Statistical Analyses
To investigate the within and among population variation in phenotypic
traits, each garden was modeled separately using linear mixed models fit
by maximum likelihood in the lme4 package in R (R Core Team 2014;
Bates et al. 2015). The tree traits were modeled as response
variables, while population and genotype were random effects. Garden
plot was included as a random variable to help account for within garden
environmental variance. Statistical significance was calculated using
likelihood ratio tests for the random effects using the packagelmerTest (Kuznetsova et al. 2015).
We compared the quantitative trait variation (QST) with
genetic variance at neutral loci (FST) in each garden.
For quantitative traits, the ratio of variances can be described as
where σP is the additive genetic variance among
populations and σG is the additive within-population
variance (Spitze 1993; McKay & Latta 2002). Each trait was analyzed
using the model described above, and population and genotype variances
were extracted to calculate QST. Parametric bootstrap
and Bayesian estimation are considered the best methods to obtain a
precision estimate around QST (O’Hara & Merilä 2005).
We performed parametric bootstrapping to obtain a 95% confidence
interval for QST, resampling the 16 populations with
replacement 1000 times, and estimating QST for each
bootstrapped data set. Resampling over the highest level
in a hierarchical experimental design (here the population) is
considered best practice (O’Hara & Merilä 2005). Variance in
QST becomes quite large as the number of populations
decreases (< 20), especially if populations are highly
differentiated (O’Hara & Merilä 2005; Goudet & Büchi 2006). Goudet &
Büchi (2006) recommend sampling many populations relative to the number
of families. Our design of 16 populations with 12 genotypes per
population comes close to their recommended sampling design of upwards
of 20 populations with 10 families (O’Hara & Merilä 2005; Goudet &
Büchi 2006). We directly compared the confidence intervals for
FST and QST to ascertain significance.
In using clonally replicated genotypes, our estimate of
σG includes both additive and non-additive genetic
effects, an approach that has been shown to lower QSTestimates and is thus a conservative test of QST> FST (Goudet & Büchi 2006). Broad-sense
heritability (H2) was also calculated for each trait
in each garden using the equation, H2 =
σG /(σG + σW), where
σW includes both within-population genotypic variance
and the error variance.
In order to test whether traits showed strong climatic relationships, we
calculated the Pearson’s product moment correlation coefficient in R (R
Core Team 2014) between population trait means and the first principal
component (PC1) from the environmental PCA. Systematic differences among
populations seen in these trait-climate correlations are another test to
rule out genetic drift (Whitlock 2008). To then test whether those
traits showing strong climatic patterns also showed evidence of stronger
selection, we fit a linear model of QST to the absolute
value of the trait-climate correlation coefficient
(|r|) with garden and trait modeled as covariates,
using the lmer package (Bates et al. 2015), following
Evans et al . (2016). Each garden was also modeled separately to
calculate garden-specific regression correlation coefficients.