A new family of standardized and symmetric indices for measuring the intensity and importance of plant neighbour effects

Measurements of competition and facilitation between plants often rely upon intensity and importance indices that quantify the net effect of neighbours on the performance of a target plant. A systematic analysis of the mathematical behaviour of the indices is lacking and leads to structural pitfalls, e.g. statistical problems detected in importance indices. We summarize and analyse the mathematical properties that the indices should display. We review the properties of the commonly used indices focusing on standardization and symmetry, which are necessary to avoid compromising data interpretation. We introduce a new family of indices ‘Neighbour‐effect Indices’ that meet all the proposed properties. Considering the commonly used indices, none of the importance indices are standardized, and only RII (Relative Interaction Index) displays all the required mathematical properties. The existing indices show two types of symmetries, namely, additive or commutative, which are currently confounded, potentially resulting in misleading interpretations. Our Neighbour‐effect Indices encompass two intensity and two importance indices that are standardized and have different and defined symmetries. Our new additive intensity index, NIntA, is the first of its kind, and it is generally more suitable for assessing competition and facilitation intensity than the widely used RII, which may underestimate facilitation. Our new standardized importance indices solve the main statistical problems that are known to affect Cimp and Iimp. Intensity and importance with the same symmetry should be used within the same study. The Neighbour‐effect Indices, sharing the same formulation, will allow for unbiased comparisons between intensity and importance, and between types of symmetry.


Introduction
Plant-plant interactions exert a major impact on the structure and composition of plant communities (Bertness & Callaway 1994;Callaway 1995;. While earlier studies focused on the changing role of competition between plants across productivity gradients (Grime 1977;Tilman 1982), recently additional emphasis on the role of facilitative interactions has been introduced (Callaway & Walker 1997;Pugnaire & Luque 2001;Bruno, Stachowicz & Bertness 2003;Callaway 2007;McIntire & Fajardo 2014;Michalet & Pugnaire 2016). Another shift in focus has occurred between earlier studies considering the intensity of the net effect of plant interactions, and more recent discussions assessing also their importance, defined as the effect of the interactions on performance, relative to the environmental effects on performance (Welden & Slauson 1986;Brooker & Kikividze 2008; but see also, e.g. Freckleton, Watkinson & Rees 2009;Damgaard & Fayolle species with and without neighbours are commonly used, including, among others: the relative competition index (RCI; Wilson & Keddy 1986), the relative neighbour effect (RNE; Markham & Chanway 1996), and the relative interaction index (RII; Armas, Ordiales & Pugnaire 2004) for intensity; the competition importance (C' imp , Seifan et al. 2010; modified from C imp Brooker et al. 2005) and I imp (Seifan et al. 2010) for importance.
Most of these indices, especially the ones conceived for assessing competition, lack some crucial properties to accurately represent the effects of facilitation. The need of comparing the incidence of negative vs. positive effects has led to the realization that indices need to be symmetric (Armas, Ordiales & Pugnaire 2004), or, in other words, they need to return the same value, but with opposite sign, for equivalent cases of competition and facilitation. This understanding has brought in an important correction to the intensity and importance indices, resulting in the introduction of the widely used RII (Armas, Ordiales & Pugnaire 2004) and I imp (Seifan et al. 2010), respectively.
An index is used to quantify an effect by placing it on a certain scale. This scale is inherent to the index, and it has been required to be bounded (Hedges, Gurevitch & Curtis 1999;Armas, Ordiales & Pugnaire 2004;G omez-Aparicio et al. 2004;Oksanen, Sammul & M€ agi 2006;Seifan et al. 2010;Mingo 2014b) between a minimum and maximum value. Usually, these values are required to be even, i.e. equal in absolute values (Armas, Ordiales & Pugnaire 2004;Seifan et al. 2010;Mingo 2014b). However, it has been shown recently that uneven boundaries are inevitable for I imp (Mingo 2014b), and therefore importance index ranges should be reported with results (Seifan & Seifan 2015). The boundary (un)evenness is connected to the index symmetries (Seifan & Seifan 2015), which can be of different types (e.g. Seifan & Seifan 2015 defined recently 'additive' and 'multiplicative' symmetries). These symmetries are used interchangeably in intensity and importance indices so far, and the implications for their ecological meaning have not been clarified. Also, the problematic nature of multiplicative importance indices has been established (Seifan & Seifan 2015), but no alternative for this symmetry has been proposed. In general, we lack a comprehensive analysis of the mathematical properties of intensity and importance indices, while importance indices still suffer from unresolved statistical problems (Rees, Childs & Freckleton 2012;Brooker et al. 2013;Mingo 2014a).
A careful understanding of the index properties, including boundaries and symmetry, is required for all ecological applications, and is especially relevant now, as the complexity of the analysis based on intensity or importance indices has progressively increased. The first studies merely asserted the existence of facilitation and/or looked for the general trend of neighbour effects over stress gradients (e.g. indices as increasing, decreasing or hump-shaped functions of stress, e.g. Callaway et al. 2002;Brooker et al. 2006). Recently, the numerical values of the indices are used in more involved calculations, such as relating them with other types of ecosystem properties (Wang et al. 2014), using them to estimate the relevance of indirect interactions (Michalet et al. 2015), to compare beneficiary effects with the feedback effect of the target species on its benefactor (Sch€ ob et al. 2014), to estimate the impact of climate change on plant communities (Metz & Tielb€ orger 2016) and as parameters to simulate plant interactions (Xiao et al. 2012;Bulleri et al. 2014) or in large-scale metastudies (He, Bertness & Altieri 2013). In several cases, studies use more than one index to analyse a single data set because they recognized that each index provided different insights on the properties of the system (e.g. Goldberg et al. 1999;le Roux & McGeoch 2010;He et al. 2012).
Here, we provide guidelines to choose the appropriate index (or indices) for a given study. To achieve our aim, we first list a set of mathematical properties that any intensity or importance index should display. We show that index symmetry, which is fundamental for determining the index ecological meaning, is still not completely defined for importance. Surprisingly, we find that all of the current indices for intensity and importance fail to show all of the necessary properties (except for the intensity index RII). To overcome these problems, we introduce a new general set of indices, encompassing intensity and importance, which display all of the properties, and we compare them in terms of their symmetry and their ecological meaning. We finally illustrate the use of these indices, and present some general guidelines to their use, with the help of examples where the indices are applied to previously published data.

