The two main purposes of regression

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The two main purposes of regression

• To describe the relationship of the predictors to
  the response
• To predict the mean response for given values of
  the predictors

2
The size of a tree crown is strongly correlated
with the growth of the tree. The crown displays
the leaves to allow capture of radiant energy for
photosynthesis. Thus, measurement of the tree
crown is often made to assist in the
quantification and understanding of the growth of
the tree. The biomass of the crown and the
quantity and quality of the branch material
however is also of direct interest to ecological
studies and research into the effects of trees on
pollution. The crown also has great visual
impact. This impact is important in urban forests
because it will affect the perceived beauty of a
tree. We wish to see if crown volume can be
predicted from basal area growth based on a
simple linear relationship. Basal area for trees
is the cross sectional area of a tree stem in
square feet commonly measured 4.5 feet above
ground, inclusive of bark. Assume measurements of
basal area are generally /-10 of the observed
value.
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Model 2
Model 1
Model 4
Model 3
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Model Assessment Deviance -2 Log
Likelihood BIC Deviance k ln(n) AIC
Deviance 2k
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• list(N62, xc(0.36, 0.09, 0.67, 0.44, 0.72,
  0.24, 0.33, 0.61, 0.66, 0.8, 0.47, 0.6, 0.51,
  0.14, 0.32, 0.21, 0.54, 0.7, 0.67, 0.42, 0.39,
  0.25, 0.06, 0.47, 0.55, 0.07, 0.69, 0.42, 0.39,
  0.09, 0.54, 0.74, 0.64, 0.5, 0.39, 0.3, 0.61,
  0.29, 0.38, 0.53, 0.58, 0.62, 0.66, 0.69, 0.71,
  0.14, 0.41, 0.66, 0.18, 0.21, 0.29, 0.56, 0.13,
  0.1, 0.18, 0.17, 0.63, 0.66, 0.18, 0.06, 0.58,
  0.42), yc(22, 6, 93, 62, 84, 14, 52, 69, 104,
  100, 41, 85, 90, 27, 18, 48, 37, 67, 56, 31, 17,
  7, 2, 53, 70, 5, 90, 46, 36, 14, 60, 103, 43, 22,
  75, 29, 76, 20, 29, 50, 59, 70, 81, 93, 99, 14,
  51, 75, 6, 20, 36, 50, 9, 2, 21, 17, 87, 97, 33,
  20, 96, 61))
• HW
• Assess the quality for each of the 4 models.
• Graph the predictive density and find a 95 CI
  for crown volume given basal area 0.41 (for each
  model).
• If throw crown volume is 20, find a 95 credible
  interval for basal area (for each model).