Quantifying neighbour effects on tree growth: Are common ‘competition’ indices biased?

Interactions among neighbouring plants are key determinants of plant growth. To characterise the cumulative effect of all neighbours on the growth of a focal plant, neighbourhoods are often described by ‘competition’ indices. Common competition indices calculate the summed size of neighbour plants (focal‐independent index [FII]) whilst others include the summed ratio of the neighbour size relative to focal plant size (focal dependent). A frequently overlooked statistical artefact is that focal‐dependent indices (FDIs) may lead to biased estimates of neighbourhood effects on plant growth when growth is size dependent. Here, we conduct a literature search to determine the most common index types used to explain neighbour effects on tree growth. We then assess the ability of two common index types—focal dependent and focal independent—to correctly infer neighbourhood effects in (1) observations of tree growth in an experimental forest in south‐east Tasmania, Australia, and (2) an artificially created dataset where tree growth is unrelated to the neighbourhood. Both indices detected the competitive neighbourhood effect on tree growth observed in our own dataset but differed in their conclusion regarding neighbour effects in the simulated data. Despite the simulated dataset being generated so there was no relationship between tree growth and their neighbourhood, the FDI detected strong, competitive neighbourhood effects when intrinsic growth was incorrectly related to tree size. In contrast, when we considered the FII as the neighbourhood metric, we correctly did not detect any neighbourhood effects in the simulated data regardless of how size‐dependent growth was described. Synthesis. ‘Competition’ indices are a useful method to characterise the cumulative neighbourhood effect on plant growth; however, we demonstrate that indices which include the size of the focal plant in their calculation can be biased by an inherent relationship between tree growth and initial size. Whilst this bias typically overstates the strength of competition in determining focal tree growth, we show that it can be mitigated by correctly describing intrinsic growth. We discuss the limitations of both index types, provide recommendations for performing statistical modelling and outline how to check for accurate neighbour inference.


