Stat 511
Homework 8 solution
Spring 2011
1. The data in range.txt come from a field study of plant response to fertilization. There are
five treatments (a control and four different fertilization regimes) randomly assigned to plots
arranged into blocks. The response is the average PO4 concentration in plant leaves.
(a) Assume blocks are a fixed effect and analyze the data. The model with fixed block effects
reads
Y
ij
=
μ
+
α
i
+
β
j
+
ij
with
ij
∼
N
(0
, σ
2
e
),
i, j
= 1
,
· · ·
5. Report:
The Fstatistic and pvalue for the test of no differences among fertilizers are reported by
R output as: 22
.
605 and 2
×
10

6
The estimated difference between between the 100lb.N
treatment and the 100lbNP treatments is given by
¯
Y
2
·

¯
Y
1
·
, and by R output it is 3
.
80
with standard error 0
.
5648.
The mean for the No fertilizer (No.fert) treatment (averaged over blocks) is 8.06 (
¯
Y
5
·
)
with standard error 0
.
3994.
(b) Assume blocks are a random effect and analyze the data.
Y
ij
=
μ
+
α
i
+
β
j
+
ij
with
ij
∼
N
(0
, σ
2
e
)
, β
j
∼
N
(0
, σ
2
b
),
i, j
= 1
,
· · ·
5 Report:
The Fstatistic and pvalue are the same as part(a).
So are the estimated difference
between between the 100lb.N treatment and the 100lbNP treatments and its standard
error.
The mean for the No fertilizer (No.fert) treatment (averaged over blocks) is 8.06 with
standard error
p
(
σ
2
e
+
σ
2
b
)
/
5 = 0
.
8678.
(c) Using the default parameterizations, R reports an estimated intercept of 4.2520 when
using a fixed block analysis and 6.5800 when using a random block analysis. Explain why
these numbers are different and how you can extract 6.5800 from the estimates provided
by the fixed block analysis.
The random block analysis calculates the intercept by averaging over all the blocks and
the fixed block analysis only using data from block 1. Thus
6
.
58 = 4
.
252 +
5
X
i
=2
ˆ
β
i
/
5 = 4
.
252 + (2
.
02 + 1
.
88 + 2
.
86 + 4
.
88 + 0)
/
5
2. The data in ryegrass.txt come from a study of pasture management in New Zealand.
The
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 Spring '08
 Staff
 Yijk Yijk Yijk, Source df trt, df trt fields

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