[R] Condition indexes and variance inflation factors

[Peter Flom]

Dear John

An interesting discussion!

I would be the last to suggest ignoring such diagnostics as Cook's D;
as you point out, it diagnoses a problem which condition indices do not:
Whether a point is influential.

OTOH, condition indices diagnose a problem which Cook's D does not:
Would shifting the data slightly change the results.

Consider the data given in Belsley (1991) on p. 5

y <-   c( 3.3979, 1.6094, 3.7131, 1.6767, 0.0419, 3.3768, 1.1661,
0.4701)
x2a <- c(-3.138, -0.297, -4.582, 0.301, 2.729, -4.836, 0.065, 4.102)
x2b <- c(-3.136, -0.296, -4.581, 0.300, 2.730, -4.834, 0.064, 4.103)
x3a <- c(1.286, 0.250, 1.247, 0.498, -0.280, 0.350, 0.208, 1.069)
x3b <- c(1.288, 0.251, 1.246, 0.498, -0.281, 0.349, 0.206, 1.069)
x4a <- c(0.169, 0.044, 0.109, 0.117, 0.035, -0.094, 0.047, 0.375)
x4b <- c(0.170, 0.043, 0.108, 0.118, 0.036, -0.093, 0.048, 0.376)

where x2a , x3a and x4a are very similar to x2b, x3b and x4b,
respecttively, and where both are generated from

y = 1.2I  - 0.4 x2 + 0.6x3 + 0.9x4 + e

e ~ N(0, 0.01)

Then 
modela <- lm(y~ x2a + x3a + x4a)
and 
modelb <- lm(y~x2b + x3b + x4b)

give radically different results, with neither having any significant
parameters other than the intercept.  Admittedly, the standard errors
for a couple of the parameters are large.  But why are they large? I
have certainly dealt with models with large standard errors that have
nothing to do with collinearity.

here, the function PI.lm (supplied by Juergen Gross) gives huge
condition indices, and indicates that the nature of the problem is that
all three of the x variables are highly collinear.

Variance-Decomposition Proportions for
Scaled Condition Indexes:

    (Intercept) x2b x3b x4b
1        0.0494   0   0   0
1        0.0009   0   0   0
3        0.8101   0   0   0
464      0.1396   1   1   1


Regards

Peter

Peter L. Flom, PhD
Assistant Director, Statistics and Data Analysis Core
Center for Drug Use and HIV Research
National Development and Research Institutes
71 W. 23rd St
www.peterflom.com
New York, NY 10010
(212) 845-4485 (voice)
(917) 438-0894 (fax)