Simplifying and Characterizing DAGs and Phylogenetic Networks via Least Common Ancestor Constraints

Adjusting for Selection Effects in Epidemiologic Studies

The Directed Acyclic Graph (DAG) is a graph representing causal pathways for informing the conduct of an observational study. The use of DAGs allows transparent communication of a causal model between researchers and can prevent over-adjustment biases when conducting causal inference, permitting greater confidence and transparency in reported causal estimates. In the era of ‘big data’ and increasing number of observational studies, the role of the DAG is becoming more important. Recent best-practice guidance for constructing a DAG with reference to the literature has been published in the ‘Evidence synthesis for constructing DAGs’ (ESC-DAG) protocol. We aimed to assess adherence to these principles for DAGs constructed within perioperative literature. Following registration on the International Prospective Register of Systematic Reviews (PROSPERO) and with adherence to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) reporting framework for systematic reviews, we searched the Excerpta Medica dataBASE (Embase), the Medical Literature Analysis and Retrieval System Online (MEDLINE) and Cochrane databases for perioperative observational research incorporating a DAG. Nineteen studies were included in the final synthesis. No studies demonstrated any evidence of following the mapping stage of the protocol. Fifteen (79%) fulfilled over half of the translation and integration one stages of the protocol. Adherence with one stage did not guarantee fulfilment of the other. Two studies (11%) undertook the integration two stage. Unmeasured variables were handled inconsistently between studies. Only three (16%) studies included unmeasured variables within their DAG and acknowledged their implication within the main text. Overall, DAGs that were constructed for use in perioperative observational literature did not consistently adhere to best practice, potentially limiting the benefits of subsequent causal inference. Further work should focus on exploring reasons for this deviation and increasing methodological transparency around DAG construction.

A method for computing the likelihood of a set of sequences assuming a phylogenetic network as an evolutionary hypothesis is presented. The approach applies directed graphical models to sequence evolution on networks and is a natural generalization of earlier work by Felsenstein on evolutionary trees, including it as a special case. The likelihood computation involves several steps. First, the phylogenetic network is rooted to form a directed acyclic graph (DAG). Then, applying standard models for nucleotide/amino acid substitution, the DAG is converted into a Bayesian network from which the joint probability distribution involving all nodes of the network can be directly read. The joint probability is explicitly dependent on branch lengths and on recombination parameters (prior probability of a parent sequence). The likelihood of the data assuming no knowledge of hidden nodes is obtained by marginalization, i.e., by summing over all combinations of unknown states. As the number of terms increases exponentially with the number of hidden nodes, a Markov chain Monte Carlo procedure (Gibbs sampling) is used to accurately approximate the likelihood by summing over the most important states only. Investigating a human T-cell lymphotropic virus (HTLV) data set and optimizing both branch lengths and recombination parameters, we find that the likelihood of a corresponding phylogenetic network outperforms a set of competing evolutionary trees. In general, except for the case of a tree, the likelihood of a network will be dependent on the choice of the root, even if a reversible model of substitution is applied. Thus, the method also provides a way in which to root a phylogenetic network by choosing a node that produces a most likely network.

Skeleton estimation of directed acyclic graphs using partial least squares from correlated data

Phylogenetic networks model the evolutionary history of sets of organisms when events such as hybrid speciation and horizontal gene transfer occur. In spite of their widely acknowledged importance in evolutionary biology, phylogenetic networks have so far been studied mostly for specific data sets. We present a general definition of phylogenetic networks in terms of directed acyclic graphs (DAGs) and a set of conditions. Further, we distinguish between model networks and reconstructible ones and characterize the effect of extinction and taxon sampling on the reconstructibility of the network. Simulation studies are a standard technique for assessing the performance of phylogenetic methods. A main step in such studies entails quantifying the topological error between the model and inferred phylogenies. While many measures of tree topological accuracy have been proposed, none exist for phylogenetic networks. Previously, we proposed the first such measure, which applied only to a restricted class of networks. In this paper, we extend that measure to apply to all networks, and prove that it is a metric on the space of phylogenetic networks. Our results allow for the systematic study of existing network methods, and for the design of new accurate ones.

Understanding causal mechanisms among variables is critical to efficient management of complex biological systems such as animal agriculture production. The increasing availability of data from commercial livestock operations offers unique opportunities for attaining causal insight, despite the inherently observational nature of these data. Causal claims based on observational data are substantiated by recent theoretical and methodological developments in the rapidly evolving field of causal inference. Thus, the objectives of this review are as follows: 1) to introduce a unifying conceptual framework for investigating causal effects from observational data in livestock, 2) to illustrate its implementation in the context of the animal sciences, and 3) to discuss opportunities and challenges associated with this framework. Foundational to the proposed conceptual framework are graphical objects known as directed acyclic graphs (DAGs). As mathematical constructs and practical tools, DAGs encode putative structural mechanisms underlying causal models together with their probabilistic implications. The process of DAG elicitation and causal identification is central to any causal claims based on observational data. We further discuss necessary causal assumptions and associated limitations to causal inference. Last, we provide practical recommendations to facilitate implementation of causal inference from observational data in the context of the animal sciences.

