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1
1
Introduction to Mixed Linear Models
2/14/2008
2
Statistical Models
±
A statistical model describes a formal
mathematical data generation mechanism from
which an observed set of data is assumed to
have arisen.
3
Example 1: TwoTreatment CRD
4
Assign 8 Plants to Each Treatment Completely at Random
222
22
2
1
1
1
1
1
1
1
1
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5
Treatment effects about yield
±
Suppose we have one measurement of yield
(weight) for each plant and the question is
whether the two treatments results in different
yields.
±
Data: y
11
, y
12
, y
13
, y
14
, y
15
, y
16
, y
17
, y
18
,
and
y
21
, y
22
, y
23
, y
24
, y
25
, y
26
, y
27
, y
28
±
Model:
Y
ij
=
μ
+
τ
i
+ e
ij
6
Model specification
±
μ
represents
overall mean
of yield.
±
τ
1
and
τ
2
represent the effects of treatments 1
and 2 on mean yield.
±
There is treatment effect
if
±
e
ij
, i=1,2, j=1, …, 8, represent residual random
effects that include any sources of variation
unaccounted for by other terms.
7
Linear Model
±
Our model is a
linear model
because the mean of the
response variable may be written as a
±
Suppose a model has parameters
θ
1
,
θ
2
, .
..,
θ
m
.
±
A linear combination of the parameters is
c
1
θ
1
+c
2
θ
2
+ .
.. +c
m
θ
m
where c
1
, c
2
, .
.., c
m
are
known constants
.
8
Randomly Pair Plants Receiving Different Treatments
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
3
9
Randomly Assign Pairs to Slides Balancing the Two Dye Configurations
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
10
Observed Normalized Signal Intensities (NSI) for
One Gene
Y
111
Y
112
Y
113
Y
114
Y
221
Y
222
Y
223
Y
224
Y
125
Y
126
Y
127
Y
128
Y
215
Y
216
Y
217
Y
218
treatment
dye
slide
11
Modeling the
Means
of the Observed Normalized
Signal Intensities (NSI)
μ
+
τ
1
+
δ
1
μ
+
τ
2
+
δ
2
μ
+
τ
1
+
δ
2
μ
+
τ
2
+
δ
1
represents overall mean of NSI.
represent the effects of treatments 1 and 2 on
mean NSI.
represents the effects of Cy3 and Cy5 dyes on
mean NSI.
A gene is differentially expressed if
τ
1
≠τ
2
.
12
Unknown
Random Effects
Underlying Observed NSI
s
1
+e
111
s
2
+e
112
s
3
+e
113
s
4
+e
114
s
1
+e
221
s
2
+e
222
s
3
+e
223
s
4
+e
224
s
5
+e
125
s
6
+e
126
s
7
+e
127
s
8
+e
128
s
5
+e
215
s
6
+e
216
s
7
+e
217
s
8
+e
218
s
1
, s
2
, s
3
, s
4
, s
5
, s
6
, s
7
, and s
8
represent slide effects.
e
111
,...,e
218
represent residual random effects that include any
sources of variation unaccounted for by other terms.
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13
We need to say more about the random effects for a
complete model.
±
We will almost always assume that random effects are
independent and normally distributed
with
mean zero
and a
factorspecific variance
.
±
s
1
, s
2
, .
.., s
8
~ N(0,
σ
s
) and independent of
e
111
, e
112
, e
113
, e
114
, e
221
, e
222
, e
223
, e
224
, e
125
,
e
126
, e
127
, e
128
, e
215
, e
216
, e
217
, e
218
~ N(0,
σ
e
).
2
iid
2
iid
(or just
to save time and space.)
14
What does s
1
, s
2
, .
.., s
8
~ N(0,
σ
s
) mean?
2
iid
15
Observed NSI are Means Plus Random Effects
Y
111
=
μ
+
τ
1
+
δ
1
+s
1
+e
111
Y
112
=
μ
+
τ
1
+
δ
1
+s
2
+e
112
Y
113
=
μ
+
τ
1
+
δ
1
+s
3
+e
113
Y
114
=
μ
+
τ
1
+
δ
1
+s
4
+e
114
Y
221
=
μ
+
τ
2
+
δ
2
+s
1
+e
221
Y
222
=
μ
+
τ
2
+
δ
2
+s
2
+e
222
Y
223
=
μ
+
τ
2
+
δ
2
+s
3
+e
223
Y
224
=
μ
+
τ
2
+
δ
2
+s
4
+e
224
Y
125
=
μ
+
τ
1
+
δ
2
+s
5
+e
125
Y
126
=
μ
+
τ
1
+
δ
2
+s
6
+e
126
Y
127
=
μ
+
τ
1
+
δ
2
+s
7
+e
127
Y
128
=
μ
+
τ
1
+
δ
2
+s
8
+e
128
Y
215
=
μ
+
τ
2
+
δ
1
+s
5
+e
215
Y
216
=
μ
+
τ
2
+
δ
1
+s
6
+e
216
Y
217
=
μ
+
τ
2
+
δ
1
+s
7
+e
217
Y
218
=
μ
+
τ
2
+
δ
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This note was uploaded on 08/26/2009 for the course STAT 416 taught by Professor Peng,l during the Spring '08 term at Iowa State.
 Spring '08
 Peng,L

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