In medical imaging, images are usually treated as deterministic, while their uncertainties are largely underexplored.This work aims at using deep learning to efficiently estimate posterior distributions of imaging parameters, which in turn can be used to derive the most probable parameters as well as their uncertainties.Our deep learning-based approaches are based on a variational Bayesian inference framework, which is implemented using two different deep neural networks based on conditional variational auto-encoder (CVAE), CVAE-dual-encoder and CVAE-dual-decoder. The conventional CVAE framework, i.e., CVAE-vanilla, can be regarded as a simplified case of these two neural networks. We applied these approaches to a simulation study of dynamic brain PET imaging using a reference region-based kinetic model.In the simulation study, we estimated posterior distributions of PET kinetic parameters given a measurement of time-activity curve. Our proposed CVAE-dual-encoder and CVAE-dual-decoder yield results that are in good agreement with the asymptotically unbiased posterior distributions sampled by Markov Chain Monte Carlo (MCMC). The CVAE-vanilla can also be used for estimating posterior distributions, although it has an inferior performance to both CVAE-dual-encoder and CVAE-dual-decoder.We have evaluated the performance of our deep learning approaches for estimating posterior distributions in dynamic brain PET. Our deep learning approaches yield posterior distributions, which are in good agreement with unbiased distributions estimated by MCMC. All these neural networks have different characteristics and can be chosen by the user for specific applications. The proposed methods are general and can be adapted to other problems. I.1, in which images are treated as deterministic. To assess the uncertainty, frequentist inference requires repeated measurements, which is impractical for medical imaging. Without uncertainty information, assessment of research results and clinical images can be challenging and, under certain circumstances, lead to incorrect conclusions and clinical decisions. However, we usually have prior knowledge of the image to be estimated before the measurement is made. Such prior knowledge can be combined with the measurement to obtain an estimation of the posterior distribution, which can then be used to assess the uncertainty of the image. This falls into Bayesian inference (BI)1, which is a coherent solution to the problem of uncertainty estimation. The posterior distribution is a full distribution on the parameter. It is possible to make all sorts of probabilistic statements about the parameter. For example, we can make a statement of credible interval  if a posterior distribution is known.Uncertainty quantification of medical imaging data is fundamentally important for clinical diagnosis and clinical trials. However, medical images presented in both research and clinical settings are usually non-statistical in the sense that they do not contain information about uncertainty. From the point of view of statistical inference, this belongs to the frequentist methodx in a parameter space given an observable measurement y. In the framework of BI, we define the problem as: given y and a prior, p(x), which represents our knowledge on x before the measurement, what is the posterior distribution, p(x|y)? The conventional method to tackle this problem is to use Markov Chain Monte Carlo (MCMC)2, which is known to produce an asymptotically unbiased estimation of the posterior distribution. MCMC does not require a full analytic posterior description as long as the ratios of probability density functions at pairs of locations  can be calculated3. Although this requirement is met for many medical imaging problems, MCMC has been rarely used in the past. One reason is that recomputing likelihood p(y|x) becomes too expensive for most problems without even accounting for the fact that a large number of burn-in steps are needed for MCMC. Using a dynamic positron emission tomography (PET) study performed on a GE Discovery MI-5 scanner as an example, a time series of sinogram set, y, has a dimension of 54×1981×415×272 , while the corresponding time series of image volumes, x, has a dimension of 54×256×256×256 . Approximate Bayesian Computing (ABC) is another well-known method for estimating posterior distributions for a given measurement4. In ABC, model parameters sampled from the prior are used to generate artificial measurement datasets. If the resulting datasets are very close to the given measurement according to a predefined discrepancy function, the corresponding parameters are then accepted as the part of the posterior. Unlike MCMC, ABC is an approximation. Also, it does not offer much advantage over MCMC in terms of computational time. Another reason is that the prior is subjective in BI. A poor prior certainly leads to a poor posterior estimation.Most medical imaging problems can be generalized as the estimation of As a large amount of training data becomes available in medical imaging, BI combined with deep learning (DL) has the potential to play an important role in posterior estimation in the future. We first define a very general problem, which is not limited to medical imaging, as:D samples, y, and a testing observable measurement, y*, what is the posterior distribution, p(x|y*)?Given a training dataset of p(x), is no longer subjective but implicitly defined by the training dataset itself. To solve the above problem using MCMC is challenging because only training data rather than the underlying analytic forward and noise models are available. This makes it difficult to compute the ratios of probability density functions at pairs of locations as required by MCMC. For such a problem, it is also difficult to use ABC from the training data without knowing the underlying model. We intend to sample the posterior distribution, p(x|y*), using a conditional variational auto-encoder (CVAE), in which the generation process is conditioned on y*. In addition, we introduce a latent multivariate random variable z to account for the information loss in the forward process from x to y5. The CVAE is trained with the paired dataset p(x|y*), which represents a complete picture of the parameter space, using a predefined distribution of the latent variable, p(z). Based on this strategy, we have derived different DL-based approaches for estimating posterior distributions using the CVAE framework 9. However, INNs require special coupling layers to achieve the normalizing flow, which can be insufficiently expressive and computationally expensive10. In addition, a CVAE can be used as as baseline for INN5. This approach, which is denoted by CVAE-vanilla is an oversimplication since y and z are assumed to be independent, and therefore cannot guarantee accurate estimation of the posterior distribution.In the past, there have been various types of DL-based approaches proposed for BI. One is to directly train a deterministic inverse mapping from p(x|y*), a ground truth is necessary, but it is not available if the only available data is the training dataset, y*. We, therefore, used a simple simulation study for dynamic brain PET imaging, in which we can not only generate a training dataset for our DL-based approaches, but also perform MCMC to produce asymptotically unbiased posterior distributions to be used as the gold standard  in a target region based on a simplified reference tissue model (SRTM12) and Gaussian noise model. For a given testing TAC, the posterior distributions of kinetic parameters using our generative DL-based approaches were compared to the unbiased distributions sampled by MCMC.In order to validate our DL-based approaches for estimating II.In this section, we first propose our DL-based methods for posterior estimation. We then describe how we performed the simulation for dynamic brain PET using SRTM and Gaussian noise model. Afterward, we explain in detail how we performed MCMC and our DL-based approaches to obtain the posterior distributions of the kinetic parameters for a given dynamic PET measurement, i.e., a TAC. Finally, we describe how we evaluated the performance of our DL-based approaches using the unbiased posterior distributions sampled by MCMC as the gold standards.II.A.14.In this work, we propose to use a CVAE framework for efficiently sampling the posterior distributions given an observed measurement. We propose different deep neural networks (DNN) for estimating posterior distributions based on the evidence lower bounds (ELBOs)p(x|y) for a given observable measurement y, we define a random multidimensional latent variable z to capture the information loss in the forward process from x to y. We intend to train a neural network (known as decoder), θ, which performs z is sampled from distribution p, i.e., z ~ p. To make such training possible, we introduce another neural network (known as encoder), ϕ, which maps x and y to z. The two neural networks, i.e., θ and ϕ, must be decoupleable after the training, which can be achieved by minimizing the following Kullback-Leibler (KL) divergence:εA and εB are two equivalent ELBOs, since logp(y) is independent to the value of z. Specifically, we have:p, p(z|y), and p(y|z) with pθ, pϕ′(z|y), and pθ′(y|z), respectively . In addition, if we assume that z is independent of y, i.e., p(z|y) = p(z), we have: εB ≈ εC + logp(y).To estimate the posterior distribution εA, εB, and εC, respectively.We therefore propose three different DNNs: CVAE-dual-encoder, CVAE-dual-decoder, and CVAE-vanilla See , which aII.A.1.εA, which consists of an encoder ϕ , a decoder KL||pϕ′(z|y)) in εA, we use the reparameterization trick13 in both the encoder neural networks, i.e., ϕ and ϕ′, with two K-dimensional multivariable normal distributions defined by 𝒩) and 𝒩) representing pϕ and pϕ′(z|y), respectively. As a result, we introduce another loss function given a pair of training sample:K is the dimension of the latent code z or μk and k-th node. In practice, the output layer of each of the two encoders has two branches (each branch consists of K nodes), which represent mean and variance, respectively. We then sample z and z = μ + σ ⊙ ϵ and ϵ is a standard multivariable normal distribution, i.e., ϵ ~ . We then we define the overall loss function as:βA is a hyperparameter to weight ℒA2, defined in Maximizing y*, we use ϕ′ to predict σ′ and μ′, followed by sampling y* to the decoder θ to generate the corresponding For inference, given the observation II.A.2.εB, which consists of an encoder ϕ , a decoder B1, which is used to maximize the first term in εB, is the same as ℒA1. Similar to CVAE-dual-encoder, we use the same reparameterization trick to handle the KL term in ℒB1 with the only difference being the use of  to represent p(z). The second loss function becomesεB:βB and λ are the hyperparameters to weight ℒB2 and ℒB3, respectively. Notably, ℒB2 is different from ℒA2.Obviously, the first loss function ℒz ~ , and concatenate z and y to the decoder to generate the corresponding For inference, we use II.A.3.