Mathematical properties of intensity and importance indices
Here, we define a minimal set of properties that any neighbour-effect index should display. We introduce these properties having in mind that indices need to be mathematically consistent and intuitive, and partly based on considerations reported in previous works (Hedges, Gurevitch & Curtis 1999;Armas, Ordiales & Pugnaire 2004;Seifan et al. 2010;Mingo 2014a,b;Seifan & Seifan 2015). Tables 2 and 3 summarize how the indices mentioned in this study perform for each criterion.

S T A N D A R D I Z A T I O N
Any index should reflect the effect of neighbours in relation to a certain species-specific quantity that allows for comparisons across a large set of conditions. For instance, the relevance of a certain increase in biomass (due to facilitation) will depend on the type of plant studied. For example, an increase of 2 biomass units should weigh more for a small plant (e.g. biomass change from 2 to 4 biomass units, Fig. 1a left) than for a large plant (biomass change from 100 to 102 biomass units, Fig. 1a right). between competition and facilitation (Hedges, Gurevitch & Curtis 1999;Armas, Ordiales & Pugnaire 2004;Seifan et al. 2010). Even though this concept of equivalence may seem ecologically difficult to define, it is, however, implicitly present in all the indices.
Let us consider a 'classical' experimental set-up, where the performance P of the target species is measured in: (i) isolation, i.e. the control case (P ÀN ; see Table 1 for symbols used in this study), and (ii) in the presence of neighbours (P +N ). Indices displaying different symmetries will differ in the following way: 1 An additively symmetric index (called additive index hereafter for brevity) provides equal but opposite values when equivalent 'amounts' of competition and facilitation modify the performance in the absence of neighbours (P ÀN ), by the same amount DP, which is defined as the difference between the performance with and without neighbours (DP = P +N À P ÀN ; e.g., 4 biomass units in Fig. 1b, top panel). An additive index assigns the same absolute value to these two cases, but with opposite sign. Additive symmetry is the most intuitive symmetry in most experimental set-ups because an equal amount of biomass increase or decrease corresponds to the same but opposite index values. 2 A commutatively symmetric index (commutative index in the following) provides equal but opposite values when the target performances alone (P ÀN ) and under a competitive (or facilitative) neighbour, (P +N ), are exchanged (Fig. 1b). For example (Fig. 1b, bottom panel), an intensity commutative index will indicate that two treatments are opposite and equivalent when one reduces biomass from 10 to 2Á5 units while the other increases biomass from 2Á5 to 10 units. Commutative symmetry coincides with multiplicative (sensu Seifan & Seifan 2015) for intensity indices (see Appendix S1, Supporting Information for a formal proof). However, commutative symmetry has the advantage of being applicable also for importance indices, while multiplicative symmetry has flows for these (Seifan & Seifan 2015). Noticeably, we show that it is not possible to define multiplicative standardized importance indices, as they would display counterintuitive values (see Appendix S2 for a proof, and an illustration in Fig. S4).  Performance of the target species with neighbours DP = P +N À P ÀN The total impact of neighbours P sum = P +N + P ÀN Sum of the performances of the target species with and without neighbours P w Generic weighting term in Neighbour-effect Indices MP ÀN The maximum value of target species without neighbours along the gradient MP sum The maximum value of the sum of the performances of the target species with and without neighbours along the gradient. NInd S Neighbour-effect Indices NInt A Neighbour-effect Intensity index with additive symmetry NInt C Neighbour-effect Intensity index with commutative symmetry NImp A Neighbour-effect Importance index with additive symmetry NImp C Neighbour-effect Importance index with commutative symmetry

