Introduction

Plants can cause changes to the physical, chemical and biological properties of the soil in which they grow that can, in turn, benefit or inhibit the performance of that plant species and its neighbours: a process known as plant-soil feedback (PSF) (Bever 1994; Bever et al. 1997). Negative PSF occurs when the soil modifications reduce plant performance, promoting species turnover and coexistence, while positive PSF increases plant performance, promoting monodominance (Bever et al. 1997). An increasing body of work is showing that spatial and temporal vegetation patterns, such as the relative abundance of plant species and plant succession, can be mediated by PSF processes (van der Putten et al. 1993; Bever 2002, 2003; Klironomos 2002; Kardol et al. 2007; Kulmatiski et al. 2008; Mangan et al. 2010; Suding et al. 2013; Burns & Brandt 2014). The success of many invasive or range- expanding plant species may also be the result of escaping soil-borne enemies that drive negative PSF in the original range (e.g., Callaway et al. 2004; Reinhart & Callaway 2004; van Grunsven et al.

2010; Maron et al. 2014). Despite this extensive literature, many aspects of PSF processes remain poorly understood, including how long alien plant species escape negative PSF, how climate change affects PSF processes and how different components of soil biota contribute to PSF responses (van der Putten 2012; van der Putten et al. 2013; Kardol et al. 2013).

Controlled PSF experiments can be conducted to quantify the effect of soil biota on plant growth, for instance, by comparing plant performance in soil previously cultivated with a host-specific

community of biota, to a control soil lacking that biota. The strength of PSF can be expressed as a feedback ratio that quantifies the performance of plants grown with soil biota relative to the control treatment. These feedback ratios can be calculated in several ways depending on the design and aim of the experiment (reviewed by Brinkman et al. 2010), however, current calculation approaches have some limitations, such as in dealing with non-independent and unbalanced data.

In PSF studies, it is common to collect rhizosphere soil from separate field populations of a target plant species and use these soils as replicates in glasshouse experiments in order to encompass spatial variation in the composition of soil biota and resultant PSF effects. However, this sampling design introduces a confounding factor, because the abiotic properties of soil from the different populations will vary and may affect plant growth independently of soil biota. This can be dealt with firstly by using a background soil into which a small inoculation of field soil is made, and secondly, by pairing plants grown in live and control soil treatments, where the soil for each pair was sourced from the same field site (e.g., half of the soil from a site used for a live soil treatment and the remainder sterilised for use in the control). A PSF ratio can then be calculated for each plant pair to quantify the strength of PSF at each site (known as ‘pairwise’ calculation). Mean PSF and the associated uncertainty can then be calculated from these pairwise ratios, having accounted for

confounding among-site variation in plant growth caused by factors other than soil biota (see Brinkman et al. 2010).

These calculations work well when plant growth observations are available for live and control treatments from each sampling site. Unfortunately, it is not unusual for some plants to die during experiments (e.g., van Grunsven et al. 2009), causing missing data and unbalanced designs. This is not handled well during the calculation of pairwise feedback ratios, as the loss of one plant in a pair means that data for the surviving plant also has to be omitted, representing a loss of information. In addition, some sampling designs do not have a natural one-to-one pairing of observations. For example, replication could be increased by including multiple replicates of the live and control treatments per site rather than having a single pair per site. It is then not clear how these within-site replicates should be handled because they are not independent. One approach to deal with this non- independence issue is to average plant growth data within the soil treatments from each site, use these to calculate a single feedback ratio for each site, and then average over sites to get an overall mean feedback value (e.g., Maron et al. 2014). Invariably, however, the resulting overall uncertainty does not include that associated with averaging data at the site level.

PSF experiments usually measure the net effect of all resident soil biota on plant performance. This net effect will depend on the balance between interactions with biota that affect plant performance positively (e.g., mycorrhizal fungi and nitrogen-fixing bacteria) and negatively (e.g., root-feeders and pathogens). Experimentally isolating different components of soil biota and quantifying their relative contribution to net PSF effects remains a significant challenge (Reinhart & Callaway 2006; Cortois & de Deyn 2011). Nevertheless, the plant growth effects of some soil biota can be quantified. For example, it is often possible to estimate the abundance of mutualistic nitrogen-fixing rhizobia bacteria by assessing the degree of root nodulation on legume hosts, and this is widely observed to correlate positively with legume performance (e.g., Thrall et al. 2007; Wandrag et al. 2013).

Statistically modelling the relationship between the degree of nodulation and plant growth can be used to estimate the growth increment associated with increasing nodulation. This can be used to statistically remove the effect of rhizobia on plant growth by calculating the rate of growth in the absence of nodules, and thus, calculate a PSF ratio that statistically excludes the contribution of this mutualist from other soil biota. A method incorporating this approach into the calculation of feedback ratios would be a step towards opening the PSF effect ‘black box’, at least in terms of teasing away the influence of quantifiable components of the soil community, such as rhizobia or mycorrhizal mutualists. To our knowledge, though, this approach has not yet been implemented.

Once PSF ratios have been calculated, it may be useful to perform further calculations with these values. For example, we might want to test for significant biogeographic differences in PSF for a

species between its native and introduced range, which would involve testing if the difference between two feedback ratios differed from zero or not. However, it can be challenging to properly incorporate all of the uncertainties involved in the calculation of differences between two feedback ratios.

In this study, we describe a new method for calculating PSF ratios that deals with the issues raised above. Our approach is to embed PSF ratio calculation into a mixed model framework that can accommodate unbalanced study designs and non-independent replicates. The framework can additionally incorporate plant growth covariates in order to statistically remove their effect from net PSF and elucidate the relative contribution of quantifiable components of soil biota. We illustrate the approach by applying it to a small simulated data set, followed by a case study from a PSF

experiment using the nitrogen-fixing species Trifolium glomeratum. The R scripts used for these examples are provided in text form in Appendix A.