Discussion

We introduce a new statistical approach to the calculation of plant-soil feedback (PSF) that can perform precisely the same PSF ratio calculations as the standard approach, but implemented in a linear mixed model framework. It offers some significant benefits over the standard approach, with improved handling of unbalanced and non-independent data. The uncertainties can also be handled using simulation methods so that they can be propagated into subsequent calculations involving PSF values. Furthermore, it offers the novel opportunity to statistically remove the influence of plant growth covariates from PSF and provide some insight into the relative contribution of quantifiable components of soil biota to plant growth.

The ability to include non-independent, within-field site replicates in PSF studies represents a large advantage of the novel approach. Given that plant growth rates would vary among a group of plants grown from a single seed source in an identical soil, it is desirable to include some level of replication at the lowest treatment level. Some PSF experiments that treat field sites as replicates, however, calculate a single PSF ratio from each field site, based on a pair of plants, i.e. one plant in each soil treatment. This is largely because multiple observations for each soil treatment from the same field site are not independent and standard PSF calculations lack a statistically robust way to account for them. The uncertainty around these non-independent observations also needs to be represented in overall mean PSF values, and therefore, averaging such within-site replicates before calculating PSF masks a degree of uncertainty. Our model approach overcomes these issues, as within-site replicates can be handled in a manner that accounts for both their non-independence and associated

uncertainties. The inclusion of these within-site replicates then offers significant value in providing more information for the model to estimate PSF from each site. In addition, when data is unbalanced as a result of plants dying, the model approach is able to make use of all available observations, whereas in standard PSF calculations, data from surviving plants would need to be omitted to balance the data.

We also applied the model approach to isolate PSF responses resulting from soil biota other than nitrogen-fixing rhizobia. Doing so accounted for the presence of rhizobia contaminants in sterilised treatments in the case study data set, while also excluding the possibility for different symbiotic benefits of rhizobia between the introduced and native range of the study species. Although the rhizobia-removed PSF response for NZ soil was still positive, representing faster plant growth in live relative to sterilised soil, this may have resulted from the growth benefits of mycorrhizal fungal mutualists in live soil. Using a more continuous variable to assess the abundance of rhizobia, such as the number of nitrogen-fixing root nodules, may also capture better estimations of rhizobia-removed PSF. The approach of removing the effect of plant growth covariates from PSF may also be applied to

other quantifiable soil biota components, such as the ubiquitous belowground mutualist mycorrhizal fungi by including a covariate that quantifies the degree of mycorrhizal colonisation in plant roots (see Vierheilig, Schweiger & Brundrett 2005 for quantification methods).

Furthermore, the actual effect of a covariate on PSF responses can be quantified by calculating the difference in PSF between a model with and without the covariate, i.e. the difference between net PSF and PSF having removed the covariate effect. We illustrated this by quantifying the relative contribution of rhizobia in both NZ and UK soil. Given that rhizobia were present in sterilised treatments in our example data, these estimates are unreliable. However, it does illustrate how the method may be used to statistically disentangle the contribution of different soil biota components. To our knowledge, this is the first description of an approach that can be used to statistically isolate the contribution of different components of net PSF. The ability to isolate such effects statistically may prove very useful, given that it is difficult to achieve this through experimental manipulation. Moreover, retaining an intact soil community in PSF experiments likely represents field effects most accurately, because the sum of effects from experimentally-separated components may not be additive (e.g., Callaway et al. 2011), likely owing to interactions among the components. Nodulation by rhizobia bacteria, for instance, can be facilitated by nematodes (Horiuchi et al. 2005). Statistically teasing away the effect of quantifiable soil biota components from net PSF may therefore be a preferred method to partially unravel the PSF ‘black box’, wherever possible.

The removal of covariate effects from PSF may have further applications, for example, investigating how aboveground factors such as the degree of aboveground herbivory influences PSF responses by quantifying and including these factors as covariates. The approach may also be useful to statistically account for other confounding factors, such as infection by unwanted opportunistic aboveground pathogens and pests. The degree of infection by common glasshouse pathogens such as powdery mildew or thrips, for example, may be scored and included as a covariate.

Relationships between response variables and covariates of varying complexity can be

accommodated in mixed models. The models that we fitted to remove the rhizobia effect from PSF (M4) assumed that the slopes of the plant growth rate x nodulation score regressions were equal in live and sterilised soil, which was a good description of the data (Figure 2.2). However, if the slope of growth rate x covariate regressions are expected to differ between the soil treatments, a varying- slope, varying-intercept model can be fitted. This could be achieved for our data by specifying an interaction between the soil treatment and covariate fixed effects in the M4 models. Additionally, if the response variable x covariate relationship is non-linear, the approach may be developed using a non-linear mixed model framework (e.g., with R package nlme; Pinheiro et al. 2014) or implemented in Bayesian hierarchical models.

Many different PSF experimental designs and ratio equations can be accommodated by the model approach. The soil treatment fixed effect variable in our example models can be easily substituted with a variable comprising the numerator and denominator of the PSF ratio in an experiment. Similarly, the random effect variable included to calculate pairwise PSF ratios can be customised to accommodate other pairwise experimental designs. More complex designs, for which mean PSF needs to be calculated for multiple units such as different plant species or environments can be accommodated in two ways. A single model could be fitted that includes a fixed effect variable referring to both the species/environment unit and the soil treatment, although this will assume a common variance term across the data and may not be appropriate in some circumstances. Alternatively, separate models could be fitted for each unit, as we did for the two soil provenances (NZ and UK) in our case study example.