Sara López-Pintado

On the Concept of Depth for Functional Data

The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new definition of depth for functional observations based on the graphic representation of the curves. Given a collection of functions, it establishes the “centrality” of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics, such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus, this new depth is also suitable for complex observations such as microarray data, images, and those arising in some recent marketing and financial studies. Natural properties of these new concepts are established and the uniform consistency of the sample depth is proved. Simulation results show that the corresponding depth based trimmed mean presents better performance than other possible location estimators proposed in the literature for some contaminated models. Data depth can be also used to screen for outliers. The ability of the new notions of depth to detect “shape” outliers is presented. Several real datasets are considered to illustrate this new concept of depth, including applications to microarray observations, weather data, and growth curves. Finally, through this depth, we generalize to functions the Wilcoxon rank sum test. It allows testing whether two groups of curves come from the same population. This functional rank test when applied to children growth curves shows different growth patterns for boys and girls.

Serial high sensitivity cardiac troponin T measurement in acute heart failure: insights from the RELAX‐AHF study

Troponin elevation is common in acute heart failure (AHF) and may be useful for prognostication; however, available data are mixed and many previous studies used older, less sensitive assays. We examined the association between serial measurements of high‐sensitivity cardiac troponin T (hs‐cTnT) and outcomes in RELAX‐AHF.

Depth-based inference for functional data

Robust inference tools for functional data are proposed. They are based on the notion of depth for curves. The ideas of trimmed regions, contours and central regions are extended to functions and their structural properties and asymptotic behavior are studied. Next, a scale curve is introduced to describe dispersion in a sample of functions. The computational burden of these techniques is not heavy, so they are also adequate to analyze high-dimensional data. These inferential methods are applied to several real data sets.

Domestic airborne black carbon and exhaled nitric oxide in children in NYC

Differential exposure to combustion by-products and allergens may partially explain the marked disparity in asthma prevalence (3–18%) among New York City neighborhoods. Subclinical changes in airway inflammation can be measured by fractional exhaled nitric oxide (FeNO). FeNO could be used to test independent effects of these environmental exposures on airway inflammation. Seven- and eight-year-old children from neighborhoods with lower (range 3–9%, n=119) and higher (range 11–18%, n=121) asthma prevalence participated in an asthma case–control study. During home visits, FeNO was measured, and samples of bed dust (allergens) and air (black carbon; BC) were collected. Neighborhood built-environment characteristics were assessed for the 500 m surrounding participants’ homes. Airborne BC concentrations in homes correlated with neighborhood asthma prevalence (P<0.001) and neighborhood densities of truck routes (P<0.001) and buildings burning residual oil (P<0.001). FeNO concentrations were higher among asthmatics with than in those without frequent wheeze (≥4 times/year) (P=0.002). FeNO concentrations correlated with domestic BC among children without seroatopy (P=0.012) and with dust mite allergen among children with seroatopy (P=0.020). The association between airborne BC in homes and both neighborhood asthma prevalence and FeNO suggest that further public health interventions on truck emissions standards and residual oil use are warranted.

Simplicial band depth for multivariate functional data

We propose notions of simplicial band depth for multivariate functional data that extend the univariate functional band depth. The proposed simplicial band depths provide simple and natural criteria to measure the centrality of a trajectory within a sample of curves. Based on these depths, a sample of multivariate curves can be ordered from the center outward and order statistics can be defined. Properties of the proposed depths, such as invariance and consistency, can be established. A simulation study shows the robustness of this new definition of depth and the advantages of using a multivariate depth versus the marginal depths for detecting outliers. Real data examples from growth curves and signature data are used to illustrate the performance and usefulness of the proposed depths.

