Australian institute of sport data

Description

Data on 102 male and 100 female athletes collected at the Australian Institute of Sport.

Format

This data frame contains the following columns:

Sex

(0 = male or 1 = female)

Ht

height (cm)

Wt

weight (kg)

LBM

lean body mass

RCC

red cell count

WCC

white cell count

Hc

Hematocrit

Hg

Hemoglobin

Ferr

plasma ferritin concentration

BMI

body mass index, weight/(height)**2

SSF

sum of skin folds

Bfat

Percent body fat

Label

Case Labels

Sport

Sport

Source

Ross Cunningham and Richard Telford

References

S. Weisberg (2005). Applied Linear Regression, 3rd edition. New York: Wiley, Section 6.4

Examples

data(ais)

Swiss banknote data

Description

Six measurements made on 100 genuine Swiss banknotes and 100 counterfeit ones.

Format

This data frame contains the following columns:

Length

Length of bill, mm

Left

Width of left edge, mm

Right

Width of right edge, mm

Bottom

Bottom margin width, mm

Top

Top margin width, mm

Diagonal

Length of image diagonal, mm

Y

0 = genuine, 1 = counterfeit

Source

Flury, B. and Riedwyl, H. (1988). Multivariate Statistics: A practical approach. London: Chapman & Hall.

References

Weisberg, S. (2005). Applied Linear Regression, 3rd edition. New York: Wiley, Problem 12.5.

Examples

data(banknote)

Internal function to find the basis of a subspace

Description

If a dimension reduction object has p dimension, returns a p \times r matrix whose columns space a subspace specified by a formula.

Usage

coord.hyp.basis(object, spec, which = 1)

Arguments

Details

The workings of this function is best explained by an example. Suppose the dr object was created with the formula y~x1+x2+x3+x4, so we have p=4 predictors. A matrix that spans the subspace of R^4 specified by Span(x1,x2,x3,x4) is simply the identity matrix of order 4.

This function will return a subset of the columns of this identity matrix, as determined by spec. For example, if spec = ~.-(x3+x4), the function returns the columns corresponding to x1 and x2 if which=+1 or the columns corresponding to x3 and x4 if which=-1. Similarly, if spec=~x1+x2, the same matrices will be returned.

Value

A matrix corresponding to the value of spec and which given.

Author(s)

Sanford Weisberg, sandy@stat.umn.edu

See Also

dr.coordinate.test

Examples

data(ais)
s1 <- dr(LBM~log(Ht)+log(Wt)+log(RCC)+log(WCC)+log(Hc)+log(Hg),
         data=ais,method="sir")
coord.hyp.basis(s1,~.-log(Wt)-log(Hg))
 

Main function for dimension reduction regression

Description

This is the main function in the dr package. It creates objects of class dr to estimate the central (mean) subspace and perform tests concerning its dimension. Several helper functions that require a dr object can then be applied to the output from this function.

Usage

dr (formula, data, subset, group=NULL, na.action = na.fail, weights, ...)
    
dr.compute (x, y, weights, group=NULL, method = "sir", chi2approx="bx",...)
 

Arguments

Details

The general regression problem studies F(y|x), the conditional distribution of a response y given a set of predictors x. This function provides methods for estimating the dimension and central subspace of a general regression problem. That is, we want to find a p \times d matrix B of minimal rank d such that

F(y|x)=F(y|B'x)

Both the dimension d and the subspace R(B) are unknown. These methods make few assumptions. Many methods are based on the inverse distribution, F(x|y).

For the methods "sir", "save", "phdy" and "phdres", a kernel matrix M is estimated such that the column space of M should be close to the central subspace R(B). The eigenvectors corresponding to the d largest eigenvalues of M provide an estimate of R(B).

For the method "ire", subspaces are estimated by minimizing an objective function.

Categorical predictors can be included using the groups argument, with the methods "sir", "save" and "ire", using the ideas from Chiaromonte, Cook and Li (2002).

The primary output from this method is (1) a set of vectors whose span estimates R(B); and various tests concerning the dimension d.

Weights can be used, essentially to specify the relative frequency of each case in the data. Empirical weights that make the contours of the weighted sample closer to elliptical can be computed using dr.weights. This will usually result in zero weight for some cases. The function will set zero estimated weights to missing.