εC, which consists of an encoder ϕ  and a decoder C1, which is the same as both ℒA1 and ℒB1, and ℒC2, which is the same as ℒB2. An overall loss function is defined as:βC is the hyperparameter to weight ℒC2. For inference, we use z ~  and concatenate z and y to the decoder to generate the corresponding II.B.x, when a measurement of TAC, y, is given in a target region. In this study, we used SRTM for tracer kinetics in a brain region, which can be formulated as:CT (t) and CR(t) are the activity concentrations in the target region and a pre-defined reference region respectively at time t, DV R is the distribution volume ratio between the target and reference region, k2 is the rate constant from free to plasma compartment, and R1 is the ratio of rate constants for transform from plasma to free compartment. The analytic solution of TAC in the target region is:x {DVR, k2, R1}, to tn = tn – tn−1, and σ is the standard deviation. We assumed that σ follows a gamma distribution, i.e., σ ~ 10−4Gamma. For this simulation study, we chose the temporal lobe and cerebellum grey as the target and reference region, respectively. We also set the number of time frames to N = 54. We used the following sequence of time frame durations: 6 × 10s, 8 × 15s, 6 × 30s, 8 × 60s, 8 × 120s, and 18 × 300s. In the simulation, kinetic parameters were randomly sampled from predefined priors , based on our knowledge before the measurement, the posterior distribution is determined as p(x|y) ∝ p(y|x)p(x). In this work, we chose to use the widely used random walk Metropolis-Hastings MCMC (MH-MCMC) to sample the posterior distributions of kinetic parameters. A symmetric proposal distribution that represents a Markov Chain transition from step l − 1 to step l, J(x(l)|xl−1) = N(xl)(|xl−1), was used. The diagonal covariance matrix ∑ was used to control the acceptance rate of MCMC.The conventional approach for sampling posterior distribution is to follow a rejection sampling scheme with MCMC15. To provide a more reliable assessment of convergence, we calculated the mean of the first 10% and last 50% steps counted after burn-in steps to check if the difference between these two means is approaching zero16.In the implementation of the MCMC, an important step is the judgment of convergence, which usually indicates whether the algorithm is drawing the sample from the true distribution and achieves balance. Trace plots of the  log-likelihood are often used as visual and subjective tool to give a hintII.D.p(x) as DV R ~ N, k2 ~ N, and R1 ~ N, based on a previous [18F]MK-6240 study across 35 subjects11. To demonstrate the effectiveness on multiple simulated kinetic parameter sets, we further increased the mean, variance, and both mean and variance of the prior by 20%, and denoted as setting 2, 3, and 4, respectively. For the purpose of quantitative evaluation of our approaches, we kept sampling x = {DV R, k2, R1} from the prior distributions until we collected a total of 200 testing x’s that satisfy i = 1, 2, 3, where α = 0.26 was chosen based on the variance of measured DV R across all subjects in the previous study11. The corresponding testing measurement of TAC, y, for each testing x was then generated using the SRTM and the Gaussian noise model (See II.B).In this study, we first (setting 1) defined the prior For each testing measurement, we first used MCMC to sample its corresponding asymptotically unbiased posterior distributions using the defined prior distributions as well as the forward and noise models as described in β = 1 and λ = 1 13. We used the same network structure in encoder ϕ and decoder θ. Specifically, in the encoder, we used four fully connected layers, which contain 128, 100, 50, and 20 nodes. The output layer of the encoder has ten nodes for mean values and ten nodes for variance values, which are in turn used to define the distribution of z, whose dimension is K = 10. In the decoder, we also used four fully connected layers, which contain 128, 100, 50, and 3 nodes. The output layer of the decoder is a three-dimensional vector, i.e., . In both the encoder and decoder, ReLU is used as an activation function. For the CVAE-dual-encoder, we have an additional encoder ϕ′, which has the same structure as encoder ϕ, though parameters in these two encoders are not shared. For the CVAE-dual-decoder, we had an additional decoder θ′, which has four fully connected layers containing 16, 16, 32, and 54 nodes.For all our DL-based approaches, we set x using the defined priors. Each sampled x was then used to generate its corresponding y using the SRTM and the Gaussian noise model. The resulting training data pairs were used to train the neural network in each one of our DL-based approaches. We used the same learning rate, i.e., 10−4, and stochastic gradient descent (SGD) optimizer with a momentum of 0.9 for all the neural networks. For each approach, the trained neural network was used to generate 45,000 samples to obtain posterior distributions for each testing y afterward.To construct the training set for our DL-based approaches, we generated 10,000 samples of M = 200 testing samples for each kinetic parameter using:m-th sample using a Gaussian function. We also computed the average KL divergence, m-th observable testing sample.To evaluate the performance of each DL-based approach using MCMC as the reference, we first computed the average relative difference of normalized mean and standard deviation, i.e., We used the PyTorch toolbox for the implementation of our DL-based approaches. We performed all the computation on a server with an NVIDIA V100 GPU (32GB graphics RAM version) and an Intel Xeon 8-core CPU alongside 24GB of RAM.III.DV R, k2, and R1 obtained from MCMC and DL-based methods for a single noisy TAC measurement y* generated using DV R = 1.0, k2 = 0.0006 min−1, and R1 = 0.74. All DL-based approaches agree reasonably well with asymptotically unbiased MCMC, while both CVAE-dual-encoder and CVAE-dual-decoder yield better agreement than the CVAE-vanilla.βA, βB and λ in CVAE-dual-encoder and CVAE-dual-decoder. We note that all of the standard deviations for βA, βB and λ) for the same range of . Therefore, we simply use βA = 1, βB = 1 and λ = 1 for all of our DL-based approaches.17, a python based MCMC implementation. All the neural networks in our DL-based approaches were trained with 200 epochs, which took ~2, 2, and 2.5 hours for CVAE-vanilla, CVAE-dual-decoder, and CVAE-dual-encoder, respectively. It took less than 15 seconds for each trained CVAE neural network to infer posterior distributions for all three parameters  for a given y*. For estimating posterior distributions for a problem where MCMC is feasible, our DL-based approaches can be, therefore, much more efficient than MCMC. In this study, we focused on a single-region dynamic brain PET, which allows us to perform both MCMC and our DL-based approaches for estimating posterior distributions. For a problem with high-dimensional data, MCMC can be computationally intractable, while a trained neural network is still feasible.It took ~10 minutes to sample 45,000 samples using PyMCIn the following subsections, we provide some detailed discussions on a few topics related to MCMC and DL-based methods for estimating posterior distributions.16. DV R values sampled by MCMC. After 15,000 burn-in steps, the sampled values become stable. For all three parameters , with 15,000 burn-in steps, the difference between the mean values from the first 10% and from the last 50% steps are less than 0.001, which indicates a good convergence16.A critical question in MCMC is to determine if the sampling has converged to a stationary distribution. Though several convergence criteria have been proposed in the past, in practice, the convergence is often determined empirically, e.g., using Geweke’s test based on the trace of temporal seriesDV R value used to generate the measurement, y*, is low based on the prior distribution of DV R used to generate the training data. DV R*, which is the DV R value used to generate y* measurements. We can see that DV R* if DV R* < 3.35, i.e., the sum of mean  and full width at half maximum (FWHM) of N . The increase of DV R* becomes much faster if DV R* > 3.35. As a result, when we apply our DL-based methods to the problem as defined in p(x), which is implicitly defined by the training data, should cover the value of x corresponding to measurement y*.For all the DL-based methods, data shift represents a mismatch between the distributions of the training and testing data, which can degrade the performance. For this simulation study, we expect that the performance, measured using As shown in Compared to CVAE-dual-encoder, CVAE-dual-decoder requires more training time because it has one more loss term. However, CVAE-dual-decoder is faster in inference because it has a simpler inference structure than CVAE-dual-encoder  that can be described by SRTM. In such a problem, we can not only generate the data for the training of deep neural networks but also perform MCMC. As a result, we are able to evaluate the performance of our deep learning approaches using MCMC posterior distributions as reference. For this particular problem, both MCMC and one of our deep learning approaches can be used to estimate posterior distributions for a given measurement  on a subject (a definition of prior of kinetic parameters is needed for both approaches). It is also worth noting that source and target domains should be the same for our deep-learning based approaches to avoid inference bias18. For example, unless some domain adaption techniques are used, it is not appropriate to apply a neural network trained for one tracer to obtain posterior distributions for a different tracer because priors, for example, can be very different for different tracers even the same kinetic model is used.We would like to point out that our goal is to use deep learning to solve the general problem as stated in y*, in dynamic brain PET. We can, for example, extend our work so that y represents dynamic sinogram data rather than a TAC. As stated in In this work, we estimated the posterior distributions of kinetic parameters given a measurement of TAC, i.e., IV.We have proposed DL-based approaches for estimating posterior distributions. Our approaches, which are based on a deep variational inference framework, are implemented using two different deep neural networks, CVAE-dual-encoder and CVAE-dual-decoder. The conventional CVAE framework, i.e., CVAE-vanilla, can be regarded as a simplified case of these two neural networks. All these neural networks have different characteristics and can be chosen by the user for specific applications. We have applied these approaches to a simulation study of dynamic brain PET and evaluated their performance using asymptotically unbiased MCMC as the reference. Both CVAE-dual-encoder and CVAE-dual-decoder yield good agreement with MCMC for estimating posterior distributions of kinetic parameters given a measurement of TAC. For our simulation study, we have also found that CVAE-vanilla can also be used for estimating posterior distributions, although it has an inferior performance to both CVAE-dual-encoder and CVAE-dual-decoder.


Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states . Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given monomial evaluation orders . As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain binomial differential inclusions. In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result.We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible  It can be seen as a discrete version of the Laplace operator defined on graphs. On the one hand, the Laplacian matrix of an ng Mohar . On the V of a simple digraph E represent transitions. Moreover, edge labels k represent transition rate constants. The dynamical system for a state variable A for the graph Laplacian and indicate its dependence on the edge labels k by a subscript.) The linear system can be called “Laplacian dynamics”, it is equivalent to the stochastic master equation, and it is studied in applications ranging from biochemistry to systems biology , and edge labels k. In particular, every edge Y collects the complexes y(i) for Y (by stoichiometry) as well as by the Laplacian matrix Now, we can introduce a mass-action system as a simple digraph entclass1pt{minimacomplex-balanced equilibrium (CBE), also known as vertex-balanced steady state. Indeed, at a CBE, the sum of all “flows” from vertex i/complex y(i) equals the sum of all to the latter. As shown by Horn . Moreover, the polynomial dynamical system is a sum of binomials. (Just note that every reversible reaction x). We show that this also holds for weakly reversible networks. To this end, we provide a new decomposition of the graph Laplacian, involving an invertible core matrix, based on an order on the vertices. Further, we extend the classical result by Horn and Jackson on the asymptotic stability of complex-balanced equilibria. In addition to a Lyapunov function , we consider regions in the positive orthant with given monomial evaluation orders . As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain binomial differential inclusions. In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result.For symmetric digraphs (“reversible” networks), core matrix, based on an order on the vertices. Depending on the particular order, the core matrix has additional properties.In Sect. monomial evaluation orders and corresponding geometric objects . In Sect. binomial differential inclusions and show that all equilibria of the latter are asymptotically stable, and in Sect. In Sect. In “Appendix A”, we provide explicit formulas for the vector of tree constants and the Laplacian matrix, using cycle decomposition. In “Appendix B”, we state auxiliary results used in the new decomposition of the graph Laplacian. In “Appendix C”, we give another proof of the asymptotic stability of complex-balanced equilibria (and the non-existence of other steady states) without using differential inclusions.Notation We denote the positive real numbers by I, we write y(j) is the column of Y with index For vectors In the following, we assume that the components of a digraph are strongly connected. For the simplicity of the presentation, we first consider one strongly connected component separately.k is the vector of transition rate constants in the continuous-time, linear process We consider a strongly connected, simple, directed graph G is connected, G is strongly connected,tree constants) can be given explicitly in terms of k,G rooted at vertex Since f of Eq. , Lemma 1Clearly, the matrixk. For a derivation of this formula and a discussion of the Birkhoff/von Neumann Theorem  or of the form star graph with root For an auxiliary digraph Let V  In the labeled digraph chain graphs. For example, for E), we findE), we findLet Explicitly, let V, That is,  plus the common entry of row i and column j. That is,A, it has positive diagonal entries and nonpositive off-diagonal entries and is (row and column) diagonally dominant.  Let mentclass2pt{minimExample (continued) In the labeled digraph star graphs. For example, for E), we findE), we findRemark In applications to mass-action systems in Sect. In general, we consider a labeled, simple digraph Propositions core matrix of the graph Laplacian, such thatLet Explicitly,We apply the graph-theoretic/algebraic results from the previous section to mass-action systems. We start with a brief summary of fundamental concepts and results.chemical reaction network  is given by a simple directed graph (reactions)complexG is “embedded” in G (the linkage classes) are strongly connected, then the network is called weakly reversible.A mass-action systemG, y) where every edge rate constantA concentrations of n molecular species) is given bystoichiometric subspace is given by stoichiometric (compatibility) class.The resulting dynamical system for complex-balanced equilibrium (CBE), also known as vertex-balanced steady state.If an equilibrium Remark In the linear setting, the Laplacian matrix captures state transitions on a graph. Let x is a CBE. If x is a general equilibrium.the mass-action system is weakly reversible  equilibrium in every stoichiometric class  Let G be weakly reversible and Remark (result 2) In Sect. In “Appendix C”, we give another proof for the asymptotic stability of complex-balanced equilibria (and the non-existence of other steady states) without using differential inclusions.Remark (result 3) Technically, result 3 states that y Fulton , where iy Fulton , where iy Fulton . For geny Fulton , Müller y Fulton , Müller y Fulton , Craciuny Fulton  and Gopay Fulton .G are strongly connected), our main graph-theoretic/algebraic result, Theorem Given that the network is weakly reversible , but truly is a binomial dynamical system.In particular, for a complex-balanced equilibrium, not just the right-hand side of  is zero,y (the matrix Y).Let total orderpartial orderG and hence, on the values of the monomials.For fixed In the following, we will consider monomials with coefficients:Let monomial evaluation order  has a different meaning in algebra). We represent the order by a chain graph We call an order on the entries of Analogously, the maximal entries of Conversely, fix an auxiliary graph If there exists a positive CBE k.  The lineality space of In the following study of complex-balanced mass-action systems , we use chain graphs stratum, cf. Siegel and Johnston  and  The positive orthant is a union of strata corresponding to monomial evaluation orders. In particular, consider the stratum given by the order G have two strongly connected components y(1), y(2), y(3) as above, and Finally, let In general, Finally, we extend a classical result by Horn and Jackson from 1972. is a strict Lyapunov function, and All proofs are based on the entropy-like Lyapunov function Previous proofs further use inequalities for the exponential function or the logarithm and cycle decomposition of the graph, cf. Horn and Jackson , Sontag In the following, we extend the stability result and provide a maximally transparent, polyhedral-geometry proof. First, we relate the dynamics in a given stratum to the corresponding polyhedral cone.C, where In Proposition Let Let l system  and Theo in Eqs.  and 10)\documentBy Theorem binomial differential inclusion asNow, let l system . ProposiLet l system . Then, tmentclasspt{minimaFinally, we extend Theorem Let mentclasspt{minimaL(x) is a strict Lyapunov function. Let recession cone of a set C.Even if a weakly reversible mass-action system cf. Eqs. . That isbinomial differential inclusions (BDIs), introduced in this work. Every positive equilibrium of a BDI is asymptotically stable, see Theorem As Horn and Jackson in 1972  with graph E) and on some positive equilibrium Given a reaction network  have been used in a proposed proof , In previous work, we have studied


Nature 10.1038/s41586-023-06551-1 Published online 24 August 2023Correction to: In the version of the article initially published, there was an error in the tenth paragraph, where in the sentence now reading “… where CO flux dominates  and, conversely, where H2 emission prevails ”, the colours “light yellow-white” and “dark orange-red” were inadvertently swapped. This has been corrected in the HTML and PDF versions of the article.


