Linear Mixed Model for Flowering Time

Model Specification

We want to model flowering time (y) as a function of replicate, donor taxon, inv4m genotype, and a random effect of donor plant.

The model equation is:

\[ y_{ijk\ell} = \mu + \text{Rep}_i + \text{DonorTaxa}_j + \text{Inv4mGenotype}_k + u_{\ell} + \varepsilon_{ijk\ell} \]

Where:

• \(y_{ijk\ell}\) = flowering time observation
  
• \(\mu\) = overall mean (grand mean)
  
• \(\text{Rep}_i\) = fixed effect of replicate
  
• \(\text{DonorTaxa}_j\) = fixed effect of donor taxon
  
• \(\text{Inv4mGenotype}_k\) = fixed effect of inv4m genotype
  
• \(u_{\ell} \sim N(0, \sigma^2_u)\) = random effect of donor plant
  
• \(\varepsilon_{ijk\ell} \sim N(0, \sigma^2)\) = residual

Implementation in R

Using lme4::lmer

library(lme4)
# Example model (assuming your dataset is called df)
mod_lmer <- lmer(
  y ~ Rep + donor_taxa + inv4m_genotype + (1 | donor_plant),
  data = df
)
summary(mod_lmer)

Using nlme::lme

library(nlme)
mod_lme <- lme(
  fixed = y ~ Rep + donor_taxa + inv4m_genotype,
  random = ~ 1 | donor_plant,
  data = df
)
summary(mod_lme)