dr.coordinate.test {dr}

Dimension reduction tests

Description

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

Usage

dr.test(object, numdir, ...)

dr.coordinate.test(object, hypothesis,d,chi2approx,...)

## S3 method for class 'ire'
dr.joint.test(object, hypothesis, d = NULL,...)

Arguments

object

The name of an object returned by a call to dr.

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 \chi^2(1) random variables. Choices are from c("bx","wood"), for the Bentler-Xie and Wood approximatations, respectively. The default is either "bx" or the value set in the call that created the dr object.

...

Additional arguments. None are currently available.

Details

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.

Value

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 \chi^2 distribution, so the value of chi2approx is not relevant.

Author(s)

Yongwu Shao for SIR and SAVE and Sanford Weisberg for all methods, <[e-mail]>

References

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.

See Also

drop1.dr, coord.hyp.basis, dr.step, dr.pvalue

Examples

#  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)