Regression Examples

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Regression Examples
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Gas Mileage 1993

• SOURCES
• Consumer Reports The 1993 Cars - Annual Auto
  Issue (April 1993), Yonkers, NY Consumers Union.
• PACE New Car Truck 1993 Buying Guide (1993),
  Milwaukee, WI Pace Publications Inc.
• Specifications are given for 93 new car models
  for the 1993 year.

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Gas Mileage vs Weight

• Several measures are available, such as price,
  mpg ratings, engine size, cylinders, weight,
  horsepower, etc.
• We consider the relationship between weight and
  highway mpg.
• Since more fuel is needed to move more weight, an
  increase in weight should result in a decrease in
  mpg.

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SAS Output

• Pearson Correlation Coefficients, N 93
• Prob gt r under H0 Rho0
• mpg weight
• mpg 1.00000 -0.81066
• lt.0001
• weight -0.81066 1.00000
• lt.0001

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• The REG Procedure
• Model MODEL1
• Dependent Variable mpg
• Number of Observations Read
  93
• Number of Observations Used
  93
• Analysis of Variance
• Sum of
  Mean
• Source DF Squares
  Square F Value Pr gt F
• Model 1 1718.69528
  1718.69528 174.43 lt.0001
• Error 91 896.61655
  9.85293
• Corrected Total 92 2615.31183
• Root MSE 3.13894
  R-Square 0.6572

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Interpretation

• From the graph it is evident that there is a
  fairly strong negative correlation between weight
  and mpg.
• The correlation output tells us that r-.81.
• The regression output, under parameter estimates,
  tells us that the equation for the least squares
  line (best fit line) is mpg51.6 - .0073weight.
• Also, there are standard errors for the estimates
  that can be used to build confidence intervals,
  and there are t-values and p-values for a test of
  the hypothesis Ho Parameter0 (vs not 0).
• Since the p-value for weight is .0001, we can
  reject Ho and conclude that there is a linear
  relationship between weight and mpg (weight
  contributes information about mpg, or helps to
  predict mpg).

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Gas Mileage vs Engine Size

• Generally speaking, larger engines burn more fuel
  (but is this due to weight?).
• We can check the relationship between liters
  (engine displacement) and mpg.
• We expect a negative relationship, since a larger
  engine would tend to decrease mpg.

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• Pearson Correlation Coefficients, N 93
• Prob gt r under H0 Rho0
• mpg liters
• mpg 1.00000 -0.62679
• lt.0001
• liters -0.62679 1.00000
• lt.0001

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• The REG Procedure
• Model MODEL1
• Dependent Variable mpg
• Number of Observations Read
  93
• Number of Observations Used
  93
• Analysis of Variance
• Sum of
  Mean
• Source DF Squares
  Square F Value Pr gt F
• Model 1 1027.48125
  1027.48125 58.89 lt.0001
• Error 91 1587.83058
  17.44869
• Corrected Total 92 2615.31183
• Root MSE 4.17716
  R-Square 0.3929

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Interpretation

• From the first graph it is evident that there is
  a negative relationship between weight and mpg.
• However, the pattern is not purely linear. It
  seems to be some kind of curve. Thus we will not
  expect a linear analysis to tell the whole story.
• The correlation output tells us that r-.63.
• The regression output, tells us that the equation
  for the least squares line is mpg37.68
  3.22liters.
• The p-value for liters is .0001, so we conclude
  that mpg has a linear relationship with engine
  size.
• Which is a better predictor of mpg, engine size
  or weight? We can use the R-square value to
  determine that. For weight it is .6572, and for
  liters it is .3929.

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More Advanced Interpretation

• Which is a better predictor of mpg, engine size
  or weight? We can use the R-square value to
  determine that. For weight it is .6572, and for
  liters it is .3929.
• R-square is an indication of the proportion of
  changes in y that are accounted for by x, so a
  larger value corresponds to a better predictor.
  Thus weight is a better predictor than engine
  size.
• The graphs show the best fit line and a best fit
  parabola (quadratic equation). The latter is
  provided for comparison purposes only.
• Note that the quadratic equation, even though it
  fits the points better, may not be a better
  model, because it shows mpg rising as engine
  sizes get very large. This does not make sense.

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Brief Look at Multiple Regression

• Now you might think, what if we wanted to use
  both weight and engine size to predict mpg?
• This idea is called multiple regression, and it
  involves making an equation with two or more x
  variables to predict y.
• The next regression output shows this.
• Compare the R-square and p-values to previous
  results.

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• The REG Procedure
• Model MODEL1
• Dependent Variable mpg
• Number of Observations Read
  93
• Number of Observations Used
  93
• Analysis of Variance
• Sum of
  Mean
• Source DF Squares
  Square F Value Pr gt F
• Model 2 1749.76356
  874.88178 90.97 lt.0001
• Error 90 865.54826
  9.61720
• Corrected Total 92 2615.31183
• Root MSE 3.10116 R-Square
  0.6690

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Interpretation

• The least squares equation is mpg53.6-1.05liter
  s-.00888weight.
• The R-square for weight alone is .6572. In the
  new model, it is .6690. It has gone up, but not
  much. This means that adding engine size to the
  equation does not improve predicted mpg very
  much.
• The p-value for weight is still very small, but
  the p-value for liters is now suspiciously large.
  Using alpha.05, we would not reject that the
  coefficient of liters is zero, which means we are
  not able to detect a contribution to mpg.