A randomization-based causal inference framework for uncovering environmental exposure effects on human gut microbiota.

Statistical analysis of microbial genomic data within epidemiological cohort studies holds the promise to assess the influence of environmental exposures on both the host and the host-associated microbiome. However, the observational character of prospective cohort data and the intricate characteristics of microbiome data make it challenging to discover causal associations between environment and microbiome. Here, we introduce a causal inference framework based on the Rubin Causal Model that can help scientists to investigate such environment-host microbiome relationships, to capitalize on existing, possibly powerful, test statistics, and test plausible sharp null hypotheses. Using data from the German KORA cohort study, we illustrate our framework by designing two hypothetical randomized experiments with interventions of (i) air pollution reduction and (ii) smoking prevention. We study the effects of these interventions on the human gut microbiome by testing shifts in microbial diversity, changes in individual microbial abundances, and microbial network wiring between groups of matched subjects via randomization-based inference. In the smoking prevention scenario, we identify a small interconnected group of taxa worth further scrutiny, including Christensenellaceae and Ruminococcaceae genera, that have been previously associated with blood metabolite changes. These findings demonstrate that our framework may uncover potentially causal links between environmental exposure and the gut microbiome from observational data. We anticipate the present statistical framework to be a good starting point for further discoveries on the role of the gut microbiome in environmental health.

Assessing environmental epidemiology questions in practice with a causal inference pipeline: An investigation of the air pollution-multiple sclerosis relapses relationship.

When addressing environmental health-related questions, most often, only observational data are collected for ethical or practical reasons. However, the lack of randomized exposure often prevents the comparison of similar groups of exposed and unexposed units. This design barrier leads the environmental epidemiology field to mainly estimate associations between environmental exposures and health outcomes. A recently developed causal inference pipeline was developed to guide researchers interested in estimating the effects of plausible hypothetical interventions for policy recommendations. This article illustrates how this multistaged pipeline can help environmental epidemiologists reconstruct and analyze hypothetical randomized experiments by investigating whether an air pollution reduction intervention decreases the risk of multiple sclerosis relapses in Alsace region, France. The epidemiology literature reports conflicted findings on the relationship between air pollution and multiple sclerosis. Some studies found significant associations, whereas others did not. Two case-crossover studies reported significant associations between the risk of multiple sclerosis relapses and the exposure to air pollutants in the Alsace region. We use the same study population as these epidemiological studies to illustrate how appealing this causal inference approach is to estimate the effects of hypothetical, but plausible, environmental interventions.

Distributional changes in gene-specific methylation associated with temperature.

Temperature has been related to mean differences in DNA methylation. However, heterogeneity in these associations may exist across the distribution of methylation outcomes. This study examined whether the association between three-week averaged of temperature and methylation differs across quantiles of the methylation distributions in nine candidate genes. We measured gene-specific blood methylation repeatedly in 777 elderly men participating in the Normative Aging Study (1999-2010). We fit quantile regressions for longitudinal data to investigate whether the associations of temperature on methylation (expressed as %5mC) varied across the distribution of the methylation outcomes. We observed heterogeneity in the associations of temperature across percentiles of methylation in F3, TLR-2, CRAT, iNOS, and ICAM-1 genes. For instance, an increase in three-week temperature exposure was associated with a longer left-tail of the F3 methylation distribution. A 5°C increase in temperature was associated with a 0.15%5mC (95% confidence interval (CI): -0.27,-0.04) decrease on the 20th quantile of F3 methylation, but was not significantly related to the 80th quantile of this distribution (Estimate:0.06%5mC, 95%CI: -0.22, 0.35). Individuals with low values of F3, TLR-2, CRAT, and iNOS methylation, as well as a high value of ICAM-1 methylation, may be more susceptible to temperature effects on systemic inflammation.

Bronchial cell epigenetic aging in a human experimental study of short-term diesel and ozone exposures.

