Understandable Multifunctionality Measures Using Hill Numbers

In ecology, multifunctionality measures the simultaneous performance of multiple ecosystem functions. If species diversity describes the variety of species that together build the ecosystem, multifunctionality attempts to describe the variety of functions these species perform. A range of methods have been proposed to quantify multifunctionality, successively attempting to alleviate problems that have been identified with the previous methods. This has led to a proliferation of more-or-less closely related metrics which, however, lack an overarching theoretical framework. Here we borrow from the comprehensive framework of species diversity to derive a new metric of multifunctionality. Analogously to the effective number of species used to quantify species diversity, the metric we propose is influenced both by the number of functions as well as, crucially, the evenness of performance levels across functions. In addition, the effective multifunctionality also considers the average level at which the functions are performed. The result is a measure of the cumulative performance of the system were all functions provided equally. The framework allows for the inclusion of the correlation structure among functions, thus allowing it to account for non-independence between functions. We show that the average metric is a special case of the newly proposed metric when all functions are uncorrelated and performed at equal levels. We hope that by providing a new metric of multifunctionality anchored in the rigorous framework of species diversity based on effective numbers, we will overcome the considerable skepticism that the larger community of ecologists has built against indices of multifunctionality. We thereby hope to help popularize this important concept which, as biological diversity, describes a fundamental property of ecosystems and thus lies at the heart of ecology.

multifunctionality also considers the average level at which the functions are performed. The 23 result is a measure of the cumulative performance of the system were all functions provided 24 equally. The framework allows for the inclusion of the correlation structure among functions, 25 thus allowing it to account for non-independence between functions. We show that the average 26 metric is a special case of the newly proposed metric when all functions are uncorrelated and 27 performed at equal levels. We hope that by providing a new metric of multifunctionality 28 anchored in the rigorous framework of species diversity based on effective numbers, we will 29 overcome the considerable skepticism that the larger community of ecologists has built against 30 indices of multifunctionality. We thereby hope to help popularize this important concept which, 31 Gamfeldt and Roger 2017). We do note that this is a difficult problem. Throwing many different 51 ecologists against this problem in a stimulating working group over multiple years (we are 52 thankful for NCEAS for providing this venue) has likely resulted in more ink impregnated on 53 foreheads hitting whiteboards than should be typical, to say nothing of others who have 54 attempted to cut this Gordian knot before and since (Hector and  First, many have opted for simplicity and taking the average of all functions (e.g., Maestre et al. 81 2012). This approach is, at first glance, appealing, particularly as it provides a metric that can be 82 put on a y-axis while a predictor is on the x-axis. However, it sacrifices crucial information about 83 the system. An arithmetic mean tells us what level of functioning we would expect were we to 84 sample any one function at random. Consider two plots -one where all functions are similar and 85 performing at half their value and one where half of the functions are at their maximum while 86 half are absent. The averaging approach says that they are identical. Geometric averaging (e.g., 87 Hensel and Silliman 2013) appears to get around this problem to some degree, as the geometric 88 mean is less than the arithmetic mean by a function of the variance of the observations (the two 89 means are the same when all observations are the identical). However, given its formulation, one 90 critically low function can dominate the measurement -e.g., if even one function is 0, the 91 geometric mean will be 0.  Figure S1). In the absence of suitable null-models, and even if the approach yields rich 114 information about multifunctionality sensu stricto, it is unwieldy for most if not all who choose 115 to use it, and we have noticed that many of those who do, often report a single threshold in their 116 main text and leave further exploration to supplementary materials anyway. Related, more 117 recent, efforts have sought to use dimensionality-reducing techniques which have yielded metrics 118 that, while useful, similarly lack easily interpretable meaning (Meyer et al. 2018). 119

