Non-Linear Regression

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Non-Linear Regression
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The data frame trees is made available in R
with gtdata(trees) These record the girth in
inches, height in feet and volume of timber in
cubic feet of each of a sample of 31 felled black
cherry trees in Allegheny National Forest,
Pennsylvania. Note that girth is the diameter
of the tree (in inches) measured at 4 ft 6 in
above the ground.
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We treat volume as the (continuous) response
variable y and seek a reasonable model describing
its distribution conditional first on the
explanatory variable girth (we will call this
x). This might be a first step to prediction of
volume based on further observations of the
explanatory variables.
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• Observation of the graph leads us to first try
  out whether there may be a linear dependence
  here.
• Thus the relationship is approximately
  yabx?, for some constants a and b
• We will use R to find a and b, their standard
  errors and the residuals.

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The fitted model is volume -36.9 5.07 girth
residual i.e. y -36.9 5.07x (
residual) To check its validity, first look at
the standard errors
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The standard errors of both a and b are low in
comparison with the actual values and the
p-values associated with the coefficients show
that neither of these may reasonably be taken as
zero. Thus there is evidence that the model is
appropriate.
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Some measure of the success of the fitted model
is also given by the residual standard error. For
a good fit this should be small in relation to
the variation in the response variable itself.
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Note ?18.1 4.252
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However, a full examination of the residuals, and
of the nature of any further dependence they may
have on the explanatory variables, is to be
preferred to reliance on any single number. All
this will require graphical analysis, the results
of which follow.
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There is a slight evidence of non random
behaviour in the residuals with perhaps the hint
of a quadratic curve. We now adapt the model.
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The residuals from Model 1 show some further,
perhaps quadratic, dependence on the explanatory
variable girth, so we try introducing a nonlinear
term. We consider the model volume a b1
girth b2 (girth)2 resid The relevant R
commands, and associated output, are now gtmodel2
lm(VolumeGirthI(Girth2)) gt summary(model.2)
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The fitted model is therefore volume 10.8 -
2.09 girth 0.255 (girth)2 residual.
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Consider now the graphs produced by the following
commands. gt plot(VolumeGirth) gt
lines(fitted(model2)Girth) gt plot(residuals(mode
l2)Girth, ylab"residuals from Model 2")
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It is clear that these residuals are both smaller
than those from Model 1 and show no further
obvious dependence on the explanatory variable
girth. Further the very small p-value (0.00015)
associated with the coefficient b2 shows that
this cannot reasonably be set equal to zero, so
that Model 2 is considerably more successful than
Model 1.
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Note also that the residual standard error in
Model 2 is 3.335 whilst in Model 1 it is
4.252. Further Analysis On physical grounds,
we might also consider the simpler model Volume
b2 (Girth)2 Residual For extra
justification look at this R output
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The R code to fit this model, and brief summary
output, are gt model3 lm(Volume I(Girth2) -
1) gt summary(model3)
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