bootcov
Bootstrap Covariance and Distribution for Regression Coefficients
bootcov
computes a bootstrap estimate of the covariance matrix for a set
of regression coefficients from ols
, lrm
, cph
,
psm
, Rq
, and any
other fit where x=TRUE, y=TRUE
was used to store the data used in making
the original regression fit and where an appropriate fitter
function
is provided here. The estimates obtained are not conditional on
the design matrix, but are instead unconditional estimates. For
small sample sizes, this will make a difference as the unconditional
variance estimates are larger. This function will also obtain
bootstrap estimates corrected for cluster sampling (intracluster
correlations) when a "working independence" model was used to fit
data which were correlated within clusters. This is done by substituting
cluster sampling with replacement for the usual simple sampling with
replacement. bootcov
has an option (coef.reps
) that causes all
of the regression coefficient estimates from all of the bootstrap
resamples to be saved, facilitating computation of nonparametric
bootstrap confidence limits and plotting of the distributions of the
coefficient estimates (using histograms and kernel smoothing estimates).
The loglik
option facilitates the calculation of simultaneous
confidence regions from quantities of interest that are functions of
the regression coefficients, using the method of Tibshirani(1996).
With Tibshirani's method, one computes the objective criterion (2 log
likelihood evaluated at the bootstrap estimate of $beta$ but with
respect to the original design matrix and response vector) for the
original fit as well as for all of the bootstrap fits. The confidence
set of the regression coefficients is the set of all coefficients that
are associated with objective function values that are less than or
equal to say the 0.95 quantile of the vector of B + 1
objective
function values. For the coefficients satisfying this condition,
predicted values are computed at a userspecified design matrix X
,
and minima and maxima of these predicted values (over the qualifying
bootstrap repetitions) are computed to derive the final simultaneous
confidence band.
The bootplot
function takes the output of bootcov
and
either plots a histogram and kernel density
estimate of specified regression coefficients (or linear combinations
of them through the use of a specified design matrix X
), or a
qqnorm
plot of the quantities of interest to check for normality of
the maximum likelihood estimates. bootplot
draws vertical lines at
specified quantiles of the bootstrap distribution, and returns these
quantiles for possible printing by the user. Bootstrap estimates may
optionally be transformed by a userspecified function fun
before
plotting.
The confplot
function also uses the output of bootcov
but to
compute and optionally plot nonparametric bootstrap pointwise confidence
limits or (by default) Tibshirani (1996) simultaneous confidence sets.
A design matrix must be specified to allow confplot
to compute
quantities of interest such as predicted values across a range
of values or differences in predicted values (plots of effects of
changing one or more predictor variable values).
bootplot
and confplot
are actually generic functions, with
the particular functions bootplot.bootcov
and confplot.bootcov
automatically invoked for bootcov
objects.
A service function called histdensity
is also provided (for use with
bootplot
). It runs hist
and density
on the same plot, using
twice the number of classes than the default for hist
, and 1.5 times the
width
than the default used by density
.
A comprehensive example demonstrates the use of all of the functions.
 Keywords
 models, methods, hplot, regression, htest
Usage
bootcov(fit, cluster, B=200, fitter, coef.reps=TRUE, loglik=FALSE, pr=FALSE, maxit=15, eps=0.0001, group=NULL, stat=NULL)
bootplot(obj, which=1 : ncol(Coef), X, conf.int=c(.9,.95,.99), what=c('density', 'qqnorm', 'box'), fun=function(x) x, labels., ...)
confplot(obj, X, against, method=c('simultaneous','pointwise'), conf.int=0.95, fun=function(x)x, add=FALSE, lty.conf=2, ...)
histdensity(y, xlab, nclass, width, mult.width=1, ...)
Arguments
 fit

a fit object containing components
x
andy
. For fits fromcph
, the"strata"
attribute of thex
component is used to obtain the vector of stratum codes.  obj

an object created by
bootcov
withcoef.reps=TRUE
.  X

a design matrix specified to
confplot
. Seepredict.rms
orcontrast.rms
. Forbootplot
,X
is optional.  y

a vector to pass to
histdensity
.NA
s are ignored.  cluster

a variable indicating groupings.
cluster
may be any type of vector (factor, character, integer). Unique values ofcluster
indicate possibly correlated groupings of observations. Note the data used in the fit and stored infit$x
andfit$y
may have had observations containing missing values deleted. It is assumed that if there were any NAs, annaresid
function exists for the class offit
. This function restores NAs so that the rows of the design matrix coincide withcluster
.  B
 number of bootstrap repetitions. Default is 200.
 fitter

