Lecture on Angelini et al. (2020)

Discussion on Bailey’s lecture

2001-04-01

The Paper in One Sentence

• Seven camelina cultivars × two sowing times × two years

• Which combination gives the best site-specific results?

Why it matters statistically: the experimental design is more complex than it looks.

Not a simple RCBD

• All 14 treatments randomized within each block

• That’s impossible: You can’t sow half a block in October and the other half in March

Split plot:

BLOCK
├── WHOLE PLOT: Autumn
│   └── V1 V2 V3 V4 V5 V6 CELINE
└── WHOLE PLOT: Spring
    └── V1 V2 V3 V4 V5 V6 CELINE

What are we seeing here?

\[ y = \mu + \text{Replicate} + \text{Year} + \text{SowingTime} + \text{WholePlotError} + \text{Cultivar} + \text{Interactions} + \text{SubplotError} \]

Conceptual model:

\[ y_{ij} = \mu + B_i + T_j + \epsilon_{ij} \]

Where:

• \(y_{ij}\) = observed response in block \(i\) receiving treatment \(j\)

• \(\mu\) = overall mean response

• \(B_i\) = effect of block \(i\)

  • Accounts for background variation we are not primarily interested in
  • Examples: lake, field, site, year
  • Typically treated as a random effect

• \(T_j\) = effect of treatment \(j\)

  • The main effect we want to study
  • Treated as a fixed effect

• \(\epsilon_{ij}\) = residual error

  • Random unexplained variation
  • Represents variability among experimental units within blocks
  • Assumed: \[ \epsilon_{ij} \sim \text{Normal}(0, \sigma^2) \]

Key idea:

There is only ONE level of experimental units and ONE source of experimental error.

Two Error Strata

Stratum	Tests	Unit
Whole-plot error	Sowing Time	Whole plot in block
Subplot error	Genotype, G×ST	Subplot in whole plot

Important

Using a single pooled error gives an anti-conservative test for ST and an over-conservative test for G. This is one of the most common errors in agronomy papers.

aov() vs. lmer() for Split-Plots

Classical approach — explicit error strata:

aov(yield ~ ST * G * Year + Error(Block/ST), data = dat)

What lmerTest does:

lmer(yield ~ G * ST * Year + (1|Block) + (1|Block:ST), data = dat)
anova(model)  # Satterthwaite df

Both are valid. But lmer output labeled as “ANOVA” obscures that this is a linear mixed model with approximated denominator df.

The Method–Question–Data Triangle

       QUESTION
   "Best genotype × 
   sowing time?"
        /\
       /  \
 DATA /    \ METHOD
-----/------\-------
Factorial    Split-plot LMM
blocked      Two error strata
cont+count   Poisson for counts

Alignment? Mostly yes - but the gap between what was done (LMM) and what was reported (“ANOVA”) is a transparency issue.

The 6 Key Takeaways

1. Split-plot because ST cannot be randomized at subplot level → two error strata
2. lmer + anova() is a linear mixed model, not classical ANOVA
3. Year as fixed is pragmatic with n=2 but limits generalizability
4. G×ST is disordinal — separate recommendations required
5. Poisson for counts was correct; response type matters
6. Data exploration is underreported — Zuur would want figures

Questions

• How did the authors control for environmental variation when comparing genotypes across different sowing dates?

• The paper mentions they used “visual inspection of model residuals was also done to check for model assumptions”