Advanced Topics in Forest
Biometrics – FOR6934
Mixed Models – Part I
Fall 2007, C. Staudhammer
Slide 2
Motivating texts
Schabenberger, O. and F.J. Pierce (2002)
Contemporary Statistical Models for the
Plant and Soil Sciences
. CRC Press, NY,
NY.
Littell, R.C., G.A. Milliken, W.W. Stroup, and
R.D. Wolfinger (1996)
SAS System for
Mixed Models
. SAS Institute, Cary,NC.
Fall 2007, C. Staudhammer
Slide 3
Outline
1.
What makes a model mixed?
2.
Examples of mixed models
Fall 2007, C. Staudhammer
Slide 4
Recall: Statistical models
Statistical models are stochastic models that
contain
unknown constants
that we estimate
from data
These unknown constants are
parameters (e.g.,
τ
1
,
τ
2
,
...
)
Statistical models describe distributional
properties of response variables
Variability can be decomposed into known and
unknown sources
Fall 2007, C. Staudhammer
Slide 5
Recall: Statistical model terminology
Response: the outcome of interest (
Y
)
Parameter: any unknown constant (e.g.,
τ
i
,
σ
2
)
Prediction: fitted value of the model
Model errors: difference between the observed
response and the fitted value
The observed number of fruits from the
i
th
treatment,
j
th tree:
)
ˆ
(
Y
ij
ij
ij
e
y
y
ˆ
ˆ
+
=
)
ˆ
(
e
Fall 2007, C. Staudhammer
Slide 6
Recall: Example model
Example: the weight of
a brazil nut fruit,
Y,
under
silvicultural treatment
i
is:
Y
=
µ
+
τ
i
+
e
where:
e
is a random variable with mean
zero and variance
σ
2
The expected value of
Y
under treatment
i
is:
E[
Y
i
] =
µ
+
τ
i
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Fall 2007, C. Staudhammer
Slide 7
Fixed effects models
An effect is fixed if all possible levels about which
inferences will be made are represented
A level of a fixed effect is an unknown constant, which does
not vary
Fixed effect models contain
only
fixed effects (apart from a
single error term)
Most regression models are fixed effects models
If treatments are fixed, then for treatments A and B:
n
y
y
y
y
n
y
n
y
y
E
y
E
B
A
B
A
B
A
B
B
A
A
/
2
)
var(
)
var(
)
var(
:
Thus
/
)
var(
/
)
var(
)
(
and
)
(
2
2
2
σ
σ
σ
τ
µ
τ
µ
=
+
=
−
=
=
+
=
+
=
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
(
)
(
)
(
)
(
)
2
1
2
1
2
1
1
1
1
1
2
2
2
2
...
var
var
)
(
var
var
)
var(
σ
σ
τ
µ
n
n
An
A
A
n
Aj
n
Aj
A
n
Aj
n
A
n
e
e
e
e
e
y
y
=
=
+
+
+
=
=
+
+
=
=
∑
∑
∑
⋅
Fall 2007, C. Staudhammer
Slide 8
Random effects models
Effects are random if the levels represent only a
random sample of possible levels
Random effect models contain
only
random effects
(apart from intercept)
Subsampling, clustering, and random selection of
treatments result in random effects in models
If treatments are random, then:
where:
σ
τ
2
is the variance of the treatment effects (which is
non
zero!
)
)
/
(
2
)
var(
/
)
var(
)
(
2
2
2
2
n
y
y
n
y
y
E
B
A
A
A
A
σ
σ
σ
σ
τ
µ
τ
τ
+
=
−
+
=
+
=
⋅
⋅
⋅
⋅
In a random effects model,
is not normally of any
interest since the only
parameter being estimated
is
µ
. If you are interested in
treatment means, then you
should have a fixed effects
model.
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 Spring '08
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 Random effects model, C. Staudhammer

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