Blood-based, observational, and cross-sectional epidemiological studies suggest that air pollutant exposures alter biological aging. In a single-blinded randomized crossover human experiment of 17 volunteers, we examined the effect of randomized 2-h controlled air pollution exposures on respiratory tissue epigenetic aging. Bronchial epithelial cell DNA methylation 24 h post-exposure was measured using the HumanMethylation450K BeadChip, and there was a minimum 2-week washout period between exposures. All 17 volunteers were exposed to ozone, but only 13 were exposed to diesel exhaust. Horvath DNAmAge [Pearson coefficient (r) = 0.64; median absolute error (MAE) = 2.7 years], GrimAge (r = 0.81; MAE = 13 years), and DNAm Telomere Length (DNAmTL) (r = -0.65) were strongly correlated with chronological age in this tissue. Compared to clean air, ozone exposure was associated with longer DNAmTL (median difference 0.11 kb, Fisher's exact P-value = .036). This randomized trial suggests a weak relationship of ozone exposure with DNAmTL in target respiratory cells. Still, causal relationships with long-term exposures need to be evaluated.

Researching COVID to enhance recovery (RECOVER) pediatric study protocol: Rationale, objectives and design.

IMPORTANCE: The prevalence, pathophysiology, and long-term outcomes of COVID-19 (post-acute sequelae of SARS-CoV-2 [PASC] or "Long COVID") in children and young adults remain unknown. Studies must address the urgent need to define PASC, its mechanisms, and potential treatment targets in children and young adults. OBSERVATIONS: We describe the protocol for the Pediatric Observational Cohort Study of the NIH's REsearching COVID to Enhance Recovery (RECOVER) Initiative. RECOVER-Pediatrics is an observational meta-cohort study of caregiver-child pairs (birth through 17 years) and young adults (18 through 25 years), recruited from more than 100 sites across the US. This report focuses on two of four cohorts that comprise RECOVER-Pediatrics: 1) a de novo RECOVER prospective cohort of children and young adults with and without previous or current infection; and 2) an extant cohort derived from the Adolescent Brain Cognitive Development (ABCD) study (n = 10,000). The de novo cohort incorporates three tiers of data collection: 1) remote baseline assessments (Tier 1, n = 6000); 2) longitudinal follow-up for up to 4 years (Tier 2, n = 6000); and 3) a subset of participants, primarily the most severely affected by PASC, who will undergo deep phenotyping to explore PASC pathophysiology (Tier 3, n = 600). Youth enrolled in the ABCD study participate in Tier 1. The pediatric protocol was developed as a collaborative partnership of investigators, patients, researchers, clinicians, community partners, and federal partners, intentionally promoting inclusivity and diversity. The protocol is adaptive to facilitate responses to emerging science. CONCLUSIONS AND RELEVANCE: RECOVER-Pediatrics seeks to characterize the clinical course, underlying mechanisms, and long-term effects of PASC from birth through 25 years old. RECOVER-Pediatrics is designed to elucidate the epidemiology, four-year clinical course, and sociodemographic correlates of pediatric PASC. The data and biosamples will allow examination of mechanistic hypotheses and biomarkers, thus providing insights into potential therapeutic interventions. CLINICAL TRIALS.GOV IDENTIFIER: Clinical Trial Registration: http://www.clinicaltrials.gov. Unique identifier: NCT05172011.

Researching COVID to enhance recovery (RECOVER) tissue pathology study protocol: Rationale, objectives, and design.

IMPORTANCE: SARS-CoV-2 infection can result in ongoing, relapsing, or new symptoms or organ dysfunction after the acute phase of infection, termed Post-Acute Sequelae of SARS-CoV-2 (PASC), or long COVID. The characteristics, prevalence, trajectory and mechanisms of PASC are poorly understood. The objectives of the Researching COVID to Enhance Recovery (RECOVER) tissue pathology study (RECOVER-Pathology) are to: (1) characterize prevalence and types of organ injury/disease and pathology occurring with PASC; (2) characterize the association of pathologic findings with clinical and other characteristics; (3) define the pathophysiology and mechanisms of PASC, and possible mediation via viral persistence; and (4) establish a post-mortem tissue biobank and post-mortem brain imaging biorepository. METHODS: RECOVER-Pathology is a cross-sectional study of decedents dying at least 15 days following initial SARS-CoV-2 infection. Eligible decedents must meet WHO criteria for suspected, probable, or confirmed infection and must be aged 18 years or more at the time of death. Enrollment occurs at 7 sites in four U.S. states and Washington, DC. Comprehensive autopsies are conducted according to a standardized protocol within 24 hours of death; tissue samples are sent to the PASC Biorepository for later analyses. Data on clinical history are collected from the medical records and/or next of kin. The primary study outcomes include an array of pathologic features organized by organ system. Causal inference methods will be employed to investigate associations between risk factors and pathologic outcomes. DISCUSSION: RECOVER-Pathology is the largest autopsy study addressing PASC among US adults. Results of this study are intended to elucidate mechanisms of organ injury and disease and enhance our understanding of the pathophysiology of PASC.