One of the major challenges for epidemiologists is to understand causal relationships between risk factors and health outcomes by analysing data from observational studies. Epidemiologists know too well that correlations, such as those in regression analysis, rarely mean causation, and it would be very desirable if there is a methodology for observational studies, analogous to randomization in experimental studies, which can discover causes and effects amongst variables or at least confirm or refute the proposed causal relationships. Randomized controlled trials (RCTs) are certainly the ‘gold standard’ of establishing causes and effects, but quite often it is either unethical or unfeasible to conduct RCTs to test causal relations in epidemiological research. Epidemiologists need a methodology, which is sort of a combination of the directed acyclic graphs (DAGs) for conceptual construction of causal models and regression analysis for testing those models. It is therefore surprising that structural equation modelling (SEM) has not been so frequently used in epidemiology as in the social sciences, given that both epidemiologists and social scientists want to delineate causes and effects from observational data. The difference between DAGs and path diagrams in SEM is almost trivial to epidemiologists: the path between two variables can only have one direction in DAGs. 1,2 An individual path in SEM is tested in the same way the regression coefficient is in regression analysis, and model fit indices provided by SEM software packages help the analysts to assess the adequacy of the proposed causal model compared with the observed associations in the sample data. 3,4 Why then is SEM still under-utilized in epidemiology? This is a question posed by a commentary 5 in this journal a few years ago. The answers cited included unfamiliar terminology (SEM theory is formulated in Greek), restriction in the assumptions of variables (outcome variables need to be continuous), difficulties in testing interaction and non-linear relationship (it can be quite tedious to set up SEM models to do these), and equivalent models (two different causal models imply the same correlation structure and consequently, it is impossible to tell which is better). 6 Recent advances in SEM theory and software development has resolved some of these issues: new estimation methods do not require the strict assumption of multivariate normality, 7 and the outcome variables can be binary, ordinal or counts. 7–9 The basic rationale behind SEM is rather simple: multiple linear equations are used to specify causal relationship between variables some of which are manifest variables (i.e. observed and collected by the researchers), whilst others are latent variables (i.e. derived from the observed variables by specifying their relations using equations), such as those in factor analysis. The multiple equations in each causal model entail a certain structure of correlational relationships between observed variables which is usually given as a correlation (or covariance) matrix . The estimation procedure is to minimize the difference between and the observed correlation/covariance matrix S formulated by a likelihood function. The 2 -test is then used to evaluate the difference between these two matrices by taking into account the number of the estimated parameters in the proposed model. When the 2 -value is large (i.e. the difference between the two matrices is large) relative to the model’s degree of freedom, the proposed model is rejected, i.e. something in the causal relationships specified by the proposed model is not quite right and require a second thought. When the 2 -value is small, we fail to reject the model or tentatively accept the model as adequate. Because of the possibility of equivalent models, our model may still be wrong in terms of the causal relations amongst variables but happen to entail the same correlation structure (the same S) of the ‘true’ model. So SEM seems to be an endeavour to search for the truth by approximation, and in this aspect, doing SEM is quite similar to the process of ‘Conjectures and Refutations’ as described by the great philosopher

We consider a binary response which is potentially affected by a set of continuous variables. Of special interest is the causal effect on the response due to an intervention on a specific variable. The latter can be meaningfully determined on the basis of observational data through suitable assumptions on the data generating mechanism. In particular we assume that the joint distribution obeys the conditional independencies (Markov properties) inherent in a Directed Acyclic Graph (DAG), and the DAG is given a causal interpretation through the notion of interventional distribution. We propose a DAG-probit model where the response is generated by discretization through a random threshold of a continuous latent variable and the latter, jointly with the remaining continuous variables, has a distribution belonging to a zero-mean Gaussian model whose covariance matrix is constrained to satisfy the Markov properties of the DAG; the latter is assigned a DAG-Wishart prior through the corresponding Cholesky parameters. Our model leads to a natural definition of causal effect conditionally on a given DAG. Since the DAG which generates the observations is unknown, we present an efficient MCMC algorithm whose target is the posterior distribution on the space of DAGs, the Cholesky parameters of the concentration matrix, and the threshold linking the response to the latent. Our end result is a Bayesian Model Averaging estimate of the causal effect which incorporates parameter, as well as model, uncertainty. The methodology is assessed using simulation experiments and applied to a gene expression data set originating from breast cancer stem cells.

