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STAT:5201 Applied Statistic II
i) RCBD with 12 blocks (litters), 4 treatments per block, N=48.
Restricted randomization done within each litter.
Diagram: LITTER12⇥ trt4
Every treatment observed for every litter (crossed terms).
To emphasize the RCBD, we can also include an “R” as...Diagram: LITTER12
R ⇥ trt4
2 4 6 8 10 12
01
23
4
12 distinct litters (N=48)
Grouping
y
trt 1trt 2trt 3trt 4
SAS Output 12 distinct litters:
Class Level Information
Class Levels Values
Grouping 12 1 2 3 4 5 6 7 8 9 10 11 12
DayLength 2 long short
Climate 2 cold warm
Number of Observations Used 48
1
Source DF Type III SS Mean Square F Value Pr > F
Grouping 11 9.06396512 0.82399683 96.77 <.0001
DayLength 1 8.27435377 8.27435377 971.73 <.0001
Climate 1 0.46745416 0.46745416 54.90 <.0001
DayLength*Climate 1 0.01488276 0.01488276 1.75 0.1952
The 4 treatments come from the ‘superfactor’ representing combi-
nations of DayLength & Climate. Three degrees of freedom are
used for that ‘superfactor’ in the model (i.e. 4 distinct cell means).
ii) Slight modification of the RCBD above where instead of 12 dis-
tinct litters all from unique families, suppose we now have 6 fam-
ilies and 2 litters from each family (so, still 12 blocking ‘units’).
Diagram: FAMILY6 ⇥ trt4
|REP2
R
Restricted randomization done within rep within each family.
Four observations from each rep (4⇥ 6 ⇥ 2 or N=48).
Within each FAMILY, we observe all treatments (crossed).
Within each litter (or REP), we observe all treatments (crossed).
A rep is unique to a family, REP is nested in a FAMILY (nested).
Or similarly, REP 1 in family 1 has a di↵erent meaning than REP
1 in family 2.
We still essentially have 12 ‘blocking units’, but how we partition the
variability among the 12 blocks is di↵erent (we will break it apart as
some among families, and some among litters within family).
2
SAS Output 6 distinct families 2 reps from each:
Class Level Information
Class Levels Values
Family 6 1 2 3 4 5 6
Rep 2 1 2
DayLength 2 long short
Climate 2 cold warm
Number of Observations Read 48
Number of Observations Used 48
Source DF Type III SS Mean Square F Value Pr > F
Family 5 8.91754806 1.78350961 209.45 <.0001
Rep(Family) 6 0.14641706 0.02440284 2.87 0.0233
DayLength 1 8.27435377 8.27435377 971.73 <.0001
Climate 1 0.46745416 0.46745416 54.90 <.0001
DayLength*Climate 1 0.01488276 0.01488276 1.75 0.1952
2 4 6 8 10 12
01
23
4
6 distinct litters, 2 reps within each (N=48)
Grouping
y
3
Diagram: i) LITTER12
R ⇥ trt4
The Mixed Procedure
Covariance Parameter
Estimates
Cov Parm Estimate
Grouping 0.2039
Residual 0.008515
Diagram: ii) FAMILY6⇥trt4
|REP2
R
The Mixed Procedure
Covariance Parameter
Estimates
Cov Parm Estimate
Family 0.2199
Rep(Family) 0.003972
Residual 0.008515
NOTE: p-values for the fixed treatment e↵ects are same either way.
4
