Agriculture Reference
In-Depth Information
The following illustrative discussion combines related examples presented in
Yuen and Hughes (2002), McRoberts
et al
. (2003) and Yuen and Mila (2003).
Imagine a forecaster that has both specificity and sensitivity equal to 0.9. That is, it
correctly distinguishes true positives from false negatives 90% of the time (or, in
other words, only one in ten 'cases' would (wrongly) not receive treatment in the
long run) and true negatives from false positives 90% of the time (only one in ten
'controls' would (wrongly) receive treatment in the long run). In this case, we have
LR
+
= 0.9/(1-0.9) = 9.0 and LR
-
= (1-0.9)/0.9 = 0.11. Referring back to Fig. 12.2,
everything we have said so far about likelihood ratios would fall inside the central
box. That is, the message about requirement for action to control disease is captured
in the likelihood ratios, which summarise 'the empirical content' of the data, and it
is this message that would be contained within the IT-based tool. We can also see
that the development and evaluation of tools in this way involves all of the steps in
Fig. 12.1 in an explicit manner that makes the development process transparent.
Consider a grower, A, who, based on previous experience, including personal
subjective factors and the known long-term prevalence of disease, has a prior
probability of 20% for the need for treatment, written as P
prior_A
(
D
+) = 0.2 (that is,
odds
prior_A
(
D
+) = 0.25). Assume that in using the forecaster, A obtains a positive
prediction of need for treatment. According to equation 12.1, A's posterior odds,
odds
post_A
(
D
+), are LR
+
0.25 = 2.25, corresponding to a
P
post_A
(
D
+) = 0.692. The conclusion from this is that A will have moved from a
position of 'thinking' there was a 20% chance of need for treatment, to thinking that
the chance is just under 70%. A second grower, B, has a prior probability
P
prior_B
(
D
+) = 0.8 (corresponding to odds
prior_B
(
D
+) = 4). Assume that in using the
forecaster, B obtains a negative prediction of need for treatment. According to
equation 12.1, odds
post_B
= LR
-
⋅
⋅
odds
prior_A
(
D
+) = 9
×
odds
prior_B
(
D
+) = 0.11
×
4 = 0.44, corresponding to
P
post_B
(
D
+)
0.31. Summarising, we can say that B's position has changed from one
in which the perceived chance of need for treatment was 80% to one in which it was
just over 30%.
Returning to Shannon's (1948) fundamental problem, the messages that the
forecaster can deliver are summarised in equation 12.1. We can paraphrase them in
natural language as follows: (i) given a prediction of need for treatment, multiply
your personal odds of need for treatment by LR
+
to find out what the odds of need
for treatment are now, in the light of the message contained in the (positive)
prediction; (ii) given a prediction of no need for treatment, multiply your personal
odds of need for treatment by LR
-
to find out what the odds of need for treatment are
now, in the light of this (negative) prediction. Messages (i) and (ii) are quite simple
and although this, in itself, is no guarantee of them being reproduced '
exactly or
approximately
' at the point of delivery, it will not decrease the chances of this
happening. Shannon (1948) pointed out that delivering a message with high fidelity
says nothing about what that message means. A little thought quickly leads to the
conclusion that the meaning of messages (i) and (ii) will depend on the user, for at
least two important reasons. First, different users might start with different prior
probabilities of need for treatment and will therefore have different posterior
probabilities of need for treatment even when they obtain the same forecast (receive
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