Outlines for today

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Outlines for today

• Examples of nonlinear relationship
• Polynomial regression
• Two statistical concepts in interpretation of
  nonlinear regressions
• -- Inflection point
• -- Derivative function for slope
• Other formats of nonlinear relationships

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Power Transformation
.. Y -gt Y4 Y -gt Y3 Y -gt Y2 Adjust the negative skew
Y -gt Y1
Y -gt Y0.5 Y -gt logY Y -gt -Y-0.5 Y -gt -Y-1 .. Adjust the positive skew
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Review of Log Transformation

• logs are exponents logbx y (the log of x to
  the base b is y) means that byx
• Some examples

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Log Transformation

• The logarithmic transformation reduces positive
  skewness. Why?
• Because the logbX function is a function whose
  return value is the power to which b must be
  raised to equal X.
• For example, log10100 2. Below shows the
  pattern for log base 10

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Examples of nonlinear relationship
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Polynomial Regression
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Quadratic Regression
The best way to interpret the nonlinear
regression results is to plot the relationship.
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?
Y BMI X age
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Inflection Point

• An inflection point is a point on a curve at
  which the curvature changes sign.

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Find the inflection point
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Calculating the inflection point of a second
order power polynomial
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Slope (using a derivative func.)

• In a nonlinear regression model, the slope of X
  is not constant but varies with the value of X.
• A derivative function provides a best
  straight-line approximation to the curve.
• The slope of X in a nonlinear relationship refers
  to a derivative function of X, which allows to
  vary by X.

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Calculating slope for selected values of X

• Y .32390 - .00446X .00013 X2
• where X marital duration in years, and
• Y of household tasks performed by husband
• Slope -.00446 2(.00013)X

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Other Nonlinear Models

• Log-linear
• Log(Y)b0b1Xe
• Linear-log
• Yb0b1log(X) e
• Log-log
• Log(Y)b0b1log(X) e
• Again, the best way to interpret the nonlinear
  regression results is to plot the relationship.

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