Urban scaling with censored data

In the realm of urban science, scaling laws are essential for understanding the relationship between city population and urban features, such as socioeconomic outputs. Ideally, these laws would be based on complete datasets; however, researchers often face challenges related to data availability and reporting practices, resulting in datasets that include only the highest observations of the urban features (top-k). A key question that emerges is: Under what conditions can an analysis based solely on top-k observations accurately determine whether a scaling relationship is truly superlinear or sublinear? To address this question, we conduct a numerical study that explores how relying exclusively on reported values can lead to erroneous conclusions, revealing a selection bias that favors sublinear over superlinear scaling. In response, we develop a method that provides robust estimates of the minimum and maximum potential scaling exponents when only top-k observations are available. We apply this method to two case studies involving firearm violence, a domain notorious for its suppressed datasets, and we demonstrate how this approach offers a reliable framework for analyzing scaling relationships with censored data.Author summary: Over the past two decades, urban scaling has become essential for understanding the rural-urban continuum by quantifying how urban characteristics depend on a city’s population size. For example, more populous cities are expected to have more patents and wages per capita, but fewer gas stations and road surfaces. Nonetheless, access to incomplete datasets about urban features systematically skews the conclusions derived from this theory. This issue is particularly relevant for features related to health outcomes, which are regularly obtained from partially censored datasets. For instance, data on firearms in the United States remain inaccessible to the public. To address this limitation, we developed a framework that enables urban researchers to draw reliable conclusions about urban scaling, even when dealing with censored datasets. We demonstrate this framework with data on firearm homicide and the number of firearms recovered by authorities in American cities.