# A Baseline Category Logit Model for Assessing Competing Strains of Rhizobium Bacteria

Abstract: In this paper we describe novel methodology for evaluating competition among strains of Rhizobium bacteria which can be found naturally occurring in or can be introduced into soil. Rhizobia can occupy nodules on the roots of legume plants allowing the plant to ‘fix’ atmospheric nitrogen. Our model defines competitive outcomes for a community (the multinomial count of nodules occupied by each strain at the end of a time period) relative to the past state of the community (the proportion of each strain present at the beginning of the time period) and incorporates this prior information in the analysis. Our approach for assessing competition provides an analogy to multivariate methods for continuous responses in competition studies and an alternative to univariate methods for discrete responses that respects the multivariate nature of the data. It can also handle zero values in the multinomial response providing an alternative to compositional data analysis methods, which traditionally have not been able to facilitate zero values. The proposed experimental design is based on the simplex design and the model is an extension of multinomial baseline category logit models that includes multiple offsets and random terms to allow for correlation among clustered responses. Supplemental materials for this article are available from the journal website.

## Summary (2 min read)

### 1. INTRODUCTION

- Competition occurs among species when a required resource is limited and the species ‘compete’ to each obtain the resource.
- Offsets have previously been used with models for discrete responses (logistic regression in Agresti 2002) but multiple offsets have not been used with multinomial models or for the purpose of assessing competition among species.
- The experimental response in their motivating example is the number of nodules acquired by each strain of rhizobia in each community and this is a multinomial vector.
- The novel features are the marrying of simplex designs with multinomial responses in a discrete modeling framework that defines competitive outcomes for a community of species relative to a previous state of the community and incorporates this prior information in the analysis.

### 2. METHODS

- The authors propose a multinomial baseline category logit model (Agresti 2002) to measure the competition between J species that will allow the assessment of competitive relationships among species and consequences for community structure.
- This model is analogous to the specification of the RGRD model in Connolly and Wayne (2005, Equation (4)).
- The authors extend this model to include a community specific random effect to allow for variation from community-to-community (Hartzel, Agresti, and Caffo 2001).
- To interpret the model the final proportions of success counts for each species can be predicted for a range of initial communities and these predictions used to determine the outcome of competition.
- Π̂ij /pij compares the predicted proportion of success counts relative to initial proportion present for an individual species, also known as Compositional change measure (1).

### 3.1. EXPERIMENTAL DESIGN

- When a Rhizobium strain has occupied a nodule on the root of a legume, it normally has the ability to ‘fix’ nitrogen (N) from the atmosphere and supply the host plant with N and provide additional N in the legume environment.
- Competition was investigated among M. loti strains Ml8,Ml19 and Ml16; named A, B and C, respectively, from here on.
- Nodule occupancy of the rhizobial strains was determined by ERIC-PCR fingerprinting (de Bruijn 1992).
- Several of the responses for individual species were zero.

### 3.2. MODEL FITTING

- The authors fitted a series of multinomial baseline category logit random effects models to the multinomial data.
- The authors maximized the log of the likelihood function given in (2.5) using the NLMIXED procedure in SAS software .
- The authors predicted from the fitted model for a range of initial compositions using Equation (2.6).
- Approximate standard errors were generated for these tests using the Delta method (Billingsley 1986).

### 3.3. RESULTS

- The final model, after extensive model selection using AIC , included the initial proportions of each strain and density in the linear predictor.
- Two interaction terms piApiB and piBpiC were also included and these interactions were found to be of similar strength and so were constrained to be equal.
- While the inclusion of the random effects to account for variation from community-to-community was not significant , this component was included in the models to respect the structure in the experimental design.
- Based on these two compositional change measures, strain C was the most competitive strain, particularly at high density, while there was no out-right winner between strains A and B. Ta bl e 2.

### 4. DISCUSSION

- In this paper the authors present an experimental design and modeling framework for assessing multinomial responses from multiple species competition studies.
- It also provides a multivariate alternative to the univariate methods used for discrete responses based on Lotka–Volterra models (Leslie 1958; May 2001) that allows for correlation among responses within a community.
- Strain C occupied a large number of the nodules even when it was least represented in the inoculum particularly at high inoculum density, which in general is an indication of a highly competitive strain (Thies, Benbohlool, and Singleton 1992).
- The authors have shown that using an appropriate simplex design allows the fitting of model (2.7) through which they can assess the relative competitiveness of species, whether species interfere with or interact with each other, and the outcome of these interspecific relationships on community composition.
- The authors model is also closely related to the Lotka–Volterra differential equations for competing species.

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##### Citations

^{1}, Imperial College London

^{2}, Maynooth University

^{3}, Teagasc

^{4}, University of Minnesota

^{5}, ETH Zurich

^{6}, University of the Republic

^{7}, Helmholtz Centre for Environmental Research - UFZ

^{8}, University of Lleida

^{9}, Leipzig University

^{10}

57 citations

### Cites background from "A Baseline Category Logit Model for..."

...They have been used in understanding the BEF relationship in a number of plant and invertebrate assemblages (Sheehan et al. 2006; Kirwan et al. 2007; Connolly et al. 2009, 2011; Frankow-Lindberg et al. 2009; Nyfeler et al. 2009; O’Hea, Kirwan & Finn 2010; Brophy et al. 2011)....

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1 citations

### Cites background from "A Baseline Category Logit Model for..."

...Recently Brophy et al. (2011) described a novel methodology for evaluating competition among strains of Rhizobium bacteria which can be found naturally occurring in or can be introduced into soil....

[...]

##### References

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### "A Baseline Category Logit Model for..." refers methods in this paper

...We propose a multinomial baseline category logit model (Agresti 2002) to measure the competition between J species (categories) that will allow the assessment of competitive relationships among species and consequences for community structure....

[...]

...Offsets have previously been used with models for discrete responses (logistic regression in Agresti 2002) but multiple offsets have not been used with multinomial models or for the purpose of assessing competition among species....

[...]

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