The network scale-up method (NSUM) is a cost-effective approach to estimating the size or prevalence of a group of people that is hard to reach through a standard survey. The basic NSUM involves two steps: estimating respondents' degrees by one of various methods (in this paper we focus on the probe group method which uses the number of people a respondent knows in various groups of known size), and estimating the prevalence of the hard-to-reach population of interest using respondents' estimated degrees and the number of people they report knowing in the hard-to-reach group. Each of these two steps involves taking either an average of ratios or a ratio of averages. Using the ratio of averages for each step has so far been the most common approach. However, we present theoretical arguments that using the average of ratios at the second, prevalence-estimation step often has lower mean squared error when a main model assumption is violated, which happens frequently in practice; this estimator which uses the ratio of averages for degree estimates and the average of ratios for prevalence was proposed early in NSUM development but has largely been unexplored and unused. Simulation results using an example network data set also support these findings. Based on this theoretical and empirical evidence, we suggest that future surveys that use a simple estimator may want to use this mixed estimator, and estimation methods based on this estimator may produce new improvements.

翻译：网络尺度法（NSUM）是一种经济高效的方法，用于估计通过标准调查难以触及的人群规模或流行率。基本NSUM包含两个步骤：首先通过多种方法估计受访者的度数（本文聚焦于探针组法，该方法利用受访者在已知规模各群体中认识的人数）；其次利用受访者估计度数及其报告的在目标难触及群体中认识的人数，估计该群体的流行率。这两个步骤均涉及比值平均值或平均值比值的选择。目前最常用的方法是两步均采用平均值比值。然而，我们提出了理论论证：当关键模型假设在实际中频繁被违反时，在第二步（流行率估计）采用比值平均值通常具有更低的均方误差。这种在度数估计中使用平均值比值、在流行率估计中使用比值平均值的混合估计量，在NSUM发展早期已被提出，但长期缺乏深入探索和应用。基于示例网络数据集的模拟结果亦支持上述发现。综合理论与实证证据，我们建议未来采用简单估计量的调查可优先考虑此混合估计量，且基于该估计量的估计方法可能产生新的改进。