Value

dr returns an object that inherits from dr (the name of the type is the value of the method argument), with attributes:

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>.

References

Bentler, P. M. and Xie, J. (2000), Corrections to test statistics in principal Hessian directions. Statistics and Probability Letters, 47, 381-389. Approximate p-values.

Cook, R. D. (1998). Regression Graphics. New York: Wiley. This book provides the basic results for dimension reduction methods, including detailed discussion of the methods "sir", "phdy" and "phdres".

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

Cook, R. D. and Nachtsheim, C. (1994), Reweighting to achieve elliptically contoured predictors in regression. Journal of the American Statistical Association, 89, 592–599. Describes the weighting scheme used by dr.weights.

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. The "ire" is described in this paper.

Cook, R. D. and Weisberg, S. (1999). Applied Regression Including Computing and Graphics, New York: Wiley. The program arc described in this book also computes most of the dimension reduction methods described here.

Chiaromonte, F., Cook, R. D. and Li, B. (2002). Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30 475-497. Introduced grouping, or conditioning on factors.

Shao, Y., Cook, R. D. and Weisberg (2007). Marginal tests with sliced average variance estimation. Biometrika. Describes the tests used for "save".

Wen, X. and Cook, R. D. (2007). Optimal Sufficient Dimension Reduction in Regressions with Categorical Predictors, Journal of Statistical Inference and Planning. This paper extends the "ire" method to grouping.

Wood, A. T. A. (1989) An F approximation to the distribution of a linear combination of chi-squared variables. Communications in Statistics: Simulation and Computation, 18, 1439-1456. Approximations for p-values.

Examples

data(ais)
# default fitting method is "sir"
s0 <- dr(LBM~log(SSF)+log(Wt)+log(Hg)+log(Ht)+log(WCC)+log(RCC)+
  log(Hc)+log(Ferr),data=ais) 
# Refit, using a different function for slicing to agree with arc.
summary(s1 <- update(s0,slice.function=dr.slices.arc))
# Refit again, using save, with 10 slices; the default is max(8,ncol+3)
summary(s2<-update(s1,nslices=10,method="save"))
# Refit, using phdres.  Tests are different for phd, and not
# Fit using phdres; output is similar for phdy, but tests are not justifiable. 
summary(s3<- update(s1,method="phdres"))
# fit using ire:
summary(s4 <- update(s1,method="ire"))
# fit using Sex as a grouping variable.  
s5 <- update(s4,group=~Sex)

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

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, <sandy@stat.umn.edu>

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)

Directions selected by dimension reduction regressiosn

Description

Dimension reduction regression returns a set of up to p orthogonal direction vectors each of length p, the first d of which are estimates a basis of a d dimensional central subspace. The function returns the estimated directions in the original n dimensional space for plotting.

Usage

dr.direction(object, which, x)
dr.directions(object, which, x)
## Default S3 method:
dr.direction(object, which=NULL,x=dr.x(object))

dr.basis(object,numdir)

## S3 method for class 'ire'
dr.basis(object,numdir=length(object$result))

Arguments

Details

Dimension reduction regression is used to estimate a basis of the central subspace or mean central subspace of a regression. If there are p predictors, the dimension of the central subspace is less than or equal to p. These two functions, dr.basis and dr.direction, return vectors that describe the central subspace in various ways.

Consder dr.basis first. If you set numdir=3, for example, this method will return a p by 3 matrix whose columns span the estimated three dimensional central subspace. For all methods except for ire, this simply returns the first three columns of object$evectors. For the ire method, this returns the three vectors determined by a three-dimensional solution. Call this matrix C. The basis is determined by back-transforming from centered and scaled predictors to the scale of the original predictors, and then renormalizing the vectors to have length one. These vectors are orthogonal in the inner product determined by Var(X).

The dr.direction method return XC, the same space but now a subspace of the original n-dimensional space. These vectors are appropriate for plotting.

Value

Both functions return a matrix: for dr.direction, the matrix has n rows and numdir columns, and for dr.basis it has p rows and numdir columns.

Author(s)

Sanford Weisberg <sandy@stat.umn.edu>

References

See R. D. Cook (1998). Regression Graphics. New York: Wiley.