the name of a function with arguments
(x,y)
that will fit bootstrap samples. Default is taken from the class offit
if it isols
,lrm
,cph
,psm
,Rq
.  coef.reps

set to
TRUE
if you want to store a matrix of all bootstrap regression coefficient estimates in the returned componentboot.Coef
.  loglik

set to
TRUE
to store 2 log likelihoods for each bootstrap model, evaluated against the originalx
andy
data. The default is to do this whencoef.reps
is specified asTRUE
. The use ofloglik=TRUE
assumes that anoos.loglik
method exists for the type of model being analyzed, to calculate outofsample 2 log likelihoods (seermsMisc
). After theB
2 log likelihoods (stored in the element namedboot.loglik
in the returned fit object), theB+1
element is the 2 log likelihood for the original model fit.  pr

set to
TRUE
to print the current sample number to monitor progress.  maxit
 maximum number of iterations, to pass to
fitter
 eps
 argument to pass to various fitters
 group

a grouping variable used to stratify the sample upon bootstrapping.
This allows one to handle ksample problems, i.e., each bootstrap
sample will be forced to select the same number of observations from
each level of group as the number appearing in the original dataset.
You may specify both
group
andcluster
.  stat

a single character string specifying the name of a
stats
element produced by the fitting function to save over the bootstrap repetitions. The vector of saved statistics will be in theboot.stats
part of the list returned bybootcov
.  which

one or more integers specifying which regression coefficients to
plot for
bootplot
 conf.int

a vector (for
bootplot
, default isc(.9,.95,.99)
) or scalar (forconfplot
, default is.95
) confidence level.  what

for
bootplot
, specifies whether a density or a qq plot is made, aggplot2
is used to produce a box plot of all coefficients over the bootstrap reps  fun

for
bootplot
orconfplot
specifies a function used to translate the quantities of interest before analysis. A common choice isfun=exp
to compute antilogs, e.g., odds ratios.  labels.

a vector of labels for labeling the axes in plots produced by
bootplot
. Default is row names ofX
if there are any, or sequential integers.  ...

For
bootplot
these are optional arguments passed tohistdensity
. Also may be optional arguments passed toplot
byconfplot
or optional arguments passed tohist
fromhistdensity
, such asxlim
andbreaks
. The argumentprobability=TRUE
is always passed tohist
.  against

For
confplot
, specifyingagainst
causes a plot to be made (or added to). Theagainst
variable is associated with rows ofX
and is used as the xcoordinates.  method

specifies whether
"pointwise"
or"simultaneous"
confidence regions are derived byconfplot
. The default is simultaneous.  add

set to
TRUE
to add to an existing plot, forconfplot
 lty.conf

line type for plotting confidence bands in
confplot
. Default is 2 for dotted lines.  xlab

label for xaxis for
histdensity
. Default islabel
attribute or argument name if there is nolabel
.  nclass

passed to
hist
if present  width

passed to
density
if present  mult.width

multiplier by which to adjust the default
width
passed todensity
. Default is 1.
Details
If the fit has a scale parameter (e.g., a fit from psm
), the log
of the individual bootstrap scale estimates are added to the vector
of parameter estimates and and column and row for the log scale are
added to the new covariance matrix (the old covariance matrix also
has this row and column).
For Rq
fits, the tau
, method
, and hs
arguments are taken from the original fit.
Value

