Page 8 - Volume 4, Issue 1, January 2011

© Benaki Phytopathological Institute
Makowski 6
A limitation of model comparison is that,
in some cases, several models show simi-
lar performance (e.g. Hernandez et al., 2006;
Roura-Pascual et al., 2009). Another issue is
that reliable data are not always available.
Model selection is then somewhat arbitrary.
Several statisticians emphasised that, in
some cases, it is better to mix all models than
to use the single selected model. The basic
idea is to use a weighted sum of the individ-
ual model predictions instead of the predic-
tion derived from the single ‘best’ model. Sev-
eral methods were developed to estimate the
weight associated to each model from a train-
ing dataset (Buckland et al., 1997; Hoeting et
al., 1999; Yang, 2003; Raftery et al., 2005; Yuan
and Yang, 2005). These methods can be ap-
plied to a great diversity of models, linear, lo-
gistic, nonlinear, or dynamic models (Raftery
et al., 1997; Viallefond et al., 2001; Raftery et al.,
2005), and statistical packages are now avail-
able to implement them. See, for example,
the BMA R package available at http://cran.r-
project.org/web/packages/BMA/index.html
and the MMIX R package available at http://
cran.r-project.org/web/packages/MMIX/in-
dex.html. Both can be freely downloaded and
applied using the R statistical software.
Model-mixing methods can improve
the accuracy of model predictions and give
more realistic confidence intervals (Chat-
field, 1995; Draper, 1995). According to a re-
cent statistical study (Yuan and Yang, 2005),
model-mixing is better than selection when
the model errors are large. Recently, mod-
el-mixing methods have been applied for
mapping species distribution (Araujo and
New, 2006; Marmion et al., 2009) and bio-
logical invasion (Roura-Pascual et al., 2009).
It is likely that this approach will be more fre-
quently applied in the future.
4. Case study
In this section, we present a simple case
study to show how uncertainty and sensi-
tivity analysis can be used in practice. We
consider the simple generic infection model
for foliar fungal plant pathogens defined by
Magarey et al . (2005):
and
if T min≤ T ≤Tmax and zero otherwise
where T is the mean temperature during
wetness period (°C), W is the wetness dura-
tion required to achieve a critical disease in-
tensity (5% disease severity or 20% disease
incidence) at temperature T. Tmin, Topt, Tmax are
minimum, optimal, and maximum temper-
atures for infection, respectively, Wmin and
Wmax are minimum and maximum possible
wetness duration requirements for critical
disease intensity, respectively. This model
was used to compute the wetness duration
requirement as a function of the tempera-
ture for many species and was included in
a disease forecast system (Magarey et al.,
2005; 2007).
T min, Topt, Tmax, Wmin and Wmax are five spe-
cies-dependent parameters whose values
were estimated from experimental data and
expert knowledge for different foliar patho-
gens (e.g. Magarey et al., 2005; EFSA 2008b).
However, for some species, these parame-
ters are uncertain due to the limited avail-
ability of data (Magarey et al., 2005) and, in
such cases, it is important to perform uncer-
tainty and sensitivity analysis.
In this case study, uncertainty and sen-
sitivity analysis techniques were applied to
the generic infection model defined above
for infection of bean foliage by the fungal
pathogen Sclerotinia sclerotiorum. All com-
putations were done using the freely avail-
able statistical software R(http://cran.r-proj-
ect.org/). Parameter values reported by
Magarey et al. (2005) for this pathogen are
Tmin =1°C, Topt=25°C, Tmax=30°C, Wmin =48 h and
Wmax=144 h but, according to the authors,
there is uncertainty about these values. The
response curve of W vs. T obtained with the
estimated parameter values is presented in
min
max min , ( )
W
W W
f T
⎧ ⎫
= ⎨ ⎬
⎩ ⎭
min max ( )/( )
max min
max min
( )
opt opt T T T T
opt opt
T T T T
f T
T T T T
− −
⎛ ⎞⎛ ⎞
− −
= ⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟
− −
⎝ ⎠⎝ ⎠