Functions to compute various tests concerning the dimension of a central subspace.
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The name of an object returned by a call to
A specification of the null hypothesis to be tested by the coordinate hypothesis. See details below for options.
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.
The maximum dimension to consider. If not set defaults to 4.
Approximation method for p.values of linear combination
of Chisq(1) random variables. Choices are from
Additional arguments. None are currently available.
dr.test returns marginal dimension tests.
dr.coordinate.test returns marginal dimension tests (Cook, 2004)
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
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
x4 could be dropped from the problem by testing
The hypothesis to be tested is determined by the argument
hypothesis = ~.-x4 would test the hypothesis of the last
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
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
For SAVE objects, two p.values are also returned.
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
For IRE and PIRE, the tests
statistics have an asymptotic Chisq distribution, so the
chi2approx is not relevant.
Yongwu Shao for SIR and SAVE and Sanford Weisberg for all methods, <firstname.lastname@example.org>
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.
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# 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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