Description Usage Arguments Details Value Author(s) References See Also Examples

Functions to compute various tests concerning the dimension of a central subspace.

1 2 3 4 5 6 | ```
dr.test(object, numdir, ...)
dr.coordinate.test(object, hypothesis,d,chi2approx,...)
## S3 method for class 'ire'
dr.joint.test(object, hypothesis, d = NULL,...)
``` |

`object` |
The name of an object returned by a call to |

`hypothesis` |
A specification of the null hypothesis to be tested by the coordinate hypothesis. See details below for options. |

`d` |
For conditional coordinate hypotheses, specify the dimension of the central mean subspace, typically 1, 2 or possibly 3. If left at the default, tests are unconditional. |

`numdir` |
The maximum dimension to consider. If not set defaults to 4. |

`chi2approx` |
Approximation method for p.values of linear combination
of |

`...` |
Additional arguments. None are currently available. |

`dr.test`

returns marginal dimension tests.
`dr.coordinate.test`

returns marginal dimension tests (Cook, 2004)
if `d=NULL`

or conditional dimension tests if `d`

is a
positive integer giving the assumed dimension of the central
subspace. The function `dr.joint.test`

tests the coordinate
hypothesis and dimension simultaneously. It is defined only for
ire, and is used to compute the conditional coordinate test.

As an example, suppose we have created a `dr`

object
using the formula
`y ~ x1 + x2 + x3 + x4`

.
The marginal coordinate hypothesis defined by Cook (2004) tests
the hypothesis that `y`

is independent of some of the
predictors given the other predictors. For example, one could test
whether `x4`

could be dropped from the problem by testing `y`

independent of `x4`

given `x1,x2,x3`

.

The hypothesis to be tested is determined by the argument `hypothesis`

.
The argument `hypothesis = ~.-x4`

would test the hypothesis of the last
paragraph. Alternatively, `hypothesis = ~x1+x2+x3`

would
fit the same hypothesis.

More generally, if `H`

is a *p times q*
rank *q* matrix, and
*P(H)* is the projection
on the column space of `H`

, then specifying `hypothesis = H`

will test the
hypothesis that *Y* is independent of *(I-P(H))X | P(H)X*.

Returns a list giving the value of the test statistic and an asymptotic
p.value computed from
the test statistic. For SIR objects, the p.value is computed in two ways. The
*general test*, indicated by `p.val(Gen)`

in the output, assumes only
that the predictors are linearly related. The *restricted test*, indicated
by `p.val(Res)`

in the output, assumes in addition to the linearity condition
that a constant covariance condition holds; see Cook (2004) for more information
on these assumptions. In either case, the asymptotic distribution is a linear
combination of Chi-squared random variables. The function specified by the
`chi2approx`

approximates this linear combination by a single Chi-squared
variable.

For SAVE objects, two p.values are also returned. `p.val(Nor)`

assumes
predictors are normally distributed, in which case the test statistic is asympotically
Chi-sqaured with the number of df shown. Assuming general linearly related
predictors we again get an asymptotic linear combination of Chi-squares that
leads to `p.val(Gen)`

.

For IRE and PIRE, the tests
statistics have an asymptotic *Chisq* distribution, so the
value of `chi2approx`

is not relevant.

Yongwu Shao for SIR and SAVE and Sanford Weisberg for all methods, <sandy@stat.umn.edu>

Cook, R. D. (2004). Testing predictor contributions in
sufficient dimension reduction. *Annals of Statistics*, 32, 1062-1092.

Cook, R. D. and Ni, L.
(2004). Sufficient dimension reduction via inverse regression: A minimum
discrrepancy approach, *Journal of the American Statistical Association*,
100, 410-428.

Cook, R. D. and Weisberg, S. (1999). *Applied Regression Including
Computing and Graphics*. Hoboken NJ: Wiley.

Shao, Y., Cook, R. D. and Weisberg, S. (2007, in press). Marginal tests with
sliced average variance estimation. *Biometrika*.

`drop1.dr`

, `coord.hyp.basis`

,
`dr.step`

,
`dr.pvalue`

1 2 3 4 5 6 7 8 9 10 | ```
# This will match Table 5 in Cook (2004).
data(ais)
# To make this idential to Arc (Cook and Weisberg, 1999), need to modify slices to match.
summary(s1 <- dr(LBM~log(SSF)+log(Wt)+log(Hg)+log(Ht)+log(WCC)+log(RCC)+log(Hc)+log(Ferr),
data=ais,method="sir",slice.function=dr.slices.arc,nslices=8))
dr.coordinate.test(s1,~.-log(Hg))
#The following nearly reproduces Table 5 in Cook (2004)
drop1(s1,chi2approx="wood",update=FALSE)
drop1(s1,d=2,chi2approx="wood",update=FALSE)
drop1(s1,d=3,chi2approx="wood",update=FALSE)
``` |

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