Background/Aim: Identifying confounders based on causal directed acyclic graphs (DAGs) is usually based solely on expert knowledge, although epidemiological studies provide observational data. We applied data-based structure learning algorithms for the specific purpose of identifying confounders, investigating the research question whether particulate matter with an aerodynamic diameter ≤2.5 µg/m3 (PM2.5) affects progression of atherosclerosis, measured as yearly change in carotid intima media thickness (ΔcIMT). Methods: We used 5-year follow-up data of the population-based Heinz Nixdorf Recall Study. Due to computational limitations, potentially relevant variables were restricted to continuously measured data and hence included age, BMI, smoking, blood pressure, HbA1c, LDL/HDL, education, income, neighborhood unemployment (nSES) rate, physical activity and alcohol consumption. Prior knowledge was based on causal chain assumptions regarding major causes of ischaemic heart disease1, and translated to prohibited arcs, i.e. given conditions (e.g. age) are unaffected by lifestyle variables (e.g. smoking) and lifestyle variables are unaffected by subclinical medical determinants (e.g. laboratory measures). We applied a revised structure learning algorithm to learn a causal DAG, including the score-based hill-climbing algorithm, a bootstrap-check to validate the conditional independence structure between variables, and a clustering of minimal sufficient adjustment set (MSAS)-equivalent subsets within the Markov equivalence-class. Results: Based on 2279 participants, the revised structure learning resulted in one MSAS-equivalent cluster, including one DAG of 59 unambiguous arcs. The only biasing path from PM2.5 to ΔcIMT emanated from nSES. Suggested adjustment consisted of four confounder-equivalent MSAS ({nSES}; {age, physical activity, blood pressure}; {age, alcohol consumption, BMI, physical activity, smoking}; {age, BMI, education, income, physical activity, smoking}), yielding an identical exposure effect estimate of 0.002 mm (95%-CI: -0.002, 0.007) increase in cIMT per year per 5 µg/m3 increase in PM2.5. Conclusions: Data-based structure learning returned one unambiguous DAG widely matching general believes. Identical exposure effect estimates of confounder-equivalent MSAS underline a stable (causal) structure.

The objective of causal discovery is to uncover the causal relationships among natural phenomena or human behaviors, thus establishing the basis for subsequent prediction and inference. Traditional ways to reveal the causal structure between variables, such as interventions or random trials, are often deemed costly or impractical. With the diversity of means of data acquisition and abundant data, approaches that directly learn causal structure from observational data are attracting more interest in academia. Unfortunately, learning a directed acyclic graph (DAG) from samples of multiple variables has been proven to be a challenging problem. Recent proposals leverage the continuous optimization method to discover the structure of the target DAG from observational data using an acyclic constraint function. Although it has an elegant mathematical form, its optimization process is prone to numerical explosion or the disappearance of high-order gradients of constraint conditions, making the convergence conditions unstable and the result deviating from the true structure. To tackle these issues, we first combine the existing work to give a series of algebraic equivalent conditions for a DAG and its corresponding brief proofs, then leverage the properties of the M-matrix to derive a new acyclic constraint function. Based on the utilization of this function and linear structural equation models, we propose a novel causal discovery algorithm called LeCaSiM, which employs continuous optimization. Our algorithm exploits the spectral radius of the adjacency matrix to dynamically adjust the coefficients of the matrix polynomial, effectively solving the above problems. We conduct extensive experiments on both synthetic and real-world datasets, the results show our algorithm outperforms the existing algorithms on both computation time and accuracy under the same settings.

BackgroundTo estimate the causal relationship between two variables it is important to first establish the potential causal pathways and to consider causes of bias. Omission of key confounders from the...

This paper concerns the assessment of direct causal effects from a combination of: (i) non-experimental data, and (ii) qualitative domain knowledge. Domain knowledge is encoded in the form of a directed acyclic graph (DAG), in which all interactions are assumed linear, and some variables are presumed to be unobserved. The paper establishes a sufficient criterion for the identifiability of all causal effects in such models as well as a procedure for estimating the causal effects from the observed covariance matrix.

Failure to account for time‐dependent treatment use when developing a prognostic model can result in biased future predictions. We reviewed currently available methods to account for treatment use when developing a prognostic model. First, we defined the estimands targeted by each method and examined their mechanisms of action with directed acyclic graphs (DAGs). Next, methods were implemented in data from 1,906 patients; 325 received selective β‐blockers (SBBs) during follow‐up. We demonstrated seven Cox regression modeling strategies: (a) ignoring SBB treatment; (b) excluding SBB users or (c) censoring them when treated; (d) inverse probability of treatment weighting after censoring (IPCW), including SBB treatment as (e) a binary or (f) a time‐dependent covariate; and (g) marginal structural modeling (MSM). Using DAGs, we demonstrated IPCW and MSM have the best properties and target a similar estimand. In the case study, compared to (a), approaches (b) and (e) provided predictions that were 1% and 2% higher on average. Performance (c‐statistic, Brier score, calibration slope) varied minimally between approaches. Our review of methods confirmed that ignoring treatment is theoretically inferior, but differences between the prediction models obtained using different methods can be modest in practice. Future simulation studies and applications are needed to assess the value of applying IPCW or MSM to adjust for treatments in different treatment and disease settings.