| INTRODUC TI ON
Plant growth describes the change in size of a plant over a given period. The rate of change can be expressed in absolute or relative terms, with both metrics influenced by several factors, the strongest of which being a plant's initial size (Coomes & Allen, 2007;Weiner & Damgaard, 2006;Zhang et al., 2017). Plant size, along with physiological and morphological characteristics, control a plant's ability to capture the essential resources required for growth (i.e. light, water and nutrients; West & Ratkowsky, 2021). The availability of those resources, however, is also strongly influenced by neighbouring plants (Coomes & Allen, 2007). Interactions among neighbouring plants are generally considered to be competitive (Connell, 1983;Grime, 1977;Trinder et al., 2013); however, facilitative interactions are also observed, where plant growth rate is positively related to the presence of neighbours (Callaway et al., 2002;Del Río et al., 2014;Gómez-Aparicio et al., 2004;Pretzsch et al., 2013). Thus, the growth rate of a plant is governed by both its own size and interactions with its neighbours. Correctly describing the relative influence of both components is critical to accurately predicting future plant productivity and plant community dynamics (Canham et al., 2006).
The importance of both plant size and its neighbourhood on plant growth is well known, particularly in forest ecology (West & Ratkowsky, 2021). Common methods of describing tree growth include both size-dependent and neighbour-dependent terms; namely: where g is the growth rate of a given tree expressed in absolute or relative terms. The first term, g 0 , is the intrinsic growth and depends only on the plant's own size (y); it describes plant growth in the absence of neighbours. The second term, n , describes how neighbours impact plant growth. This term depends on a neighbourhood index, denoted k, that represents any measure of a plant's neighbourhood, such as neighbourhood density, summed basal area of neighbours or more complex 'competition indices'. Because the net neighbourhood effect is not necessarily competitive, and may also be facilitative (Vogt et al., 2014), we refer to such indices as neighbourhood indices hereafter. Most neighbourhood indices that have been proposed are calculated by summing the influence of each individual neighbour across all neighbours within a specified distance of a nominated focal plant (Contreras et al., 2011). The influence of each neighbour is commonly determined by its absolute or relative size (with respect to the focal tree) and the distance between the neighbour and the focal tree (Bella, 1971;Boyden et al., 2005;Hegyi, 1974;Luo et al., 2020;Rouvinen & Kuuluvainen, 1997). Neighbours have no impact on growth when k = 0, in which case, n (0) = 1.
Several studies have compared the ability of various neighbourhood indices to explain variation in growth rates among individual plants (Contreras et al., 2011;Rivas et al., 2005), and while no universally applicable index has emerged, perhaps the most commonly adopted is the Hegyi index (Hegyi, 1974). The Hegyi index, and other similar indices (e.g. Bella, 1971;Rouvinen & Kuuluvainen, 1997) include both the size of each neighbour and the size of the focal tree, as well as the distance between them (see Equation 2 below). We refer to such indices as focal-dependent indices (FDIs). Other, less commonly used indices do not include any information on the size of the focal tree (e.g. Boivin et al., 2010;Forrester et al., 2013), instead relying only on neighbour sizes (Equation 3). We refer to these indices as focal-independent indices (FII). Here, we consider the following specific forms for both types of neighbourhood indices: where n is the number of neighbouring trees, y j is the size of neighbouring tree j, y i is the size of the focal tree i (i.e. the tree for which we are predicting growth) and d i,j is the distance between the focal i and the neighbour j.
To test for evidence of neighbourhood effects on plant growth, a common approach is to perform a linear regression on annual growth (absolute or relative) with respect to the chosen neighbourhood metric (e.g. k = FDI or FII). A negative or positive slope of the regression coefficient is interpreted as evidence of competition or facilitation, respectively, with the magnitude of the coefficient indicating the relative strength of the neighbourhood effect. However, this simple approach is potentially problematic if the calculation of the neighbourhood metric includes the size of the focal tree, which is true whenever we use a FDI. The problem arises because the neighbourhood index is negatively related to the size of the focal plant, whereby the effect of each neighbour (y j ) is scaled by the focal plant size (y i ; Equation 2). Thus, if plant growth depends on plant size (i.e. g 0 is not a constant), which we often expect, then size-dependent plant growth will be correlated with the neighbourhood index, even in the absence of neighbour effects. This effect can be observed by considering how annual growth, g, changes with the neighbourhood index, k. Applying calculus and the chain rule to the growth function, g, given by Equation 1, we get: (1) g(y, k) = g 0 (y)g n (k), The true neighbourhood effect is indicated by the sign of dg n /dk; however, this may differ from the sign of dg/dk, which is given by the coefficient associated with the neighbourhood metric reported by a regression analysis. Mis-interpretation of neighbour effects when using a regression analysis can occur under size-dependent growth (i.e. dg 0 /dy ≠ 0), which is commonly reported in a range of plant communities, and strong relationships are typically found in studies of tree growth (Bowman et al., 2013;Weiskittel et al., 2011). The exact allometric relationship between initial tree size and growth rate depends on the size metric, but in general, absolute growth is positively related to tree size (Herault et al., 2011;Weiskittel et al., 2011); whereas relative growth is usually negatively related to tree size (Rees et al., 2010 masking a positive interaction such as facilitation, which is increasingly recognised as a common and important process operating in a range of plant communities (Bertness & Callaway, 1994;Bimler et al., 2018;Britton et al., 2021;Fajardo & McIntire, 2011).
Here, we determine the suitability of incorporating FDIs and FIIs into regression analyses of tree growth to correctly infer neighbour effects. First, we conduct a literature search to determine the prevalence of both index types among studies of tree growth. Second, we assess the ability of both index types to correctly infer neighbour effects in (1) observations of tree growth in an experiment forest and (2) an artificially created dataset where tree growth is unrelated to neighbour effects. We demonstrate the consequences of inadequately describing size-dependent growth when considering the two types of neighbourhood indices. Our dataset from the experimental forest is particularly insightful as it includes focal trees not under the influence of neighbours (k = 0), which allows us to directly characterise patterns of intrinsic growth g 0 (y). Finally, we provide recommendations of how to correctly infer neighbour effects on tree growth.

| Literature search
We conducted a search of peer-reviewed literature using Scopus to identify the most common neighbourhood indices and whether their prevalence had changed over time. Studies until the year 2021 inclusive were obtained using the search string: 'tree' and 'competition ind*' (search date: 06-12-2021). The Scopus search method returns any studies that contained search terms in their title, abstract or keywords. This method returned 497 studies and the title and abstract of each paper were then screened to ensure the relevance of each study.
Using a random subset of 100 studies from the relevant Scopus search results, we classified the types of neighbourhood indices used in each study. Indices were classified as 'focal dependent' if they explicitly included the size of the focal tree in any part of its calculation or if they included only neighbours that were bigger than the focal tree. Indices that only included neighbours bigger than the focal tree were considered as focal dependent because the index value is determined, in part, by the size of the focal tree. Indices that did not include any measure of the focal tree, and were therefore related to the neighbourhood alone, were classified as 'focal independent'. We also determined the growth response (i.e. absolute, relative or both) used in each study, and whether the size of the focal tree was included as a separate, independent predictor in a regression analysis.