High-dimensional randomization-based inference capitalizing on classical design and modern computing.

A common complication that can arise with analyses of high-dimensional data is the repeated use of hypothesis tests. A second complication, especially with small samples, is the reliance on asymptotic p-values. Our proposed approach for addressing both complications uses a scientifically motivated scalar summary statistic, and although not entirely novel, seems rarely used. The method is illustrated using a crossover study of seventeen participants examining the effect of exposure to ozone versus clean air on the DNA methylome, where the multivariate outcome involved 484,531 genomic locations. Our proposed test yields a single null randomization distribution, and thus a single Fisher-exact p-value that is statistically valid whatever the structure of the data. However, the relevance and power of the resultant test requires the careful a priori selection of a single test statistic. The common practice using asymptotic p-values or meaningless thresholds for "significance" is inapposite in general.

A permutation-based approach using a rank-based statistic to identify sex differences in epigenetics.

Epigenetic sex differences and their resulting implications for human health have been studied for about a decade. The objective of this paper is to use permutation-based inference and a new ranked-based test statistic to identify sex-based epigenetic differences in the human DNA methylome. In particular, we examine whether we could identify separations between the female and male distributions of DNA methylation across hundred of thousands CpG sites in two independent cohorts, the Swedish Adoption Twin study and the Lamarck study. Based on Fisherian p-values, we set a threshold for methylation differences "worth further scrutiny". At this threshold, we were able to confirm previously-found CpG sites that stratify with respect to sex. These CpG sites with sex differences in DNA methylation should be further investigated for their possible contribution to various physiological and pathological functions in the human body. We followed-up our statistical analyses with a literature review in order to inform the proposed disease implications for the loci we uncovered.

Researching COVID to enhance recovery (RECOVER) pediatric study protocol: Rationale, objectives and design.

Importance: The prevalence, pathophysiology, and long-term outcomes of COVID-19 (post-acute sequelae of SARS-CoV-2 [PASC] or "Long COVID") in children and young adults remain unknown. Studies must address the urgent need to define PASC, its mechanisms, and potential treatment targets in children and young adults. Observations: We describe the protocol for the Pediatric Observational Cohort Study of the NIH's RE searching COV ID to E nhance R ecovery (RECOVER) Initiative. RECOVER-Pediatrics is an observational meta-cohort study of caregiver-child pairs (birth through 17 years) and young adults (18 through 25 years), recruited from more than 100 sites across the US. This report focuses on two of five cohorts that comprise RECOVER-Pediatrics: 1) a de novo RECOVER prospective cohort of children and young adults with and without previous or current infection; and 2) an extant cohort derived from the Adolescent Brain Cognitive Development (ABCD) study ( n =10,000). The de novo cohort incorporates three tiers of data collection: 1) remote baseline assessments (Tier 1, n=6000); 2) longitudinal follow-up for up to 4 years (Tier 2, n=6000); and 3) a subset of participants, primarily the most severely affected by PASC, who will undergo deep phenotyping to explore PASC pathophysiology (Tier 3, n=600). Youth enrolled in the ABCD study participate in Tier 1. The pediatric protocol was developed as a collaborative partnership of investigators, patients, researchers, clinicians, community partners, and federal partners, intentionally promoting inclusivity and diversity. The protocol is adaptive to facilitate responses to emerging science. Conclusions and Relevance: RECOVER-Pediatrics seeks to characterize the clinical course, underlying mechanisms, and long-term effects of PASC from birth through 25 years old. RECOVER-Pediatrics is designed to elucidate the epidemiology, four-year clinical course, and sociodemographic correlates of pediatric PASC. The data and biosamples will allow examination of mechanistic hypotheses and biomarkers, thus providing insights into potential therapeutic interventions. Clinical Trialsgov Identifier: Clinical Trial Registration: http://www.clinicaltrials.gov . Unique identifier: NCT05172011.

Researching COVID to Enhance Recovery (RECOVER) adult study protocol: Rationale, objectives, and design.