B O U N D E D N E S S
The limits of the indices should be finite (Hedges, Gurevitch & Curtis 1999;Armas, Ordiales & Pugnaire 2004;G omez-Aparicio et al. 2004;Oksanen, Sammul & M€ agi 2006), providing a fixed frame so that values of different experiments can be compared. Finite boundaries are usually standardized to À1 and 1, implicitly assuming that limits need to be even as well (e.g. Mingo 2014b). We argue here that for both intensity and importance defining indices with even boundaries is fundamentally not always possible because of the intrinsic differences between competitive and facilitative effects. In the classical experimental set-up described above ('Symmetry' section), neighbours can reduce the performance of the target species to the point of competitive exclusion. Thus, the difference between the performance with and without neighbours (DP) can vary between zero and a finite negative value (ÀP ÀN ≤ DP ≤ 0). Facilitative neighbours, however, can a priori increase the performance of a target species to a non-finite larger value (i.e. P +N < ∞ and 0 ≤ DP < ∞), which is not as clearly bounded as 0, as is the case with competitive neighbours. For this fairly common experimental set-up, competitive and facilitative effects are not additively symmetric (as defined by Seifan & Seifan 2015 and above). Therefore, for additive indices we do not consider equal limits for facilitation and competition as a necessary requirement for a neighbour-effect index.

E R R O R E S T I M A T I O N S A N D S T A T I S T I C A L S I G N I F I C A N C E
To be used in statistical analyses, indices should allow the calculation of their confidence interval, which requires continuity and differentiability. For this reason, indices that include terms that vary in case of facilitation or competition (e.g. max or min functions as in RNE or C' imp ) are not preferred (Armas, Ordiales & Pugnaire 2004).
In addition to the formal properties just mentioned, which can be mathematically defined and checked (see Appendix S1), we maintain that indices should pass a more subjective test: they should have an intuitive interpretation. An index transforms data by simplifying the information included. The formulas used to calculate these indices besides satisfying the abovementioned properties should have a fully understandable ecological meaning. In particular, index sensitivity, i.e. how it responds to small changes in the performances, should be checked to verify that it corresponds to its expected behaviour (see Appendix S1).

Current indices and their properties
By inventorying the properties of the intensity and importance indices most commonly used in the literature, we found that only one of the intensity indices (RII) showed all of the abovelisted properties, and that none of the importance indices did (see Table 2). All of the intensity indices were standardized, but, strikingly, none of the importance indices (C' imp and I imp ) were. This problem in the importance indices resulted from the definition of importance as the ratio between the neighbourdriven change in performance, and the change in performance driven by 'all the factors in the environment that influence plant success' (Brooker & Kikividze 2008). Using this approach, the total impact was not expressed relatively to the performance of the target species. Lack of standardization in importance indices means that one cannot use them to compare across situations where species, or other conditions, differ. Referring to the previous example (Fig. 1a, and also Figs 2a,b), an equal change in biomass has a very different relevance for a small or a large plant. This is completely not captured by the current indices, which instead give the same importance value to both situations (see, e.g. I imp in Fig. 2a,b). This is a key limitation of the current importance indices that has not been recognized so far, and for which a solution is urgently needed.
The indices originally designed to assess competition (such as RCI and lnRR for intensity, and C imp for importance) were not bounded as they did not have a finite limit for extreme cases of facilitation (Armas, Ordiales & Pugnaire 2004;Seifan et al. 2010). The unboundedness of RCI and lnRR is problematic (Callaway 2007) and, for example, made RCI inapplicable in a large number of cases (G omez- Aparicio et al. 2004;Oksanen, Sammul & M€ agi 2006). Some corrections have been proposed a posteriori for some of these indices which, however, introduced new issues. The use of terms such as the maximum or minimum performance added, for example, to correct RNE and C' imp prevented the calculation of error propagation (as pointed out by Armas, Ordiales & Pugnaire 2004 for RNE; see also Seifan et al. 2010). In fact, RNE can be rewritten to avoid such problem, as we show below. All the bounded indices have even boundaries, À1 and 1, with the exception of I imp whose practical boundaries are À1 and 1/2 (Mingo 2014a,b).
In summary, this examination of current index properties indicated that we were currently lacking a standardized importance index, and that only one intensity index (RII) showed the four required mathematical properties. It is noteworthy that RII is bounded and has a commutative (and thus multiplicative) symmetry, but that none of the additive symmetric intensity indices is bounded, even though, as we noted above, additive symmetry provides the most intuitive interpretation for most standard experimental set-ups (Fig. 1b). To fulfil the need for indices that display all the above-listed properties, in the following section we propose a new family of indices for intensity and importance, including intensity indices with additive or commutative symmetry, and standardized importance indices.