See Also

dr

Examples

data(ais)
#fit dimension reduction using sir
m1 <- dr(LBM~Wt+Ht+RCC+WCC, method="sir", nslices = 8, data=ais)
summary(m1)
dr.basis(m1)
dr.directions(m1)

Permutation tests of dimension for dr

Description

Approximates marginal dimension test significance levels for sir, save, and phd by sampling from the permutation distribution.

Usage

dr.permutation.test(object, npermute=50,numdir=object$numdir)

Arguments

Details

The method approximates significance levels of the marginal dimension tests based on a permutation test. The algorithm: (1) permutes the rows of the predictor but not the response; (2) computes marginal dimension tests for the permuted data; (3) obtains significane levels by comparing the observed statsitics to the permutation distribution.

The method is not implemented for ire.

Value

Returns an object of type ‘dr.permutation.test’ that can be printed or summarized to give the summary of the test.

Author(s)

Sanford Weisberg, sandy@stat.umn.edu

References

See www.stat.umn.edu/arc/addons.html, and then select the article on dimension reduction regression or inverse regression.

See Also

dr

Examples

data(ais)
attach(ais)  # the Australian athletes data
#fit dimension reduction regression using sir
m1 <- dr(LBM~Wt+Ht+RCC+WCC, method="sir", nslices = 8)
summary(m1)
dr.permutation.test(m1,npermute=100)
plot(m1)

Compute the Chi-square approximations to a weighted sum of Chi-square(1) random variables.

Description

Returns an approximate quantile for a weighted sum of independent \chi^2(1) random variables.

Usage

dr.pvalue(coef,f,chi2approx=c("bx","wood"),...)

bentlerxie.pvalue(coef, f)

wood.pvalue(coef, f, tol=0.0, print=FALSE)

Arguments

Details

For Bentler-Xie, we approximate f by c \chi^2(d) for values of c and d computed by the function. The Wood approximation is more complicated.

Value

Returns a data.frame with four named components:

Author(s)

Sanford Weisberg <sandy@stat.umn.edu>

References

Peter M. Bentler and Jun Xie (2000), Corrections to test statistics in principal Hessian directions. Statistics and Probability Letters, 47, 381-389.

Wood, Andrew T. A. (1989) An F approximation to the distribution of a linear combination of chi-squared variables. Communications in Statistics: Simulation and Computation, 18, 1439-1456.

Divide a vector into slices of approximately equal size

Description

Divides a vector into slices of approximately equal size.

Usage

dr.slices(y, nslices)

dr.slices.arc(y, nslices)

Arguments

Details

If y is an n-vector, order y. The goal for the number of observations per slice is m, the smallest integer in nslices/n. Allocate the first m observations to slice 1. If there are duplicates in y, keep adding observations to the first slice until the next value of y is not equal to the largest value in the first slice. Allocate the next m values to the next slice, and again check for ties. Continue until all values are allocated to a slice. This does not guarantee that nslices will be obtained, nor does it guarantee an equal number of observations per slice. This method of choosing slices is invariant under rescaling, but not under multiplication by -1, so the slices of y will not be the same as the slices of -y. This function was rewritten for Version 2.0.4 of this package, and will no longer give exactly the same results as the program Arc. If you want to duplicate Arc, use the function dr.slice.arc, as illustrated in the example below.

If y is a matrix of p columns, slice the first column as described above. Then, within each of the slices determined for the first column, slice based on the second column, so that each of the “cells” has approximately the same number of observations. Continue through all the columns. This method is not invariant under reordering of the columns, or under multiplication by -1.

Value

Returns a named list with three elements as follows:

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>

References

R. D. Cook and S. Weisberg (1999), Applied Regression Including Computing and Graphics, New York: Wiley.

See Also

dr

Examples

 
data(ais)
summary(s1 <- dr(LBM~log(SSF)+log(Wt)+log(Hg)+log(Ht)+log(WCC)+log(RCC)+
                 log(Hc)+log(Ferr), data=ais,method="sir",nslices=8))
# To make this idential to ARC, need to modify slices to match.
summary(s2 <- update(s1,slice.info=dr.slices.arc(ais$LBM,8)))

Estimate weights for elliptical symmetry

Description

This function estimate weights to apply to the rows of a data matrix to make the resulting weighted matrix as close to elliptically symmetric as possible.

Usage

dr.weights(formula, data = list(), subset, na.action = na.fail,
    sigma=1, nsamples=NULL, ...)

