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Sliced inverse regression
Sliced inverse regression, or sir, was proposed by Li (1991); see Cook
(1998, Chapter 11).  In sir, we
make use of the fact that given certain assumptions on the marginal
distribution of  1, the inverse regression problem
1, the inverse regression problem 
 .  The general computational outline for sir is as follows:
.  The general computational outline for sir is as follows:
- Examine  by dividing the range of by dividing the range of into into slices,
    each with approximately the same number of observations.  With a
    multivariate response ( slices,
    each with approximately the same number of observations.  With a
    multivariate response ( has has columns), divide the range of columns), divide the range of into into cells.  For example,
    when cells.  For example,
    when , and we slice , and we slice into 3 slices, into 3 slices, into 2 slices, and into 2 slices, and into 4 slices, we will have into 4 slices, we will have cells.
    The number
    of slices or cells cells.
    The number
    of slices or cells is a tuning parameter of the procedure. is a tuning parameter of the procedure.
- Assume that within each slice or cell  is approximately
    constant.  Then the expected value of the within-slice vector of
    sample means will be a vector in is approximately
    constant.  Then the expected value of the within-slice vector of
    sample means will be a vector in . .
- Form the  matrix whose matrix whose -th row is
    the vector of weighted sample means in the -th row is
    the vector of weighted sample means in the -th slice.
    The matrix -th slice.
    The matrix is the is the sample covariance matrix of these sample mean
    vectors. sample covariance matrix of these sample mean
    vectors.
sir thus concentrates on the mean function , and ignores any
other dependence.
The output given in the last section is an example of typical output for
sir.  First is given the eigenvectors and eigenvalues of
, and ignores any
other dependence.
The output given in the last section is an example of typical output for
sir.  First is given the eigenvectors and eigenvalues of  ; the
eigenvectors have been back-transformed to the original
; the
eigenvectors have been back-transformed to the original  -scale.
Assuming that the dimension is
-scale.
Assuming that the dimension is  , the estimate of
, the estimate of 
 is given by
the first
 is given by
the first  eigenvectors.  Also given along with the eigenvectors is the
square of the correlation between the ols fitted values and the first
 eigenvectors.  Also given along with the eigenvectors is the
square of the correlation between the ols fitted values and the first  principal directions.  The first direction selected by sir is almost
always about the same as the first direction selected by ols, as is the case
in the example above.
For sir, Li (1991) provided asymptotic tests of dimension based on partial
sums of eigenvalues, and these tests are given in the summary.  The tests
have asymptotic Chi-square distributions, with the number of degrees of
freedom shown in the output.
Examining the tests shown in the final output, we see that the
test of
principal directions.  The first direction selected by sir is almost
always about the same as the first direction selected by ols, as is the case
in the example above.
For sir, Li (1991) provided asymptotic tests of dimension based on partial
sums of eigenvalues, and these tests are given in the summary.  The tests
have asymptotic Chi-square distributions, with the number of degrees of
freedom shown in the output.
Examining the tests shown in the final output, we see that the
test of  versus
 versus  has a very small
 has a very small  -value, so we would
reject
-value, so we would
reject  .  The test for
.  The test for  versus
 versus  has
 has  -value near
-value near
 , suggesting that
, suggesting that  is at least 2.  The test for
 is at least 2.  The test for  versus
versus  has
 has  -value of about
-value of about  , so we suspect that
, so we suspect that
 for this problem.  This suggests that further analysis of
this regression problem can be done based on the 3D graph of the
response versus the linear combinations of the predictors
determined by the first two eigenvectors, and the dimension of the
problem can be reduced from 4 to 2 without loss of information.
See Cook (1998), and Cook and Weisberg (1994, 1999), for further
examples and interpretation.
When the response is multivariate, the format of the call is:
 for this problem.  This suggests that further analysis of
this regression problem can be done based on the 3D graph of the
response versus the linear combinations of the predictors
determined by the first two eigenvectors, and the dimension of the
problem can be reduced from 4 to 2 without loss of information.
See Cook (1998), and Cook and Weisberg (1994, 1999), for further
examples and interpretation.
When the response is multivariate, the format of the call is:
m1 <- dr(cbind(LBM,RCC)~Ht+Wt+WCC))
The summary for a multivariate response is similar:
> summary(m1)
Call:
dr(formula = cbind(LBM, RCC) ~ Ht + Wt + WCC)
Terms:
cbind(LBM, RCC) ~ Ht + Wt + WCC
Method:
sir with 9 slices, n = 202, using weights.
Slice Sizes:
24 23 23 23 22 21 22 22 22
Eigenvectors:
      Dir1    Dir2    Dir3
Ht  0.4857  0.3879  0.1946
Wt  0.8171 -0.2238 -0.1449
WCC 0.3105 -0.8941  0.9701
              Dir1    Dir2    Dir3
Eigenvalues 0.7076 0.05105 0.02168
R^2(LBM|dr) 0.9911 0.99124 1.00000
R^2(RCC|dr) 0.9670 0.97957 1.00000
Asymp. Chi-square tests for dimension:
              Stat df p-value
0D vs >= 1D 157.63 24  0.0000
1D vs >= 2D  14.69 14  0.3995
2D vs >= 3D   4.38  6  0.6254
The test statistics are the same as in the univariate response case, as is
the interpretation of the eigenvalues and vectors.  The output gives the
squared correlation of each of the responses with the eigenvectors.
 
 
 
 
 
   
 Next: Sliced average variance estimation
 Up: Methods available
 Previous: Methods available
Sandy Weisberg
2002-01-10