Multimodal Imaging of Central Retinal Disease Progression in a 2-Year Mean Follow-up of Retinitis Pigmentosa

PURPOSE To determine the rate of progression and optimal follow-up time in patients with advanced-stage retinitis pigmentosa (RP) comparing the use of fundus autofluorescence imaging and spectral-domain optical coherence tomography. DESIGN Retrospective analysis of progression rate. METHODS Longitudinal imaging follow-up in 71 patients with retinitis pigmentosa was studied using the main outcome measurements of hyperautofluoresent ring horizontal diameter and vertical diameter along with ellipsoid zone line width from spectral-domain optical coherence tomography. Test-retest reliability and the rate of progression were calculated. The interaction between the progression rates was tested for sex, age, mode of inheritance, and baseline measurement size. Symmetry of left and right eye progression rate was also tested. RESULTS Significant progression was observed in >75% of patients during the 2-year mean follow-up. The mean annual progression rates of ellipsoid zone line and hyperautofluorescent ring horizontal diameter and vertical diameter were 0.45 degree (4.9%), 0.51 degree (4.1%), and 0.42 degree (4.0%), respectively. The ellipsoid zone line width and hyperautofluorescent ring horizontal diameter and vertical diameter had low test-retest variabilities of 8.9%, 9.5%, and 9.6%, respectively. This study is the first to demonstrate asymmetrical structural progression rate between right and left eye, which was found in 19% of patients. The rate of progression was significantly slower as the disease approached the fovea, supporting the theory that RP progresses in an exponential fashion. No significant interaction between progression rate and patient age, sex, or mode of inheritance was observed. CONCLUSIONS Fundus autofluorescence and optical coherence tomography detect progression in patients with RP reliably and with strong correlation. These parameters may be useful alongside functional assessments as the outcome measurements for future therapeutic trials. Follow-up at 1-year intervals should be adequate to efficiently detect progression.

Analyzing growth trajectories.

Growth trajectories play a central role in life course epidemiology, often providing fundamental indicators of prenatal or childhood development, as well as an array of potential determinants of adult health outcomes. Statistical methods for the analysis of growth trajectories have been widely studied, but many challenging problems remain. Repeated measurements of length, weight and head circumference, for example, may be available on most subjects in a study, but usually only sparse temporal sampling of such variables is feasible. It can thus be challenging to gain a detailed understanding of growth patterns, and smoothing techniques are inevitably needed. Moreover, the problem is exacerbated by the presence of large fluctuations in growth velocity during early infancy, and high variability between subjects. Existing approaches, however, can be inflexible because of a reliance on parametric models, require computationally intensive methods that are unsuitable for exploratory analyses, or are only capable of examining each variable separately. This article proposes some new nonparametric approaches to analyzing sparse data on growth trajectories, with flexibility and ease of implementation being key features. The methods are illustrated using data on participants in the Collaborative Perinatal Project.

Robust depth-based tools for the analysis of gene expression data

Microarray experiments provide data on the expression levels of thousands of genes and, therefore, statistical methods applicable to the analysis of such high-dimensional data are needed. In this paper, we propose robust nonparametric tools for the description and analysis of microarray data based on the concept of functional depth, which measures the centrality of an observation within a sample. We show that this concept can be easily adapted to high-dimensional observations and, in particular, to gene expression data. This allows the development of the following depth-based inference tools: (1) a scale curve for measuring and visualizing the dispersion of a set of points, (2) a rank test for deciding if 2 groups of multidimensional observations come from the same population, and (3) supervised classification techniques for assigning a new sample to one of G given groups. We apply these methods to microarray data, and to simulated data including contaminated models, and show that they are robust, efficient, and competitive with other procedures proposed in the literature, outperforming them in some situations.

