lect18.pdf - 3 4 Xij = µ τi #ij . Note how this...

18.0 Two-Factor Designs
•
Answer Questions
•
RCBD
•
Concrete Example
1
18.1 RCBD
The
Randomized Complete Block Design
is also known as the two-way
ANOVA without interaction. A key assumption in the analysis is that the e
ff
ect of each level of the treatment factor is the same for each level of the
blocking factor. That assumption would be violated if, say, a particular fertilizer worked well for one stain but poorly for another; or if one cancer therapy were better for lung cancer but a di
ff
erent therapy were better for stomach cancer.
In RCBD, there is one observation for each combination of levels of the treatment and block factors.
2
RCBD notation:
•
I
is the number of treatments;
J
is the number of blocks
•
X
ij
is the measurement on the unit in block
j
that received treatment
i
.
•
n
..
=
I
*
J
is the total number of experimental units.
•
n
i.
=
J
and
n
.j
=
I
, the number of observations for a given treatment
level or block level, respectively.
•
X
i.
is the sum of all measurements for units receiving treatment
i
, and
X
.j
is the sum of all measurements for units in the
j
block.
•
¯
X
i.
is the average of all measurements for units receiving treatment
i
, or
1 J
∑
J j
=1
X
ij
.
•
¯
X
.j
is the average of all measurements for units in the
j
th block, or
1 I
∑
I i
=1
X
ij
.
•
¯
X
..
is the average of all measurements.
We continue to use the dot and bar conventions.
3
Some assumptions: The model for an RCBD (or two-way ANOVA without interactions) is:
X
ij
=
μ
+
τ
i
+
β
j
+
ij
where
μ
is the overall mean of all experimental units,
τ
i
is the e
ff
ect of
treatment
i
,
β
j
is the e
ff
ect of block
j
, and the
ij
are random errors that are:
•
normally distributed with mean zero and unknown standard deviation
σ
•
independent of the values for all other errors.
Note how this generalizes the one-way ANOVA model:
X
ij
=
μ
+
τ
i
+
ij
.