Analysis with Multiple Observations Per Experimental Unit
(Balanced Data Case)
A soil scientist studied the growth of barley plants under three different levels of salinity in a controlled growth
medium. Each salinity treatment was assigned to two containers using a completely randomized design. Three
plants were measured in each container. The data on the dry weight of the plants in grams follow.
Salinity
control
control
6 bars
6 bars
12 bars
12 bars
Container
1
2
3
4
5
6
i
1
1
2
2
3
3
Weight in grams
11.29, 11.08, 11.10
7.37, 6.55, 8.50
5.64, 5.98, 5.69
4.20, 3.34, 4.21
4.83, 4.77, 5.66
3.28, 2.61, 2.69
j
1
2
1
2
1
2
k
1, 2, 3
1, 2, 3
1, 2, 3
1, 2, 3
1, 2, 3
1, 2, 3
1. What are the experimental units and observational units in this example?
Consider the following model for the data.
yijk = µ + τi + eij + dijk
where...
yijk is dry weight of plant k in container j of treatment i,
µ denotes the average of the treatment means,
τi denotes the effect of treatment i (mean for treatment i minus average of all treatmentmeans, i.e., τi = µi − µ),
eij is the random effect of container j of treatment i,
dijk is the random effect of plant k in container j of treatment i,
eij ∼ N (0, σe2 ), dijk ∼ N (0, σd2 ), and all eij and dijk are independent.
2. In general we use t to denote the number of treatments, r to denote the number of experimental units per
treatment, and n to denote the number of observations per experimental unit. What are t, r, and n in this
example?
We can partition the total variation as follows
r X
n
t X
X
(yijk − ȳ··· )2 =
n
t X
r X
X
(ȳi·· − ȳ··· )2 +
i=1 j=1 k=1
i=1 j=1 k=1
= rn
t
X
ȳ··· =
(ȳi·· − ȳ··· )2 + n
t X
r X
n
1 X
yijk
trn i=1 j=1 k=1
(ȳij· − ȳi·· )2 +
r
t X
X
(ȳij· − ȳi·· )2 +
i=1 j=1
ȳi·· =
r X
n
1 X
yijk
rn j=1 k=1
n
t X
r X
X
(yijk − ȳij· )2
i=1 j=1 k=1
i=1 j=1 k=1
i=1
where
r X
n
t X
X
n
t X
r X
X
(yijk − ȳij· )2
i=1 j=1 k=1
ȳij· =
n
1X
yijk .
n k=1
We can organize the sums of squares in an ANOVA table:
Source
TRT
EU(TRT)
OU(EU TRT)
Total
d.f.
Sum of Squares
t−1
t(r − 1)
rn
n
Pt
i=1 (ȳi··
Pt
i=1
Mean Squares
SST
t−1
− ȳ··· )2
Pr
j=1 (ȳij·
tr(n − 1)
Pt
Pr
Pn
− ȳij· )2
trn − 1
Pt
Pr
Pn
− ȳ··· )2
i=1
i=1
j=1
j=1
k=1 (yijk
k=1 (yijk
σd2 + nσe2 + rnθt2
SSE
t(r−1)
− ȳi·· )2
Expect Mean Squares
σd2 + nσe2
SSO
tr(n−1)
σd2
where θt2 =
1
t−1
Pt
2
i=1 τi .
3. By examining the expected mean squares, can you suggest how to test for differences among treatment means?
4. ȳi·· is the estimated mean for treatment i. It can be show that
Var(ȳi·· ) =
σe2
σ2
+ d.
r
rn
Give the formula for the standard error of ȳi·· .
5. It can be show that the variance of a linear combination of treatment means
Var
t
X
i=1
ci ȳi··
!
=
Pt
i=1 ci ȳi··
is
t
σ2 X
σe2
+ d
c2 .
r
rn i=1 i
!
Give the formula for the standard error of a linear combination of estimated treatment means.