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In-Depth Information
Itisnowalsonatural toconsiderwhetherasimplelinearmodelmightbeadequate
todescribe theunderlying relationship, with thecurvature exhibited bythenonpara-
metric estimate attributable tosampling variation. Variability bands provideoneway
of approaching this. However, a more direct way of assessing the evidence is through
a reference band, which indicates where a nonparametric estimate is likely to lie if
the underlying regression is indeed linear. Since bias in the nonparametric estimate
dependsonthe curvature ofthe underlyingregression function, itfollows that anon-
parametric estimate, fitted by the local linear method, is unbiased in the special case
of data fromalinear model.Ifthe fitted linear modelat the covariate value x is repre-
sented as
n
i
=
l
i
y
i
,then it is straightforward tosee that the variance of the difference
between the linear and nonparametric models is simply
n
i
=
σ
.Onsubsti-
tuting an estimate of σ
, a reference band extending for a distance of two standard
errors above and below the fitted linear model can then easily be constructed.
his is illustrated in the right-hand panel of Fig.
.
, where the evidence against
asimplelinear modelisconfirmed inthis graphical manner.hisaddition totheplot
has therefore identified an important feature which is not easily spotted from plots
of the raw data.
Aformal,globaltestcanalso becarried out,asdescribedbyBowmanandAzzalini
(
), but the discussion here will be restricted to graphical aspects. It should also
be noted that the calculations for the reference band have been adjusted to account
forthe correlations in the residuals fromthe fitted cosine model.However,this effect
is a very small one.
he idea of local fitting of a relevant parametric model is a very powerful one
which can be extended to a wide variety of settings and types of data. For example,
nonparametric versions of generalised linear models can be constructed simply by
addingsuitable weightstotherelevantlog-likelihoodfunction.Underanassumption
ofindependentobservations, thelog-likelihoodforageneralised linearmodelcan be
represented as
(
v
i
−
l
i
)
n
i
=
l
. A local likelihood for nonparametric estimation at the
covariate value x can then be constructed as
n
i
=
(
α, β
)
l
(
α, β
)
w
(
x
i
−
x; h
)
andthefittedvalueofthemodelatx extractedtoprovidethenonparametric estimate
m
.
Anexample isprovidedbythe data displayedinthe toplet-hand panelofFig.
.
,
which indicate whether the dissolved oxygen in water samples, taken fromtwo well-
separated monitoring points on the river, was below (
) or above (
) a threshold of
mg/l,whichisthelevelrequiredforhealthyresidentandmigratoryfishpopulations.
In order to standardise for the year and seasonal effects which were noted in the ear-
lier analysis, the measurements considered here are restricted to the years from
onwards and to the summer months of May and June. he level of dissolved oxygen
is likely to be related to temperature, which has therefore been used as a covariate. It
is particularly di
cult to assess the nature of any underlying relationship when the
response variable is binary, even when some random variation has been added to the
response values to allow the density of points to be identified more clearly.
(
x
)