Arguments

Details

The basic outline is: (1) Estimate a mean m and covariance matrix S using a possibly robust method; (2) For each iteration, obtain a random vector from N(m,sigma*S). Add 1 to a counter for observation i if the i-th row of the data matrix is closest to the random vector; (3) return as weights the sample faction allocated to each observation. If you set the keyword weights.only to T on the call to dr, then only the list of weights will be returned.

Value

Returns a list of n weights, some of which may be zero.

Author(s)

Sanford Weisberg, sandy@stat.umn.edu

References

R. D. Cook and C. Nachtsheim (1994), Reweighting to achieve elliptically contoured predictors in regression. Journal of the American Statistical Association, 89, 592–599.

See Also

dr

Examples

data(ais)
w1 <- dr.weights(~ Ht +Wt +RCC, data = ais)
m1 <- dr(LBM~Ht+Wt+RCC,data=ais,weights=w1)

Accessor functions for data in dr objects

Description

Accessor functions for dr objects.

Usage

dr.x(object)
dr.y(object)
dr.z(object) 

Arguments

Value

Returns a component of a dr object. dr.x returns the predictor matrix reduced to full rank by dropping trailing columns; dr.y returns the response vector/matrix, and dr.z returns the centered and scaled predictor matrix.

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>

See Also

dr.

Sequential fitting of coordinate tests using a dr object

Description

This function implements backward elimination using a dr object for which a dr.coordinate.test is defined, currently for SIR SAVE, IRE and PIRE.

Usage

dr.step(object,scope=NULL,d=NULL,minsize=2,stop=0,trace=1,...)

## S3 method for class 'dr'
drop1(object, scope = NULL,  update=TRUE,
test="general",trace=1,...)

Arguments

Details

Suppose a dr object has p=a+b predictors, with a predictors specified in the scope statement. drop1 will compute either marginal coordinate tests (if d=NULL) or conditional marginal coordinate tests (if d is positive) for dropping each of the b predictors not in the scope, and return p.values. The result is an object created from the original object with the predictor with the largest p.value removed.

dr.step will call drop1.dr repeatedly until \max(a,d+1) predictors remain.

Value

As a side effect, a data frame of labels, tests, df, and p.values is printed. If update=TRUE, a dr object is returned with the predictor with the largest p.value removed.

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>, based on the drop1 generic function in the base R. The dr.step function is also similar to step in base R.

References

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

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

See Also

dr.coordinate.test

Examples

data(ais)
# To make this idential to ARC, need to modify slices to match by
# using slice.info=dr.slices.arc() rather than nslices=8
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.method=dr.slices.arc,nslices=8)) 
# The following will almost duplicate information in Table 5 of Cook (2004).
# Slight differences occur because a different approximation for the
# sum of independent chi-square(1) random variables is used:
ans1 <- drop1(s1)
ans2 <- drop1(s1,d=2)
ans3 <- drop1(s1,d=3)
# remove predictors stepwise until we run out of variables to drop.
dr.step(s1,scope=~log(Wt)+log(Ht))

Mussels' muscles data

Description

Data were furnished by Mike Camden, Wellington Polytechnic, Wellington, New Zealand. Horse mussels, (Atrinia), were sampled from the Marlborough Sounds. The response is the mussels' Muscle Mass.

Format

This data frame contains the following columns:

H

Shell height in mm

L

Shell length in mm

M

Muscle mass in g

S

Shell mass in g

W

Shell width in mm

Source

R. D. Cook and S. Weisberg (1999). Applied Statistics Including Computing and Graphics. New York: Wiley.

Basic plot of a dr object

Description

Plots selected direction vectors determined by a dimension reduction regression fit. By default, the pairs function is used for plotting, but the user can use any other graphics command that is appropriate.

Usage

## S3 method for class 'dr'
plot(x, which = 1:x$numdir, mark.by.y = FALSE, plot.method = pairs, ...)

Arguments

Value

Returns a graph.

Author(s)

Sanford Weisberg, <sandy@stat.umn.edu>.

Examples

data(ais)
# default fitting method is "sir"
s0 <- dr(LBM~log(SSF)+log(Wt)+log(Hg)+log(Ht)+log(WCC)+log(RCC)+
  log(Hc)+log(Ferr),data=ais)
plot(s0)
plot(s0,mark.by.y=TRUE)