Agriculture Reference
In-Depth Information
5
Statistical Issues in Nutritional Modelling
N. St-Pierre*
The Ohio State University, Columbus, Ohio, USA
Abstract
Most models in biology have a deterministic structure: they make definite predictions for quantities
without any associated probability distribution. A stochastic model, on the other hand, contains ran-
dom elements and probability distributions within the model itself. Historically, the predominant
perspective was that the stochasticity of a model represented the unknown part of a system, with the
implication that stochasticity should be absent in a model of sufficient complexity. This characteriza-
tion of stochasticity is at best naive, and at worst plain wrong. In general, a univariate model can be
represented as:
Y
=
f
(
X
,
b
) +
e
, where
Y
is a vector of observations,
f
is some unknown function or set
of functions,
X
is a matrix of input variables (observed),
b
is a vector of parameters to the function
f
and
e
is a vector of errors. Stochasticity enters the model at four locations. First,
e
represents the outcome
from many sources of error. Measurement errors on the
Y
appear in
e
; this component is generally
referred to as pure error. But
e
also contains some of the errors associated with having an incorrect
model, what has been called the lack-of-fit component. A proper experimental design allows separate
estimates of these two components. The second source of stochasticity is related to
b
. In a frequentist
framework,
b
are fixed, errorless parameters. Their estimates
B
, however, are stochastic variables with
a multivariate distribution (often considered multivariate normal in application). Thus, the prediction
error of any model includes components related to
e
, but also the variances and covariances associated
with the estimated
b
. A simulation of the model errors, even in a frequentist paradigm, must include
the variance in the
B
as well as in the
e
. In a Bayesian framework, the
b
are themselves random param-
eters. Thus their stochastic property is evident and explicitly considered during estimation and must
therefore be retained during model evaluation and simulation. A third source of stochasticity relates
to the
X
elements. Inputs to models are never known with certainty. The error in the measurements
of the variables must be accounted for during parameterization. This is a very complicated problem
for dynamic models expressed in the form of differential equations. Last, the function
f
is itself an
approximation to the true but unknown function. Thus, the uncertainty surrounding
f
adds uncer-
tainty to model predictions. This stochasticity can be included in the analysis in instances when there
is more than one competing model (functions). Additionally, global as opposed to local sensitivity
analysis techniques must be used. The probability distributions of the model outcomes can be com-
pared to the probability distributions of the experimental outcomes, thus answering the question of
whether the two are from the same process. Viewed this way, stochasticity is an inherent characteristic
of biological models the same way that errors and probability distributions are inherent characteristics
of experiments.