Hill numbers and effective diversity
120 Over the last 70 years, ecologists studying how to measure species diversity have shown that the 121 vast majority of previous diversity metrics can be organized into a common framework 122 (Macarthur 1965, Hill 1973, Jost 2006, Chao et al. 2014a, 2019. This is true for indices such as 123 Shannon entropy, all Simpson measures, all Renyi entropies, all HCDT or "Tsallis" entropies and 124 species richness (Jost, 2006) and more. All can be expressed as generalized entropies that can be 125 converted to an effective number of species of "order" q which specifies the weighting of 126 proportional abundances. The general formula for the diversity of order q for S species is the 127 following: 128 Here, pi is the relative abundance of the i th species and q is the weight given to the species' 130 relative abundances. Species richness, the effective number of species based on Shannon 131 entropy, the effective number of species based on the Simpson index, and the Berger-Parker 132 dominance index are all effective numbers of species of order = 0, 1, 2 and ∞, respectively. 133 (Note that the formula is undefined for q = 1, but its limit q → 1 is exp(-Σpi log pi)). The 134 effective number of species of order q is also often referred to as Hill numbers. To define the effective number of functions, we begin with a set of measurements on k functions 146 (Table 1) that have been standardized to a common scale (i.e., between 0 and 1 where 0 means 147 no function and 1 means maximum level of function). Let " , ∈ 1, 2, . . .
show the level of 148 function for function i ( Table 1). The relative proportion a function contributes to the whole is 149 defined as 150 (2) 151 We can now substitute the relative proportion into the formula for the effective number of types 152 given in Eq 1 153 where q N is the effective number of functions for some order q ( Table 1) the same, a system with high average function and low average function will look the same. 183 Indeed, for some values of q (e.g., q = 1), under some scenarios if one function goes up, the 184 effective number of functions can actually drop. To achieve the translation to a metric of 185 multifunctionality, we need to take into account the level at which the functions are performed: 186 the arithmetic mean of the function values standardized to a common scale, which we define as A 187 (Table 1). As we are using standardized values as before, A will range from 0 to 1. 188

