Stat%20112:%20Lecture%208%20Notes

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Stat 112 Lecture 8 Notes

• Homework 2 Due on Thursday
• Assessing Quality of Prediction (Chapter 3.5.3)
• Comparing Two Regression Models (Chapter 4.4)
• Prediction Intervals for Multiple Regression
  (Chapter 4.5)

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Assessing Quality of Prediction (Chapter 3.5.3)

• R squared is a measure of a fit of the regression
  to the sample data. It is not generally
  considered an adequate measure of the
  regressions ability to predict the responses for
  new observations.
• One method of assessing the ability of the
  regression to predict the responses for new
  observations is data splitting.
• We split the data into a two groups a training
  sample and a holdout sample (also called a
  validation sample). We fit the regression model
  to the training sample and then assess the
  quality of predictions of the regression model to
  the holdout sample.

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Measuring Quality of Predictions
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Comparing Two Regression Models

• Multiple Regression Model for automobile data
• We use t test to test if one variable, for
  example, cargo is useful after putting the rest
  of the three variables into the model.
• How to test whether cargo and/or seating are
  useful predictors once weight and hp are taken
  into account, i.e., test

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Full vs. Reduced Model

• General setup for testing whether any of the
  variables are useful for predicting
  y after taking into account variables
• Full model
• Reduced model
• Is the full model better than the reduced model?

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Partial F test

• Test statistic
• Under H0, F has an
  distribution. Round both degrees of freedom down
  when using Table B.4.
• Decision rule for test with significance level
• Reject H0 if
• Accept H0 if
• p-value Prob (F(K-L, n-K-1) gtF)

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Cargo and Seating are not useful
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Automobile Example

• Test whether cargo and seating are useful
  predictors once hp and weight are taken into
  account.
• From Table B.4, F(.05 2,60)3.15.
• Because 10.49gt3.15, we reject H0. There is
  evidence that cargo and/or seating are useful
  predictors once hp and weight are taken into
  account.

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Test of Usefulness of Model

• Are any of the variables useful
  for predicting y?
• Multiple Linear Regression model

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F Test of Usefulness of Model

• Under , F has F(K,n-K-1) distribution.
• Decision rule Reject if
  see Appendix B.3-B.5
• F test in JMP in Analysis of Variance table.
  ProbgtF is the p-value for the F test.

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Prediction in Automobile Example

• The design team is planning a new car with the
  following characteristics horsepower 200,
  weight 4000 lb, cargo 18 ft3, seating 5
  adults.
• What is a 95 prediction interval for the GPM1000
  of this car?

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Prediction with Multiple Regression Equation

• Prediction interval for individual with x1,,xK

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Finding Prediction Interval in JMP

• Enter a line with the independent variables
  x1,,xK for the new individual. Do not enter a y
  for the new individual.
• Fit the model. Because the new individual does
  not have a y, JMP will not include the new
  individual when calculating the least squares
  fit.
• Click red triangle next to response, click Save
  Columns
• To find , click Predicted Values. Creates
  column with
• To find 95 PI, click Indiv Confid Interval.
  Creates column with lower and upper endpoints of
  95 PI.

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Prediction in Automobile Example

• The design team is planning a new car with the
  following characteristics horsepower 200,
  weight 4000 lb, cargo 18 ft3, seating 5
  adults.
• From JMP,
• 95 prediction interval (37.86, 52.31)