| Study site and data collection
We used tree growth data from the Australian Forest Evenness 2 m radius neighbourhood (12.6 m 2 total area). We refer to the focal trees in these neighbourhoods as 'non-isolated'. The 2 m radius was substantially larger than a previous recommendation of 40 times the mean focal tree diameter (0.95 cm; Sutherland et al., 1991) and was selected as the largest neighbourhood within which we could feasibly sample all neighbouring plants. For every tree in the neighbourhood sampling area (focal and neighbours), we recorded the height (m), diameter at breast height (DBH; cm) and GPS location (accuracy of <10 cm). In total, 114 non-isolated focal trees and 7854 neighbours were mapped and measured, capturing a range of community compositions ranging in density from 0.8 to 13.5 plants m −2 (Supporting Information Figure S1). In addition, in winter 2018 we measured the height and DBH of 34 focal trees that had no neighbours within 2 m, which we refer to as 'isolated' focal trees. All focal trees (non-isolated and isolated) were remeasured in winter 2019, 2020 and 2021 to determine 3 years of annual incremental growth.

| Modelling growth
We modelled annual change in tree basal area (BA; cm 2 ), which we calculated from DBH measurements. Let y i,t to denote the observed size (BA) of focal tree i in year t. For each tree, we calculated annual absolute basal area increment (BAIa = y i,t + 1 -y i,t ; cm 2 year −1 ) and relative basal area increment (BAIr = y i,t + 1 /y i,t ; year −1 ). In the instances where we detected negative BAIa and BAIr values, we assumed this to be a result of measurement error associated with repeated sampling rather than negative growth because trees of this age are growing rapidly. Observed negative growth, despite being uncommon, is problematic when performing our regression analyses (see below), and thus, we dealt with negative growth in two ways.
First, tree sizes that implied negative growth were removed from the dataset. Second, we replaced the negative values with the lowest positive absolute growth observed for each species in each year.
Importantly, our main conclusions did not differ depending on which approach we adopted (Supporting Information Table S1) and thus, here we present simpler option of excluding negative growth data.

| Model of intrinsic growth
Isolated focal tree growth data were used to determine the growth metric (BAIa or BAIr) and the regression model that could adequately describe intrinsic growth, g 0 (y). These data correspond to our general growth model (Equation 1) when the neighbour index, k, is set to zero. Annual growth was determined for 20 isolated E. delegatensis and 14 isolated E. regnans trees. It was clear in the raw data that both species exhibited similar size-dependent growth across years (see Figure 2a). For both growth metrics we considered the possibility of an exponential or power relationship with BA; however, neither of these relationships can describe BAIr for our study as many BAIr observations were near their lower bound of 1, which is not imposed by either relation. Instead, we found that a power relationship well described the relation between BAIa and BA (see Figure 2c). Therefore, our statistical analyses consider BAIa as the growth response hereafter.

| Observed neighbourhoods
To identify whether annual growth rates were impacted by neighbours in the observed dataset, the isolated focal plant data were extended to include the non-isolated focal plant observations and the same mixed effect linear model described above was fitted to this extended dataset. Additionally, a second model was fit that also included an additional binary fixed factor indicating whether the focal tree was isolated or not. We compared Akaike information criterion (AIC) between the two models to determine whether the presence of neighbours improved the model's ability to predict BAIa.
Next, we assessed the ability of the two indices (FDI and FII) to infer the true neighbour effects on growth we observed in the extended dataset. We compared the AIC of three models: a null model that did not include a neighbourhood metric, and two models that differed only in their fixed effect neighbourhood metric, FDI or FII (both log-transformed). The growth response in all three models was log-transformed BAIa and included log-transformed BA as a fixed effect, species and year as fixed factors and a neighbourhood-level random effect to account for repeated sampling.

