We describe a new residual for general regression models, defined as pr(Y* < y) − pr(Y* > y), where y is the observed outcome and Y* is a random variable from the fitted distribution. This probability-scale residual can be written as E {sign(y, Y*)} whereas the popular observed-minus-expected residual can be thought of as E(y − Y*). Therefore, the probability-scale residual is useful in settings where differences are not meaningful or where the expectation of the fitted distribution cannot be calculated. We present several desirable properties of the probability-scale residual that make it useful for diagnostics and measuring residual correlation, especially across different outcome types. We demonstrate its utility for continuous, ordered discrete, and censored outcomes, including current status data, and with various models including Cox regression, quantile regression, and ordinal cumulative probability models, for which fully specified distributions are not desirable or needed, and in some cases suitable residuals are not available. The residual is illustrated with simulated data and real datasets from HIV-infected patients on therapy in the southeastern United States and Latin America.

Summary Increased droughts impair tree growth worldwide. This study analyzes hydraulic and carbon traits of conifer species, and how they shape species strategies in terms of their growth rate and drought resilience. We measured 43 functional stem and leaf traits for 28 conifer species growing in a 50‐yr‐old common garden experiment in the Netherlands. We assessed: how drought‐ and carbon‐related traits are associated across species, how these traits affect stem growth and drought resilience, and how traits and drought resilience are related to species’ climatic origin. We found two trait spectra: a hydraulics spectrum reflecting a trade‐off between hydraulic and biomechanical safety vs hydraulic efficiency, and a leaf economics spectrum reflecting a trade‐off between tough, long‐lived tissues vs high carbon assimilation rate. Pit aperture size occupied a central position in the trait‐based network analysis and also increased stem growth. Drought recovery decreased with leaf lifespan. Conifer species with long‐lived leaves suffer from drought legacy effects, as drought‐damaged leaves cannot easily be replaced, limiting growth recovery after drought. Leaf lifespan, rather than hydraulic traits, can explain growth responses to a drier future.

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Single-cell RNA sequencing (scRNA-seq) has become a powerful tool for the systematic investigation of cellular diversity. As a number of computational tools have been developed to identify and visualize cell populations within a single scRNA-seq dataset, there is a need for methods to quantitatively and statistically define proportional shifts in cell population structures across datasets, such as expansion or shrinkage or emergence or disappearance of cell populations. Here we present sc-UniFrac, a framework to statistically quantify compositional diversity in cell populations between single-cell transcriptome landscapes. sc-UniFrac enables sensitive and robust quantification in simulated and experimental datasets in terms of both population identity and quantity. We have demonstrated the utility of sc-UniFrac in multiple applications, including assessment of biological and technical replicates, classification of tissue phenotypes and regional specification, identification and definition of altered cell infiltrates in tumorigenesis, and benchmarking batch-correction tools. sc-UniFrac provides a framework for quantifying diversity or alterations in cell populations across conditions and has broad utility for gaining insight into tissue-level perturbations at the single-cell resolution.

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