Stat 6601 Project: Bootstrapping Linear Models (V&R 6.6)

Stat 6601 Project: Bootstrapping Linear Models (V&R 6.6) Jaimie Kwon Statistics, CSUH

Goal • How do we apply bootstrap to linear regression models?

Data • ‘Phones’ data: the annual numbers of telephone calls, in Belgium • ‘year’ : The last two digits of the year. • ‘calls’ : The number of telephone calls made (in millions of calls).

Model • Linear models of the form: Y=X + in which only  is considered random.

Method • Most obvious form of bootstrapping: randomly sample pairs (xi, yi) with replacement. (called “case-based resampling” in Davison and Hinkley (1997)) • Might not be appropriate • Alternative: “Model-based resampling” – resample the residulas.

Method (Continued) • Procedure: • After fitting the linear model to get:yi=xib + ei • Create a new dataset by yi=xib + ei where ei* are resample with replacement from the residuals (ei). • Some issues:

Codes • library(MASS) • library(boot) • plot(phones$year, phones$calls) • fit <- lm(calls ~ year, data=phones) • ph <- data.frame(phones, res=resid(fit), fitted=fitted(fit)) • ph.fun <- function(data, i){ • d <- data • d$calls <- d$fitted + d$res[i] • coef(update(fit, data=d)) • } • (ph.lm.boot <- boot(ph, ph.fun, R=999)) • plot(ph.lm.boot)

Results • Bootstrap Statistics : • original bias std. error • t1* -260.059246 -3.5164095 96.730498 • t2* 5.041478 0.0690514 1.567871

Summary • Phones data • Bootstrap linear models

Some Tips • Don’t try to cover too much. • Keep it structured. • Don’t put too much in the slides. • Let them listen to you. • Plan to use blackboards as well. • Don’t agonize over equations (or equation editor). • Make slides look neat & pleasant. • Practice and time your presentation.

Stat 6601 Project: Bootstrapping Linear Models (V&R 6.6)