Neighbour-effect Indices
We introduce a family of symmetrical and standardized neighbour-effect indices that we name Neighbour-effect Indices, in short NInd S (where the subscript S highlights their symmetric nature). We choose the word 'effect' to underline that the indices measure the net effects of all the ongoing plant interactions. We define them to have an S-shape (i.e. a sigmoidal function) as: where DP is the total impact of neighbours (P +N À P ÀN ) that is positive for facilitation and negative for competition, and |DP| is its absolute value (i.e. without the sign). P w is a weighting, positive term, which will vary case by case as we discuss below (see Table 1 for notation). *MP AEN is the maximum value of target species with and without neighbours along the gradient (Seifan et al. 2010).
In bold we underline important shortcomings of the indices. For symbols see Table 1. For the expression of indices errors of RII see Table S1 and Note S1. For the sake of a common notation, the signs of RCI, RNE and lnRR have been changed (negative for competition, positive for facilitation).
-6  (a-b) and of a small grass (c-d) are measured. The biomasses are MP ÀN when maximal (100 and 10 for the tree and grass, respectively), P ÀN in a certain stressful environment (98 and 8, respectively) and P +N with neighbours in the same stressful environment (92 and 2, respectively). The same total impacts of stress (decline of 2) and of neighbours (decline of 6) are observed for both species. The standardized NImp A yields to a lower effect of neighbours on the larger plant (a-b) than on the small plant (c-d), whereas the non-standardized I imp yields to equal values for both plants. The weighing factors, the denominators of the indices, are displayed in panels (b) and (d), black arrows for I imp (equal length in (b-d)) and grey arrows for NImp A (longer arrows in (b) than in (d)). All the biomass values are here expressed in a generic biomass unit. See also Table 1 for symbols.
Thanks to their sigmoidal formula (eqn 1), the neighbour-effect indices have one general formulation, and show all of the properties listed in the previous section, namely, standardization, boundedness, symmetry and allow for error estimation (see Appendix S1). Other indices proposed before also use the sigmoidal function either explicitly (as in I imp , see Table 2 for a rewriting of the index according to our notation) or implicitly (RNE, as we show below). Our indices are bounded and differentiable functions of DP (as we define P w as positive). Thus, our indices overcome all of the common limitations of previous intensity and importance indices. We must note here that the NInd S indices are defined only when plant performance is evaluated with traits that have non-negative values (P ≥ 0), as is the case for most of the other common indices (Kikvidze & Armas 2010).
The NInd S have a general formulation that, depending on the definition of P w , can lead to intensity or importance indices, showing either additive or commutative symmetry. For intensity indices the weighting term P w only includes the standardization for size, whereas for importance indices P w consists of two weighting terms: the total impact of the environment (stress) and the standardization for size. These weighting terms are specified in the next sections. The general formulation of all the indices within the same family allows for comparing intensity and importance, or indices with different symmetries without introducing spurious differences due to different mathematical formulations of the individual indices.

D E F I N I T I O N O F N E I G H B O U R -E F F E C T I N T E N S I T Y I N D I C E S
The effect of neighbours on the performance of a target species is captured within intensity indices by weighting the total impact of neighbours across a wide variety of experimental conditions with respect to a reference performance of the target species. Different choices for this reference performance lead us to define here below an additive index and a commutative (and multiplicative) index.

Additive intensity index
We follow here the common practice of using the performance without neighbours as a reference performance (as, e.g. RCI). In our notation, this leads to setting P w = P ÀN , thus obtaining NInt A (Neighbour-effect Intensity index with Additive symmetry): This index will likely be the most common choice for experiments (for reasons that we will illustrate below). This index is À1 for competitive exclusion, and +2 for obligate facilitation and for an infinite increase in the performance of the target species with neighbours.

Commutative intensity index
Here, we will use the average of plant performances measured with and without neighbours as a reference term for the impacts of neighbours across studies (similarly to former commutative and multiplicative intensity indices, e.g. RII). In our notation we can choose P w = P sum = P ÀN + P +N (i.e. twice the average of the plant performances with and without neighbours), thus obtaining NInt C (Neighbour-effect Intensity index with commutative symmetry): NInt C can be reduced to the known RNE (see Appendix S1) with the advantage that our new formulation (eqn 3) solves its main drawback (Armas, Ordiales & Pugnaire 2004). The values of NInt C are À1 for competitive exclusion, and +1 for obligate facilitation and for an infinite increase in performance of the target species with neighbours.

