. Author manuscript; available in PMC: 2019 Jun 1.

Published in final edited form as: Oecologia. 2018 Mar 30;187(2):427–445. doi: 10.1007/s00442-018-4116-4

Table 1.

Coefficient estimates for models of plant foliar phenotypes from field common garden experiment.

Term type

Coefficient estimate

sym¹

(log) leaf area (cm2)⁴

petiole length (mm) A⁵

petiole length (mm) B⁶

PC1

PC2

Main effects

Intercept [shade]

α₀

0.02 (−0.04, 0.08)

1.2 (−7.7, 10.0)

−1.0 (−7.3, 5.3)

−1.94 (−2.6, –1.3)^***

0.44 (−0.08, 0.96)

source [sun]

α_k

0.08 (0.03, 0.13)^***

11.8 (6.4, 17.2)^***

4.0 (−0.4, 8.4)^•

1.29 (0.6, 2.0)^***

0.39 (0.15, 0.64)^***

garden [sun]

β₀

−0.08 (−0.16, 0)^•

−11.6 (−23.8, 0.60)

−4.8 (−14.0, 4.0)

2.3 (1.4, 3.1)^***

−0.92 (−1.65, −0.2)^•

source * garden [sun, sun]

β_k

−0.09 (−0.14, –0.05)^***

−6.5 (−12.4, −0.8)^*

0.9 (−4.4, 6.2)

−0.9 (−1.5, −0.3)^***

−0.44 (−0.77, −0.12)^*

Covariate terms

γ₀

0.77 (0.7, 0.84)^***

23.6 (10.3, 37.1)^***

50 (24, 76)^***

x × source [sun]

γ₁

0.12 (0.03, 0.21)^*

15.6 (−2.8, 33.8)^•

39 (5.4, 73)^*

x × garden [sun]

γ₂

–

2.4 (−16.09, 20.45)

7.8 (−29.0, −44.0)

x × source [sun] × garden [sun]

γ₁₂

–

−16.3 (−40, 7.15)

−36.5 (−83.61-9.48)

Random effects²

source ID

τ_α(j)

0.04 (0.02, 0.06)

3.1 (1.1, 5.3)

1.6 (0, 3.16)

0.5 (0.2, 0.8)

0 (0, 0.16)

garden ID

τ_β(l)

0.04 (0.02, 0.08)

7.3 (3.5, 12.3)

5.2 (2.3, 8.7)

0.45 (0.2, 0.8)

0.44 (0.2, 0.7)

residual error

σ_e

0.08 (0.08, 0.09)

10.5 (9.5, 11.3)

9.2 (8.4, 10)

1.2 (1.1, 1.3)

0.65 (0.6, 0.7)

Model statistics

R_{m}^{2}

0.79

0.33

0.47

0.31

0.38

R_{c}^{2}

0.86

0.58

0.61

0.47

0.57

p-values indicated by asterisks:

^•

0.05<p<0.1,

p<0.05,

^**

p<0.01,

^***

p<0.001 from t tests with Kenward–Roger approximated ddf; confidence intervals calculated via profile likelihood.

see ESM section 2.1 for symbolic representation of model coefficients in context of regression equations.

group-level random effect estimates are presented on the standard deviation scale.

⁴

response = residual(log leaf area ~ log leaf number); covariate = residual(log leaf mass ~ log leaf number).

⁵

response = residual(petiole length ~ log leaf number); covariate = residual(log leaf mass ~ log leaf number)

⁶

response = residual(petiole length ~ log leaf number + log leaf mass); covariate = residual(log leaf area ~ log leaf number + log leaf mass)