Ecological Archives E093-207-A3

H. George Wang, Robert J. Marquis, and Christina S. Baer. 2012. Both host plant and ecosystem engineer identity influence leaf-tie impacts on the arthropod community of Quercus. Ecology 93:2186–2197. http://dx.doi.org/10.1890/11-1838.1

Appendix C. GLMMs for occupancy comparison.

Methods

We used generalized linear mixed models (GLMMs) to compare tie occupancy across the leaftie treatments, host tree species, and the two censuses. Tie occupancy was recorded as a binary response with 0 representing an empty leaftie or untied leaf, and 1 representing a sample containing at least one individual arthropod. A set of GLMMs was constructed to examine the random effect of individual tree (hereafter tree), and the fixed effects of leaftie treatment (hereafter treatment), host tree species (hereafter species), and census period (hereafter census). All GLMMs were fitted by the Laplace approximation of the true GLMM likelihood and with binomial error distribution using the glmer function of the lme4 package of R 2.12.1 (R Development Core Team 2010).

We first selected the random effects by fitting eight GLMMs with different random effects (Table C1). Random effects of the first model were vector-valued, containing the intercept and correlated slopes of treatment and census within tree. This model estimated the potential correlation of leaftie treatments and the two censuses within each tree. The second model contained a vector-valued, correlated (intercept and slope) random effect of treatment within tree and an uncorrelated (slope only) random effect of census within tree. The third model contained a scalar (intercept only) random effect of tree, and separate uncorrelated (slope only) random effects of treatment and census within tree. The fourth model contained a correlated, vector-valued random effect of treatment within tree, whereas the fifth model contained a scalar random effect of tree and uncorrelated random effect of treatment within tree. The sixth model contained a correlated, vector-valued random effect of census within tree, while the seventh model contained a scalar random effect of tree and an uncorrelated random effect of census within tree. The eighth model contained a simple scalar (intercept only) random effect of individual tree, which accounts for the tree-to-tree variability of the response variable. All eight abovementioned models contained additive fixed effects of treatment, species, and census. The best model was selected to determine the best random effect in describing our data based on Akaike's information criterion (AIC) values of the models, and the variances and correlations of the random effects. A random effect was dropped if its estimated variance was 0 or if it had 1 or -1 correlation with another random effect.

We then fit nine additional GLMMs with the selected random effects and different fixed effects (Table C2). The first model contained the additive fixed effects of treatment and species. The second model contained the additive fixed effects of treatment and census. The third model contained the additive fixed effects of species and census. The fourth model contained the additive and interacting fixed effects of treatment and species. The fifth model contained the additive and interacting fixed effects of treatment and census. The sixth model contained the additive fixed effect of species, and the additive and interacting fixed effects of treatment and census. The seventh model contained the additive fixed effect of treatment, and the additive and interacting fixed effects of species and census. The eighth model contained the additive fixed effect of census, and the additive and interacting fixed effects of treatment and species. The ninth model contained the additive and interacting fixed effects of treatment, species, and census. The best model was selected based on AIC values.

Table C1. GLMMs used to determine the random effects for leaftie occupancy by all arthropods across leaftie treatments, host tree species, and census periods. The models are ranked in increasing AIC values. Terms in GLMMs: Treatment: artificial leaftie treatment; T-Species: host tree species; Census: census period; Tree: individual tree; 1: intercept (random or fixed effects); 0: no intercept (random effect only). Therefore, (1 | Tree) = scalar random effect of individual tree; (1 + Treatment | Tree) = intercept and leaftie treatment by individual tree (i.e., correlated, vector-valued random effect); (0 + Treatment | Tree) = no intercept, and leaftie treatment by individual tree (i.e., uncorrelated random effect). The format of the terms follows that of Bates (2010).

Model (selection for random effects) d.f. AIC ΔAIC
1 + Treatment + T-Species + Census + (1 | Tree) 13 315.9 0.0
1 + Treatment + T-Species + Census +
(1 + Census | Tree)
15 319.9 4.0
1 + Treatment + T-Species + Census +
(1 | Tree) + (0 + Census | Tree)
16 321.9 6.0
1 + Treatment + T-Species + Census +
(1 + Treatment | Tree)
22 324.3 8.4
1 + Treatment + T-Species + Census +
(1 | Tree) + (0 + Treatment | Tree)
23 325.7 9.8
1 + Treatment + T-Species + Census + (1 +
Treatment | Tree) + (0 + Census | Tree)
25 328.2 12.3
1 + Treatment + T-Species + Census +
(1 + Treatment + Census | Tree)
27 330.8 14.9
1 + Treatment + T-Species + Census + (1 | Tree) +
(0 + Treatment | Tree) + (0 + Census | Tree)
26 332.3 16.4

Table C2. GLMMs used to determine the fixed effects for leaftie occupancy by all arthropods across leaftie treatments, host tree species, and census periods. The models are ranked in increasing AIC values. GLMM terms are the same as those in Table C1.

Model (selection for fixed effects) d.f. AIC ΔAIC
1 + Treatment + Census + (1 | Tree) 6 313.8 0.0
1 + Treatment × Census + (1 | Tree) 9 315.8 2.0
1 + Treatment + T-Species + Census + (1 | Tree) 13 315.9 2.1
1 + Treatment + T-Species × Census + (1 | Tree) 20 316.4 2.6
1 + Treatment × Census + T-Species + (1 | Tree) 16 318.0 4.2
1 + Treatment + T-Species + (1 | Tree) 12 329.2 15.4
1 + Treatment × T-Species + (1 | Tree) 33 353.6 39.8
1 + Treatment × T-Species + Census + (1 | Tree) 34 340.1 26.3
1 + Treatment × T-Species × Census + (1 | Tree) 65 380.2 66.4
1 + T-Species + Census + (1 | Tree) 10 400.5 86.7

Results

The random effect selected was the scalar (intercept only) random effect of individual tree (Table C1). The lack of a leaftie treatment slope in the random effect suggests that an increase in occupancy rates in artificial leafties was not correlated with occupancy of non-tied leaves within individual trees. The four highest ranked models for fixed effects are essentially equally valid (all ΔAIC < 3, Table C2). The second ranked model (ΔAIC = 2.0) included a treatment × census interaction, suggesting that leaftie treatment effects on occupancy differed between the two census periods (Fig. C1).

Fig. C1. Percent occupancy of artificial leafties or single leaves by all arthropods across the leaftie treatments and two census periods.


Literature Cited

Bates, D. M. 2010. lme4: Mixed-effects modeling with R. Page 103. Springer.

R Development Core Team. 2010. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.


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