D E F I N I T I O N O F N E I G H B O U R -E F F E C T I M P O R T A N C E I N D I C E S
Although defined in a more general way (Welden & Slauson 1986), importance has been mainly used for experimental setups including two treatments, where the performance is measured along an environmental gradient and in the absence or presence of neighbours. Importance indices weigh the total impact of neighbours against the total impact of treatments, which include the impact of neighbours and stress (Brooker et al. 2005). We here define two symmetric importance indices, following this definition but taking into account that they also need to be standardized for size (Figs 2 and 3b). The NInd S already include the impact of neighbours in the denominator (|DP| in eqn 1), thus the weighting term, P w , needs to include: (i) an estimation of the impact of the environment on performances and (ii) a term for standardization.

Additive importance index
We define the two components of P w mentioned above as: (i) the difference in reference performances (P ÀN , as for the additive intensity index) between the maximum (MP ÀN ) and a given level of stress (P ÀN ); and (ii) the maximum of the reference performance along the gradient (MP ÀN ). We obtain NImp A (Neighbour-effect Importance with Additive symmetry): (see Appendix S2 for a derivation of the formula). We note that NImp A is closely related to the C imp importance index (Brooker et al. 2005) and even more to the I imp importance index (Seifan et al. 2010) which is also a sigmoidal additive importance index. Our index is a relevant improvement of the former two as it includes an extra term in the denominator which includes standardization for size for the first time.
As importance includes the effect of the environment, we will give the values for competitive exclusion and obligate facilitation when the environment is optimal. In this case, the index is À1 for competitive exclusion (when the performance of the plant alone is equal to the maximum performance, P ÀN = MP ÀN ), and 2/3 for obligate facilitation (when the performance of the plant with neighbours is equal to the maximum performance, P +N = MP ÀN ). Independently of the environment, NImp A is equal to +2 for an infinite increase in the performance of the target plant with neighbours.

Commutative importance index
As for the definition of NInt C , we define the new commutative importance index using as a reference value the sum of the performances with and without neighbours, P sum . The two parts of the weighting term P w are thus (i) the difference in reference performance between the maximum (MP sum ) and a given level of stress (P sum ); and (ii) the maximum of the reference performance along the gradient (MP sum ). In other words, MP sum is defined as the maximum value of the sum of the performances of the target species with and without neighbours along the gradient. We obtain the Neighbour-effect Importance index with commutative symmetry, NImp C , as: As for NImp A , we illustrate here the index values in some simple cases where the effect of the environment can be discarded. In these conditions, NImp C is equal to À1 for competitive exclusion (when the performance of the plant alone is equal to the maximum of the sum of the performances with and without neighbours, P ÀN = MP sum ) and to +1 for obligate facilitation (when the plant performance with neighbours equals the maximum performance without neighbours P +N = MP sum ). NImp C is also equal to +1 for an infinite increase in the performance of the target plant with neighbours (P +N , P sum and MP sum tend to infinite).

Intensity indices
The formulae for the defined intensity indices and their key symmetry boundaries are summarized in Table 3 (first row). The formulae for their variance (calculated by error propagation, Ku 1966) are given in Appendix S1. Here, we only highlight the most important aspects. NInt A is additively symmetric (Fig. 3a) with respect to fixed P ÀN . The different terms in the formula of NInt A depend on the changes in performance related to either standardization or to the effect of facilitation or competition, as we illustrate with a quantitative bar graph in Fig. 3a. This index, being additive, displays uneven boundaries. As we explained above, this is a consequence of the intrinsic asymmetry of competition and facilitation. For NInt A , the additive symmetry leads to considering as equivalent a case with extreme competition (P +N = 0 and thus DP = ÀP ÀN ), and a case where facilitation has the effect of doubling the performance (P +N = 2 P ÀN and DP = P ÀN ). Within this limited region (DP < P ÀN , grey area in Fig. 4), competition and facilitation are equivalent. As a consequence, very large facilitation cases (DP > P ÀN , on the right of the grey region in Fig. 4a) have no equivalently large competition situations as P ÀN cannot be smaller than zero (DP < ÀP ÀN , left of the grey region in Fig. 4). An additive index necessarily translates such asymmetric nature of the neighbours impact into uneven boundaries. The absolute value of the index for a facilitative effect that tends to infinite (NInt A (max) = +2 when P +N ? ∞) is twice the absolute value of the index for maximum competition (NInt A (min) = À1 when P +N ? 0). While this boundary unevenness is a direct consequence of additive symmetry, we must note that the specific maximum value of the index is due to the choice of a sigmoidal formula for the NInd S , which was chosen as it allows for a general formulation of the four indices. In both panels, the total impact (DP, continuous line) is the difference between the performance with (void bars) and without (light grey bars) neighbours. The competitive ('comp') and facilitative ('fac') treatments shown are additively symmetric and, consequently, the quantities P w and the index denominators are equal for both of them and shown only once for each index. (a) Intensity: in NInt A (eqn 2) the weighing value is the performance without neighbours, P ÀN (light grey bar), and the denominator is the former value plus the total impact of neighbours, DP (continuous line). (b) Importance: in NImp A (eqn 4) the weighing performance is the environmental impact (MP ÀN À P ÀN , dashed line) plus the total impact of neighbours, DP (continuous line); the denominator is the sum of the former weighing value and the performance in the optimal environment (MP ÀN , dark grey bar).