| Simulated neighbourhoods
To assess the ability of the two neighbourhood indices (FII and FDI) to correctly infer the absence of neighbour impacts on growth, we used our observed data to construct a simulated dataset where there was no relationship between neighbourhood community composition and focal tree growth. Simulated datasets were generated by assigning a new, random set of neighbours to each non-isolated focal tree as follows. First, the number of neighbours assigned to each focal tree was determined by bootstrapping the observed distribution of neighbourhood sizes. Next, the size and distance to the focal tree, for each randomly generated neighbour, was determined by bootstrapping these traits across all observed neighbours. Thus, in each simulated dataset, the characteristics of the neighbourhoods (i.e. tree density, tree sizes and distances) were consistent with those observed; however, in this case, the observed growth of each focal tree should now be unrelated to their neighbourhood. Using the randomly generated neighbourhoods, we calculated the two neigh- We tested our prediction that focal-dependent neighbourhood indices may imply neighbour effects, even when absent, by comparing models fit to the randomly simulated neighbourhood data.
Specifically, we considered three descriptions of growth, and for each description we compared three linear models that differed in that they either ignored neighbours or included one of either FDI or FII as a log-transformed predictor. The three descriptions of growth differed in how tree size was included in the linear model describing log(BAIa): (1) tree BA was ignored, (2) addition of a BA term, or (3) addition of a log(BA) term. The three model descriptions correspond to the assumptions that BAIa is (1) constant with respect to tree size (BA), (2) exponentially related to size, and (3) a power relation with size. All models included a fixed effect for year and a neighbourhoodlevel random effect to account for repeated measurements on each focal plant.
For each of the three assumptions of growth, we fit the three neighbourhood models to 1000 randomly simulated neighbourhood datasets. For each dataset, we performed an AIC analysis to determine which of the three neighbourhood models (no neighbourhood metric, FII or FDI) best described the data. For all AIC analyses, we ranked our models based on AIC, disregarding all models with △AIC > 6 from the minimum AIC calculated, and we avoided selecting overly complex models by disregarding models having a higher AIC than a simpler nested model (Richards, 2008). All data analysis was performed using the R programming language (v4.0.4; R Development Core Team, 2021).

| Literature search
The annual number of studies referencing neighbourhood indices has increased steadily over time ( Figure 1a). Since 1974-the year that the Hegyi index was first presented-484 relevant papers have been published that met our Scopus search criteria (Figure 1a). In the random subset of 100 studies that were summarised in detail (20.7% of the total returned studies), 93 explicitly used neighbourhood indices to assess neighbourhood effects on tree biology. In these 93 studies, the neighbourhood index type used, and growth response examined, varied ( Figure 1b,c). FDIs were the most common (47.3%) index type, followed by studies that used both FDIs and FIIs (30.1%, Figure 1b). A relatively small proportion of studies used FIIs alone (19.4%) or other index types that did not fit into either FDI or FII categories (3.2%; Figure 1b). The response each study attempted to explain or predict was predominantly absolute growth, followed by other tree responses to competition (e.g. tree mortality or crown architecture; Figure 1c). Relative growth and a combination of relative and absolute growth were comparatively uncommon ( Figure 1c). 72% of the studies that related neighbour indices to tree biology included initial focal tree size as a predictor in their model; however, more than one-quarter of these studies (28%; Figure 2) omitted size as a predictor or provided insufficient information to determine their model structure. Thus, most tree growth studies explored how absolute growth was influenced by a neighbourhood metric that included the size of the focal tree in its calculation (FDI) despite such indices being potentially problematic when size-dependent growth is incorrectly described.

| Intrinsic growth
The ability of a FDI to correctly infer neighbour effects on growth hinges on whether intrinsic growth is size dependent and adequately modelled.
Therefore, we examined isolated focal tree growth (i.e. growth in the absence of neighbours) to determine the pattern of intrinsic growth in E. delegatensis and E. regnans in the AFEX. Intrinsic growth of both eucalypts was similar and strongly size dependent (Figure 2), whereby absolute and relative basal area increment (BAIa and BAIr, respectively) were both highly correlated with tree size (BA) at the start of each sampling period (Figure 2). BAIa was positively related to BA, whereas BAIr showed a negative relationship. Importantly, the data show that BAIa can be well described by a power relationship with BA ( Figure 2; Supporting Information

| Isolated vs non-isolated focal tree growth relationship
The growth rate of focal trees surrounded by neighbours (nonisolated) differed from that of isolated trees such that the inclusion of the binary fixed factor of neighbour presence considerably improved model fit (△AIC 48.55; Figure 3). Like isolated focal trees, size-dependent growth of non-isolated trees was well described by a power relationship between BAIa and BA ( Figure 3). However, nonisolated focal tree growth was suppressed by the presence of neighbours, whereby the growth of focal trees with neighbours was lower than that of similarly sized, isolated focal trees (Figure 3).

