CRD, RBD, LSD essentials

The three basic designs

Design	Model	Error d.f.	Controls
CRD	$y_{ij} = \mu + \tau_{i} + \varepsilon_{ij}$	N − k	nothing
RBD	$y_{ij} = \mu + \tau_{i} + \beta_{j} + \varepsilon_{ij}$	(k−1)(r−1)	one nuisance factor
LSD (p×p)	$y_{ijk} = \mu + \tau_{i} + \rho_{j} + \gamma_{k} + \varepsilon$	(p−1)(p−2)	two nuisance factors

In an LSD the number of treatments equals the number of rows and columns (p), and $p^{2}$ units are used. If the error d.f. is 42, then (p−1)(p−2) = 42 gives p = 8, so treatments have 7 d.f.

ANOVA table shape

Total d.f. = N − 1, split into treatment, block(s) and error. The F statistic is MS_treatment/MS_error with (k−1, error) degrees of freedom.

Principles

• Randomisation removes bias; replication gives an error estimate; local control (blocking) removes a known source of variability.
• Blocking increases precision only if the blocks really differ — otherwise you have merely lost error degrees of freedom.
• Relative efficiency of RBD to CRD = (MSE_CRD/MSE_RBD), adjusted for d.f.

Factorial experiments

$A 2^{n}$ factorial estimates main effects and interactions; confounding sacrifices a high-order interaction to reduce block size. Effects are estimated by contrasts of the treatment totals.

Missing plot

For one missing value in an RBD: ŷ = (rB + kT − G)/((r−1)(k−1)), and the error d.f. drops by one.

Key takeaways

• Memorise the three error d.f. formulas — they are asked directly.
• LSD: p treatments, $p^{2}$, two nuisance directions.
• Blocking trades error d.f. for reduced error variance; it only pays if blocks differ.