Fitting curves through points: Regression

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Chapter 8

• Fitting curves through points Regression

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Introduction

• This chapter considers data comprising a
  continuous independent variable x and a
  continuous response variable y
• Simplest model is a straight line
• Use the model to test the significance of the
  linear relationship and make predictions
• Least squares most common way to choose the best
  line
• demoLeastSquares.jsl

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Example

• Data from 232 branches of an East Coast bank
• First look at Number of New Accounts versus
  Number of Sales Staff during a particular one
  year period

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• Similar to ANOVA except the fitted means are
  constrained to fall on a line

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Linear Regression
ANOVA
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Regression Model

• Number of new accounts
• 179.4 (78.9 x sales staff) residual

• Parameter estimates subject to sampling
  variability
• Can test hypotheses, compute confidence
  intervals, etc.

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Confidence Curves Fit
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Regression Model Assumptions

• At each level of x, have a random sample of ys
• At each level of x, the ys are normally
  distributed
• All those normal distributions have the same
  variance

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Residual Plots

• Use residual plots to check assumptions
• Residuals should be approximately normally
  distributed
• Residuals when plotted against x or y should show
  no particular pattern

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Forensic Example (FSI, 1994)
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Example Growth.jmp
Exclude points to the left???
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Polynomial Models
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Transformations
fit special Splines? Piecewise?
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Tukeys Bulging Rule
x2 y2
y2, sqrt x log x, 1/x
x2, sqrt y log y, 1/y
sqrt x, sqrt y, log x, log y, 1/x, 1/y
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Another example Cell Phone Use
Which transformation? polynomial?
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Always look at the data!
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Prediction Cottages Example
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Prediction for 3,000 sq. feet?

• Two components of uncertainty
• Dont know where the true line is
• Even if we know where the true line is, the data
  are normally distributed about this line
• Confidence curves fit only show uncertainty
  about the true line
• Confidence curves indiv include both
  components of uncertainty

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Fit
Indiv
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Revisit Cell Phone Use
Prediction for Dec, 1999 (period 31)?