| Neighbourhood effects in the observed dataset
Modelling the neighbour effect on growth in the observed dataset using either neighbourhood index (FDI and FII), and including focal

| Neighbourhood effects in the simulated datasets
When we analysed 1000 simulated datasets, where we ensured that there was no relationship between focal tree growth and their neighbours, we frequently and incorrectly detected evidence of neighbourhood effects depending on how the growth model was described and which neighbourhood index was used ( and year t , (b) and (c) show the relationship between tree size and relative and absolute basal area increment (BAIa), respectively. All plants were isolated from neighbours by at least 2 m. Solid line depicts the best fitting statistical model of growth, which assumes annual change in absolute growth in basal area (BAIa) is a power relation with size (BA) and growth is the same for both species.

F I G U R E 3 Observed size-dependent absolute annual growth for Eucalyptus delegatensis (filled points) and
Eucalyptus regnans (open points) for three annual periods: 2018-2019 (circles), 2019-2020 (squares) and 2020-2021 (diamonds). Isolated focal trees (without neighbours; Figure 2c) are shown by the light grey points, and nonisolated focal trees (with neighbours) are shown by the dark points.
FDI was always selected as the model that best described the data in all 1000 datasets (FDI; Table 1). In every instance, the sign of the FDI effect was negative, indicating a strong, consistent competitive effect on focal growth (Figure 4b). In contrast, neighbourhood effects were never detected using the FII, as the FII model was never selected as a model that best fit the data, as should never have been the case (FII ; Table 1).
Second, including tree size as a separate predictor, but incorrectly assuming that BAIa was exponentially related to focal tree size rather than via a power relationship, also often incorrectly showed a strong, competitive effect of neighbours on focal tree growth (  Table 1). The selected model set also occasionally included the model without neighbour effects, and the model that described neighbour effects via the focal-independent neighbour index (FII ;   Table 1); however, it was comparatively rare for these models to be selected. Similarly to when growth was assumed to be size independent, when intrinsic growth was exponentially related to focal tree size, the FDI effect was always estimated to be negative (Table 1) indicating a competitive neighbour effect (Figure 4b). Although it was very rare to select the model that included FII, when selected, neighbour effects were estimated to be positively related to focal tree growth (Table 1), although this was a much weaker relationship ( Figure 4a).
In contrast to the two previous situations, we correctly found no evidence that neighbours impacted focal tree growth when intrinsic growth was modelled as a power relation with size ( Table 1, bottom row). In this case, AIC analyses always correctly selected the model that ignored neighbour indices (None) and only selected a model that incorporated a neighbour index less than 5% of the time ( Table 1).

| DISCUSS ION
Here, we show that the use of neighbourhood indices to investigate the effects of plant neighbour interactions on tree growth is increasingly common in the literature, and the most frequent type of index includes the size of the focal tree in their calculation, which we termed FDIs (e.g. Hegyi, 1974). Indices that excluded the size of the focal plant in their calculation, FIIs, were comparatively rare. Whilst both indices were able to detect observed competitive neighbour effects on focal tree growth in our own dataset, the two index types (FDI and FII) differed in their ability to correctly identify the absence of neighbour effects in the simulated dataset. When using the FDI, our analyses of the simulated datasets demonstrated that reliably and correctly inferring the absence of neighbour effects relied on correctly describing intrinsic growth. In both cases where growth was incorrectly related to tree size, the FDI detected strong, competitive neighbourhood effects where they were truly absent. In contrast, when the model considered FII as the neighbourhood metric, we did not detect strong neighbourhood effects regardless of how size-dependent growth was described and hence, the FII was more robust to the manner in which growth was modelled.
Our findings demonstrated that adequately describing the sizedependent growth relationship firstly required the inclusion of a size term and secondly, correctly relating the size term to the subsequent growth of each tree. First, most studies that used FDIs in the random subset of our literature search included a size term in their regression analysis; however, 28% of studies excluded the term or did not explicitly say either way. Second, including a size-dependent term in the regression analyses did not necessarily lead to correct neighbour inference in our simulated dataset, and we found vastly different conclusions depending on whether growth was modelled as an exponential or power relationship with tree size. The correct sizegrowth relationship is therefore vital when using FDIs to determine TA B L E 1 Summary of replicated AIC analyses assessing evidence for neighbourhood effects when neighbour effects were removed from the datasets. Three sets of AIC analyses are presented that differ in how intrinsic growth was modelled: size independent, and exponential and power relationship with size. For each treatment of intrinsic growth, three neighbourhood models were considered that differed in how they described the effect of neighbours: no neighbour effect (None), and the two neighbour indices: FII and FDI (both log-transformed). For each intrinsic growth treatment, 1000 datasets were randomly simulated so that neighbours had no effect on growth, and for each simulation, AIC was used to select among the three models that differed in how they incorporated neighbour effects. Bold values indicate the number of times each model was included in the AIC selected set. Correct inference occurs when the None model (no neighbour effect) is in the selected set. Bracketed terms indicate how often the neighbour index was estimated to be positive (+) or negative (−) if selected. Model formulae that were used to describe the intrinsic component are provided in the table footnote.