189
We can then calculate effective multifunctionality of order q (Table 1)  imply both a high level of function and high functional evenness (i.e., pi is close to 1/K for all 229 functions). Low values imply that, even if a single function is being optimized, the assemblage of 230 functions as a whole is not performing well. The relationship can also be easily decomposed into 231 its constituent parts for a more detailed examination of its behavior. 232 Last, from a convenience standpoint, having a single metric allows us to begin to examine it as 233 any other response variable. In the Biodiversity and Ecosystem Functioning world, we might 234 look at additive partitioning in addition to complementary overlap approaches. In global change 235 biology, we can look at the stability, resistance, and resilience of this metric in ecosystems 236 confronting human stressors. This metric can be used just as any other univariate metric in any 237 field, leading to easy adoption of the multifunctionality concept across many fields of endeavor. 238 The options are open. contributions to services. Fortunately, choosing an 'optimal' level of function to link to 1 can 247 alleviate this (e.g., if 25% of function is sufficiently high for the provision of a service, 25% or 248 higher can be considered a '1'). Functions can also be upweighted or downweighted in the 249 calculation of q N so long as ∑ " = 1. Choices for standardization are often best made in the 250 context of a specific system or application and must be transparently justified. and that a metric of true multifunctionality should identify "variables that represent independent 257 aspects of ecosystem functioning" (Manning et al. 2018). As an illustration, consider a scenario 258 where we measure a set of functions, several of which result from a shared mechanism so that 259 they are inextricably linked (for example, growth rate and final biomass). If we want to study the 260 circumstances under which overall multifunctionality is maximized, the results will be 261 disproportionately driven by circumstances maximizing this mechanism influencing the set of 262 For this, we need a distance matrix of some sort (Table 1). If this is based on a priori 286 mechanistic information about the underlying processes driving functions (e.g., physiological or 287 evolutionary constraints), so much the better. We often do not have such a matrix. Other options 288 include a matrix derived from principal components or other methods of constructing distance 289 matrices (see Manning et al. 2018). In the absence of such information, a practical choice could 290 be to look at the correlation matrix among functions, R. Defining D = (1 -R )/2 would create a 291 distance matrix where a dij value of 0 means two functions are inextricably linked (perfectly 292 correlated) while 1 means they trade-off completely (rij = -1). This is one choice. We note that 293 some functions could be correlated for non-biological reasons, and as such a biologically-based 294 distance matrix might be wiser. We suggest that determining the proper way to create a distance 295 matrix between functions for the estimation of effective diversity is an exciting area of research. 296 297 With a distance matrix in hand, we need to ask ourselves, how distinct must two functions be 298 before we consider them completely different? This incorporates the threshold τ (Table 1)  including metrics in this paper, that shows some concordance between the two. The threshold 360 approach seeks to remedy some of the drawbacks of using q = 0, and can be useful for managers 361 that have specific targets of functional performance they need to meet. However, it is too 362 sensitive to a number of decisions, as well as complicated to interpret (Gamfeldt and Roger 363 2017). Other metrics in the literature are often targeted at questions related to multifunctionality, 364 but do not directly address a univariate measure of simultaneous function (e.g., the approach of 365 Dooley et al. 2015 seeks to look at a multivariate response and quantify tradeoffs rather than 366 provide a single univariate measure). Rather, we have sought to provide a metric that builds on 367 past work while providing a robust foundation for the future of multifunctionality research. 368 369 Ultimately, we feel that the proliferation of univariate multifunctionality metrics without strong 370 theoretical underpinning has caused a great deal of confusion about how to measure 371 multifunctionality. We hope that this piece will provide the field of multifunctionality with a way 372 out of its current state of division and confusion. Further, we hope it provides food for additional 373 theory that addresses the causes and consequences of ecosystem multifunctionality, something 374 that is currently sorely lacking but highly relevant to policy and management (e.g., the efforts of 375 the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services). We 376 have been heartened by the idea, leaving the cradle of the field of biodiversity and ecosystem 377 function, and feel that it has the promise to provide a holistic unifying concept for anyone 378 interested in capturing a snapshot of system dynamics in a single meaningful metric with direct 379 ties to the beautifully developing field of diversity partitioning. Much is to be done on honing the 380 particulars of this approach, but we feel it offers a strong theory-driven unified approach that will 381 In addition to the appendices, see https://github.com/jebyrnes/new_multifunc_metric/ for 394 additional code, notes, and co-author discussions related to this manuscript. The code to 395 implement all methods here is part of the multifunc package in R and can be found at 396 https://jebyrnes.github.io/multifunc/ with tutorials. 397 398 Table 1: Definitions of parameters used to define the effective multifunctionality of an 500 ecosystem. 501 Parameter Description k The number of functions measured.

Fi
Level of function i, standardized by dividing by its maximum observed or theoretical value. Quantifies the effective number of functions were all functions independent and provided at the same level.

A
The arithmetic mean of the k measured functions.
q Mef Effective multifunctionality of order q. The cumulative performance of the system were it composed of functions all performing at equal levels.
q M Standardized effective multifunctionality of order q. Expresses effective multifunctioning as a fraction of the maximum attainable value when all functions are performing at their maximum (i.e., all functions are 1). It is obtained by dividing q Mef by k to provide a metric comparable across systems, although care is required when comparing different sets of functions.
D A dissimiliarity matrix describing the independence of functions from each other. Based on a priori mechanistic information (e.g., physiological or evolutionary constraints) or empirical estimates of among functions correlation matrix, R ( = 0.5(1 − )). Empirical estimates of R can be obtained from the data set being analyzed or additional data.
τ Threshold dissimilarity between two functions for them to be considered independent in the calculation of q N. The dissimilarity matrix is truncated, so that any values above τ are converted to 1. The average weighted pairwise distance among functions, dmean is the recommended default, following Chao et al. (2019). If τ = 1, no truncation occurs; if τ = 0.5, all negatively correlated functions are considered independent. If τ = 0, all functions are treated as completely independent and we recover the value of q N uncorrected for correlations among functions. 502