IMPORTANCE: SARS-CoV-2 infection can result in ongoing, relapsing, or new symptoms or other health effects after the acute phase of infection; termed post-acute sequelae of SARS-CoV-2 infection (PASC), or long COVID. The characteristics, prevalence, trajectory and mechanisms of PASC are ill-defined. The objectives of the Researching COVID to Enhance Recovery (RECOVER) Multi-site Observational Study of PASC in Adults (RECOVER-Adult) are to: (1) characterize PASC prevalence; (2) characterize the symptoms, organ dysfunction, natural history, and distinct phenotypes of PASC; (3) identify demographic, social and clinical risk factors for PASC onset and recovery; and (4) define the biological mechanisms underlying PASC pathogenesis. METHODS: RECOVER-Adult is a combined prospective/retrospective cohort currently planned to enroll 14,880 adults aged ≥18 years. Eligible participants either must meet WHO criteria for suspected, probable, or confirmed infection; or must have evidence of no prior infection. Recruitment occurs at 86 sites in 33 U.S. states, Washington, DC and Puerto Rico, via facility- and community-based outreach. Participants complete quarterly questionnaires about symptoms, social determinants, vaccination status, and interim SARS-CoV-2 infections. In addition, participants contribute biospecimens and undergo physical and laboratory examinations at approximately 0, 90 and 180 days from infection or negative test date, and yearly thereafter. Some participants undergo additional testing based on specific criteria or random sampling. Patient representatives provide input on all study processes. The primary study outcome is onset of PASC, measured by signs and symptoms. A paradigm for identifying PASC cases will be defined and updated using supervised and unsupervised learning approaches with cross-validation. Logistic regression and proportional hazards regression will be conducted to investigate associations between risk factors, onset, and resolution of PASC symptoms. DISCUSSION: RECOVER-Adult is the first national, prospective, longitudinal cohort of PASC among US adults. Results of this study are intended to inform public health, spur clinical trials, and expand treatment options. REGISTRATION: NCT05172024.

Bridging observational studies and randomized experiments by embedding the former in the latter.

Consider a statistical analysis that draws causal inferences from an observational dataset, inferences that are presented as being valid in the standard frequentist senses; i.e. the analysis produces: (1) consistent point estimates, (2) valid p-values, valid in the sense of rejecting true null hypotheses at the nominal level or less often, and/or (3) confidence intervals, which are presented as having at least their nominal coverage for their estimands. For the hypothetical validity of these statements, the analysis must embed the observational study in a hypothetical randomized experiment that created the observed data, or a subset of that hypothetical randomized data set. This multistage effort with thought-provoking tasks involves: (1) a purely conceptual stage that precisely formulate the causal question in terms of a hypothetical randomized experiment where the exposure is assigned to units; (2) a design stage that approximates a randomized experiment before any outcome data are observed, (3) a statistical analysis stage comparing the outcomes of interest in the exposed and non-exposed units of the hypothetical randomized experiment, and (4) a summary stage providing conclusions about statistical evidence for the sizes of possible causal effects. Stages 2 and 3 may rely on modern computing to implement the effort, whereas Stage 1 demands careful scientific argumentation to make the embedding plausible to scientific readers of the proffered statistical analysis. Otherwise, the resulting analysis is vulnerable to criticism for being simply a presentation of scientifically meaningless arithmetic calculations. The conceptually most demanding tasks are often the most scientifically interesting to the dedicated researcher and readers of the resulting statistical analyses. This perspective is rarely implemented with any rigor, for example, completely eschewing the first stage. We illustrate our approach using an example examining the effect of parental smoking on children's lung function collected in families living in East Boston in the 1970s.

Particulate air pollution and survival in a COPD cohort.

BACKGROUND: Several studies have shown cross-sectional associations between long term exposure to particulate air pollution and survival in general population or convenience cohorts. Less is known about susceptibility, or year to year changes in exposure. We investigated whether particles were associated with survival in a cohort of persons with COPD in 34 US cities, eliminating the usual cross-sectional exposure and treating PM10 as a within city time varying exposure. METHODS: Using hospital discharge data, we constructed a cohort of persons discharged alive with chronic obstructive pulmonary disease using Medicare data between 1985 and 1999. 12-month averages of PM10 were merged to the individual annual follow up in each city. We applied Cox's proportional hazard regression model in each city, with adjustment for individual risk factors. RESULTS: We found significant associations in the survival analyses for single year and multiple lag exposures, with a hazard ratio for mortality for an increase of 10 microg/m(3) PM10 over the previous 4 years of 1.22 (95% CI: 1.17-1.27). CONCLUSION: Persons discharged alive for COPD have substantial mortality risks associated with exposure to particles. The risk is evident for exposure in the previous year, and higher in a 4 year distributed lag model. These risks are significantly greater than seen in time series analyses.