Growth model
Neighbourhood index
neighbour effects on tree growth, but it is difficult, if not impossible, to determine whether the correct growth relationship is used in published analyses of neighbour effects on tree growth. Determining the relationship between tree size and growth in the absence of neighbours (i.e. intrinsic growth) can be extremely challenging in natural forest ecosystems, not least because size-dependent growth relationships differ with tree age (Bowman et al., 2013) and neighbour effects may also vary with plant age (Miriti, 2006). Thus, it is not always clear how growth should be modelled with respect to tree size, which risks incorrectly inferring neighbour effects when using an FDI. For example, when investigating the possibility of neighbourhood effects on tree growth in our simulated dataset, and exponentially relating initial tree size with growth, using an FDI would lead to the conclusion that there was a strong, suppressive effect of neighbours on absolute growth when in fact, no relationship was present. In contrast, when growth was adequately described by a power relationship with initial tree size, neighbourhood influences were correctly attributed using either the FDI or FII. These findings highlight the importance of correctly modelling size-dependent intrinsic growth, as well as the risk of poor mechanistic inference when incorporating a predictor that includes focal size, such as FDIs.
We have shown that modelling neighbour effects using FDIs can lead to false inference. The reason why model fitting may incorrectly support a model that includes FDI as the neighbourhood metric, even when neighbours have no effect, is because this metric includes focal plant size. Mis-specified-size-dependent growth can be compensated by the size dependence of the FDI term, in which size effects will be incorrectly attributed to the neighbourhood term rather than the size-dependent growth term. We recommend that if FDIs are to be used to determine plant growth, then it is important to first establish patterns of size-dependent growth and adequately incorporate such growth into all fitted models. In contrast, FIIs never lead to incorrect neighbour inference because their calculation does not include focal plant size and thus, cannot incorrectly attribute size-dependent growth to the neighbourhood term. However, FIIs ignore the fact that the size of a plant not only influences its competitive effect on neighbours, but also the strength of competition it experiences from those neighbours (i.e. competitive response; Goldberg & Barton, 1992;Goldberg, 1996). For example, consider a situation where the dominant interaction is competition for light.
In this instance, the size of the focal plant is critical in determining the competitive effect of a neighbouring plant: a focal plant twice as tall as its neighbour will likely experience a negligible shading effect from that neighbour, whereas a focal plant that is half the size of the same neighbour which will likely experience a strong, competitive shading effect. Therefore, depending on the study system and the type of competition driving the neighbour effect on plant growth, the use of an FDI may be most suitable but must be correctly modelled with size-dependent growth to ensure accurate neighbour inference.
The consequence of incorrectly describing growth and using FDIs to assess neighbour effects is dependent on the growth response. In this study, we used absolute growth as our response metric. Absolute growth is typically positively size dependent, at least in tree species (Bowman et al., 2013;Weiskittel et al., 2011), and so incorrectly attributing size effects to the FDI creates a negative relationship between absolute growth and the FDI (see from the literature demonstrates a preference in tree growth studies to use absolute growth as the response metric, suggesting that the most common consequence of using FDIs and inadequately describing size-dependent tree growth is to overestimate competition among neighbouring trees. This bias could have contributed to the focus on competition, and under-appreciation of facilitation, as the dominant interaction type in forest systems (but see Baumeister & Callaway, 2006;Callaway et al., 1991;Fajardo & McIntire, 2011). This conclusion is supported by the fact that the neighbourhood indices are commonly referred to as 'competition indices' despite purportedly being able to detect both negative and positive neighbourhood effects (Vogt et al., 2014).
Our study focused on the Hegyi index as our nominated FDI, which was the most common, or variations thereof, in our detailed literature search (Hegyi, 1974;Luo et al., 2020). However, we did find a range of FDIs in our search, including simple indices such as the summed basal area of all neighbours larger than the focal tree (BAL; Adame et al., 2008;Collet & Chenost, 2006;Schröder & Gadow, 1999). Unlike other FDIs, the BAL method does not explicitly include the size of the focal tree in its calculation, but the same potential issues exist as with the Hegyi index because only neighbours larger than the focal tree are included. Thus, in any given neighbourhood, the larger the focal tree is, the fewer larger neighbours it will have, and hence the lower its BAL value. Our simulation example highlights the potential issue with using any neighbourhood index that is calculated in a way that is influenced by the size of the focal tree to explain or predict size-dependent growth.
Our literature search also showed that FDIs were used to predict other measures of plant performance such as survival or mortality (e.g. Das et al., 2011;Eid & Tuhus, 2001). Size-dependent mortality relationships are much more variable when compared with size-dependent growth and there is conflicting evidence of whether small or large trees have the greatest probability of dying (Bennett et al., 2015;Keith et al., 2012;Peng et al., 2011;Phillips et al., 2010).
Thus, the likelihood of FDIs to incorrectly infer neighbour effects on tree mortality is far less likely but remains possible if sizemortality relationships exist. Beyond this specific example, interest in neighbour effects extends beyond plant growth and mortality, including plant biomass allocation, pest and disease incidence and reproductive output or success (Connell, 1971;Janzen, 1970;Mayfield & Stouffer, 2017;Yang et al., 2019). Future studies using FDIs to capture neighbour effects on these other plant metrics must be cautious of how the response in question is related to the size of the focal plant. In addition, future studies should also be aware that graphically relating the FDI to the response variable without including a size covariate can bias interpretations of observed neighbour effects.

