Anna L. W. Sears and Peter Chesson. 2007. New methods for quantifying the spatial storage effect: an illustration with desert annuals. Ecology 88:2240–2247.

Appendix B. Statistical methods for estimating cov(E,C).

In the following, Y represents plant response on the natural scale, and y is the log of this response; where E = response to the environment in the absence of neighbors, C = plant response to neighbors, in T = removal and U = non-removal treatments, at the xth growing location, ε = error, a = a constant, b = the regression coefficient of plant response, n_B is the number of blocks, and n_k is the number of replicates within blocks.

Following the model given in the text, Eqs. 6 and 7, the plant response for non-removal treatments is determined by

(B.1)

nd as given previously,

(B.2)

The Cov(E,C) is estimated as b × Var(E), using orthogonal regression (Carroll et al. 1995) to estimate the model parameters from experimental data. In the estimation below, and are the sample variances of y_T and y_U, and is their sample covariance. Here represents the ratio of non-removal to removal within-block error variance, , which cannot be estimated directly from unreplicated block designs. The orthogonal regression test assumes that  = 1, but the robustness of results can be evaluated for a range of values. These can be derived directly when experimental designs have within-block replication:

(B.3)

   
This estimation tends to underestimate the value of the error variance ratio, because it does not include equation-error variance associated with non-removal treatments (Carroll et al. 1995), which would inflate the numerator.

  
The slope, b, is estimated as

(B.4)

  
and the intercept as

(B.5)

Under the null hypothesis, b = 0, so the response of plants to the environment is given by

(B.6)

and the error sum of squares for the null, SSE₀, is given by

(B.7)

Under the alternative hypothesis, assuming that b ≠0, and that there is a relationship between plant responses to the environment and competition, the response to the environment is given by,

(B.8)

and the error sum of squares, SSE₁, for the alternative hypothesis, is given by

(B.9)

The F statistic for the significance test is then

(B.10)

Using the test above, we can evaluate the covariance estimate for a range of values. However, bootstrap simulations show that this test is somewhat biased, and tends to give too liberal P values (Type I error), especially when sample sizes are small, or if the variance of E is not much greater than the error variance. In these cases, the tabulated P values should be replaced by those indicated by simulation. In certain cases, extreme values of in the presence of low variance in E lead to severe overfitting of the model and erroneous results. For Fig. 2, we need to estimate Var(E), Cov(E, C) and Var(C). Var(E) is estimated as ,which then leads to as the estimate of Cov(E,C), and  as the estimate of Var(C).

LITERATURE CITED

Carroll, R. J., D. Ruppert, and L. A. Stefanski.1995. Measurement Error in Nonlinear Models, First edition. Chapman and Hall, London, UK.