Diagnostics

1
Diagnostics Part II

• Using statistical tests to check to see if the
  assumptions we made about the model are realistic

2
Diagnostic methods

• Some simple (but subjective) plots. (Then)
• Some formal statistical tests. (Now)

3
Simple linear regression model
The response Yi is a function of a systematic
linear component and a random error component
with assumptions that

• Error terms have mean 0, i.e., E(?i) 0.
• ?i and ?j are uncorrelated (independent).
• Error terms have same variance, i.e., Var(?i)
  ?2.
• Error terms ?i are normally distributed.

4
Why should we keep NAGGING ourselves about the
model?

• All of the estimates, confidence intervals,
  prediction intervals, hypothesis tests, etc. have
  been developed assuming that the model is
  correct.
• If the model is incorrect, then the formulas and
  methods we use are at risk of being incorrect.
  (Some are more forgiving than others.)

5
Summary of the tests well learn

• Durbin-Watson test for detecting correlated
  (adjacent) error terms.
• Modified Levene test for constant error variance.
• (Ryan-Joiner) correlation test for normality of
  error terms.

6
The Durbin-Watson test for uncorrelated
(adjacent) error terms
Durbin-Watson test statistic

• Compare D to Durbin-Watson test bounds in Table
  B.7
• If D gt upper bound (dU), conclude no
  correlation.
• If D lt lower bound (dL), conclude positive
  correlation.
• If D is between the two bounds, the test is
  inconclusive.

7
Example Blaisdell Company
Seasonally adjusted quarterly data, 1988 to 1992.
Reasonable fit, but are the error terms
positively auto-correlated?
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9
Blaisdell Company Example Durbin-Watson test

• Stat gtgt Regression gtgt Regression. Under
  Options, select Durbin-Watson statistic.
• Durbin-Watson statistic 0.73
• Table B.7 with level of significance a0.01,
  (p-1)1 predictor variable, and n20 (5 years, 4
  quarters each) gives dL 0.95 and dU1.15.
• Since D0.73 lt dL0.95, conclude error terms are
  positively auto-correlated.

10
For completeness sake one more thing about
Durbin-Watson test

• If test for negative auto-correlation is desired,
  use D4-D instead. If D lt dL, then conclude
  error terms are negatively auto-correlated.
• If two-sided test is desired (both positive and
  negative auto-correlation possible), conduct both
  one-sided tests, D and D, separately. Level of
  significance is then 2a.

11
Modified Levene Test for nonconstant error
variance

• Divide the data set into two roughly equal-sized
  groups, based on the level of X.
• If the error variance is either increasing or
  decreasing with X, the absolute deviations of the
  residuals around their group median will be
  larger for one of the two groups.
• Two-sample t to test whether mean of absolute
  deviations for one group differs significantly
  from mean of absolute deviations for second group.

12
Modified Levene Test in Minitab

• Use Manip gtgt Code gtgt Numeric to numeric to
  create a GROUP variable based on the values of X.
• Stat gtgt Regression gtgt Regression. Under Storage
  , select residuals.
• Stat gtgt Basic statistics gtgt 2 Variances Specify
  Samples (RESI1) and Subscripts (GROUP). Select
  OK. Look in session window for Levene P-value.

13
Example How is plutonium activity related to
alpha particle counts?
14
A residual versus fits plot suggesting
non-constant error variance
15
Plutonium Alpha Example Modified Levenes Test
Levene's Test (any continuous distribution)
Test Statistic 9.452 P-Value 0.006
It is highly unlikely (P0.006) that wed get
such an extreme Levene statistic (L9.452) if the
variances of the two groups were equal. Reject
the null hypothesis at the 0.01 level, and
conclude that the error variances are not
constant.
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(Ryan-Joiner) Correlation test for normality of
error terms in Minitab

• H0 Error terms are normally distributed vs. HA
  Error terms are not normally distributed
• Stat gtgt Regression gtgt Regression. Under
  storage, select residuals.
• Stat gtgt Basic statistics gtgt Normality Test.
  Select residuals (RESI1) and request Ryan-Joiner
  test. Select OK.

17
100 chi-square (1 df) data values
18
Normal probability plot and test for 100
chi-square (1 df) data values
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100 normal(0,1) data values
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Normal probability plot and test for 100
normal(0,1) data values
21
Normal probability plot for Tree diameter (X) and
C-dating Age (Y)
22
Tree diameter and Age Example Ryan-Joiner
Correlation Test
23
Some closing comments

• Checking of assumptions is important, but be
  aware of the robustness of your methods, so you
  dont get too hung up.
• Model checking is an art as well as a science.
• Do not think that there is some definitive
  correct answer in the back of the book.
• Use your knowledge of the subject matter.