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 the random mixing assumption is violated, which seems likely 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.

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