Article Figures and data Abstract Editor's evaluation Introduction Materials and methods Results Discussion Data availability References Decision letter Author response Article and author information Metrics Abstract Background: The risk of adult onset cardiovascular and metabolic (cardiometabolic) disease accrues from early life. Infection is ubiquitous in infancy and induces inflammation, a key cardiometabolic risk factor, but the relationship between infection, inflammation, and metabolic profiles in early childhood remains unexplored. We investigated relationships between infection and plasma metabolomic and lipidomic profiles at age 6 and 12 months, and mediation of these associations by inflammation. Methods: Matched infection, metabolomics, and lipidomics data were generated from 555 infants in a pre-birth longitudinal cohort. Infection data from birth to 12 months were parent-reported (total infections at age 1, 3, 6, 9, and 12 months), inflammation markers (high-sensitivity C-reactive protein [hsCRP]; glycoprotein acetyls [GlycA]) were quantified at 12 months. Metabolic profiles were 12-month plasma nuclear magnetic resonance metabolomics (228 metabolites) and liquid chromatography/mass spectrometry lipidomics (776 lipids). Associations were evaluated with multivariable linear regression models. In secondary analyses, corresponding inflammation and metabolic data from birth (serum) and 6-month (plasma) time points were used. Results: At 12 months, more frequent infant infections were associated with adverse metabolomic (elevated inflammation markers, triglycerides and phenylalanine, and lower high-density lipoprotein [HDL] cholesterol and apolipoprotein A1) and lipidomic profiles (elevated phosphatidylethanolamines and lower trihexosylceramides, dehydrocholesteryl esters, and plasmalogens). Similar, more marked, profiles were observed with higher GlycA, but not hsCRP. GlycA mediated a substantial proportion of the relationship between infection and metabolome/lipidome, with hsCRP generally mediating a lower proportion. Analogous relationships were observed between infection and 6-month inflammation, HDL cholesterol, and apolipoprotein A1. Conclusions: Infants with a greater infection burden in the first year of life had proinflammatory and proatherogenic plasma metabolomic/lipidomic profiles at 12 months of age that in adults are indicative of heightened risk of cardiovascular disease, obesity, and type 2 diabetes. These findings suggest potentially modifiable pathways linking early life infection and inflammation with subsequent cardiometabolic risk. Funding: The establishment work and infrastructure for the BIS was provided by the Murdoch Children's Research Institute (MCRI), Deakin University, and Barwon Health. Subsequent funding was secured from National Health and Medical Research Council of Australia (NHMRC), The Shepherd Foundation, The Jack Brockhoff Foundation, the Scobie & Claire McKinnon Trust, the Shane O'Brien Memorial Asthma Foundation, the Our Women's Our Children's Fund Raising Committee Barwon Health, the Rotary Club of Geelong, the Minderoo Foundation, the Ilhan Food Allergy Foundation, GMHBA, Vanguard Investments Australia Ltd, and the Percy Baxter Charitable Trust, Perpetual Trustees. In-kind support was provided by the Cotton On Foundation and CreativeForce. The study sponsors were not involved in the collection, analysis, and interpretation of data; writing of the report; or the decision to submit the report for publication. Research at MCRI is supported by the Victorian Government's Operational Infrastructure Support Program. This work was also supported by NHMRC Senior Research Fellowships to ALP (1008396); DB (1064629); and RS (1045161) , NHMRC Investigator Grants to ALP (1110200) and DB (1175744), NHMRC-A*STAR project grant (1149047). TM is supported by an MCRI ECR Fellowship. SB is supported by the Dutch Research Council (452173113). Editor's evaluation This paper provides data from a population-based cohort study on early life infection and proinflammatory, atherogenic metabolomic and lipidomic profiles at 12 months of age. The authors generated matched infection, metabolomics and lipidomics data from 555 infants in a pre-birth longitudinal cohort and they showed that frequent infant infections are associated with adverse metabolomic and lipidomic profiles. They also report that similar profiles are noted with higher glycoprotein acetyls (GlycA), but not hsCRP. The paper is interesting and assesses the role of infection and markers of inflammation on lipid and metabolic profile of patients. It provides a comprehensive analysis of lipids and metabolites in infants in response to infection. https://doi.org/10.7554/eLife.75170.