Ambulatory assessments of psychological and peripheral stress-markers predict birth outcomes in teen pregnancy

OBJECTIVE Pregnant adolescents have high rates of poor birth outcomes, but the causes are unclear. We present a prospective, longitudinal study of pregnant adolescents assessing associations between maternal psychobiological stress indices and offspring gestational age at birth and birthweight. METHOD Healthy nulliparous pregnant adolescents were recruited (n=205) and followed during pregnancy. Ambulatory assessments over 24h of perceived psychological stress (collected every 30 min) and salivary cortisol (6 samples) and a summary questionnaire, the Perceived Stress Scale, were collected at three time points (13-16, 24-27, and 34-37 gestational weeks). Corticotropin-releasing hormone, C-reactive protein, and interleukin 6 were assayed from blood taken at the latter 2 sessions. A final sample of 119 participants was selected for analyses. RESULTS The ambulatory assessment of perceived psychological stress was positively correlated with the Perceived Stress Scale (r=.20, p=.03) but neither was associated with any of the biological assays (all ps>.20). Based on backward selection regression models that included all stress variables and relevant covariates, the ambulatory assessments of perceived psychological stress and cortisol - though not the Perceived Stress Scale - were negatively associated with gestational age at birth (F(4, 107)=3.38, p=.01) while cortisol was negatively related to birthweight (F(5, 107)=14.83, p<.0001). CONCLUSIONS Targeted interventions to reduce psychological and biological indicators of heightened stress during pregnancy may have positive public health benefits for the offspring given the associations of shortened gestation and lower birthweight with risk for poor mental and physical health outcomes.

Continuous positive airway pressure for children with undifferentiated respiratory distress in Ghana: an open-label, cluster, crossover trial

BACKGROUND In low-income and middle-income countries, invasive mechanical ventilation is often not available for children at risk of death from respiratory failure. We aimed to determine if continuous positive airway pressure (CPAP), a form of non-invasive ventilation, decreases all-cause mortality in children with undifferentiated respiratory distress in Ghana. METHODS This open-label, cluster, crossover trial was done in two Ghanaian non-tertiary hospitals where invasive mechanical ventilation is not routinely available. Eligible participants were children aged from 1 month to 5 years with a respiratory rate of more than 50 breaths per min in children 1-12 months old, or more than 40 breaths per min in children older than 12 months, and use of accessory muscles or nasal flaring. CPAP machines were allocated to one hospital during each study block, while the other hospital served as the control site. The initial intervention site was randomly chosen using a coin toss. 5 cm of water pressure was delivered via CPAP nasal prongs. The primary outcome measure was all-cause mortality rate at 2 weeks after enrolment in patients for whom data were available after 2 weeks. We also did post-hoc regression analysis and subgroup analysis of children by malaria status, oxygen saturation, and age. This study is registered with ClinicalTrials.gov, number NCT01839474. FINDINGS Between Jan 20, 2014, and Dec 5, 2015, 2200 children were enrolled: 1025 at the intervention site and 1175 at the control site. Final analysis included 1021 patients in the CPAP group and 1160 patients in the control group. 2 weeks after enrolment, 26 (3%) of 1021 patients in the CPAP group, and 44 (4%) of 1160 patients in the control group, had died (relative risk [RR] of mortality 0·67, 95% CI 0·42-1·08; p=0·11). In children younger than 1 year, all-cause mortality was ten (3%) of 374 patients in the CPAP group, and 24 (7%) of 359 patients in the control group (RR 0·40, 0·19-0·82; p=0·01). After adjustment for study site, time, and clinically important variables, the odds ratio for 2-week mortality in the CPAP group versus the control group was 0·4 in children aged up to 6 months, 0·5 for children aged 12 months, 0·7 for children aged 24 months, and 1·0 for those aged 36 months. 28 patients (3%) in the CPAP group and 24 patients (2%) in the control group had CPAP-related adverse events, such as vomiting, aspiration, and nasal, skin, or eye trauma. No serious adverse events were observed. INTERPRETATION In the unadjusted analysis the use of CPAP did not decrease all-cause 2-week mortality in children 1 month to 5 years of age with undifferentiated respiratory distress. After adjustment for study site, time, and clinically important variables, 2-week mortality in the CPAP group versus the control group was significantly decreased in children 1 year of age and younger. CPAP is safe and improves respiratory rate in a non-tertiary setting in a lower-middle-income country. FUNDING General Electric Foundation.