Regression Models with Nonlinear Transformations

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Regression Models with Nonlinear Transformations
Topics Motivational Example Modeling Curvature
with Square Term Log Transformations Interpreting
Models with Log Transforms Implementing
Transforms in StatTools
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Problem Scenario

• A restaurant chain wants to investigate the
  relationship between shop revenue (000s) and the
  median household income (000s) in the
  neighborhood
• Is there a better fit to the data than a linear
  model?

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Exploring the Best Model Fit to Data with Excel
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Exploring the Best Model Fit to Data with Excel
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Exploring the Best Model Fit to Data with Excel
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Modeling Curvature with Quadratic (square) Term

• The Quadratic fit is the best with the highest
  rsquared for the model (81.3)
• The main downside to a quadratic regression
  equation is that there is no easy interpretation
  of the coefficients of Units and Sqr_Units for
  median income

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Modeling Curvature with Quadratic (square) Term

• We can say the terms in the equation combine to
  explain the nonlinear relationship between
  revenue and median income

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Modeling Curvature with Quadratic (square) Term

• Note the coefficient of Income Sqr is negative to
  model the downward bend of the parabola curve.
  This produces the decreasing marginal revenue,
  where every extra unit of median household income
  is associated with a smaller revenue

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Modeling Curvature Logarithmic X Variable

• The Log transform model is the next best fit to
  the data with rsquared 76.5 although the simple
  linear model is not much worse with rsquared
  73.1
• The log model is easier to interpret than the
  quadratic model.

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Interpretation of Log X Slope Coefficient

• In general if the log transformed equation is Y
  a b logX
• For a 1 increase in X, Y is expected to increase
  by b/100 units

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Interpretation of Log X Regression Slope
coefficient

• Revenue 558.17 Log (Income) 630.52
• When Median income increases by 1 the restaurant
  can expect an increase in revenue of 5.5817
  (000s) or 5,582

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Interpretation of Log X Regression Slope
coefficient

• Note that for larger values of median income, a
  1 increase represents a larger absolute
  increase. But each such 1 increase entails the
  same 5,582 increase in revenue. This is another
  way of describing the decreasing marginal revenue
  property observed in the plot of the data.

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Other Log Transformations in Regression Models

• Whenever the response (Y) variable in regression
  is highly skewed to the right a log transform
  helps to normalize the distribution.
• This is often done for Salary Data
• E.g. Regression of CEO Salary against Company
  Profit

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Interpretation of X Variable Coefficient in
Regression Models with Log Y

• In general if the log transformed equation is Log
  Y a b X where b is expressed as a percent,
  then
• For a 1 unit increase in X, Y is expected to
  increase by approximately b

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Comparing R2 and Se for Regression Models with
Log Y

• Since the Y variable in a log transformed model
  of the form is Log Y a b X is converted to a
  log scale, R2 and Se are also measuring
  variations on a log scale and cannot be compared
  with R2 and Se for a regular Y

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Creating Log Variables in StatTools

• Name the data set in the usual way
• Place the cursor anywhere in the spreadsheet and
  click on the Data Utilities icon (3rd from left)
• Select Transform and by clicking, place a check
  in the box next to the variable (s) to be
  transformed

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Creating Log Variables in StatTools

• Accept the default log function in the
  transformation box as well as the other defaults
  then click O.K.
• Click yes when StatTools warns if you wish to
  continue to insert a new column
• StatTools will insert the new column with the log
  transformed variable next to your data