Comparing apples to apples: an environmental criminology analysis of the effects of heat and rain on violent crimes in Boston.

Weather characteristics have been suggested by many social scientists to influence criminality. A recent study suggested that climate change may cause a substantial increase in criminal activities during the twenty-first century. The additional number of crimes due to climate have been ethoroughly discussed the first draft of the paper. Allstimated by associational models, which are not optimal to quantify causal impacts of weather conditions on criminality. Using the Rubin Causal Model and crime data reported daily between 2012 and 2017, this study examines whether changes in heat index, a proxy for apparent temperature, and rainfall occurrence, influence the number of violent crimes in Boston. On average, more crimes are reported on temperate days compared to extremely cold days, and on dry days compared to rainy days. However, no significant differences in the number of crimes between extremely hot days versus less warm days could be observed. The results suggest that weather forecasts could be integrated into crime prevention programs in Boston. The weather-crime relationship should be taken into account when assessing the economic, sociological, or medical impact of climate change. Researchers and policy makers interested in the effects of environmental exposures or policy interventions on crime should consider data analyses conducted with causal inference approaches.

Long-Term Exposure to Ambient Fine Particulate Matter and Renal Function in Older Men: The Veterans Administration Normative Aging Study.

BACKGROUND: It is unknown if ambient fine particulate matter (PM2.5) is associated with lower renal function, a cardiovascular risk factor. OBJECTIVE: We investigated whether long-term PM2.5 exposure was associated with estimated glomerular filtration rate (eGFR) in a cohort of older men living in the Boston Metropolitan area. METHODS: This longitudinal analysis included 669 participants from the Veterans Administration Normative Aging Study with up to four visits between 2000 and 2011 (n = 1,715 visits). Serum creatinine was measured at each visit, and eGFR was calculated according to the Chronic Kidney Disease Epidemiology Collaboration equation. One-year exposure to PM2.5 prior to each visit was assessed using a validated spatiotemporal model that utilized satellite remote-sensing aerosol optical depth data. eGFR was modeled in a time-varying linear mixed-effects regression model as a continuous function of 1-year PM2.5, adjusting for important covariates. RESULTS: One-year PM2.5 exposure was associated with lower eGFRs; a 2.1-µg/m3 interquartile range higher 1-year PM2.5 was associated with a 1.87 mL/min/1.73 m2 lower eGFR [95% confidence interval (CI): -2.99, -0.76]. A 2.1 µg/m3-higher 1-year PM2.5 was also associated with an additional annual decrease in eGFR of 0.60 mL/min/1.73 m2 per year (95% CI: -0.79, -0.40). CONCLUSIONS: In this longitudinal sample of older men, the findings supported the hypothesis that long-term PM2.5 exposure negatively affects renal function and increases renal function decline. CITATION: Mehta AJ, Zanobetti A, Bind MC, Kloog I, Koutrakis P, Sparrow D, Vokonas PS, Schwartz JD. 2016. Long-term exposure to ambient fine particulate matter and renal function in older men: the VA Normative Aging Study. Environ Health Perspect 124:1353-1360; http://dx.doi.org/10.1289/ehp.1510269.

Particulate Air Pollution and Fasting Blood Glucose in Nondiabetic Individuals: Associations and Epigenetic Mediation in the Normative Aging Study, 2000-2011.