ADDITIVE SYMMETRY NInt
NInt C displays commutative (and multiplicative) symmetry and has even boundaries, NInt C (max) = +1 and NInt C (min) = À1. NInt C uses the performance with and without neighbours as a reference (P sum in eqn 3), thereby levelling off the asymmetry of competition and facilitation. We note here that RII has similar characteristics (Armas, Ordiales & Pugnaire 2004) for analogous reasons (see Fig. S1), but see below ('Recommendation and guidelines') for a more detailed comparison between NInt C and RII.

Importance indices
NImp A and NImp C are the first standardized importance indices (Figs 2 and 3b). We report in Table 3 (last row) a summary of the definition and key properties of the Neighboureffect importance indices (the formulae for variance are given in Table S1).
The standardization for size, which we introduce for the first time in importance indices, not only makes them comparable between different experiments but also it corrects one major flaw of previous importance indices. As pointed out by Rees, Childs & Freckleton (2012) and Mingo (2014a), in a system where environmental conditions are almost optimal, but the effect of neighbours is negligible with respect to the performance of the target plant, all former importance indices (e.g. I imp , C' imp ) reach maximum absolute values. Our new importance indices solve this issue, and NImp A and NImp C tend to zero in such a system (Fig S4). This is a consequence of the standardization for size, which weighs the impact of small neighbour effects with respect to the maximum performance of the target plant (see Fig. S5 and Appendix S2 for further details). As a consequence, NImp S standardization is a step forward towards solving the issue of spurious patterns possibly emerging in the importance indices (see Appendix S2).
Overall, Neighbour-effect Importance indices share the same properties as their associated intensity indices. NImp A and NImp C display additive and commutative symmetry (see Appendix S1), respectively. As a consequence of their symmetries, they have uneven (see Appendix S2) and even boundaries, respectively.
We should acknowledge that it may be difficult to assess the theoretical maximum performance of the species with standard experimental set-ups, and thus these values should be assessed using all available information (including other data sets or previously published literature) to find their best proxy. For most relevant experimental designs on plant-plant interactions, the optimal performance without neighbours is also the superior limit for the performance with neighbours (P +N ≤ MP ÀN ) and MP ÀN can be estimated using also the maximum performance, regardless of the presence or absence of neighbours (following Seifan et al. 2010). For example, when facilitation is mediated by the same resource whose availability defines the stress treatment, and the study includes nonstressed conditions, the maximum performance along a gradient always occurs without neighbours, although it might not be experimentally observed. However, this 'practical' upper limit of additive importance indices can be surpassed if the performance with neighbour exceeds the optimal performance without neighbours (P +N > MP ÀN ). The absolute boundaries Table 3. The new family of Neighbour-effect Indices (NInd S ), with the formulas and limit values of the two intensity and the two importance indices, with additive or commutative symmetry NInd S Additive symmetry Commutative symmetry Min À1; Max 1 *Note that for most experimental set-ups MP ÀN can be estimated with the maximum performance available. All the indices are standardized, bounded and allow for error estimation (for the expression of indices errors see Table S1 and Note S1). For symbols see Table 1. See also Note S2 for boundaries of Neighbour-effect Importance indices.  Fig. 4. Values of different intensity indices (NInt A , NInt C and RII) as a function of the total impact of neighbours. The performance of the target species is fixed at 1 unit (P ÀN = 1, or equivalently the horizontal axis represents DP/P ÀN ), and the impact of neighbours ranges from maximum competition (P +N = 0, and thus DP = À1) to large values of facilitation (DP = 20). The index NInt A is additively symmetric (as evident by the symmetric shape) with respect to DP = 0 in the grey square (À1 < DP < 1) and has uneven limit values (minimum equal to À1 and maximum equal to +2). The multiplicative character of commutative symmetry of NInt C and RII is not apparent in this figure because of the axes chosen (see Fig. S1 with a logarithmic scale for comparison). RII and, to a lesser extent, NInt C are further from NInt A at facilitation than at competition, and thus commutative indices, especially RII, underestimate facilitation with respect to competition. All the performance values are here expressed in a generic unit. and the upper 'practical' boundary of the additive importance index (see Appendix S2 for details) are reported in Table 3.