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Infectious diseases are ubiquitous in infancy and childhood, with potential long-term impacts on health across the life course. Infection has been recognised as a potential contributor to atherosclerotic cardiovascular disease (CVD), one of the leading causes of adult morbidity and mortality, since the 19th century (Nieto, 1998). More recent adult studies link previous infection with long-term risks of disease (Bergh et al., 2017; Cowan et al., 2018; Wang et al., 2017). The mechanisms are largely unknown, but likely include immune activation and heightened inflammation (Shah, 2019), which are pathways central to CVD pathogenesis (Donath et al., 2019b; Ferrucci and Fabbri, 2018) and therefore offer potentially druggable targets in high-risk individuals (Donath et al., 2019a; Ridker et al., 2017). High-sensitivity C-reactive protein (hsCRP) has been extensively used as a marker of chronic inflammation in adult studies but is an acute phase reactant in children and may not reflect chronic inflammation in early life. Glycoprotein acetyls (GlycA) is a nuclear magnetic resonance (NMR) composite measure that is suggested to better reflect cumulative, chronic inflammation (Connelly et al., 2017). GlycA is an emerging biomarker for cardiometabolic risk (Connelly et al., 2017) that outperforms hsCRP as a predictor of CVD events and mortality (Akinkuolie et al., 2014; Duprez et al., 2016), and of infection-related morbidity and mortality (Ritchie et al., 2015). For example, in the Multi-Ethnic Study of Atherosclerosis (n = 6523), higher GlycA was associated with increased risk of incidence CVD and death, even after adjustment for hsCRP and other inflammatory markers. Conversely, prediction of these outcomes by hsCRP attenuated to null after mutual adjustment (Duprez et al., 2016). Cardiovascular and metabolic (cardiometabolic) disease pathogenesis begins in early life and accrues across the life course (Nakashima et al., 2008). Infections occur disproportionally in early childhood (Cromer et al., 2014; Grüber et al., 2008; Troeger et al., 2018; Tsagarakis et al., 2018), and there is a dose-response relationship between childhood infections, adverse cardiometabolic phenotypes (Burgner et al., 2015c), and CVD events (Burgner et al., 2015a) in adulthood. Infection is linked to proatherogenic metabolic perturbations in later childhood and adulthood (Feingold and Grunfeld, 2019; Khovidhunkit et al., 2000), including higher triglycerides and oxidised low-density lipoprotein (LDL), and lower high-density lipoprotein (HDL) cholesterol and apolipoprotein A1 (ApoA1) (Liuba et al., 2003; Pesonen et al., 1993), and to acute and chronic inflammation (Burgner et al., 2015b; Ritchie et al., 2015), but little is known about these relationships in early life, when most infections occur. We therefore aimed to characterise metabolomic and lipidomic profiles at 6 and 12 months of age and their relationship to infection burden during the first year of life. We also investigated the extent to which inflammation mediated the relationship between infection burden and metabolomic and lipidomic differences. Materials and methods Study cohort Request a detailed protocol This study used available data from 555 mother-infant dyads in the Barwon Infant Study (BIS), a population-based pre-birth longitudinal cohort (n = 1074 mother-infant dyads). The cohort details and inclusion/exclusion criteria have been detailed elsewhere (Vuillermin et al., 2015); in brief, mothers were eligible if they were residents of the Barwon region in south-east Australia and planned to give birth at the local public or private hospital. Mothers were recruited at approximately 15 weeks' gestation and provided informed consent. They were excluded if they were not a permanent Australian resident, aged <18 years, required an interpreter to complete questionnaires, or had previously participated in BIS. Infants were excluded if they were very preterm (<32 completed weeks gestation) or had a serious illness or major congenital malformation identified during the first few days of life. Ethics approval was granted by the Barwon Health Human Research Ethics Committee (HREC 10/24). Parent-reported infections Request a detailed protocol At the 4-week, 3-month, 6-month, 9-month, and 12-month time points following birth, mothers were asked to report each episode of infant illness or infection since the previous time point using standardised online questionnaires. The number of parent-reported infections from birth to 12 months was defined as the total number of respiratory tract infections, gastroenteritis, conjunctivitis, and acute otitis media episodes. In secondary analyses, numbers of parent-reported infections from birth to 6 months and from 6 to 12 months were considered. It was not possible to identify the proportion of parent-reported infections that lead to health service utilisation (Rowland et al., 2021). Other maternal and infant measures Request a detailed protocol Questionnaires during pregnancy and at birth were used to collect self-reported data on maternal age, household income, maternal education, and prenatal smoking (considered here as a dichotomous any/none exposure). Residential postcode was used to determine neighbourhood disadvantage using the Index of Relative Socio-Economic Disadvantage (IRSD) from the 2011 Socio-Economic Indexes for Areas (SEIFA) (Pink, 2013), with a lower score corresponding to greater socioeconomic disadvantage. Pre-eclampsia (based on International Association of Diabetes and Pregnancy Study Groups criteria; Tranquilli et al., 2014) and gestational diabetes (based on International Society for the Study of Hypertension in Pregnancy criteria; Nankervis et al., 2013) diagnoses were extracted from hospital records. Infant gestational age, birth weight, and mode of delivery (categorised as vaginal, planned caesarean section, or unplanned caesarean section delivery) were collected from birth records, and the age- and sex-standardised birth