2.6 Within-Family Broad-Sense Heritability (H2) Calculation
Surveys of the Melampsora sp. leaf rust, S. musiva leaf spot and counts of the Phyllocolpa sp. galls had within-garden microsite variation removed using thin-plate spline regression (fields R package). Residuals of the spatial correction were added to the mean of each survey dataset for each tree observation to spatially correct and rescale the values. These corrected values were then used to calculate the proportion of variance in the fungal and insect distributions that was due to genotype using a linear-mixed model (lmer function of the lme4 R package) with insect counts and fungal scores as the response, tree genotype as the predictor, and stem diameter biomass estimates as a covariate where applicable. Broad-sense heritability was calculated as\(\ H^{2}=\ \delta_{g}^{2}/{(\delta}_{g}^{2}+\delta_{e}^{2})\), where \(\delta_{g}^{2}\) is the genetic variance due to genotype and \(\delta_{e}^{2}\) is the residual variance. Rapid, simulation-based exact likelihood ratio tests were used to evaluate the significance of variation due to genotype for each linear model (exactRLRTfunction of the RLRsim R package).
SAS software version 9.4 (2013) was used to test for the normality of all data sets and for all subsequent statistical analyses, transformations were conducted when necessary. Data from all chemical assays were analyzed by single-factor ANOVAs (PROC GLM) to test for differences between low and high Phyllocolpa sp. abundance.