Graphics Reference
In-Depth Information
Figure
.
.
he top let-hand panel shows data on the occurrence of very low (
%) levels of dissolved
oxygen in the River Clyde, related to the temperature of the water. he top right-hand panel shows
a smooth nonparametric regression curve to display the pattern of change in the probability of very low
dissolved oxygen as a function of temperature. he bottom let-hand panel displays standard error
bands to indicate the variability in the nonparametric estimate. he bottom right-hand panel shows
nonparametric regression curves simulated from the fitted logistic model, together with the
nonparametric curve from the observed data
<
A natural model in this setting is a simple logistic regression where the proba-
bility p
of observing a measurement below the threshold is assumed to be de-
scribed by exp
(
x
)
,whereα and β are unknown parame-
ters. he log-likelihood contribution for an observation
(
α
+
βx
)(
+
exp
(
α
+
βx
))
(
x
i
, y
i
)
can be written as
. Maximising the weighted log-likelihood, as
describedabove, andusingasmoothing parameter h
y
i
log
p
(
x
i
) + (
−
y
i
)
log
−
p
(
x
i
)
,producesthe nonparamet-
ricestimate showninthetopright-hand panelofFig.
.
.hetechnical details ofthis
process are described by Fan et al. (
).
he generally increasing nature of the relationship between low measurements
and temperature is apparent. However,the downturn in the estimate of the probabil-
ityforthe highesttemperature isasurprise.Again,itishelpfultoaddinformation on
variability to the plot. he bottom let hand panel of Fig.
.
shows variability bands
=