The factorial structure and the reliability of the Observer Report of HEXACO-MSI-E were confirmed. Convergent and divergent validity were successfully established with a version of the inventory filled out by parents. Convergent and divergent validity were also established with BFQ-C notwithstanding some only apparent anomalies. Criterion validity was established with respect to four specific groups of criteria collected in both self-report and observer form. HEXACO-MSI-E traits were more predictive with respect to self-report than to observer criteria and the majority were common. Together with the positive results of this study, implications and issues for future studies are discussed.In this cross-sectional study we aimed at: 1) validating the observer (Obs) version of HEXACO Medium School Inventory Extended (MSI-E); 2) establishing convergent and divergent construct validity of the HEXACO-MSI-E; 3) establishing criterion validity of HEXACO-MSI-E. We administered the HEXACO-MSI-E, the Big Five Questionnaire-Children (BFQ-C), the Internalizing and Externalizing scales of Youth Self Report (YSR), some items of the 2019 Middle School Youth Risk Behavior Survey (YRB) and some items about adolescents’ values, beliefs, behaviors, and desired features of a possible future job to 1175 adolescents (M During the past two decades, there emerged a near-consensus that human personality variation can be summarized by five broad dimensions that are collectively known as the Big Five –3. TheseResearch suggests that the HEXACO model has practical and theoretical advantages over the Big Five for predicting individually, organizationally, and academically relevant outcomes , 24. ForIn conclusion, this brief review of the predictive capacity of the HEXACO with respect to many criteria together with many other systematic review on the topic –41 showsth, 8th and 9th grade), construct validity with respect to an observer report and to BFQ-C and, finally, criterion validity with respect to school grades. Indeed, three factors, Conscientiousness, Honesty-Humility, and eXtraversion proved good predictors of school grades. Despite several merits, this preliminary version of the inventory had some weaknesses regarding the extent of content validity and the balance among reversed and not reversed items. To solve these problems, Gnisci and colleagues set up two studies ; CFI = .929, DWLSAs shown in Table A in All dimensions of Obs HEXACO-MSI-E had at least excellent levels of internal consistency, as shown by alpha indexes reported in the Skewness and kurtosis provide evidence of normality for each factor . Note, hThe multitrait–multimethod matrix of self-report HEXACO-MSI-E (N = 1175), Obs HEXACO-MSI-E (N = 343) and the BFQ-C (N = 400), reporting observed  and latent traits (below) correlations is shown in As regard self-report and Obs HEXACO-MSI-E, first, correlations between the same self-report and observer traits of HEXACO-MSI-E showed a very good convergent validity: homotrait-heteromethod correlations ranged from .529 to .705. Second, correlations between different self-report and observer traits–that is, heterotrait-heteromethod coefficients–were, in general, low (max r = |.372|) and always lower than homotrait-heteromethod coefficients. Therefore, convergent and divergent construct validity of HEXACO-MSI-E with respect to the observer form of HEXACO-MSI was established.As regards the correlation coefficients between the HEXACO-MSI-E self-report traits and their BFQ-C counterparts, three of them were strong , one was adequate (O-Openness), one negligeable (between E and Neuroticism). Taking aside this last correlation (and the extra-trait H), 17 out of 20 correlations between different traits assessed by different methods were lower than the above-described correlations. The three higher correlations were due to A correlating strongly and negatively with Neuroticism and moderately and positively with Conscientiousness as well as C correlating moderately with Openness. In general, out of the 20 correlations between different traits assessed by different methods, in absolute value, two are negligible (.00-.09), 12 weak (.10-.39), 5 moderate (.40-.69) and one strong (.70-.89). As mentioned, E does not correlate with Neuroticism and has null or weak correlations with the other BFQ-C factors. As far as the H trait is concerned, it shows weak but consistent co-variation with Neuroticism, Agreeableness and Conscientiousness.Finally, as expected, correlations between observed and latent dimensions are very strong within each method (min .953), indicating that factor scores are representative of the estimated constructs.In this section, we will present data on criterion validity of the self-report HEXACO-MSI-E traits with respect to a particular group of criteria connected to: 1) relevant dimensions connected to internalizing and externalizing problems of adolescents (YSR and YOR); 2) risk factors in adolescence (YRB); 3) personal passions, values, beliefs and behaviors (PVB&B); 4) desired features of a possible future job (FFJ). Each group of criteria has been measured in two different forms, self-report and observer , and we have verified how the six traits predict indicators of each group considered in both versions. For each group, zero-order correlations between the criterion indices and the six HEXACO traits of adolescents were preliminarily calculated and are shown in Correlations between the same items but in self-report (YSR) and observer (YOR) form for specific and general internalizing and externalizing problems ranged between .24 and .47  as target were performed for both self-report and observer criteria : H negatively predicted all the YSR criteria  apart from one; E positively all the YSR criteria but three, delinquent behavior is the only one that correlates negatively (4); X negatively predicted all the YSR criteria (7); A negatively predicted only two YSR scores (2); C negatively all the YSR criteria (6) but one; finally, O positively predicted three YSR criteria (3).The following factors significantly predicted the following YOR criteria (N = 373): H negatively predicted three criteria ; E negatively one (1); X negatively four (4); A negatively two and one positively (3); C negatively two (2); finally, O negatively one criterion (1).Significant observer traits predictions (YOR) were few with respect to the self-report analogous (YSR); however, in 11 cases out of 14 they shared the sign.MIN = 80, NMAX = 124; refer to Table Q in Correlations between the same items of Youth Risk Behavior (YRB) in self-report and observer form ranged between .17 and .63  as target. All the models were significant  and observer N = 373) PVB&B criteria. All the models were significant  and three negatively . E predicted seven criteria, five positively  and two negatively (being engaged and playing sports). Out of the nine criteria predicted by X, seven were positive  and two negative . A predicted four criteria in all, three positively  and one negatively (listening to music). C predicted six behaviors, four positively  and two negatively (being engaged and sleeping late). Finally, O predicted five behaviors, three positively  and two negatively .The six HEXACO traits predicted 27 observed criteria. H predicted four criteria, two positively (respecting older people and using polite words) and two negatively . E predicted five criteria, three positively  and two negatively . X predicted only two behaviours positively (attending religious services and feeling good). A predicted three criteria, one positively (preferring science to art) and two negatively (writing a diary and joking). Out of the nine criteria predicted by C, seven were positive  and two negative . Finally, O predicted four criteria, three positive  and one negative .Traits predicted more self-report (38) than observed (27) PVB&B criteria. Out of the 27 observed predictions, 17 were shared with self-report ones.Correlations between the same items of the features of a possible future job of the participant (FFJ) in self-report and observer form ranged between .18 and .36  and by the parent/legal guardian . All the models were significant  than observer (8) features of a future job. Out of the eight predicted observed, six were shared with the self-report ones.In the next sections, we will briefly discuss the most important outcomes of this contribution with respect to the declared aims and delineate possible implications of the results of the research as well as its limitations and future developments.One important outcome is having provided a validation of an Observer Report of the HEXACO-MSI-E that we make public and available to interested researchers. This result can be considered as a relevant contribution especially for the fields of child and adolescent psychology, where reports from significant others are particularly important. Indeed, the factorial structure and the reliability of the Observer Report of the HEXACO-MSI-E was confirmed. Therefore, it is possible to affirm that the observer version of the HEXACO-MSI-E provides the same valid and reliable measure of the adolescent HEXACO traits as the self-report version . HoweverAnother important result of the research is having reported strong evidence of construct validity, contrasting the self-report HEXACO-MSI-E inventory, first, with the analogous Observer Report and then, with BFQ-C, meant to measure personality in children from the perspective of FFM.While construct convergent and divergent validity of the HEXACO-MSI-E with respect to its observer counterpart was excellent and straightforward, correlations between BFQ-C needs more elaboration. Indeed, all the dimensions converged with their BFQ-C homologue apart from E, and, in general, the dimensions with convergent validity also showed a divergent validity almost satisfying with some interesting ‘anomalies’. We want to discuss now these ‘anomalies’ taking into consideration some important references: the theoretical difference between the two models (the HEXACO and the FFM), the correlations found in adult literature between the two instruments, in particular, the recent meta-analysis by Thielmann et al. , on 152 As stated in the results, apart from the correlation between A and BFQ-C Neuroticism, the highest discriminant correlations were found between A and BFQ-C