Aggregation of macroparasites among hosts is a nearly universal phenomenon, and as such it is one of the few “laws” of parasite ecology (Poulin, 2007a). This widespread pattern of most hosts harbouring few or no infecting macroparasites (hereafter parasites), while a minority of hosts experience much higher intensity infections, has important consequences for the impacts of disease-causing organisms on hosts (Hudson et al., 1992, Poulin, 2007b, Poulin, 2007), the probability and intensity of co-infection of hosts by different parasite strains or species (Morrill et al., 2017), and the stability of host-parasite associations (Anderson and May, 1978, Rosà and Pugliese, 2002). Such a common and consequential phenomenon has attracted much attention from researchers – principally ecologists, wildlife researchers and parasitologists – who have sought to measure and explain aggregation, as well as predict its consequences (Shaw and Dobson, 1995, Johnson and Hoverman, 2014, Cox et al., 2017). Understanding aggregation of parasites is a fundamental problem in the study of parasite evolutionary ecology, owing to its impact on, as examples, parasite population genetic structure (Cornell et al., 2003), infection intensity-dependent population regulation (Møller, 2005), and both parasite reproduction and mating systems (Criscione et al., 2005; Cox et al., 2017). It is an important problem in applied ecology and wildlife management as well, since treating only the small subset of heavily infected hosts can be an efficient and cheaper approach to parasite control in managed populations (Perkins et al., 2003). The advances in the study of parasite aggregation also have the potential to inform other branches of science (e.g. contaminant biology), where measures of aggregation or skew are used (Morrill et al., 2014).

Researchers may agree – or, at least, take it for granted that they agree – on a general definition of parasite aggregation, but have adopted and/or developed various measures for its quantification. When discussing measures of aggregation in ecology more broadly, Pielou (1977) offered the following observations on how these different methods highlight contrasting interpretations (quoted also by Hurlbert, 1990, McVinish and Lester, 2020):

“. . . the phrase ‘degree of aggregation’ describes a vague, undefined notion that is open to several interpretations. If aggregation is to be measured, we must first choose from a number of possibilities some measurable property of a spatial pattern that is to be called its aggregation, and the method of measurement is then implicit in the chosen definition. Thus the several existing ways of measuring aggregation are not different methods of measuring the same thing: they measure different things.”

It is incumbent on the research community using, and choosing between, various measures of aggregation to know that common measures can capture contrasting properties, and that these disparities point to differences in the meanings of aggregation between such measures. Having discussions about the underlying and general causes of aggregation, for example, are most fruitful if researchers are measuring the same thing, or at least know when they are discussing different attributes of aggregated distributions. Different measures of aggregation might be more or less amenable to specific research questions and analyses (as, for example, when researchers are interested in wholly different research questions such as intraspecific competition between parasites versus parasite-mediated effects on hosts).

A simple expectation is that measures less similar in their implicit meanings should be less strongly correlated with one another; they may also show distinctly different relationships with other sample-level parasitological indices such as prevalence (the proportion of hosts that are infected in a sample) and/or mean abundance (the mean number of parasite individuals per host, including uninfected hosts). Naturally, as the different measures of aggregation all attempt to encapsulate an emergent, population-level general property of the “clustering” of infecting parasites, they should be expected to correlate with one another to some degree. Nevertheless, departures from near perfect correlations, or losses of correlations in certain contexts, would help to identify differences between the various aggregation measures in their meanings. By extension, wherever near perfect correlations do arise, researchers can be more confident that the choice of one measure over another is of less consequence.

As other sample-level measures also describe how parasites are distributed among hosts (e.g. infection prevalence, mean abundance), it is not surprising that these too show varying levels of correlation with degree of parasite aggregation (Gregory and Woolhouse, 1993, Poulin, 1993, McVinish and Lester, 2020), and with each other. Pielou (1977) proposed an insightful question that highlights a simple, but informative, apparent dichotomy in measures of aggregation in terms of their relationship with the mean number of individuals per unit area/volume: should aggregation change or remain unchanged when a random proportion of individuals are removed from a distribution (e.g. random removals of parasites from a sample of hosts)? As she describes, both answers are reasonable. One could consider that random removals would be expected to result, simply by chance, in more parasites being removed from larger infrapopulations (i.e. groups of conspecific parasites infecting a single host) than smaller ones, thereby making those infrapopulations less “dense.” From this perspective, as the overall number of parasites decreases, so too should aggregation. On the other hand, as removals would be random, the only resulting difference in the distribution would be in its overall “density,” while its shape would otherwise remain unchanged; therefore, one could consider that the essential property of aggregation relating to the distribution’s shape similarly should remain unchanged. Importantly, this simple thought experiment highlights two contrasting expected consequences of random removals: either aggregation should decrease, or remain the same.

The present study has three objectives. The first two are to address the related questions: how do the implicit meanings of parasite aggregation differ among the common methods used for its quantification? And how do these measures compare in terms of their relationship with mean abundance, prevalence, and each other? Such a synthesis is, to our knowledge, lacking in the study of distributions of infecting parasites among hosts, and where previous analyses have partially addressed these questions, conclusions have been inconsistent. Our third objective is to provide a “user’s guide” to help researchers choose the measure(s) of aggregation most relevant to their studies, based on the properties and context-specific advantages and disadvantages identified in simulations, and in previous research. To accomplish these objectives, we first summarize the various commonly used measures of aggregation and consider explicitly their mathematical expressions. We then compare measures in terms of their responses to random parasite removals, following from the thought experiment posited by Pielou (1977). With a second series of simulations, we explore their relationships to one another and to mean abundance and prevalence. We propose a framework that simplifies and synthesizes these perspectives, clarifies previous results relating to the attributes of some measures, and highlights important and previously unrecognized properties of some of the popular aggregation metrics.

More specifically, we illustrate that random removals of infecting parasites results in three groupings of measures either showing no change in response to removals, a negative effect, or a positive effect on their estimate of aggregation. We then show strong correlations between certain common measures of aggregation, and that these pairs of measures align with the same three groups seen in the initial simulation. Importantly, the three pairs of measures also differ in their relationships with sample prevalence and mean abundance. This work will help guide approaches to, and interpretations of, studies of parasite aggregation, and help researchers report or calculate degrees of aggregation in such a way as to ensure inclusion of their studies in future syntheses.