weight z-score was calculated using the 2009 revised British United Kingdom World Health Organisation (UK-WHO) growth charts (Cole et al., 2011). Postnatal smoking data was collected from questionnaire data, with mothers asked the average number of hours each day someone smoked near or in the same room as the child (Gray et al., 2019). This was dichotomised as any postnatal smoke exposure if >0 hr reported at any time point up to 12 months of age, or no postnatal smoke exposure. Breastfeeding duration up to 12 months of age was collected from maternal questionnaire data. As most evidence for the protective effect of breastfeeding on early life infection is from comparisons between any breastfeeding and no breastfeeding (Victora et al., 2016), and in light of previous evidence in BIS for an association between even a short duration of breastfeeding and lower odds of infection in early infancy (Rowland et al., 2021), we first looked at breastfeeding as a binary (any/none) measure in models (presented in the main text). As most infants (98.2%) were breastfed to some extent, and it is unknown the degree to which breastfeeding, and the timing of breastfeeding, might affect infant metabolomics and lipidomics, we also considered duration of breastfeeding as a continuous variable for sensitivity analyses. Metabolomic and lipidomic profiling Request a detailed protocol Venous peripheral blood was collected from infants at the 6- and 12-month time points in sodium heparin and generally processed within 4 hr, with a minority (197 of 555) of 12-month samples processed after 4 hr (median time for those 197 samples = 19.9 hr, inter-quartile range [IQR] [18.7, 21.4]). The time interval between collection and post-processing storage of samples was included as a covariate in analyses. Due to the bimodal distribution of 12-month sample collection times as samples were either processed same day of collection or the following day, sensitivity analysis excluding participants with a plasma storage time greater than 4 hr (197 out of 555 infants, predominantly processed the following day) was performed, as described in the Statistical analysis section below. Plasma was stored at –80°C, and aliquots were shipped on dry ice to Nightingale Health (Helsinki, Finland) for NMR metabolomic quantification and Baker IDI (Melbourne, Australia) for liquid chromatography/mass spectrometry (LC/MS) lipidomic quantification, as described below. For secondary analyses investigating possible 'reverse causality', that is, whether metabolomic or lipidomic profile at birth was associated with number of parent-reported infections from birth to 6 months of age, metabolomics and lipidomics data using the same platforms from venous cord blood collected at birth, as previously described (Burugupalli et al., 2022; Mansell et al., 2021), was used. The NMR-based metabolomics platform has been described in detail (Kettunen et al., 2016; Soininen et al., 2015), and quantified a broad range of metabolic measures including lipoprotein size subclasses, triglycerides, cholesterols, fatty acids, amino acids, ketone bodies, glycolysis metabolites, and GlycA. In brief, plasma samples were mixed with a sodium phosphate buffer prior to NMR measurements with Bruker AVANCE III 500 MHz and Bruker AVANCE III HD 600 MHz spectrometers (Bruker, Billerica, MA). Samples were kept at 6°C using the SampleJet sample changer (Bruker) to prevent degradation. After initial measurements, samples went through a multiple-step lipid extraction procedure using saturated sodium chloride solution, methanol, dichloromethane, and deuterochloroform. The lipid extracts were then analysed using the 600 MHz instrument (Soininen et al., 2015). The utility of this platform in epidemiological research has been detailed elsewhere (Würtz et al., 2017). Using the Nightingale Health 2016 bioinformatics protocol, 228 metabolomic measures were generated for the 12-month samples. From a subset of participants, replicate samples were quantified, and these showed a low percentage coefficient of variation (<10%). Subsequently, the Nightingale Health 2020 bioinformatics protocol was used to generate 250 metabolomic measures for the 6-month samples, with 224 of these measures also present in the 12-month data. As a large proportion of the NMR metabolomic measures are ratios and are strongly correlated with each other in children and adults (Ellul et al., 2019), an informative subset of 51 measures that captured the majority of variation in the metabolomic dataset, primarily absolute metabolite concentrations, were included in analysis presented in the main text. Analyses for excluded metabolomic measures are presented as supplementary data. To complement GlycA as a measure of inflammation, hsCRP was also quantified in 6- and 12-month plasma using enzyme-linked immunosorbent assay (ELISA) (R&D Systems, Minneapolis, MN, cat. no. DY1707), as per the manufacturer's instruction. The details of the high-performance LC/MS lipidomics platform have been described elsewhere (Beyene et al., 2020). In addition, we used medronic acid to passivate the LC/MS system to avoid peak tailing for acidic phospholipids (Hsiao et al., 2018). In brief, this platform quantified 776 lipid features in 36 lipid classes, including sphingolipids, glycerophospholipids, sterols, glycerolipids, and fatty acyls. Analysis was performed on an Agilent 6490 