BACKGROUND: Among nondiabetic individuals, higher fasting blood glucose (FBG) independently predicts diabetes risk, cardiovascular disease, and dementia. Ambient PM2.5 (particulate matter with aerodynamic diameter ≤ 2.5 µm) is an emerging determinant of glucose dysregulation. PM2.5 effects and mechanisms are understudied among nondiabetic individuals. OBJECTIVES: Our goals were to investigate whether PM2.5 is associated with an increase in FBG and to explore potential mediating roles of epigenetic gene regulation. METHODS: In 551 nondiabetic participants in the Normative Aging Study, we measured FBG, and DNA methylation of four inflammatory genes (IFN-γ, IL-6, ICAM-1, and TLR-2), up to four times between 2000 and 2011 (median = 2). We estimated short- and medium-term (1-, 7-, and 28-day preceding each clinical visit) ambient PM2.5 at each participant's address using a validated hybrid land-use regression satellite-based model. We fitted covariate-adjusted regression models accounting for repeated measures. RESULTS: Mean FBG was 99.8 mg/dL (SD = 10.7), 18% of the participants had impaired fasting glucose (IFG; i.e., 100-125 mg/dL FBG) at first visit. Interquartile increases in 1-, 7-, and 28-day PM2.5 were associated with 0.57 mg/dL (95% CI: 0.02, 1.11, p = 0.04), 1.02 mg/dL (95% CI: 0.41, 1.63, p = 0.001), and 0.89 mg/dL (95% CI: 0.32, 1.47, p = 0.003) higher FBG, respectively. The same PM2.5 metrics were associated with 13% (95% CI: -3%, 33%, p = 0.12), 27% (95% CI: 6%, 52%, p = 0.01) and 32% (95% CI: 10%, 58%, p = 0.003) higher odds of IFG, respectively. PM2.5 was negatively correlated with ICAM-1 methylation (p = 0.01), but not with other genes. Mediation analysis estimated that ICAM-1 methylation mediated 9% of the association of 28-day PM2.5 with FBG. CONCLUSIONS: Among nondiabetics, short- and medium-term PM2.5 were associated with higher FBG. Mediation analysis indicated that part of this association was mediated by ICAM-1 promoter methylation. Citation: Peng C, Bind MA, Colicino E, Kloog I, Byun HM, Cantone L, Trevisi L, Zhong J, Brennan K, Dereix AE, Vokonas PS, Coull BA, Schwartz JD, Baccarelli AA. 2016. Particulate air pollution and fasting blood glucose in nondiabetic individuals: associations and epigenetic mediation in the Normative Aging Study, 2000-2011. Environ Health Perspect 124:1715-1721; http://dx.doi.org/10.1289/EHP183.

Beyond the Mean: Quantile Regression to Explore the Association of Air Pollution with Gene-Specific Methylation in the Normative Aging Study.

BACKGROUND: Air pollution has been related to mean changes in outcomes, including DNA methylation. However, mean regression analyses may not capture associations that occur primarily in the tails of the outcome distribution. OBJECTIVES: In this study, we examined whether the association between particulate air pollution and DNA methylation differs across quantiles of the methylation distribution. We focused on methylation of candidate genes related to coagulation and inflammation: coagulation factor III (F3), intercellular adhesion molecule 1 (ICAM-1), interferon gamma (IFN-γ), interleukin-6 (IL-6), and toll-like receptor 2 (TRL-2). METHODS: We measured gene-specific blood DNA methylation repeatedly in 777 elderly men participating in the Normative Aging Study (1999-2010). We fit quantile regressions for longitudinal data to investigate whether the associations of particle number, PM2.5 (diameter ≤ 2.5 µm)black carbon, and PM2.5 mass concentrations (4-week moving average) with DNA methylation [expressed as the percentage of methylated cytosines over the sum of methylated and unmethylated cytosines at position 5 (%5mC)] varied across deciles of the methylation distribution. We reported the quantile regression coefficients that corresponded to absolute differences in DNA methylation (expressed in %5mC) associated with an interquartile range increase in air pollution concentration. RESULTS: Interquartile range increases in particle number, PM2.5 black carbon, and PM2.5 mass concentrations were associated with significantly lower methylation in the lower tails of the IFN-γ and ICAM-1 methylation distributions. For instance, a 3.4-µg/m3 increase in PM2.5 mass concentration was associated with a 0.18%5mC (95% CI: -0.30, -0.06) decrease on the 20th percentile of ICAM-1 methylation, but was not significantly related to the 80th percentile (estimate: 0.07%5mC, 95% CI: -0.09, 0.24). CONCLUSIONS: In our study population of older men, air pollution exposures were associated with a left shift in the lower tails of the IFN-γ and ICAM-1 methylation distributions.