show the results for the light and heavy sbottom decays in Fig. 5. Here, the results are very similar to those of the stop decays. We do not add figures for stau or τ -sneutrino decays, or the decays of first and second generation sfermions; they would look very similar to the ones for stop and sbottoms, only the overall size of the loop corrections being smaller. Thus, in total we found a very good agreement between SARAH and SFOLD for all kinds of two-body decays of sfermions. In this section we compare the decays of gluinos in the MSSM obtained with SARAH and SPheno against the results generated with the code FVSFOLD.