| Recommendations
If growth is expected to be influenced by the relative size of neighbours, meaning FDIs are the most suitable neighbourhood metric, a greater emphasis must be placed on determining the correct pattern of size-dependent growth in the absence of neighbours (i.e. intrinsic growth). Unfortunately, directly quantifying intrinsic growth is often difficult in field situations where trees are rarely considered isolated.
Here, we have shown that the relation between tree basal area and BAIa can be well described using a power relation for both species, and we expect this to often be the case. To gain confidence that an analysis has resulted in correct inference of the effect of neighbours (i.e. net competitive versus facilitative), we also recommend that the statistical analysis be repeated using bootstrapped neighbourhoods, as performed here. The subsequent analysis should not infer any neighbourhood effects if growth is adequately modelled.

| CON CLUS IONS
Here, we show that correct inference of neighbour effects on tree growth is challenging when using a common class of neighbourhood index, FDI. We demonstrate that analyses of size-dependent growth data, which is particularly common, can be biased when a FDI is used as a neighbourhood predictor. Typically, this bias overstates competitive neighbour effects on absolute tree growth. We show that such biases can be mitigated when the statistical model can correctly describe intrinsic growth, and we provide some recommendations for how to perform statistical modelling and how to check for consistency. Recognising the issues and solutions highlighted here will ultimately increase our ability to accurately predict future tree size and hence, ecosystem productivity across a range of forest systems.

ACK N O WLE D G E M ENTS
We acknowledge the traditional custodians of the land on which this research was undertaken, the palawa people of lutruwita/Tasmania, and pay respects to elders past and present.
Specifically, we acknowledge the lairmairrener people as the traditional custodians of the land on which the fieldwork component of this study was completed. We thank Melissa R. Gerwin for establishing the neighbourhood plot data used in this study, Rose E. Brinkhoff for assistance with data collection and Sustainable Timbers Tasmania for providing access to the study site. T.G.B.   year. Three growth models were considered and for each, the two neighbour indices were considered as well as no neighbour effect. For each growth model, 1000 data sets were randomly simulated so that neighbours had no effect on growth, and for each simulation, AIC was used to select the models that differed