Transforming relationships

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Transforming relationships

• A scatterplot might show a clear relationship
  between two quantitative variables, but issues of
  influential points or non linearity prevent us
  from using correlation and regression tools.
• Transforming the data changing the scale in
  which one or both of the variables are expressed
  can make the shape of the relationship linear
  in some cases.

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Transforming Relationships How is the weight
of an animals brain related to the weight of its
body?

• What can we say about the labeled points?
• Humans and dolphins are smart.
• Hippos are dumb.
• Elephants are outliers in the x-direction
  (weight).

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Transforming Relationships How is the weight
of an animals brain related to the weight of its
body?
The correlation between brain weight and body
weight is 0.86 or 74 of the data is
explained. What are the influential observations
here?
If we remove the influential observations, what
will happen?
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Transforming Relationships How is the weight
of an animals brain related to the weight of its
body?

• Note that removing the x-outlier changed the
  scale significantly.
• Does this look like a linear function?
• r is now less than 0.50 or less than 25 of the
  data is explained.
• What should we be looking at here if not a linear
  form?

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Transforming Relationships How is the weight
of an animals brain related to the weight of its
body?

• Biologists know that when comparing animals of
  different weights, it is often helpful to take
  the log of both variables before analysis.
• This is now a function where r 0.96.

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What is transforming a function?

• Transforming or Reexpressing the data by taking
  the square root or logarithms of one or more
  variables can help us see the relationship
  better.
• We may want to change both the explanatory and
  the response variables (x and y) so we will call
  the transformation t.

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First Steps in Transforming

• A linear change (such as changing slope or
  y-intercept) have been discussed in previous
  chapters.
• A linear change can not straighten out a curved
  relationship between two variables.
• To do this, we need to resort to functions that
  are not linear.
• The logarithm function is applied in the previous
  example.

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Monotonic Functions

• A monotonic function f(t) moves in one direction
  as its argument t increases.
• A monotonic increasing function preserves the
  order of the data.
• This means that if a gt b before the
  transformation, then f(a) gt f(b).
• A monotonic decreasing function reverses the
  order of the data.
• This means that if a gt b before the
  transformation, the f(a) lt f(b).

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Visualizing the Graphs of somePositive Functions
a bt, slope b gt 0
t2
log t
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Visualizing the Graphs of some Negative Functions
a bt, slope b lt 0
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What is a power function?

• There is a ladder of hierarchy that can help us
  know what transformation to choose.
• Power functions tp for positive powers are
  monotonic increasing for values tgt0. They
  preserve the order of the observations. This is
  also true from logarithms.
• Power functions tp for negative powers are
  monotonic decreasing for values tgt0. They
  reverse the order of the observations.

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Power Functions

• For tp pgt1 gives powers that bend upward.
• Powers plt1 give powers that bend downward.

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Concavity of Power Functions

• Power transformations for pgt1 are concave upward.
  The transformation pushes out the right tail of
  a distribution and pulls in the left tail.
• Power transformations for plt1 are concave
  downward. The transformation pushes out the left
  tail of the distribution and pulls in the right
  tail.
• These effects get stronger as p gets further from
  1

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Concavity of Power Functions
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Concavity of Power Functions

• Example 4.2
• Examining if life expectancy changes as the
  wealth of a nation changes. Figure a is a plot
  of the data life expectancy vs GDP.

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Concavity of Power Functions

• We can see the data is not linear, so it might be
  better if we transform the data to make it fit a
  linear pattern. Because GDP is right skewed and
  very spread out we will just try to transform it.
• Figure b transforms the data by taking the square
  root, but it doesnt help much
• Figure c transforms the data by taking the log,
  it is a better model but still bends to the
  right.
• Figure d takes the reciprocal of the square root,
  it produces the most linear graph with the
  highest r value. To avoid reversing the values
  use

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Concavity of Power Functions

• As we moved down the ladder with this function
  the data got straighter.
• The try and see approach isnt very helpful and
  it doesnt tell us anything about the
  relationship between GDP and life expectancy. We
  learn a better way to find a mathematical model.