QQQ mass spectrometer with an Agilent 1290 series high-performance liquid chromatography system and two ZORBAX eclipse plus C18 column (2.1 × 100 mm 1.8 mm) (Agilent, Santa Clara, CA) with the thermostat set at 45°C. Mass spectrometry analysis was performed in both positive and negative ion mode with dynamic scheduled multiple reaction monitoring. Quantification of lipid species was determined by comparison to the relevant internal standard. Lipid class total concentrations were calculated as the sum of individual lipid species concentrations, except in the case of triacylglycerols (TGs) and alkyl-diacylglycerols, where we measured both neutral loss and single ion monitoring (SIM) peaks, and subsequently used the SIM species concentrations for summation purposes. Statistical analysis Request a detailed protocol Analyses were performed in R (version 3.6.3) (R Development Core Team, 2018). All metabolomic and lipidomic measures had their lowest observed non-zero value (considered the lower limit of detection) added to their value before they were natural log-transformed and scaled to a standard distribution (standard deviation units). Pearson's correlations were calculated for number of infections from birth to 12 months with 12-month GlycA and hsCRP. The estimated effect of number of parent-reported infections from birth to 12 months as an exposure on 12-month metabolomic and lipidomic profile was investigated using linear regression models for each metabolomic/lipidomic measure. Standard errors were used to calculate 95% confidence intervals for estimated effects. All models were adjusted for infant sex, exact age at 12-month time point, birth weight z-score, gestational age, maternal household income, exposure to maternal smoking during pregnancy, breastfeeding, and time from collection to storage for the plasma sample. Linear regression models adjusted for the same covariates were used to investigate 12-month GlycA and hsCRP as exposures and each metabolomic/lipidomic measure as an outcome. Two-tailed p-values were adjusted for multiple comparisons within each dataset (NMR metabolomics, LC/MS lipidomic species, and LC/MS lipidomic classes) using the Benjamini-Hochberg method (Benjamini and Hochberg, 1995). To investigate the robustness of the estimates, mean model coefficients and bias-corrected accelerated percentile bootstrap confidence intervals were calculated from nonparametric bootstrap resampling (1000 iterations) using the 'boot' package (Davison and Hinkley, 1997) (version 1.3–25) in R, included in Source Data. The assumption of linearity was investigated post hoc using plots of residual values for the top 10 metabolomic and lipidomic differences (ranked by p-value) for each of number of infections, GlycA, and hsCRP. To further investigate the other sources of potential confounding and variation that could affect findings from the primary models, several sensitivity analyses were performed. These were: (i) additional adjustment of the primary model for postnatal smoking exposure, gestational diabetes, and pre-eclampsia; (ii) analyses excluding twins (five infants); (iii) analyses excluding infants with hsCRP >5 mg/L (24 infants) as a marker of acute infection (Lemiengre et al., 2018; et al., analyses excluding plasma samples with hr from collection to storage (197 and analyses for breastfeeding duration of any Analyses of the primary models adjusted for measures of socioeconomic or maternal of household were also considered. These models are presented as supplementary plots analyses to investigate 6-month of infections were performed. The birth to 6-month models investigated the relationship between infections up to 6 months of age, 6-month inflammation, and 6-month metabolomic/lipidomic The 6- to 12-month models investigated the relationship of infections between 6 and 12 months of age, 12-month inflammation, and 12-month metabolomic/lipidomic with adjustment for the corresponding 6-month For secondary analyses to investigate regression models et al., were either for (i) each metabolomic/lipidomic measure from cord blood at birth as exposure and total number of infections from birth to 6 months of age as or (ii) each 6-month metabolomic/lipidomic measures as exposure and total number of infections from 6 to 12 months of age as the with adjustment for number of infections from birth to 6 months of age. All secondary models were adjusted for the same covariates as the primary models described that is, infant sex, gestational age, exact age 6- or 12-month time birth weight z-score, maternal household income, exposure to maternal smoking during pregnancy, and sample time between collection and post-processing with birth metabolomic/lipidomic measures as the exposure were adjusted for mode of which is associated with differences across NMR metabolomic measures in cord in this cohort et al., 2021). To investigate the potential role of inflammation in mediating associations between infection and metabolic the package et al., 2017) (version in R was then used in a to the natural effect mediated by and natural effect by of infection. 