Recommendation and guidelines
We strongly recommend using indices that fulfil all four mathematical properties as previously discussed. Therefore, we cannot recommend the use of indices outside the NInd S family (with the exception of RII), as they fail to show these basic properties. Most importantly, none of the currently used importance indices is standardized, significantly hindering cross-species or cross-studies comparisons. We also recommend for the first time to use pair of indices with the same symmetry for intensity and importance, to avoid spurious effects in the comparison: NInt A with NImp A and NInt C with NImp C (see also Appendix S2 for a standardized commutative importance index related to the intensity index RII). We thus discourage the common practice of using together commutative intensity and additive importance indices (Armas, Rodr ıguez-Echeverr ıa & Pugnaire 2011;Howard, Eldridge & Soliveres 2012;Maalouf et al. 2012;Soliveres, Torices & Maestre 2012;le Bagousse-Pinguet et al. 2014;Bulleri & Piazzi 2015;de Castanho et al. 2015).
Furthermore, in general, we advise to use additive indices (NInt A and NImp A ). In the following, we will motivate this preference, discussing in details how to choose between the two types of symmetry and between NInt C and RII. We will focus on comparing across the intensity indices because they are generally easier to understand (although their properties are shared by their importance counterparts).
To illustrate the relevance of symmetry, we compare NInt A (eqn 2), NInt C (eqn 3) and RII for a fixed control performance (P ÀN ) and a variable impact of a neighbour-adding treatment, ranging from competitive to facilitative neighbours (Fig. 4, from left to right on the horizontal axis). From this visual comparison it is apparent that if the facilitative neighbour impact is within the (additive) range of symmetry (which we define as |DP| ≤ P ÀN , grey area in Fig. 4), the additive NInt A reports symmetrical values for equal competitive or facilitative effects (as it descends from its definition). Within this range, the two commutative indices are more (RII) or less (NInt C ) far from NInt A (see Appendix S3 for a mathematical calculation of this distance). In other words, the commutative indices, and especially RII, tend to underestimate facilitation with respect to equivalent values of competition. This effect is especially relevant for|DP| values larger than 10% of P ÀN (Figs 4, S4 and Appendix S3). If a commutative index is chosen, we recommend using NInt C over RII.
These differences between indices can influence the conclusions extracted from quantitative analysis of experimental data. All the indices will correctly identify the same sign for the neighbour effects (facilitation or competition), and they will order the index values from low to high neighbour effects in the same way (as they all show an increase from left to right, see Fig. 4). However, when averaging several experiments or treatments, including both competitive and facilitative situations, the average net outcome can be biased towards competition when using commutative indices, as a consequence of the abovementioned underestimation of facilitation (especially large for RII). See Appendix S3 for such an example with real data (Metz & Tielb€ orger 2016).
Although we generally recommend additive indices, commutative indices can still be used outside the range of symmetry, when there is a much larger facilitative than competitive effect (DP ≤ P ÀN ). The use of additive indices is instead the only advisable option when the effect of facilitation is within the same order of magnitude as the effect of competition (|DP| ≤ P ÀN ). In the following, we illustrate these two cases with the help of real examples.

