In regression analysis, bootstrapping is a method for statistical

inference, which focused on building a sampling distribution with the key idea

of resampling the originally observed data with replacement. The term

bootstrapping, proposed by Bradley Efron in his “Bootstrap methods:

another look at the jackknife” published in 1979, is extracted from the cliché

of ‘pulling oneself up by one’s bootstraps’. So, from the meaning of this

concept, sample data is considered as a population and repeated samples are

drawn from the sample data, which is considered as a population, to generate

the statistical inference about the sample data. The essential bootstrap analogy states that “the

population is to the sample as the sample is to the bootstrap samples”.

The bootstrap falls into two types, parametric and nonparametric. Parametric

bootstrapping assumes that the original data set is drawn from some specific

distributions, e.g. normal distribution. And the samples generally are pulled as the same size

as the original data set. Nonparametric bootstrapping is just the one described

in the beginning, which draws a portion of bootstrapping samples from the

original data. Bootstrapping is quite useful in non-linear regression and

generalized linear models. For small sample size, the parametric bootstrapping

method is highly preferred. In large sample size, nonparametric bootstrapping

method would be preferably utilized. For a further clarification of nonparametric

bootstrapping, a sample data set, A = {x1, x2, …, xk} is randomly drawn from

a population B = {X1, X2, …, XK} and K is much larger than k. The statistic T

= t(A) is considered as an estimate of the corresponding population parameter P

= t(B). Nonparametric bootstrapping generates the estimate of the sampling

distribution of a statistic in an empirical way. No assumptions of the form of the population

is necessary. Next, a sample of size k is drawn from the elements of A with replacement,

which represents as A?1 =

{x?11,

x?12,

…, x?1k}.

In the resampling, a * note is added to distinguish resampled data from

original data. Replacement is mandatory and supposed to be repeated typically

1000 or 10000 times, which is still developing since computation power develops,

otherwise only original sample A would be generated. And for each bootstrap estimate of these samples, mean is

calculated to estimate the expectation of the bootstrapped statistics. Mean minus T is the estimate of T’s bias. And

T?, the bootstrap variance estimate,

estimates the sampling variance of the

population, P. Then bootstrap confidence intervals can be constructed using

either bootstrap percentile interval approach or normal theory interval

approach. Confidence intervals by bootstrap percentile method is to use the empirical

quantiles of the bootstrap estimates, which is written as T?(lower) < P < T?(upper). More specifically, it can be written as Tˆ ?
(Tˆ ?
upper – T*ˆ) ? P ? Tˆ + (T*ˆ + Tˆ ?lower).
Bootstrapping is an effective
method to doublecheck the stability of the model estimation results. It is much
better than the intervals calculated by sample variance with normality
assumption. And simplicity is bootstrapping's another important benefit. For
complicated estimators, such as correlation coefficients, percentile points,
for complex parameters in the distribution, it is a pretty simple way to generate
estimates of confidence intervals and standard errors. However, simplicity can also
bring up disadvantage for bootstrapping, which makes the important assumptions for
the bootstrapping easy to neglect. And bootstrapping is often over-optimistic
and doesn't assure finite sample size.
There are several types of bootstrapping schemes in the regression
problems. One typical approach is to resample residuals in the regression
models. The main procedure is firstly fit the original data set with the model,
and generate model estimates, ?ˆ and calculate residuals, ?ˆ; secondly randomly
and repeatedly sample the residuals (typically 1000 or 10000 times) to get K
sets residuals of size k and add each resampled residual to the original
equation, generating bootstrapped Y*; Finally use bootstrapped Y* to refit the
model and get bootstrap estimate ?ˆ?.
Another typical approach in the regression context is random-x
resampling, which is also called case resampling. We can either apply Monte
Carlo algorithm, which is to repeatedly resample the data of the same size as
the original data set with replacement, or identify any possible resampling of
the data set. In our case, before fitting regression model with the original predictor
variable and response pairs (xi, yi), for i = 1, 2, . . ., k, these data pairs
are resampled to get K new data pairs of size k. Then the regression model is
fit to each of these K new data sets. ?ˆ? is generated from K parameter estimates.
In the next section, I'm going to review the nonparametric bootstrapping
package in R with some examples in my research area-----population
pharmacokinetics analysis. In R, a package is called "boot", which provides various
sources for bootstrapping either a single statistic or a vector. To run the
boot function in the boot library, there are 3 necessary parameters:
1)
data, which can
be a vector, matrix, or data frame for bootstrap resampling;
2)
statistic, the
function that produces the statistic for bootstrapping. This function should
include the data set and an indices parameter, giving the selection of cases
for each resampling;
3)
R, the number of resampling
times.
The function boot() runs the statistic function for R
times. In each call, it generates a group of random indices with replacement to
select a sample. Then calculated statistics for each sample are collected in
the bootobject function. So the function boot() is used as bootobject <- boot(data= , statistic= , R=,
...). After seeing the satisfying plot, we use boot.ci(bootobject, conf=, type=
) to get confidence intervals.
Bootstrapping is prevalently used in
the population analysis of clinical trials in pharmaceutical/biotech
industries. It is a pretty useful tool to assess and control the model analysis
stability. A good example is how bootstrapping validates population
pharmacokinetic model for Triptan, a vasopressor used for the treatment of
migraine attack. A single oral dose of 50 mg was administered to 26 healthy
Korean male subjects. Plasma concentration data were obtained from pre-dose through
12 hours post-dose. Population pharmacokinetics analysis of Triptan was
performed using plasma concentration data by the software called NONMEM building
models using differential equations. Total 364 observations of plasma concentrations
were successfully described by a one compartment model with first-order of both
absorption with lag time and elimination, and a combined transit compartment.
The model scheme is shown as Figure 1 as below:
Figure 1: The scheme of the final PK model of Triptan
The final model was validated through a 1000-time
resampling bootstrapping, which was to conduct with 1000 datasets resampled from
the original dataset with replacement. The median and 90% prediction intervals
of parameters were shown in the Table 1 to compare with the final parameter
estimates. Results from the visual prediction check with 1000
Table 1: NONMEM estimated Parameters
and Bootstrap Results
simulations were assessed by visual comparison of the gray area of 90%
prediction interval from the simulated data with an overlay of the circled raw
data. Any observed circled data going outside the gray area indicates that the
estimates were not legitimate.
Figure 2:
Visual predictive check plot of the model from time 0 to 12 h after a single
oral administration of 50 mg Triptan. Circles represent the raw data set: the
90% prediction interval of the 1000 times simulations (gray area), and observed
concentration (solid line) of the 5th, median, and 95th percentiles.
Our conclusion is that the
final model and its estimated parameter were sufficiently robust and stable by
the assessment of the bootstrapping. All estimated parameter from the final
model were within the 95% bootstrap confidence intervals.