SPlus Interlude contd

1
S-Plus Interlude (contd)

• Beer Calories vs. Alcohol Light-ness
• What does X look like?
• Is this model any better?
• Use options(contrasts contr.treatment)
• How about Calories Alcohol Light?
• How about Country? What models best?

2
Partial Regression Plots

• Or Partial regression leverage plots
• (residuals of y, regressed on all xs but xi)
  vs. (residuals of xi, regressed on all xs but
  xi)
• Slope of regression line is bi intercept 0
• Interpretation relationship of y and xi, after
  removing the effect of other xs
• Shows how individual points affect fit

3
Checking the Assumptions

• Everything depends on the assumptions
• especially with small sample sizes
• Check all that you can
• Usually wrong to some degree
• Some cant be checked at all
• Consequences of being wrong biased estimates,
  bad ests. of SDs, invalid t- and F-tests,
  inefficiency (Ham. p. 113)

4
The Assumptions

• 1. X is fixed each yi depends on its
  corresponding xs repeating the same experiment
  gives same X, different y
• If not, the analysis is conditional on the
  observed values of the Xs
• Experiment vs. observational study
• Test assumption by finding out how the data
  were collected

5
Assume Model is correct

• 1A The linear model is correct Ey is a linear
  function of the xs in the model
• If we have irrelevant xs in the model, our SEs
  are too big
• If we have relevant xs not in the model, forget
  it estimates are biased, SEs and t-tests are
  wrongand no way to check
• If the truth is non-linear, forget it, too

6
Assume Eei 0

• 2. Errors have mean 0 Eei 0, for all i
• If not, b0 is a biased est. of b0
• No other damage done
• Impossible to check from data (why?)
• Conceivably, we can check by examining how data
  are collected?
• If X random, require all cov (Xi, ei) 0 this,
  too, cannot be checked

7
Assume Homoscedasticity

• 3. Errors have constant variance
• varei s2, for all i
• If not, bs are unbiased, but SEs, t-tests, etc.
  fall apart
• Check by plotting residuals against fitted
  values, or abs(resids) vs. fitted.
• Try lm(), l1fit(), lowess() to look for trend in
  that plot

8
S-Plus Interlude

• Abs (residuals) vs. fitted values
• lm () is already suspect
• l1fit() minimizes the sum of the absolute
  residuals, instead of sum of squared ones
• Handy in general as a description, but doesnt
  give t-tests and the like
• lowess() adds a smooth curve

9
Weighted Least Squares

• What if you know the variances of the ei arent
  equal?
• Example yi is the mean of ni observationsIf
  each of the ni have the same variance, then yi
  has variance s2/ni
• Maybe the ys come from different pieces of
  equipment, each with its own SD
• Assume that the variances are known

10
WLS and GLS

• So we have y Xb e, e N(0, W) where W might
  be diagonal or not
• What if you pre-multiply by W-1/2?
• Then W-1/2 y W-1/2 Xb W-1/2 e, ory Xb
  e, and e N(0, I), so we can do OLS on y and
  X b (XTX)-1XTy
• In original units, b (XTW-1X)-1XTW-1y
• In Splus, call lm() with weights
• W need not be diagonal estimation hard

11
Assume No autocorrelation

• 4 Cov (ei , ej) 0 if i is not equal to j
• If not, bs are unbiased, but SEs, t-tests, etc.
  fall apart (again)
• Almost inevitable in data collected at points
  adjacent in time or space requires time series
  analysis
• Durbin-Watson test (H. p. 118), acf()
• Basic idea correlation of all pairs (ei, ei-1)
• Pictures (smoothed ri vs. time) help

12
Assume Errors are Normal

• 5 Each error ei N(0, s2)
• If not, cant use t- or F-tests, especially in
  small samples
• Can test with histograms, qqplots, symmetry
  plots, K-S or c2 g-o-f tests
• Can be helped by transformation
• Assumptions 2-5 e MVN (0, s2I)

13
The Moral

• Your model is not 100 correct, but often it
  produces believable results
• Its (maybe a good) approximation
• The quality of the approximation depends on the
  degree to which the assumptions are met
• The proof of quality is in prediction
• Try cross-validation

14
Cross-validation

• Ideally, you could fit the model on one set of
  data (the training set), evaluate its quality
  on another (the test set)
• What if the fit isnt so good? If you do this
  process again, youve already seen the test set
• Another good plan divide data into, say, 10
  parts

15
Cross-Validation contd

• Leave out one part, fit model with other nine
  then predict the missing tenth
• Do this ten times so that each part has been in
  nine times and out once
• Measure average RSS (or RSE or ) across all 10
  models xval()
• You can repeat this (maybe using common random
  numbers for the split)

16
Leverage and Influence

• Some cases more important than others
• Predicted y yhat Xb X(XTX)-1XTy
• X(XTX)-1XT puts the hat on y, so its called
  the hat matrix, H
• Diagonal entries of H, hi, are leverages
• Measures the ability to have influence its a
  consequence of the expts design

17
Leverage

• Leverages (hi) fall between 1/n and 1
• Rule of thumb beware of hi gt 2p/n
• Another beware when max(hi) gt .2
• Compute with lm.influence(), hat()
• Estimated variance of i-th residuals2 (1 - hi)
• High leverage -gt good fit, small variance

18
Influence

• A point with high leverage may or may not
  actually have high influence
• A point is influential if it has a big effect on
  the regression
• We can compute the bs, and s2, leaving each case
  out in turn, one at a time
• These come from lm.influence()

19
DFBETA

• DFBETA the change in a b that comes from
  omitting a case, expressed in SDs
• DFBETA for case i and column k isDFBETAi, k bk
  -- b(no i)k s(no i) /sqrt (RSSk)
• RSSk is the RSS from regressing col. k on all
  other columns (and including pt. i)
• By how many SDs does b change w/o i?
• Beware when DFBETA gt 2/sqrt(n)

20
Residuals

• Not only the bs, but yhats and residuals,
  change when a case is omitted
• Standardized residual zi ri .
  s sqrt(1-hi)
• Studentized residual ti use s(no i) for s
• ti is a little t-test of does case i shift the
  intercept significantly? use level a/n

21
Cooks Distance

• Cooks distance measures influence
• Di zi2 hi p (1 - hi)
• Di increases as zi increases
• When hi is small, Di is smallas hi -gt 1, Di
  gets huge
• This effect is on all the bs, not just one
• Look for Di gt 1or maybe Di gt 4/n

22
Splus Interlude

• plot.lm() gives Cooks distance plot
• lm.influence() gives leverages, new bs, new ss
  dfbetas() available
• Q What to do about influential points?
• A Do they change your conclusions?

23
Proportional Leverage Plot

• This is the leverage plot (relationship of y to
  one x, adjusting for other xs) but...
• Points have areas proportional to DFBETA
  (adjusted to be between 1 and 100)
• That area is the of SDs by which that case
  affects the line
• Shows influence problems, maybe curvilinearity or
  heteroscedasticity