Agriculture Reference
In-Depth Information
Tan and Iglewicz (1999) showed that confi-
dence intervals on β can be constructed by
transforming parameters to polar coordinates:
β = tan θ, and θ = arctan β
The linear functional relationship ap-
proach is appealing because the hypothesis
being tested is not one of equivalence but
one of practical equivalence. For example,
practical equivalence could be stated for
slope as π/
4
- ψ < θ < π/
4
+ ψ, with ψ deter-
mined before the validation based on the in-
tended usage of the model. For example,
one could use ψ = π/
30,
which roughly cor-
responds to 0.8 < β < 1.25.
Much work is needed in the area of sto-
chastic model validation to compare the
methods outlined previously.
φ = τ/cos θ, and τ = sgn (β) α/(
1
+ β
2
)
1/
2
(5.20)
Using Eqn 5.20, practical equivalence can be
tested as:
H
o
: θ ≤ θ
o
- ψ
1
or θ ≥ θ
0
+ ψ
2
vs
H
A
: θ
o
- ψ
1
≤ θ ≤ θ
o
+ ψ
2
(5.21)
This is in contrast to a test of equivalence, which
would translate to the following hypotheses:
H
o
: θ = θ
o
vs
H
A
: θ ≠ θ
o
. (5.22)
Tan and Iglewicz (1999) showed that Eqn
5.21 is the right set of hypotheses and not Eqn
5.22. In this structure, practical equivalency
forms the alternate hypothesis, whereas
non-equivalency forms the null hypothesis.
Thus we will conclude that the outcomes of
the stochastic model are not equivalent to
the outcomes of the real system (the meas-
urements) unless we can show that the two
are within ψ
1
and ψ
2
in the polar system.
Because tan(45°) =
1
and arctan(1) = 45°, and
that the 45° slope would indicate perfect cor-
respondence between the two methods Eqn
5.21 has an inherent intuitive interpretation.
Conclusions
Stochasticity is an inherent characteristic
of biological models in the same way that
errors and probability distributions are inher-
ent characteristics of experiments. Although
more complex in their formulation and cali-
bration than deterministic models, stochas-
tic models capture the inherent uncertainty
characteristics of biological systems. If sto-
chastic effects are present in the system being
studied, one cannot understand the true sys-
tem dynamics using a deterministic frame-
work. One cannot simply fit a deterministic
model to available experimental data and
then use the inferred rate constants in a sto-
chastic simulation.
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