Regression: Confidence and Prediction Intervals

1
Regression Confidence and Prediction Intervals

• MSIT3000
• Lecture 19

2
Objectives

• Create interpret Confidence Intervals for the
  conditional mean (Yhat).
• Create interpret Confidence Intervals for a
  given prediction ( also Yhat).
• Text reference section 9.8.

3
Two interpretations of Yhat

• Given x, what is the average value that y will
  take (Ey)?
• Our model suggests Yhat.
• We dont expect our model to be perfect, so there
  is an error.
• What is the predicted height of a specific
  10-year old (y)?

4
How do we create confidence intervals for the
conditional mean and the prediction?

• Find standard errors (see p 519 the next slide)
• Note the standard error is slightly different
  for a confidence interval as opposed to a
  prediction interval.
• Notation
• Standard error for a confidence interval for a
  mean value
• Standard error for a prediction interval for an
  individual new value
• Find the appropriate t to create a MOE.
• dfn-2
• Determine a confidence level.

5
Standard Errors
Note xp is the specific value of x we are
predicting for.
6
What will our CI and PI look like for different
values of x?
7
Key points to remember

1. Depending on whether you are predicting a mean or
   one specific instance (prediction) of y, you are
   estimating two different things.
2. These two, confidence intervals for y and E(y),
   will have two different standard errors.
3. Unfortunately, both point estimates are
   represented by
4. Details
5. prediction intervals are always wider than
   confidence intervals
6. both types of intervals are at their narrowest
   for xxbar.

8
How do we create CI PI?

• By hand, using the formulas on p 519.
• Using software packages.
• See e.g. Figure 9.19 SAS output (p 520) where CI
  and PI both are reported automatically.

9
Conclusion

• Objectives addressed
• Create interpret Confidence Intervals for the
  conditional mean (Yhat).
• Create interpret Confidence Intervals for a
  given prediction ( also Yhat).
• Problem 9.57, 9.65