a new fit object with class of the original object and with the element
orig.var
added. orig.var
is
the covariance matrix of the original fit. Also, the original var
component is replaced with the new bootstrap estimates. The component
boot.coef
is also added. This contains the mean bootstrap estimates
of regression coefficients (with a log scale element added if
applicable). boot.Coef
is added if coef.reps=TRUE
.
boot.loglik
is added if loglik=TRUE
. If stat
is
specified an additional vector boot.stats
will be contained in
the returned object. B
contains the number of successfully fitted
bootstrap resamples. A component
clusterInfo
is added to contain elements name
and n
holding the name of the cluster
variable and the number of clusters.bootplot
returns a (possible matrix) of quantities of interest and
the requested quantiles of them. confplot
returns three vectors:
fitted
, lower
, and upper
.
Side Effects
bootcov
prints if pr=TRUE
References
Feng Z, McLerran D, Grizzle J (1996): A comparison of statistical methods for clustered data analysis with Gaussian error. Stat in Med 15:17931806.
Tibshirani R, Knight K (1996): Model search and inference by bootstrap "bumping". Department of Statistics, University of Toronto. Technical report available from
http://wwwstat.stanford.edu/~tibs/. Presented at the Joint Statistical Meetings, Chicago, August 1996.
See Also
robcov
, sample
, rms
,
lm.fit
, lrm.fit
,
survivalinternal
,
predab.resample
, rmsMisc
,
Predict
, gendata
,
contrast.rms
, Predict
, setPb
,
cluster.boot
Examples
set.seed(191)
x < exp(rnorm(200))
logit < 1 + x/2
y < ifelse(runif(200) <= plogis(logit), 1, 0)
f < lrm(y ~ pol(x,2), x=TRUE, y=TRUE)
g < bootcov(f, B=50, pr=TRUE)
anova(g) # using bootstrap covariance estimates
fastbw(g) # using bootstrap covariance estimates
beta < g$boot.Coef[,1]
hist(beta, nclass=15) #look at normality of parameter estimates
qqnorm(beta)
# bootplot would be better than these last two commands
# A dataset contains a variable number of observations per subject,
# and all observations are laid out in separate rows. The responses
# represent whether or not a given segment of the coronary arteries
# is occluded. Segments of arteries may not operate independently
# in the same patient. We assume a "working independence model" to
# get estimates of the coefficients, i.e., that estimates assuming
# independence are reasonably efficient. The job is then to get
# unbiased estimates of variances and covariances of these estimates.
set.seed(2)
n.subjects < 30
ages < rnorm(n.subjects, 50, 15)
sexes < factor(sample(c('female','male'), n.subjects, TRUE))
logit < (ages50)/5
prob < plogis(logit) # true prob not related to sex
id < sample(1:n.subjects, 300, TRUE) # subjects sampled multiple times
table(table(id)) # frequencies of number of obs/subject
age < ages[id]
sex < sexes[id]
# In truth, observations within subject are independent:
y < ifelse(runif(300) <= prob[id], 1, 0)
f < lrm(y ~ lsp(age,50)*sex, x=TRUE, y=TRUE)
g < bootcov(f, id, B=50) # usually do B=200 or more
diag(g$var)/diag(f$var)
# add ,group=w to resample from within each level of w
anova(g) # clusteradjusted Wald statistics
# fastbw(g) # clusteradjusted backward elimination
plot(Predict(g, age=30:70, sex='female')) # clusteradjusted confidence bands
# Get design effects based on inflation of the variances when compared
# with bootstrap estimates which ignore clustering
g2 < bootcov(f, B=50)
diag(g$var)/diag(g2$var)
# Get design effects based on pooled tests of factors in model
anova(g2)[,1] / anova(g)[,1]
# Simulate binary data where there is a strong
# age x sex interaction with linear age effects
# for both sexes, but where not knowing that
# we fit a quadratic model. Use the bootstrap
# to get bootstrap distributions of various
# effects, and to get pointwise and simultaneous
# confidence limits
set.seed(71)
n < 500
age < rnorm(n, 50, 10)
sex < factor(sample(c('female','male'), n, rep=TRUE))
L < ifelse(sex=='male', 0, .1*(age50))
y < ifelse(runif(n)<=plogis(L), 1, 0)
f < lrm(y ~ sex*pol(age,2), x=TRUE, y=TRUE)
b < bootcov(f, B=50, loglik=TRUE, pr=TRUE) # better: B=500