Conscientiousness, C and BFQ-C Openness and Neuroticism, and, finally, O and BFQ-C Conscientiousness. We are going to provide a rationale for the associations found.In the 30-item BFQ-C we have utilized in this research, Openness is better defined as Intellect because it expresses the capability of succeeding in a school setting  while most of the HEXACO-MSI-E items partaking to C refer to effectively organizing and successful doing and monitoring schoolwork and assignments. Therefore, given the emphasis on similar contents applied to school setting, it is no surprise if they share variance, as our results show, and also that they share more variance than the one shared in the adult versions of the two instruments (r = .16) . AnalogoAs far as regard the extra factor H, as stated in the results, it correlated negatively with BFQ-C Neuroticism, positively with BFQ-C Agreeableness and Conscientiousness. We should underline that the sign of the three correlations is the same found in adults while the intensity is superimposable for H and Agreeableness (.40 and .47), close for H and Conscientiousness (.37 and .24), and farther for H and Neuroticism (-.41 and -.13). Indeed, H shares variance with FFM’s Conscientiousness but also with FFM’s Agreeableness –63. PartAll in all, if we consider the correlations higher than .30 between cousin traits of HEXACO-MSI-E and BFQ-C found in the present research, nine out of 14 showed the same sign and similar intensity with respect to the ones we found in 2020 . This isThe third important outcome of this research is having established criterion validity too in the four areas investigated.As far as the YSR, the most predictive HEXACO factors of many emotional and behavioral youth problems were X, H and C. They all predicted the behavioral problems in a negative way: the more a person was honest and humble, extroverted and conscientious, the less they reported such kind of problems. Therefore, these factors, in the direction of their polarity, seem broad-spectrum protective factors in adolescence. The E dimension was found as a predictive factor as well but, predictably, mainly for internalizing problems: lower E was linked to less anxiety and somatic problems. These results partially confirm what Barbaranelli and colleagues’ study  found: EAs regards the risk factors in the teenage years (self-report YRB), data showed that the stronger predictors were C, H and X. These adolescents’ personality traits were related to their participation in risky behaviors in a fairly intuitive pattern. Unconscientious, dishonest, not humble, and introverted adolescents were more likely to participate in health-compromising behaviors than their conscientious, honest, humble, and extroverted peers. In other words, adolescents who were irresponsible, not planning , unfair, pretentious , shy and unsociable  were most susceptible to engage in healthy risky behaviors. These results are consistent with Weller and Tikir  findingsThe traits more predictive of self-report PVB&B in adolescence were X, H and E. Moderately less predictively resulted C, O and A. Many significant predictions show coherence between the predictive trait and the passions, values, beliefs or behaviors. For example, adolescents high in O preferred science and music and used to write art or diary; adolescents high in C, H and E adopted respectful behaviors. Of course, we also found few predictions that can be regarded as neutral, that is, neither favorable nor contrary to the semantic of the traits significantly associated .Most HEXACO traits made coherent predictions with respect to the questions asked to the adolescents participating in the research on their possible future job (self-report FFJ): for example, H predicted positively an honest job and negatively a good income, O predicted job allowing to be in contact with nature and art, X people wanted to collaborate with other people, C people wanted an organized and planned job. Some of the arisen patterns were not coherent .In their self-report form, the most predictive factors with respect to all the four categories of criteria, were H, X and C, particularly for the YSR and YRB. For the other criteria (PVB&B and FFJ) also E emerged as predictive. All in all, many of the previsions of the criteria were coherent with the trait involved.In general, values of the correlations between traits and criteria can depend on same factors that we want to highlight. For example, due to the content of some of the chosen criteria , we expected that many respondents scored to the extreme endpoints of the measurement scale and this is what happened for some items within the criteria .Correlations between self-report and observer criteria showed, for some items, a good level of covariance; however, many other items showed little covariance. Therefore, parents/legal guardians’ ratings not necessarily reflect self-report ratings by their children, highlighting the need for external evaluations in adolescence. In any case, we should consider this result with caution because, for methodological reasons connected to the way we formed the samples of adolescents and parents/guardians, sample sizes for these analyses were relatively small. Future studies should confirm these results.In general, and within each of the four categories of criteria utilized, the self-report criteria significantly associated with traits in adolescents  were always more than the analogous observer criteria . What is more, almost two thirds of the significant ones with observer criteria coincide with the self-report ones (about 67%). This could be due, of course, to a similarity of methods  but also indicates that traits often predict criterion from an inside and outside perspective.The HEXACO traits provided specific predictions of the YOR criteria (observed), differently by the wide-spectrum predictions of many of the YSR criteria (self-report): indeed, X predicted mainly observerd internalizing problems whereas H, A and C predicted mainly externalizing problems, a pattern just found by Mottola et al.  when criHEXACO traits predicted few observed risk behaviors (YRB), the most predictive were C (5 predictions), X and O (both 3) while A did not predict any YRB. Also, for these results, similarities have been found in the same study . Indeed,This paper adds significantly to the one on content and dimensional validity, invariance across males and females and across the three classes of middle school, and reliability as internal consistency and 1-year stability , providiThis study, as any, has also limitations that indicate future prospects of research. For example, given that the inventory was validated only in Italy, we consider it important to adapt the inventory to different languages and replicate the study to verify the psychometric properties of the inventory in different countries and cultures. Moreover, given that we were from the beginning very concerned about the representativeness of the facets and the dimensions of the inventory, we settled on an extended version of the inventory that would guarantee that representativeness. However, this could, in some conditions, be impractical for its length. Therefore, we think it important to develop a validated short form of the HEXACO-MSI-E that might come in handy under certain testing conditions and practical constraints. Finally, as suggested by one of the reviewers of this paper, for some items in the Social Self-Esteem scale of Extraversion factor, within the observer report from, i.e., “Others enjoy spending time with her/him“, “Nobody likes talking to her/him” and “People like her/him”, it would be more appropriate to express them in this way: "S/he believes others enjoy spending time with her/him“, “S/he believes nobody likes talking to her/him”, and “S/he believes people like her/him”. This is because they aim at measuring levels of a person’s perception of popularity or attractiveness and not measuring objective levels of it. Therefore, we hope that, in future studies, this inventory may be used in view of these suggestions.S1 File(DOCX)Click here for additional data file. 9 Aug 2023 <div> PONE-D-23-19901 </div>  <div> Construct and criterion validity of the HEXACO Medium School Inventory Extended (MSI-E) </div>  <div> PLOS ONEDear Dr. Gnisci,Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process .As an absolute minimum, column 17 must be added. And just to avoid any mistake, could you go back to your data and check that neuroticism (BFQ-C) was correctly calculated?plosone@plos.org. 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For instructions see: We look forward to receiving your revised manuscript.Kind regards,Frantisek SudzinaAcademic EditorPLOS ONEJournal Requirements:When submitting your revision, we need you to address these additional requirements.1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf2. We note that you have stated that you will provide repository information for your data at acceptance. Should your manuscript be accepted for publication, we will hold it until you provide the relevant accession numbers or DOIs necessary to access your data. If you wish to make changes to your Data Availability statement, please describe these changes in your cover letter and we will update your Data Availability statement to reflect the information you provide.3. We note you have included a table to which you do not refer in the text of your manuscript. Please ensure that you refer to Table 7 in your text; if accepted, production will need this reference to link the reader to the Table.Additional Editor Comments:Reviewers noticed several issues that could be improved, and one column that must be added.Just to avoid any mistake, could you go back to your data and check that neuroticism (BFQ-C) was correctly calculated?Personally, I would not omit information that were already provided in the original submission.[Note: HTML markup is below. Please do not edit.]Reviewers' comments:Reviewer's Responses to Questions <font color="black"> Comments to the Author1. 