Community Stability Metrics

Calculating the variance ratio

The variance ratio was one of the first metrics to characterize patterns of species covariance (Schluter 1984) and was used in an early synthesis paper of species covariance in long time series (Houlahan et al. 2007). The metric compares the variance of the community (C) as a whole relative to the sum of the individual population (x_i) variances:

$$ VR = \frac{Var(C)}{\sum_{i}^{N} Var(x_i)} $$

where:

$$ Var(C) = \sum_{i = 1}^{N} Var(x_i) + 2\left(\sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N} Covar(x_i, x_j)\right) $$

If species vary independently then the variance ratio will be close to 1. A variance ratio < 1 indicates predominately negative species covariance, whereas a variance ratio > 1 indicates that species generally positively covary.

Significance testing

The variance ratio remains widely used but has been subject to a number of criticisms. Importantly, early uses of the variance ratio either did not include significance tests, or tested significance by comparing observed values to those returned by scrambling each species’ time series. Null models using fully-scrambled species time series can generate spurious null expectations of covariance because the process disrupts within-species autocorrelation. Phase-scrambling (Grman et al. 2010) and cyclic shifts (Hallett et al. 2014; adapted from Harms et al. 2001) have been used to address this issue.

variance_ratio uses a cyclic shift permutations to conduct null modeling for significance tests. In this method a starting time point is randomly selected for each species’ time series. This generates a null community matrix in which species abundances vary independently but within-species autocorrelation is maintained (for each species, the time series is disrupted only once).

If a replicate column is specified, the default variance_ratio setting calculates a null variance ratio for each replicate in the dataset, averages these values, and repeats as many times as specified by bootnumber. This vector of averaged, null variance ratios is then sampled for lower and upper confidence intervals, which are returned along with the average observed variance ratio.

KNZ_variance_ratio <- variance_ratio(df = knz_001d, 
                                   species.var = "species", 
                                   time.var = "year",
                                   abundance.var = "abundance", 
                                   bootnumber = 10, 
                                   replicate.var = "subplot")

kable(KNZ_variance_ratio)

Alternatively, if a replicate column is specified and average.replicates is set to FALSE, the function will return a vector of null variance ratios for each replicate in the dataset, and return the subsequent confidence intervals and observed variance ratios for each replicate.

KNZ_variance_ratio_avgrep <- variance_ratio(knz_001d, 
                                          time.var = "year",
                                          species.var = "species",
                                          abundance.var = "abundance",  
                                          bootnumber = 10, 
                                          replicate.var = "subplot", 
                                          average.replicates = FALSE)

kable(head(KNZ_variance_ratio_avgrep))

If replicate.var is left as NA the function assumes that there is a single observation for each species within a given time period.

KNZ_A1_variance_ratio <- variance_ratio(df = subset(knz_001d, subplot=="A_1"), 
                                      time.var = "year", 
                                      species.var = "species",  
                                      abundance.var = "abundance",  
                                      bootnumber = 10) 
kable(KNZ_A1_variance_ratio)

General cyclic shift function

codyn also includes the option to apply a cyclic shift permutation on other test statistics:

• cyclic_shift returns an S3 object of test statistics from a user-specified function when applied to a null community generated via a cyclic shift permutation. It requires a dataframe with a species.var, time.var and abundance.var column, and optional replicate.var column. The user-specified function should operate on a community matrix (e.g., cov).

The length of the “out” parameter in the object is the number of null iterations as specified by bootnumber). If multiple replicates are specified, null values are averaged among replicates for each iteration, but a different cyclic shift permutation is applied to each replicate within an iteration.

• confint returns the confidence intervals from the S3 object produced by cyclic_shift.

Variance ratio and species richness

A second criticism of the variance ratio is that it is sensitive to species richness. This is a particular concern when the metric is used to compare communities that have different levels of species richness. The most conservative approach is to restrict use of the variance ratio to two-species communities (Hector et al. 2010). Comparing the effect size of the observed versus null variance ratio can also account for richness differences between communities. Two alternative metrics that quantify species asynchrony have been developed in part to respond to this issue.