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quan-
tifies the linear effect of the variable ater the linear effects of the other variables are
accounted for.On the other hand,the correlation of log
Toresolvethecontradiction,recallthattheregressioncoe
cientoflog
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)
with theresponsevari-
able ignores the effects of the other variables. Since it is important to take the other
variables intoconsideration, theregression coe
cient maybeabettermeasureofthe
effect of log
(
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)
. But this conclusion requires that the linear model assumption be
correct. Nonetheless, it is hard to explain the negative linear effect of log
(
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)
(
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)
when
we are faced with Fig.
.
.
heproblemofcontradictorysignsvanisheswhenthereisonlyoneregressorvari-
able. Although it can occur with two regressor variables, the di
culty is diminished
because the fitted model can be visualized through a contour plot. For datasets that
contain more than two predictor variables, we propose a divide-and-conquer strat-
egy. Just as a prospective buyer inspects a house one room at a time, we propose to
partition the dataset into pieces such that a visualizable model involving one or two
predictors su
ces for each piece. One di
culty is that, unlike a house, there are no
predefined“rooms”or“walls”inadataset. Arbitrarily partitioning adataset makes as
much sense as arbitrarily slicing a house into several pieces. We need a method that
gives interpretable partitions of the dataset. Further, the number and kind of parti-
tions should bedictated bythe complexity of the dataset aswellas the typeof models
tobe fitted. Forexample, if a dataset isadequately described bya nonconstant simple
linear regression involving one predictorvariable and we fit a piecewise linear model
to it, then no partitioning is necessary. On the other hand, if we fit a piecewise con-
stant model to the same dataset, the number of partitions should increase with the
sample size.
he GUIDE regression tree algorithm (Loh,
) provides a ready solution to
these problems. GUIDE can recursively partition a dataset and fit a constant, best
polynomial, or multiple linear model to the observations in each partition. Like the
earlier CART algorithm (Breiman et al.,
), which fits piecewise constant models
only, GUIDE first constructs a nested sequence of tree-structured models and then
uses cross-validation to select the smallest one whose estimated mean prediction de-
viance lies within a short range of the minimum estimate. But unlike CART, GUIDE
employslack-of-fittestsoftheresidualstochooseavariable topartition ateachstage.
Asaresult, itdoes nothave the selection bias ofCART andother algorithms that rely
solely on greedy optimization.
To demonstrate a novel application of GUIDE, we use it to study the linear ef-
fect of log
ater controlling for the effects of the other variables, without mak-
ing the linear model assumption. We do this by constructing a GUIDE regression
tree in which log
(
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)
is the sole linear predictor in each partition or node of the
tree. he effects of the other predictor variables, which need not be transformed, can
be observed through the splits at the intermediate nodes. Figure
.
shows the tree,
which splits the data into
nodes. he regression coe
cients are between
(
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)
.
and
.
in all but four leaf nodes. hese nodes are colored red (for slope less than
−
−
.
)
and blue (for slope greater than
.
). We choose the cutoff values of
.
because
the coe
cient of log
in Table
.
is
.
. he tree shows that the linear effect of
(
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)
log
is neither always positive nor always negative - it depends on the values of
(
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)
