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for the estimate, constructed by carrying the weights through the usual process of
deriving standard errors in generalised linear models. hese indicate a high degree
of variability at high temperatures.
he suitability of a linear logistic model can be assessed more directly by con-
structing a reference band. A simple way of doing this here is to simulate data from
the fitted linear logistic model and construct a nonparametric estimate from each
set of simulated data. he results from repeating this
times are displayed in the
bottom right-hand panel of Fig.
.
. he appearance of some other estimates with
curvature similar tothat exhibited in the original estimate offersreassurance that the
data are indeed consistent with the linear logistic modeland prevents an inappropri-
ate interpretation of a feature in the nonparametric estimate which can reasonably
be attributed to sampling variation.
Smoothing in Two Dimensions
9.3
he implementation of smoothing techniques with two covariates is a particularly
important application because of the wide variety of types of data where it is help-
ful to explore the combined effects of two variables. Spatial data provide an im-
mediate example, where the local characteristics of particular regions lead to mea-
surement patterns which are oten not well described by simple parametric shapes.
As in the univariate case, a variety of different approaches to the construction of
a smooth estimate of an underlying regression function is available. In particular,
the extension of the local linear approach is very straightforward. From a set of data
(
,wherey denotes a response variable and x
, x
are
covariates, an estimate of m at the covariate value x arises from minimising the
weighted least squares
x
i
, x
i
, y
i
)
i
=
,...,n
n
i
=
w
y
i
−
α
−
β
(
x
i
−
x
)−
β
(
x
i
−
x
)
(
x
i
−
x
; h
)
w
(
x
i
−
x
; h
)
over α, β
and β
.heestimate m
isthefittedvalueoftheregressionatx,namely
α.Morecomplexformsofweighting arepossibleandHärdleetal.(
)give amore
general formulation. However, the product form shown above is particularly attrac-
tive in the simplicity of its construction.
From the form of this weighted sum-of-squares, it is immediately obvious that
the estimator m
(
x
)
again has a linear form v
y, for a vector of known constants v.
he concept of approximate degrees of freedom then transfers immediately, along
with automatic methods of smoothing parameter selection such as the AIC method
described in (
.
). Estimation of the underlying error variance needs more specific
thought, although the same principles of local differencing apply, as described by
Munk et al. (
). For the particular case of two covariates, a method based on
a very small degree of local smoothing is also available, as described by Bock et al.
(
), and this is used in the illustrations of this chapter.
(
x
)