A D D I T I V E S Y M M E T R Y F O R I N T E R A C T I O N S I N S I D E T H E A D D I T I V E R A N G E O F S Y M M E T R Y
When |DP| ≤ P ÀN , we strictly recommend the use of additively symmetric indices (NInt A and NImp A ) because in this range there is a correspondence between competitive and facilitative values and, arguably, because additive symmetry is more intuitive and easier to interpret than commutative symmetry. This is apparent by, for instance, re-examining the results of Mont es et al. (2008) Fig. 5. The effect of symmetry when comparing competitive and facilitative treatments on a control species in an illustrative experimental setup. The illustrations represent the experimental design and show the performance of the target species growing with and without neighbours. The table below contains the magnitudes of the total impact of neighbours, |DP|, and the corresponding value for three intensity indices, |NInt A |, |NInt C | and |RII|. The target plant is Ulex parviflorus, with performance when alone (P ÀN = 100). In absolute terms (|DP|) the facilitative ('fac') impact of Quercus coccifera is larger than the competitive ('comp') impact of Cistus albidus (97 > 77). However, the only index which provides larger strength for facilitative effect of Q. coccifera is the additive index NInt A , whereas commutative indices (|NInt C |, and |RII|) indicate a larger competitive than facilitative effect (contrary to the expectations from DP). The performance values have been calculated from the values of RII obtained through digitalization of Fig. 3 in Mont es et al. (2008); all performance values are given by a multiplicatively constant arbitrarily fixed as P ÀN = 100 units, whose value does not influence the results shown. the competitor Cistus albidus and the facilitator Quercus coccifera, has the largest impact on the productivity of the target species, Ulex parviflorus. When looking at the total effect (DP), using the performance of U. parviflorus alone as the reference (i.e. we fix P ÀN = 100 units) the change induced by Q. coccifera (increase of 97 units) is larger than the change due to C. albidus (decrease of 77, units, see Fig. 5). Noticeably, this relationship is captured only by NInt A (|NInt A | equal to 0Á98 and 0Á87 for Q. coccifera and C. albidus, respectively), whereas it is reversed by commutative indices, with a particularly large difference when using RII (with |RII| equal to 0Á32 and 0Á63 for Q. coccifera and C. albidus, respectively; Fig. 5). Commutative indices, and especially RII, can thus be a misleading choice when competition and facilitation cases are comparable in terms of absolute total impacts (|DP|).
Although this recommendation is generally valid, the researchers should check that the equivalence between symmetric values of facilitation and competition is sensible in their experiment. For example, the additive symmetrical indices are the right choice if complete displacement of the target species by the neighbour is not expected, or if this total displacement is considered as relevant as a facilitative effect that doubles the target species performance, but not as relevant as a obligate facilitative effect (if observed).

A D D I T I V E O R C O M M U T A T I V E S Y M M E T R Y F O R I N T E R A C T I O N S O U T S I D E T H E A D D I T I V E R A N G E O F S Y M M E T R Y
In studies with large differences between plants growing alone and with a neighbour, i.e. when DP > P ÀN , commutative indices can substitute, or to use in combination with, additive indices. We show this with an example of cushion plants in alpine environments, which can display a largely facilitative effect on other plants (Cavieres et al. 2014). The nurse plants can increase the abundance of other species even ninefold with respect to open sites (RII $ 0Á8) (Cavieres et al. 2014) and can increase richness up to 2Á9 times (Cavieres & Badano 2009). The additively symmetric NInt A assigns impact values close to 2 for the very large facilitative effects found for cushion plants, which is twice the magnitude it would assign to exclusive competition (i.e. disappearance of the target species). Using the additive index in this case implies that alpine conditions are highly severe for non-cushion species in the open, and competition has a much smaller range than facilitation. On the other hand, the use of a commutative index, such as NInt C or RII (which both have a maximum facilitation value of 1), implies that the total removal of the target species is considered as equivalent to extremely large facilitative impacts and to obligate facilitation. Also, commutative indices might be applied in experiments where different performance estimators rather than biomass are used, such as count data (e.g. seed production, Cavieres et al. 2014), which can span different orders of magnitude. For this type of variable, an increase from 10 to 40 seeds and a decrease from 10 to 2Á5 seeds might be considered as equivalent, if they have, for example, similar and opposite impacts on species survival, and thus a commutative symmetric index could be used.
Finally, we note that our family of indices with the same mathematical properties allows also for impartial comparisons of the impact of different index symmetries as, for example, NInt A and NInt C differ only in their symmetry.

Conclusions
As standardized importance indices and properly defined additively symmetric intensity indices are lacking, we filled these gaps introducing a new family of Neighbour-effect Indices, encompassing standardized intensity and importance indices, with additive (NInt A and NImp A ) and commutative (NInt C and NImp C ) symmetry. We recommend a widespread use of these additively symmetric indices, as their correspondence between competition and facilitation with equal net effect matches common experimental set-ups studying plant interactions. In most of these cases, current commutative intensity indices (mostly RII) are inappropriately applied, and RII underestimates the strength of facilitation when compared to the additive intensity index, NInt A . The use of NInt A will thus help clarifying the actual strength of positive net effects in plant communities. The standardization introduced in the neighbour-effect importance indices makes them a robust tool for studying plant neighbour effects along stress gradient and for cross-study comparisons, solving most of the statistical problems of previous importance indices (Rees, Childs & Freckleton 2012; but see also Brooker et al. 2013), although whether or not NImp S allow the formulation of null models to test plant neighbour effects (Mingo 2014a) is still an open question.
Our family of indices allows for unbiased comparisons of different index symmetries, and of intensity and importance, which then need to be chosen with the same symmetry within the family.
Concluding, the methodological definition of a common family of Neighbour-effect Indices introduced in this study provides a general framework to organize and compare the different measurements devised for net effects among plants. We expect that this precise methodological design will strongly help the development of new perspectives in the field, by overcoming the limitations of the current tools. feedbacks on a previous version of the manuscripts which greatly contributed to the improvement of this study.

Data accessibility
This manuscript does not include any data.