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You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. Reviewer #1: Review of PONE-D-23-19901, “Construct and criterion validity of the HEXACO Medium School Inventory Extended (MSI-E)”As its title suggests, this manuscript reports on the validity of a questionnaire for assessing the HEXACO personality characteristics in middle-school students. The research is based on a fairly large sample of adolescents (average age about 12.0 years) for whom self-reports (and in most cases parent reports) were obtained; a variety of outcome variables were also assessed in self-report or (for some of the sample) parent reports.The results generally showed good psychometric properties for the instrument. For example, the expected factor structure of the instrument was largely recovered, self/parent convergent correlations were generally strong , correlations with a measure of the Big Five were largely consistent with expectations, and correlations with a variety of outcome variables .I think that this manuscript provides a great deal of useful information about this rather new instrument, which should be of much value for researchers who study personality in children of middle-school age. I believe it is suitable for publication, but I have several recommendations for revision, all of which I think should be easy to incorporate.1. It would be useful to report frequency distributions for the items of the various criterion outcomes. In many cases it seems likely that most respondents will report a zero level. If this information won’t fit into the existing tables, it could be given in additional tables, perhaps as a supplement.2. Also, some mention of these frequency distributions will be warranted in cases where the distribution is highly skewed and/or has a large fraction of responses at the highest or lowest possible value. In general, this will attenuate correlations with the personality scale scores, which presumably will be roughly normally distributed.3. One difference between the HEXACO-MSI-E and the HEXACO-PI-R is that the HEXACO-MSI-E shows much higher correlations between the scales assessing the C and O dimensions. The authors discuss on page 26 the fact that O correlates with Big Five C in this study, noting the prominence of school-related content in the items of these scales. But the authors could also note the high correlation between C and O within the HEXACO-MSI-E itself, presumably for the same reason.4. The Discussion section seems to be a very detailed recapitulation of the Results; I think it could be greatly shortened.5. This is not really a recommendation for revision, but I did notice that the authors emphasize those correlations that reached the threshold of statistical significance, in some cases counting those correlations. There is nothing wrong with this, but it should be kept in mind that this threshold depends on the sample size as well as the effect size. Some statistically significant correlations are rather small, and some correlations that do not reach statistical significance in a medium-size sample might have population values that are non-trivial in size. I wanted to mention this only because the authors appeared to emphasize the statistical significance threshold; again, though, there is not necessarily any revision needed on this point.6. Points about Information in Tables:The descriptive statistics for the HEXACO-MSI-E scales (self and observer) should be reported somewhere.In correlation and regression tables, perhaps the F-statistic results could be deleted, because the F values don’t tell the reader directly about the size of the effect.For all of the outcome variables for which self-reports and observer reports are available, it would be useful to report the self/observer correlations.Regarding Table 2, the correlations given are based on different samples or subsamples. It would be useful to show (either in the main text or as a supplement) a version of this table based on only those respondents for whom all of these variables were assessed.Also for Table 2, I didn’t understand why the correlations between HEXACO-MSI-E scales in self-report were not reported. I might have misunderstood something, but otherwise these should be added.In Table 2, alpha reliabilities should be reported for the scale scores. Also in relation to Table 2, the column for BFQ-C Openness is missing; it would be number 17.In the title of Table 3, could specify “Self-Report HEXACO-MSI-E”7. Various Minor CorrectionsThe abbreviation “Obs” can be useful when referring to the name of the instrument, but when writing a full sentence it’s better to say “Observer Report”. This point applies in several places throughout the manuscript; one example is on page 2, line 438.In several places the authors use the word “tutor” where I think they mean “guardian” In several places “Consciousness” is used where “Conscientiousness” is intended; this could be fixed with a “find-and-replace”p. 3, line 65: “has showed” should be “has shown”pp. 4-5, line 92-93: “Longitudinal studies showed” should be “Longitudinal studies have shown”p. 5: “scale” is sometimes used in referring to the entire instrument, but “instrument” or “inventory” or “questionnaire” would be betterp. 5, line 117: “provided” should be “provide” p. 6, line 122: “filled up” should be “filled out” (one fills up a cup or bowl but fills out a questionnaire)p. 6, line 127: “BFF” has not yet been declared; it could be given as “Big Five” insteadp. 6, line 143: “internalization and externalization problems” are usually stated as “internalizing and externalizing problems”p. 10, line 233: could specify “low C” for the secondary loading of Dependence on Cp. 14, line 289: could specify as “six self-report HEXACO scales”p. 14, line 294: “the unique” should be “the only one”p. 25, line 455: “homologous” should be “counterpart”p. 25, line 451: “Tielman” should be “Thielmann”p. 27, line 506: “homo- and hetero-trait” should be “mono- and hetero-trait” p. 28, line 516: “more emotional stability” would probably be better as “lower E” or “less emotionality”, because E is not exactly opposite to emotional stability p. 28, line 521: “in teenage age” would sound better as “in the teenage years”p. 29, lines 548-549: should “YOS” and “YRS” be “YOR” and “YSR”, respectively?p. 29, line 560: “we have proved evidence” should be “we have provided evidence”p. 29, line 562: “reconducted” should be “attributed” (I think)Reviewer #2: The present research reports on psychometric properties of the observer report form of HEXACO-MSI-E as well as on further evidence for construct validity of self-report form of the inventory. The HEXACI-MSI-E was shown to have favorable psychometric properties in terms of factor structure, internal consistency reliability, and criterion-related validity. Given the importance of parents/caretakers’ reports in adolescent psychology, developing and validating the observer report form of the HEXACO-MSI-E seems to make an important contribution to the literature.Below, I provide some suggestions that the authors may consider in revising the current manuscript.1. On p. 12, the relationships between the BFQ-C and HEXACO-MSI-E variables are described using the MTMM terms . As the authors mentioned in the discussion section, the FFM variables should not be considered the same as the HEXACO variables. In this sense, it may be incorrect to call some of the correlations involving the BFQ-C variables to be homotrait-heteromethod coefficientst; as such, I would remove BFQ-C variables from the MTMM matrix (Table 2). Relationships between the HEXACO-MSI-E and BFQ-C can be described in another table . This way, the authors might be able to streamline the results presentation on page 12 and the associated discussions on pages 25—27.2. In Table 2, I wasn’t sure why the values above the diagonal for self-reports HEXACO-MSI-E variables are all blank.3. In Table 2, column 17 is missing.4. It would be helpful to mention somewhere in the text  that self-reports of the HEXACO-MSI-E were used in these analyses (as opposed to observer reports). 5. I have some suggestions for Tables 3, 4, 5, 6, and 7, which the authors may or may not accept: (1) I would omit F values and dfs from the tables for the sake of simplicity. (2) I am wondering if it is more straightforward to conduct standard multiple regression analyses instead of stepwise multiple regression analyses. (3) it might be informative to provide zero-order correlations in addition to (or instead of) beta coefficients.6. The discussions involving criterion-related validity (from p. 27) seem to be a bit repetitive to what was stated in the results section, and it might be helpful if the authors could streamline it by discussing some implications of the results.7. I have one suggestions for some items in the Social Self-Esteem scale within the observer report from; e.g., “Others enjoy spending time with her“, “Nobody likes talking to her” and “People like her”. These items are not intended to measure objective levels of the target person’s popularity or attractiveness, but to measure target person’s perception of it. As such, it might be more appropriate to add “S/he believes” or “S/he thinks” to those items.8. The manuscript may benefit from a more thorough proofreading. The followings are what I spotted while reading the manuscript, and they are not meant to be a thorough list of typos.   “shown”Line 65, “showed”�Line 66, “to a lesser extent” may sound better than “to little extent”Line 127, “BFF” may need to be spelled out.   “never had access”?Line 160, “had never access” �   In the literature it is known as”?Line 187, “in literature is known as” �Throughout, “As regard” should be “As regards”   “Thielmann”Line 461, “Tielman” �   “we have provided evidence”?Line 560, “we have proved evidence” �********** <font color="black"> 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.If you choose “no”, your identity will remain anonymous but your review may still be made public.Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. </font> Reviewer #1: NoReviewer #2: No**********https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool,  20 Sep 2023Reviewer #1: Review of PONE-D-23-19901, “Construct and criterion validity of the HEXACO Medium School Inventory Extended (MSI-E)”As its title suggests, this manuscript reports on the validity of a questionnaire for assessing the HEXACO personality characteristics in middle-school students. The research is based on a fairly large sample of adolescents (average age about 12.0 years) for whom self-reports (and in most cases parent reports) were obtained; a variety of outcome variables were also assessed in self-report or (for some of the sample) parent reports.The results generally showed good psychometric properties for the instrument. For example, the expected factor structure of the instrument was largely recovered, self/parent convergent correlations were generally strong , correlations with a measure of the Big Five were largely consistent with expectations, and correlations with a variety of outcome variables .I think that this manuscript provides a great deal of useful information about this rather new instrument, which should be of much value for researchers who study personality in children of middle-school age. I believe it is suitable for publication, but I have several recommendations for revision, all of which I think should be easy to incorporate.A: We thank R1 for his/her words and for the provided suggestions that improved the paper very much. As detailed below, we have followed almost all the suggestions by R1.1. It would be useful to report frequency distributions for the items of the various criterion outcomes. In many cases it seems likely that most respondents will report a zero level. If this information won’t fit into the existing tables, it could be given in additional tables, perhaps as a supplement.A: We thank R1 for this suggestion. We have reported frequency distributions for the items of the various criterion outcomes in additional tables .2. Also, some mention of these frequency distributions will be warranted in cases where the distribution is highly skewed and/or has a large fraction of responses at the highest or lowest possible value. In general, this will attenuate correlations with the personality scale scores, which presumably will be roughly normally distributed.A: We thank R1 for this good point. We have mentioned this aspect in the conclusions of our paper (lines 1120-1123).3. One difference between the HEXACO-MSI-E and the HEXACO-PI-R is that the HEXACO-MSI-E shows much higher correlations between the scales assessing the C and O dimensions. The authors discuss on page 26 the fact that O correlates with Big Five C in this study, noting the prominence of school-related content in the items of these scales. But the authors could also note the high correlation between C and O within the HEXACO-MSI-E itself, presumably for the same reason.A: As we reported in the paper, the BFQ-C Openness is better defined as Intellect because it expresses the capability of succeeding in a school setting . In the HEXACO-MSI-E, Openness to Experience is better defined as the attraction toward beauty of art and nature, inquisitiveness about various domains of knowledge and about ideas that may seem radical or unconventional, while most of the HEXACO-MSI-E items partaking to C refer to effectively organizing and successful doing and monitoring schoolwork and assignments. For this reason, we think that it is difficult to explain the high correlation between C and O within the HEXACO-MSI-E in the same way as we explained the high correlation between C and BFQ-C Openness.4. The Discussion section seems to be a very detailed recapitulation of the Results; I think it could be greatly shortened.A: We thank R1 for this suggestion. We have streamlined the Discussion, avoiding repetitions, and provided more keys to understanding results and their implications, also based on the few existing literature. 5. This is not really a recommendation for revision, but I did notice that the authors emphasize those correlations that reached the threshold of statistical significance, in some cases counting those correlations. There is nothing wrong with this, but it should be kept in mind that this threshold depends on the sample size as well as the effect size. Some statistically significant correlations are rather small, and some correlations that do not reach statistical significance in a medium-size sample might have population values that are non-trivial in size. I wanted to mention this only because the authors appeared to emphasize the statistical significance threshold; again, though, there is not necessarily any revision needed on this point.A: We thank R1 for this observation. In the revised conclusions of our paper (lines 1128-1129) we have highlighted that despite the use of a conservative procedure such as False Discovery Rate correction, the statistical significance threshold may have been affected by the sample size. 6. Points about Information in Tables:The descriptive statistics for the HEXACO-MSI-E scales (self and observer) should be reported somewhere.A: As correctly noted by R1, we did not report descriptive statistics in the first version of the paper. Now, in the Table 1, we have added descriptive statistics for the Obs HEXACO-MSI-E. The descriptive statistics for the self HEXACO-MSI-E are reported in the Table 7 of the already published work of Gnisci et al. (2023).In correlation and regression tables, perhaps the F-statistic results could be deleted, because the F values don’t tell the reader directly about the size of the effect.A: We thank you for this suggestion. We have moved the F-statistic results from regression tables in the manuscript to the tables of frequency distribution in the Supplementary Materials. For all of the outcome variables for which self-reports and observer reports are available, it would be useful to report the self/observer correlations.A: This is a very important information to be provided. Unfortunately, for methodological reasons due to the way we sampled the adolescents and the parents/legal guardians (see Method), we have relatively few participants who had self-report and observer ratings. Normally, they were 124-125 but, in some cases, even 80. Therefore, we decided to provide in the text, for each of the four groups of criteria, the minimum and maximum value of the correlations (and N), providing the detailed tables of correlations in the Supplementary Materials. Regarding Table 2, the correlations given are based on different samples or subsamples. It would be useful to show (either in the main text or as a supplement) a version of this table based on only those respondents for whom all of these variables were assessed.A: We thank R1 for this suggestion. We prefer not to apply it because we have too few respondents, which would make the results unreliable. However, we have realized that we did not clearly report subsample sizes on which we computed correlations. We have added this information in a note of Table 2.Also for Table 2, I didn’t understand why the correlations between HEXACO-MSI-E scales in self-report were not reported. I might have misunderstood something, but otherwise these should be added.A: In the first version of the paper, we decided to not report the correlations between HEXACO-MSI-E scales in self-report because these had already been reported and published in a previous work . However, as also suggested by the Editor and by R2, as well as by R1, in this new version of the paper, we have added these data and specified that they are already published in a previous work (see note 2 of Table 2). In Table 2, alpha reliabilities should be reported for the scale scores. A: We thank R1 for this observation. The alpha coefficients for self HEXACO-MSI-E were just published in Gnisci et al. (2023) while the ones for Obs HEXACO-MSI-E are in the Table 1 of the present paper. As far BFQ-C is concerned, in the paragraph "Big Five Questionnaire-Children" of Measures section, we have now added information on the factorial structure of the BFQ-C and the alphas for each of the BFQ-C factors.Also in relation to Table 2, the column for BFQ-C Openness is missing; it would be number 17.A: We thank R1 for noting this, we have added the column for BFQ-C Openness (column 17) in the Table 2 and the related correlations.In the title of Table 3, could specify “Self-Report HEXACO-MSI-E”A: Done. Thanks.7. Various Minor CorrectionsThe abbreviation “Obs” can be useful when referring to the name of the instrument, but when writing a full sentence it’s better to say “Observer Report”. This point applies in several places throughout the manuscript; one example is on page 2, line 438.A: We thank R1 for this consideration that we have applied in all pertinent places throughout the manuscript.In several places the authors use the word “tutor” where I think they mean “guardian” A: We thank R1 for this specification, we replaced the word “tutor” with “guardian” or “legal guardian”, depending on the case, in all the manuscript.In several places “Consciousness” is used where “Conscientiousness” is intended; this could be fixed with a “find-and-replace”A: Done. Thanks. p. 3, line 65: “has showed” should be “has shown”A: Done.pp. 4-5, line 92-93: “Longitudinal studies showed” should be “Longitudinal studies have shown”A: Done.p. 5: “scale” is sometimes used in referring to the entire instrument, but “instrument” or “inventory” or “questionnaire” would be betterA: We thank R1 for this suggestion. In all the manuscript, we have replaced the term “scale” with “inventory” when referring to entire instrument. p. 5, line 117: “provided” should be “provide” A: Done. Thanks. p. 6, line 122: “filled up” should be “filled out” (one fills up a cup or bowl but fills out a questionnaire)A: Done. p. 6, line 127: “BFF” has not yet been declared; it could be given as “Big Five” insteadA: We have removed and replaced it with “Big Five”. Thanks for this observation. p. 6, line 143: “internalization and externalization problems” are usually stated as “internalizing and externalizing problems”A: We have replaced “internalization and externalization problems” with “internalizing and externalizing problems” in all the manuscript. Thanks. p. 10, line 233: could specify “low C” for the secondary loading of Dependence on CA: Done.p. 14, line 289: could specify as “six self-report HEXACO scales”A: Done.p. 14, line 294: “the unique” should be “the only one”A: Done.p. 25, line 455: “homologous” should be “counterpart”A: Done. p. 25, line 451: “Tielman” should be “Thielmann”A: Done. p. 27, line 506: “homo- and hetero-trait” should be “mono- and hetero-trait” A: Done.p. 28, line 516: “more emotional stability” would probably be better as “lower E” or “less emotionality”, because E is not exactly opposite to emotional stability A: We thank R1 for this suggestion. We have