12-month GlycA or hsCRP were considered as for the effect of number of infections from birth to 12 months of age on metabolomic and lipidomic differences at 12 months, for metabolomic and lipidomic measures associated with number of infections with an adjusted from the linear regression models described mediation was calculated as the estimated natural effect by the total effect effect plus natural of the mediation model is in for model investigated in this The natural effect by glycoprotein acetyls or C-reactive protein and natural effect mediated by of parent-reported infections on metabolomic and lipidomic measures were with adjustment for were considered to for associations to exposure to and to As models investigating described suggested that 6-month GlycA or lipidomic measures may number of infections from 6 to 12 months of age, we performed mediation analyses to the percentage mediation of 12-month GlycA and hsCRP for the effect of number of infections from 6 to 12 months of age on 12-month metabolomic and lipidomic with additional adjustment for the corresponding 6-month measures of infections from birth to 6 months of age, 6-month GlycA or and the 6-month metabolomic/lipidomic Results The for the 555 infants included in this study is in and the cohort for these infants are in The number of total parent-reported infections from birth to 12 months of age was = infections from birth to 6 months of age was 2 and infections from 6 to 12 months of age was 12-month hsCRP and GlycA were mg/L and number of parent-reported infections between birth and 12 months of age was more strongly correlated with 12-month GlycA = than hsCRP = The of metabolomic and lipidomic measures at each time point for the cohort are in and (n = age at delivery smoking during pregnancy diabetes (n = (n = household to to to to (n = than year 10 of 10 of or 12 of or of birth caesarean caesarean postnatal smoke exposure to 12 months (n = of duration to weeks age weight weight at 6-month time point at 6 months z-score at 6 at 12-month time point at 12 months z-score at 12 of parent-reported infections from birth to 12 Infections from birth to 6 Infection from 6 to 12 GlycA at 6 months at 6 months at 12 months at 12 months All = 555 infants had complete covariate data for primary models, data of secondary exposures is to the relevant measure. lower value greater socioeconomic disadvantage on 2 of Barwon Infant Study participants included this study participants had complete infection data from time points between birth and 12 months of age, and 12-month plasma nuclear magnetic resonance (NMR) metabolomics data. included participants (n = out of 555) had 12-month plasma liquid chromatography/mass spectrometry (LC/MS) lipidomics data. Infection and inflammation burden and plasma NMR metabolomic profile at 12 months was evidence for higher number of infections with higher inflammatory markers per infection, 95% and hsCRP lower HDL to to and to cholesterols, lower to lower to higher and to a extent with higher triglycerides and lower to In models with GlycA as the marker of inflammation metabolomic differences observed for higher GlycA were largely similar but more those for parent-reported including HDL per in GlycA to higher and lipoprotein to higher apolipoprotein higher total fatty higher total triglycerides and amino and lower to metabolites lower to and lower to were generally similar to standard regression for models. effect were generally similar across most sensitivity analyses excluding samples with time greater than 4 hr the of estimated of parent-reported infections on HDL and hsCRP was associated with lower HDL cholesterol per in to to and to and higher to as observed for GlycA, and with lower of most other amino and to and higher triglycerides with 2 in 12-month plasma nuclear magnetic resonance (NMR) metabolomic measures for each in parent-reported infection to 12 and for each in 12-month glycoprotein acetyls (GlycA) (n = plots of the estimated 12-month metabolomic differences for each additional parent-reported infection from birth to 12 months or 12-month GlycA from adjusted linear regression models, and the

Evolutionary trees have the assumption that evolution and phylogeny can be represented in a strictly bifurcating manner. Firmly speaking, from one ancestral taxon, two descendant taxa emerge. Nevertheless, hybridization, recombination and horizontal gene transfer is in conflict with this straightforward concept. In such cases, evolutionary lines do not only separate from each other, but have the possibility of melting again and are called reticulations. Consequently, networks can represent evolutionary events more realistically than phylogenetic trees. Networks can display alternative topologies and co-existence of ancestors and descendants, which are otherwise not obvious when a comparison is done on several single trees or a consensus tree. Therefore, networks have the ability to visualize the conflicting information in a given data set. Moreover, the distribution, frequencies and arrangement of haplotypes in populations can reveal the phylogenetic histories of the taxa, regarding predictions from the coalescent theory. This review aims to: (1) give a brief comparison between phylogenetic trees and networks, (2) provide the overall concept of the coalescent theory, (3) clarify how phylogenetic networks can be used to display conflict data and evaluate phylogenetic histories, and (4) offer a useful starting point and guide for sequence analysis, with the aim to discover population dynamics. Key words: Phylogenetic networks, reticulation, coalescent theory, population history, character conflict.