Week 9 Flashcards
Primary idea behind blocking
Increase precision of estimated featment comparisons by identifying blocks of homogeneous units and accounting for difference between them in modelling
Reduced variances of estimators
Block comparison of treatments
Where the treatment comparison is same as with mean effects, but we are weighting each block by half
Reduced normal equations for estimating treatment effects from a block design (and model)
For block treatment model:
Yijl = μ + βi + τj + εijl
Where i = 1,…, b ; j = 1, …, t ; l = 1, …, nij
Balance in block treatment model
Where nij is # of j’th treatment in block i
If below eq is constant for all pairs of j, j’ then reduced normal equations are symmetric in treatments
This means all estimated pairwise comparisons between treatments have equal variance
ANOVA for block treatment model
qi = Y.j. - Σbi nijYi../ki
j = 1, … , t
Use reduced normal equations for RCBD to derive treatment variance
Binary design
The number of replications of treatment j in ever block i is 0 or 1
nij = 0 or 1
BIBD
Reduced normal equations for BIBD
Balance in BIBD?
Necessary but insufficient condition for BIBD
rt = bk
And
λ(t - 1) = r(k - 1)
Where k is block size
b is number of blocks
r is number of units each treatment is applied to
t is number of treatments
λ is number of times each treatment appears with each other
Unreduced BIBD
With rt = bk and λ(t-1) = r(k-1) still holding
Treatment concurrent matrix
For a general binary design, λ need not be constant therefore
Standardised variance of a pairwise comparison of treatments in BIBD
A per run measure of variance independent of the background error variance. Equal for BIBD
Var of estimator of pairwise comparison for RCBD
Var of estimator of pairwise comparison for BIBD
