Advanced Topics in Forest
Biometrics – FOR6934
Mixed Models – Part I
Fal 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.
Fal 2007, C. Staudhammer
Slide 3
Outline
1.
What makes a model mixed?
2.
Examples of mixed models
Fal 2007, C. Staudhammer
Slide 4
Recall: Statistical models
±
Statistical models are stochastic models that
contain
unknown constants
that we estimate
from data
²
These unknown constants are
parameters (e.g.,
τ
1
,
2
,
...
)
±
Statistical models describe distributional
properties of response variables
²
Variability can be decomposed into known and
unknown sources
Fal 2007, C. Staudhammer
Slide 5
Recall: Statistical model terminology
±
Response: the outcome of interest (
Y
)
±
Parameter: any unknown constant (e.g.,
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
e
y
ˆ
ˆ
+
=
)
ˆ
(
e
Fal 2007, C. Staudhammer
Slide 6
Recall: Example model
±
Example: the weight of
a brazil nut fruit,
Y,
under
silvicultural treatment
i
is:
Y
=
µ
+
i
+
e
where:
e
is a random variable with mean
zero and variance
2
±
The expected value of
Y
under treatment
i
is:
E[
Y
i
] =
+
i
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Slide 7
Fixed effects models
±
An effect is fixed if all possible levels about which
inferences will be made are represented
²
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)
²
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
τ
µ
=
+
=
−
=
=
+
=
+
=
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
(
)
()
2
1
2
1
2
1
1
1
1
1
2
2
2
2
...
var
var
)
(
var
var
)
var(
n
n
An
A
A
n
Aj
n
Aj
A
n
Aj
n
A
n
e
e
e
e
e
y
y
=
=
+
+
+
=
=
+
+
=
=
∑
∑
∑
⋅
Fal 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)
²
Subsampling, clustering, and random selection of
treatments result in random effects in models
±
If treatments are random, then:
where:
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
+
=
−
+
=
+
=
⋅
⋅
⋅
⋅
In a random effects model,
is not normally of any
interest since the only
parameter being estimated
is
. If you are interested in
treatment means, then you
should have a fixed effects
model.
⋅
A
y
(
)
(
)
2
1
2
2
1
2
2
1
1
1
1
1
1
1
2
2
...
var
var
var
var
)
(
var
var
)
var(
n
n
An
A
A
n
A
n
Aj
n
A
n
Aj
A
n
Aj
n
A
n
e
e
e
n
e
e
y
y
+
=
+
=
+
+
+
+
=
+
=
+
+
=
=
∑
∑
∑
∑
⋅
Fal 2007, C. Staudhammer
Slide 9
Mixed effects models
±
Mixed effect models contain some random and
some fixed effects (apart from intercept and
error)
²
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