replaced “more emotional stability” with “lower E”. p. 28, line 521: “in teenage age” would sound better as “in the teenage years”A: Done.p. 29, lines 548-549: should “YOS” and “YRS” be “YOR” and “YSR”, respectively?A: We thank R1 for noticing this error. We have replaced “YOS” and “YRS” with “YOR” and “YSR” respectively. p. 29, line 560: “we have proved evidence” should be “we have provided evidence”A: Done. Thanks.p. 29, line 562: “reconducted” should be “attributed” (I think)A: We thank R1 for this suggestion. We have replaced “reconducted” with “attributed”.Reviewer #2: The present research reports on psychometric properties of the observer report form of HEXACO-MSI-E as well as on further evidence for construct validity of self-report form of the inventory. The HEXACI-MSI-E was shown to have favorable psychometric properties in terms of factor structure, internal consistency reliability, and criterion-related validity. Given the importance of parents/caretakers’ reports in adolescent psychology, developing and validating the observer report form of the HEXACO-MSI-E seems to make an important contribution to the literature.Below, I provide some suggestions that the authors may consider in revising the current manuscript.A: We thank R2 for his/her words and for the provided suggestions that improved the paper very much. Based on his/her positive consideration, we have emphasized the importance of having a validated observer report form, especially in child and adolescent psychology, in the definition of aims (lines 139-141) and in the discussions of our paper (lines 956-957). As detailed below, we have followed almost all the suggestions by R2.1. On p. 12, the relationships between the BFQ-C and HEXACO-MSI-E variables are described using the MTMM terms . As the authors mentioned in the discussion section, the FFM variables should not be considered the same as the HEXACO variables. In this sense, it may be incorrect to call some of the correlations involving the BFQ-C variables to be homotrait-heteromethod coefficientst; as such, I would remove BFQ-C variables from the MTMM matrix (Table 2). Relationships between the HEXACO-MSI-E and BFQ-C can be described in another table . This way, the authors might be able to streamline the results presentation on page 12 and the associated discussions on pages 25—27.A: We thank R2 for this point. At the beginning of this study, we were not yet aware of the systematic differences between the BFQ-C and the HEXACO-MSI-E, indeed they emerged later just in the present study. Therefore, we still consider it appropriate to talk about convergent and divergent validity with respect to the BFQ-C but, when describing these results, we abandoned the terminology associated with the MTMM matrix, as R2 suggests. In particular, we specified this aspect in the research questions  and "heterotrait-heteromethod" with the periphrasis "between different traits assessed by different methods" (Lines 354 and 357). Having adopted these solutions, we think Table 2 can remain in the manuscript as it is because it is very informative.2. In Table 2, I wasn’t sure why the values above the diagonal for self-reports HEXACO-MSI-E variables are all blank.A: It is because these data are already reported and published in a previous work. Despite this, in the revised version of paper, following the Editor and R1’s suggestions, we have decided to report the results on correlations of observed score within the self-reports HEXACO-MSI-E specifying in the note 2 of Table 2 that those data have already been published in a previous work and also indicating the reference of the latter.3. In Table 2, column 17 is missing.A: We thank R2 for rising up this point. Also following the Editor and R1’s suggestions, we have added, in the Table 2, the column 17 for BFQ-C Openness.4. It would be helpful to mention somewhere in the text  that self-reports of the HEXACO-MSI-E were used in these analyses (as opposed to observer reports). A: We thank R2 for noticing this missing information. First of all, we confirm that R2 is right; in these analyses, we used the self-reports of the HEXACO-MSI-E. Therefore, following his/her suggestion, we have better specified in different places of the manuscript and also in the captions of Tables 3, 4, 5, 6, 7, and 8 that we used the self-reports of the HEXACO-MSI-E. Specifically, in the text, we specified this aspect in the following paragraphs:- "Data analysis" ;- "Criterion Validity with respect to Self-Report and Observer Criteria" ;- “Criterion Validity with respect to Dimensions Connected to Internalizing and Externalizing Problems of Adolescents in Self-report and Observer Form”;- "Criterion Validity with respect to Features of a Future Job in Self-report and Observer form" .5. I have some suggestions for Tables 3, 4, 5, 6, and 7, which the authors may or may not accept: (1) I would omit F values and dfs from the tables for the sake of simplicity. (2) I am wondering if it is more straightforward to conduct standard multiple regression analyses instead of stepwise multiple regression analyses. (3) it might be informative to provide zero-order correlations in addition to (or instead of) beta coefficients.A: We thank R2 for all the suggestions. We answer below for each point.1) Following R1’s suggestion, we have removed F values and dfs from regression tables and we have reported them in frequency distribution tables of Supplementary Materials (Tables F to O).2) Results are not much different. We ran a standard regression (with beta not corrected for multiple testing) on each of the 92 criteria, as R2 suggests, and out of 166 significant correlations that emerged with stepwise in our study, 145 were confirmed and 11 new emerged. Given these similar patterns, we confirmed the stepwise given that it makes possible comparisons with other studies in literature which used also stepwise method.3) For the sake of simplicity, we have reported the zero-order correlations in Supplementary Materials .6. The discussions involving criterion-related validity (from p. 27) seem to be a bit repetitive to what was stated in the results section, and it might be helpful if the authors could streamline it by discussing some implications of the results.A: We thank R2 for this suggestion. We have streamlined the discussions involving criterion-related validity and, in general, the whole discussions section, avoiding repetitions. We added more interpretations of the results based on the small existing literature and some theoretical and practical implications of the results.7. I have one suggestions for some items in the Social Self-Esteem scale within the observer report from; e.g., “Others enjoy spending time with her“, “Nobody likes talking to her” and “People like her”. These items are not intended to measure objective levels of the target person’s popularity or attractiveness, but to measure target person’s perception of it. As such, it might be more appropriate to add “S/he believes” or “S/he thinks” to those items.A: We thank R2 for this interesting suggestion. In the revised version of this paper, we have included this aspect as a limitation (lines 1165-1171) of our study suggesting the use of this inventory, in future studies, in the light of these tips. To this end, we have also uploaded on our webpage a revised version of Obs HEXACO-MSI-E with a note specifying that, after the validation of the inventory, we realized that three items of the Social Self-Esteem scale would be better expressed in the format that includes "S/he believes".8. The manuscript may benefit from a more thorough proofreading. The followings are what I spotted while reading the manuscript, and they are not meant to be a thorough list of typos.   “shown”Line 65, “showed”�A: Done.Line 66, “to a lesser extent” may sound better than “to little extent”A: Done. Line 127, “BFF” may need to be spelled out.A: We thank R2 for noticing this. Also following R1’s suggestion, we have removed “BFF” and replaced it with “Big Five”.   “never had access”?Line 160, “had never access” �A: Done.   In the literature it is known as”?Line 187, “in literature is known as” �Throughout, “As regard” should be “As regards”A: Done.   “Thielmann”Line 461, “Tielman” �A: Done. Thanks.   “we have provided evidence”?Line 560, “we have proved evidence” �A: Done.________________________________________6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.If you choose “no”, your identity will remain anonymous but your review may still be made public.Do you want your identity to be public for this peer review? 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Please note that Supporting Information files do not need this step.While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, AttachmentResponsesToEditor&Reviewers_R1_DEF.docxSubmitted filename: Click here for additional data file. 28 Sep 2023Construct and criterion validity of the HEXACO Medium School Inventory Extended (MSI-E)PONE-D-23-19901R1Dear Dr. Gnisci,We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.Within one week, you’ll receive an e-mail detailing the required amendments. 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For more information, please contact Kind regards,Frantisek SudzinaAcademic EditorPLOS ONEAdditional Editor Comments :Reviewers' comments: 6 Oct 2023PONE-D-23-19901R1 Construct and criterion validity of the HEXACO Medium School Inventory Extended (MSI-E) Dear Dr. Gnisci:I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. onepress@plos.org.If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact plosone@plos.org. If we can help with anything else, please email us at Thank you for submitting your work to PLOS ONE and supporting open access. 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Scientific Reportshttps://doi.org/10.1038/s41598-023-30574-3, published online 02 March 2023Correction to: The original version of this Article contained errors in Equations 7 and 8.In the original version of this article, the value “55.5” was incorrectly placed as a denominator, wherenow reads:The original Article has been corrected.Additionally, the original PDF version of this Article contained errors in Equation 8, where the minus sign is missing:now reads:This error has now been corrected in the PDF version of the Article; the HTML version was correct from the time of publication.