par(mfrow=c(2,3))
# Assess normality of regression estimates
bootplot(b, which=1:6, what='qq')
# They appear somewhat nonnormal
# Plot histograms and estimated densities
# for 6 coefficients
w < bootplot(b, which=1:6)
# Print bootstrap quantiles
w$quantiles
# Show box plots for bootstrap reps for all coefficients
bootplot(b, what='box')
# Estimate regression function for females
# for a sequence of ages
ages < seq(25, 75, length=100)
label(ages) < 'Age'
# Plot fitted function and pointwise normal
# theory confidence bands
par(mfrow=c(1,1))
p < Predict(f, age=ages, sex='female')
plot(p)
# Save curve coordinates for later automatic
# labeling using labcurve in the Hmisc library
curves < vector('list',8)
curves[[1]] < with(p, list(x=age, y=lower))
curves[[2]] < with(p, list(x=age, y=upper))
# Add pointwise normaldistribution confidence
# bands using unconditional variancecovariance
# matrix from the 500 bootstrap reps
p < Predict(b, age=ages, sex='female')
curves[[3]] < with(p, list(x=age, y=lower))
curves[[4]] < with(p, list(x=age, y=upper))
dframe < expand.grid(sex='female', age=ages)
X < predict(f, dframe, type='x') # Full design matrix
# Add pointwise bootstrap nonparametric
# confidence limits
p < confplot(b, X=X, against=ages, method='pointwise',
add=TRUE, lty.conf=4)
curves[[5]] < list(x=ages, y=p$lower)
curves[[6]] < list(x=ages, y=p$upper)
# Add simultaneous bootstrap confidence band
p < confplot(b, X=X, against=ages, add=TRUE, lty.conf=5)
curves[[7]] < list(x=ages, y=p$lower)
curves[[8]] < list(x=ages, y=p$upper)
lab < c('a','a','b','b','c','c','d','d')
labcurve(curves, lab, pl=TRUE)
# Now get bootstrap simultaneous confidence set for
# female:male odds ratios for a variety of ages
dframe < expand.grid(age=ages, sex=c('female','male'))
X < predict(f, dframe, type='x') # design matrix
f.minus.m < X[1:100,]  X[101:200,]
# First 100 rows are for females. By subtracting
# design matrices are able to get Xf*Beta  Xm*Beta
# = (Xf  Xm)*Beta
confplot(b, X=f.minus.m, against=ages,
method='pointwise', ylab='F:M Log Odds Ratio')
confplot(b, X=f.minus.m, against=ages,
lty.conf=3, add=TRUE)
# contrast.rms makes it easier to compute the design matrix for use
# in bootstrapping contrasts:
f.minus.m < contrast(f, list(sex='female',age=ages),
list(sex='male', age=ages))$X
confplot(b, X=f.minus.m)
# For a quadratic binary logistic regression model use bootstrap
# bumping to estimate coefficients under a monotonicity constraint
set.seed(177)
n < 400
x < runif(n)
logit < 3*(x^21)
y < rbinom(n, size=1, prob=plogis(logit))
f < lrm(y ~ pol(x,2), x=TRUE, y=TRUE)
k < coef(f)
k
vertex < k[2]/(2*k[3])
vertex
# Outside [0,1] so fit satisfies monotonicity constraint within
# x in [0,1], i.e., original fit is the constrained MLE
g < bootcov(f, B=50, coef.reps=TRUE, loglik=TRUE)
bootcoef < g$boot.Coef # 100x3 matrix
vertex < bootcoef[,2]/(2*bootcoef[,3])
table(cut2(vertex, c(0,1)))
mono < !(vertex >= 0 & vertex <= 1)
mean(mono) # estimate of Prob{monotonicity in [0,1]}
var(bootcoef) # varcov matrix for unconstrained estimates
var(bootcoef[mono,]) # for constrained estimates
# Find secondbest vector of coefficient estimates, i.e., best
# from among bootstrap estimates
g$boot.Coef[order(g$boot.loglik[length(g$boot.loglik)])[1],]
# Note closeness to MLE
## Not run:
# # Get the bootstrap distribution of the difference in two ROC areas for
# # two binary logistic models fitted on the same dataset. This analysis
# # does not adjust for the bias ROC area (Cindex) due to overfitting.
# # The same random number seed is used in two runs to enforce pairing.
#
# set.seed(17)
# x1 < rnorm(100)
# x2 < rnorm(100)
# y < sample(0:1, 100, TRUE)
# f < lrm(y ~ x1, x=TRUE, y=TRUE)
# g < lrm(y ~ x1 + x2, x=TRUE, y=TRUE)
# set.seed(3)
# f < bootcov(f, stat='C')
# set.seed(3)
# g < bootcov(g, stat='C')
# dif < g$boot.stats  f$boot.stats
# hist(dif)
# quantile(dif, c(.025,.25,.5,.75,.975))
# # Compute a ztest statistic. Note that comparing ROC areas is far less
# # powerful than likelihood or Brier scorebased methods
# z < (g$stats['C']  f$stats['C'])/sd(dif)
# names(z) < NULL
# c(z=z, P=2*pnorm(abs(z)))
# ## End(Not run)