Statistics 102H: Predictions and CI (cont'd)

Can't the preditions just deal with bad data?

• Try a linear fit to curved data (look at prediction intervals)
• Try a linear fit to hetroskadastic data (look at prediction intervals)
• Coverage variaries depending on where you are in the data
• sometimes 100% (i.e. too wide)
• sometimes 50% or less (i.e. too narrow)
• Hence looking at graph is VERY important

At least some of these we can fix

• Transformations for hetroskadasticity
• What is a transformation?
  • Suppose "log(Y) vs X" and and "Y vs exp(X)" both fit equally well.
  • Which to use?
  • Different prediction bounds
  • Do actual transformation
• Example of picking a transformation:
  • picking between log-log and 1/X model in housing data
  • Generate all the new variables
  • Plot on new axis
  • Which is less hetroskadastic?
• Example useful for linear case:
  • Cleaning crews data
  • linear-linear or should we use log-log or sqrt-sqrt?
  • Plot prediction intervals on orginal graph to see
  • Save residuals
• The log-log transformation (elasticities)
  • consider the asteroid data
  • raw plot is very ugly (sketch picture)
  • too extream in both directions
  • log-log tames a whole range of values

Homework

What about that statistical error?

• A little background:
  • Regression does least squares
  • solution to least squares is linear combination of erorr + truth
  • Hence normality sets in and CLT sets in
  • E(alpha-hat) = alpha (This is called unbiased)
• Thus accuracy of alpha and beta depend on where the data is collected
• more spread out X's generate better slope estimates
• But might introduct unwanted data to problem. Hence design of where to put the X's must take care.
• Error looks like sigma/sqrt(n)sigma(X)

Cell phones

• Current statistics: here.
• Looks like March 2002 = 180 Million?