Bootstrap Repeated Measurements Model

Description

For a dataset containing a time variable, a scalar response variable, and an optional subject identification variable, obtains least squares estimates of the coefficients of a restricted cubic spline function or a linear regression in time after adjusting for subject effects through the use of subject dummy variables. Then the fit is bootstrapped B times, either by treating time and subject ID as fixed (i.e., conditioning the analysis on them) or as random variables. For the former, the residuals from the original model fit are used as the basis of the bootstrap distribution. For the latter, samples are taken jointly from the time, subject ID, and response vectors to obtain unconditional distributions.

If a subject id variable is given, the bootstrap sampling will be based on samples with replacement from subjects rather than from individual data points. In other words, either none or all of a given subject's data will appear in a bootstrap sample. This cluster sampling takes into account any correlation structure that might exist within subjects, so that confidence limits are corrected for within-subject correlation. Assuming that ordinary least squares estimates, which ignore the correlation structure, are consistent (which is almost always true) and efficient (which would not be true for certain correlation structures or for datasets in which the number of observation times vary greatly from subject to subject), the resulting analysis will be a robust, efficient repeated measures analysis for the one-sample problem.

Predicted values of the fitted models are evaluated by default at a grid of 100 equally spaced time points ranging from the minimum to maximum observed time points. Predictions are for the average subject effect. Pointwise confidence intervals are optionally computed separately for each of the points on the time grid. However, simultaneous confidence regions that control the level of confidence for the entire regression curve lying within a band are often more appropriate, as they allow the analyst to draw conclusions about nuances in the mean time response profile that were not stated apriori. The method of Tibshirani (1997) is used to easily obtain simultaneous confidence sets for the set of coefficients of the spline or linear regression function as well as the average intercept parameter (over subjects). Here one computes the objective criterion (here both the -2 log likelihood evaluated at the bootstrap estimate of beta but with respect to the original design matrix and response vector, and the sum of squared errors in predicting the original 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 curves are computed at the time grid, and minima and maxima of these curves are computed separately at each time point to derive the final simultaneous confidence band.

By default, the log likelihoods that are computed for obtaining the simultaneous confidence band assume independence within subject. This will cause problems unless such log likelihoods have very high rank correlation with the log likelihood allowing for dependence. To allow for correlation or to estimate the correlation function, see the cor.pattern argument below.

Usage

rm.boot(time, y, id=seq(along=time), subset,
        plot.individual=FALSE,
        bootstrap.type=c('x fixed','x random'),
        nk=6, knots, B=500, smoother=supsmu, 
        xlab, xlim, ylim=range(y), 
        times=seq(min(time),max(time),length=100),
        absorb.subject.effects=FALSE, 
        rho=0, cor.pattern=c('independent','estimate'), ncor=10000,
        ...)

## S3 method for class 'rm.boot':
plot(x, obj2, conf.int=.95,
     xlab=x$xlab, ylab=x$ylab, 
     xlim, ylim=x$ylim,
     individual.boot=FALSE,
     pointwise.band=FALSE,
     curves.in.simultaneous.band=FALSE,
     col.pointwise.band=2,
     objective=c('-2 log L','sse','dep -2 log L'), add=FALSE, ncurves,
     multi=FALSE, multi.method=c('color','density'),
     multi.conf   =c(.05,.1,.2,.3,.4,.5,.6,.7,.8,.9,.95,.99),
     multi.density=c( -1,90,80,70,60,50,40,30,20,10,  7,  4),
     multi.col    =c(  1, 8,20, 5, 2, 7,15,13,10,11,  9, 14),
     subtitles=TRUE, ...)

Arguments

Details

Observations having missing time or y are excluded from the analysis.

As most repeated measurement studies consider the times as design points, the fixed covariable case is the default. Bootstrapping the residuals from the initial fit assumes that the model is correctly specified. Even if the covariables are fixed, doing an unconditional bootstrap is still appropriate, and for large sample sizes unconditional confidence intervals are only slightly wider than conditional ones. For moderate to small sample sizes, the "x random" method can be fairly conservative.

If not all subjects have the same number of observations (after deleting observations containing missing values) and if bootstrap.type="x fixed", bootstrapped residual vectors may have a length m that is different from the number of original observations n. If m > n for a bootstrap repetition, the first n elements of the randomly drawn residuals are used. If m < n, the residual vector is appended with a random sample with replacement of length n - m from itself. A warning message is issued if this happens. If the number of time points per subject varies, the bootstrap results for "x fixed" can still be invalid, as this method assumes that a vector (over subjects) of all residuals can be added to the original yhats, and varying number of points will cause mis-alignment.

For bootstrap.type="x random" in the presence of significant subject effects, the analysis is approximate as the subjects used in any one bootstrap fit will not be the entire list of subjects. The average (over subjects used in the bootstrap sample) intercept is used from that bootstrap sample as a predictor of average subject effects in the overall sample.

Once the bootstrap coefficient matrix is stored by rm.boot, plot.rm.boot can be run multiple times with different options (e.g, different confidence levels).

See bootcov in the Design library for a general approach to handling repeated measurement data for ordinary linear models, binary and ordinal models, and survival models, using the unconditional bootstrap. bootcov does not handle bootstrapping residuals.

Value

an object of class rm.boot is returned by rm.boot. The principal object stored in the returned object is a matrix of regression coefficients for the original fit and all of the bootstrap repetitions (object Coef), along with vectors of the corresponding -2 log likelihoods are sums of squared errors. The original fit object from lm.fit.qr is stored in fit. For this fit, a cell means model is used for the id effects.
plot.rm.boot returns a list containing the vector of times used for plotting along with the overall fitted values, lower and upper simultaneous confidence limits, and optionally the pointwise confidence limits.

Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University School of Medicine
f.harrell@vanderbilt.edu

References

Feng Z, McLerran D, Grizzle J (1996): A comparison of statistical methods for clustered data analysis with Gaussian error. Stat in Med 15:1793–1806.

Tibshirani R, Knight K (1997):Model search and inference by bootstrap "bumping". Technical Report, Department of Statistics, University of Toronto.
http://www-stat.stanford.edu/~tibs. Presented at the Joint Statistical Meetings, Chicago, August 1996.

Efron B, Tibshirani R (1993): An Introduction to the Bootstrap. New York: Chapman and Hall.

Diggle PJ, Verbyla AP (1998): Nonparametric estimation of covariance structure in logitudinal data. Biometrics 54:401–415.

Chapman IM, Hartman ML, et al (1997): Effect of aging on the sensitivity of growth hormone secretion to insulin-like growth factor-I negative feedback. J Clin Endocrinol Metab 82:2996–3004.

See Also

rcspline.eval, lm, lowess, supsmu, bootcov, units, label, polygon, reShape

Examples

# Generate multivariate normal responses with equal correlations (.7)
# within subjects and no correlation between subjects
# Simulate realizations from a piecewise linear population time-response
# profile with large subject effects, and fit using a 6-knot spline
# Estimate the correlation structure from the residuals, as a function
# of the absolute time difference

# Function to generate n p-variate normal variates with mean vector u and
# covariance matrix S
# Slight modification of function written by Bill Venables
# See also the built-in function rmvnorm
mvrnorm <- function(n, p = 1, u = rep(0, p), S = diag(p)) {
  Z <- matrix(rnorm(n * p), p, n)
  t(u + t(chol(S)) %*% Z)
}

n     <- 20         # Number of subjects
sub   <- .5*(1:n)   # Subject effects

# Specify functional form for time trend and compute non-stochastic component
times <- seq(0, 1, by=.1)
g     <- function(times) 5*pmax(abs(times-.5),.3)
ey    <- g(times)

# Generate multivariate normal errors for 20 subjects at 11 times
# Assume equal correlations of rho=.7, independent subjects

nt    <- length(times)
rho   <- .7

        
set.seed(19)        
errors <- mvrnorm(n, p=nt, S=diag(rep(1-rho,nt))+rho)
# Note:  first random number seed used gave rise to mean(errors)=0.24!

# Add E[Y], error components, and subject effects
y      <- matrix(rep(ey,n), ncol=nt, byrow=TRUE) + errors + 
          matrix(rep(sub,nt), ncol=nt)

# String out data into long vectors for times, responses, and subject ID
y      <- as.vector(t(y))
times  <- rep(times, n)
id     <- sort(rep(1:n, nt))

# Show lowess estimates of time profiles for individual subjects
f <- rm.boot(times, y, id, plot.individual=TRUE, B=25, cor.pattern='estimate',
             smoother=lowess, bootstrap.type='x fixed', nk=6)
# In practice use B=400 or 500
# This will compute a dependent-structure log-likelihood in addition
# to one assuming independence.  By default, the dep. structure
# objective will be used by the plot method  (could have specified rho=.7)
# NOTE: Estimating the correlation pattern from the residual does not
# work in cases such as this one where there are large subject effects

# Plot fits for a random sample of 10 of the 25 bootstrap fits
plot(f, individual.boot=TRUE, ncurves=10, ylim=c(6,8.5))

# Plot pointwise and simultaneous confidence regions
plot(f, pointwise.band=TRUE, col.pointwise=1, ylim=c(6,8.5))

# Plot population response curve at average subject effect
ts <- seq(0, 1, length=100)
lines(ts, g(ts)+mean(sub), lwd=3)

## Not run: 
#
# Handle a 2-sample problem in which curves are fitted 
# separately for males and females and we wish to estimate the
# difference in the time-response curves for the two sexes.  
# The objective criterion will be taken by plot.rm.boot as the 
# total of the two sums of squared errors for the two models
#
knots <- rcspline.eval(c(time.f,time.m), nk=6, knots.only=TRUE)
# Use same knots for both sexes, and use a times vector that 
# uses a range of times that is included in the measurement 
# times for both sexes
#
tm <- seq(max(min(time.f),min(time.m)),
          min(max(time.f),max(time.m)),length=100)

f.female <- rm.boot(time.f, bp.f, id.f, knots=knots, times=tm)
f.male   <- rm.boot(time.m, bp.m, id.m, knots=knots, times=tm)
plot(f.female)
plot(f.male)
# The following plots female minus male response, with 
# a sequence of shaded confidence band for the difference
plot(f.female,f.male,multi=TRUE)

# Do 1000 simulated analyses to check simultaneous coverage 
# probability.  Use a null regression model with Gaussian errors

n.per.pt <- 30
n.pt     <- 10

null.in.region <- 0

for(i in 1:1000) {
  y    <- rnorm(n.pt*n.per.pt)
  time <- rep(1:n.per.pt, n.pt)
#  Add the following line and add ,id=id to rm.boot to use clustering
#  id   <- sort(rep(1:n.pt, n.per.pt))
#  Because we are ignoring patient id, this simulation is effectively
#  using 1 point from each of 300 patients, with times 1,2,3,,,30 

  f <- rm.boot(time, y, B=500, nk=5, bootstrap.type='x fixed')
  g <- plot(f, ylim=c(-1,1), pointwise=FALSE)
  null.in.region <- null.in.region + all(g$lower<=0 & g$upper>=0)
  prn(c(i=i,null.in.region=null.in.region))
}

# Simulation Results: 905/1000 simultaneous confidence bands 
# fully contained the horizontal line at zero
## End(Not run)