Advanced Topics in Forest
Biometrics – FOR6934
Review of General Linear Models
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.
What is a model?
A
scientific
model is an abstraction of reality
Models can be further classified:
Mathematical models
Stochastic models
Statistical models
Mathematical models
Mathematical models are mechanistic (deterministic)
devices
A given set of inputs gives an output
that is predicted with certainty
Example: the weight of
a brazil nut fruit,
Y,
under
silvicultural treatment
i
is:
Y
=
µ
+
τ
i
where:
τ
i
is the
i
th treatment effect
Deterministic models have
no random components
Stochastic models
Mathematical models ignore uncertainty, whereas
stochastic models include random effects
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
Why do we need stochastic elements?
(Adapted from Schabenberger and Pierce 2002)
The model is not correct for a particular observation,
but correct on average
Assumptions are necessary to abstract
phenomenon
It is often impossible to measure/observe all
variables
Stochastic models are often more parsimonious
When you have real data, you have
sampling error
What is sampling
error?
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Statistical models
Statistical models are stochastic models that
contain
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 Spring '08
 Staff
 General Linear Models, Yij, Contemporary Statistical Models

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