This R package provides functions for computing bootstrap p-values based on boot
objects, and convenience functions for bootstrap confidence intervals and p-values for various regression models.
To install the package from CRAN:
install.packages("boot.pval")
To install the development version from Github:
library(devtools)
install_github("mthulin/boot.pval")
p-values can be computed by inverting the corresponding confidence intervals, as described in Section 14.2 of Thulin (2024) and Section 3.12 in Hall (1992). This package contains functions for computing bootstrap p-values in this way. The approach relies on the fact that:
Summary tables with confidence intervals and p-values for the coefficients of regression models can be obtained using the boot_summary
(most models) and censboot_summary
(models with censored response variables) functions. Currently, the following models are supported:
lm
,glm
or glm.nb
,nls
,MASS::rlm
,MASS:polr
,lme4::lmer
or lmerTest::lmer
,lme4::glmer
.survival::coxph
(using censboot_summary
).survival::survreg
or rms::psm
(using censboot_summary
).residuals(object, type="pearson")
returns Pearson residuals; fitted(object)
returns fitted values; hatvalues(object)
returns the leverages, or perhaps the value 1 which will effectively ignore setting the hatvalues. In addition, the data
argument should contain no missing values among the columns actually used in fitting the model.A number of examples are available in Chapters 8 and 9 of Modern Statistics with R.
Here are some simple examples with a linear regression model for the mtcars
data:
# Bootstrap summary of a linear model for mtcars:
model <- lm(mpg ~ hp + vs, data = mtcars)
boot_summary(model)
# Use 9999 bootstrap replicates and adjust p-values for
# multiplicity using Holm's method:
boot_summary(model, R = 9999, adjust.method = "holm")
# Use case resampling instead of residual resampling:
boot_summary(model, method = "case")
# Export results to a gt table:
boot_summary(model, R = 9999) |>
summary_to_gt()
See Davison and Hinkley (1997) for details about residual resampling (the default) and case resampling.
# Export results to a Word document:
library(flextable)
boot_summary(model, R = 9999) |>
summary_to_flextable() |>
save_as_docx(path = "my_table.docx")
And a toy example for a generalised linear mixed model (using a small number of bootstrap repetitions):
library(lme4)
model <- glmer(TICKS ~ YEAR + (1|LOCATION),
data = grouseticks, family = poisson)
boot_summary(model, R = 99)
For complex models, speed can be greatly improved by using parallelisation. For lmer
and glmer
models, this is set using the parallel
(available options are "multicore"
and "snow"
). The number of CPUs to use is set using ncpus
.
model <- glmer(TICKS ~ YEAR + (1|LOCATION),
data = grouseticks, family = poisson)
boot_summary(model, R = 999, parallel = "multicore", ncpus = 10)
For other models, use ncores
:
model <- lm(mpg ~ hp + vs, data = mtcars)
boot_summary(model, R = 9999, ncores = 10)
Survival regression models should be fitted using the argument model = TRUE
. A summary table can then be obtained using censboot_summary
. By default, the table contains exponentiated coefficients (i.e. hazard ratios, in the case of a Cox PH model).
library(survival)
# Weibull AFT model:
model <- survreg(formula = Surv(time, status) ~ age + sex, data = lung,
dist = "weibull", model = TRUE)
# Table with exponentiated coefficients:
censboot_summary(model)
# Cox PH model:
model <- coxph(formula = Surv(time, status) ~ age + sex, data = lung,
model = TRUE)
# Table with hazard ratios:
censboot_summary(model)
# Table with original coefficients:
censboot_summary(model, coef = "raw")
To speed up computations using parallelisation, use the parallel
and ncpus
arguments:
censboot_summary(model, parallel = "multicore", ncpus = 10)
Traditional versions of Student's t-test (t.test
in R) rely on the assumption of normality. For non-normal data, this can lead to misleading p-values and confidence intervals. In such cases, it is often recommended to use the Wilcoxon-Mann-Whitney test (wilcox.test
in R) instead. Despite being described as a test of location, or a test for differences of medians, the Wilcoxon-Mann-Whitney test is actually a test of equivalence of distributions, unless strict assumptions are met. In addition, wilcox.test
does not provide a confidence interval for the difference of the medians.
In many cases, a better option is to use a bootstrap t-test (for inference about means) or a bootstrap median test (for inference about medians). These can be used without the normality assumption, and will provide confidence intervals for the parameters of interest.
To illustrate the use of bootstrap t-tests, we'll use the classic sleep
data, which "show the effect of two soporific drugs (increase in hours of sleep compared to control) on 10 patients" (see ?sleep
for details).
We wish to test whether the mean value of the extra
(increase in hours of sleep) variable differs between the two groups described by the group
variable. The syntax for this is identical to that for t.test
:
boot_t_test(extra ~ group, data = sleep)
If you prefer, you can also use the |>
pipe as follows:
sleep |> boot_t_test(extra ~ group)
By default, the confidence interval and p-value are based on the studentized bootstrap confidence interval. Other options available are normal, basic, percentile and BCa intervals; see Chapter 5 of Davison and Hinkley (1997) for details. You can choose the method used using the type
argument.
sleep |> boot_t_test(extra ~ group, type = "perc") # Percentile interval
sleep |> boot_t_test(extra ~ group, type = "bca") # BCa interval
You can control the number of bootstrap replicates used (argument R
; the default is 9999) or the direction of the alternative hypothesis (argument alternative
):
sleep |> boot_t_test(extra ~ group, R = 999, alternative = "less")
In this case, the data is actually paired, so it would make sense to perform a paired bootstrap t-test instead. We reshape the data to a wide format, so that the first measurements ends up in the variable extra.1
, and the second measurement ends up in the variable extra.2
. We can then run the test as follows:
# Reshape to wide format:
sleep2 <- reshape(sleep, direction = "wide",
idvar = "ID", timevar = "group")
# Traditional interface:
boot_t_test(sleep2$extra.1, sleep2$extra.2, paired = TRUE)
# Using pipes:
sleep2 |> boot_t_test(Pair(extra.1, extra.2) ~ 1)
For one sample bootstrap t-tests, we only need to provide a single vector containing the measurements. We can also specify the null value of the mean (argument mu
):
# Traditional interface:
boot_t_test(sleep$extra, mu = 1)
# Using pipes:
sleep |> boot_t_test(extra ~ 1, mu = 1)
Running a bootstrap median test with the boot_median_test
function is completely analogously to running a bootstrap t-test. The only difference is under the hood - medians are used instead of means. Because the studentized and BCa versions of this test use an inner bootstrap to estimate the variance of the statistic, these takes longer to run than other tests presented here.
boot_median_test(extra ~ group, data = sleep, type = "perc")
sleep |> boot_median_test(extra ~ group, R = 999, alternative = "less")
boot_median_test(sleep$extra, mu = 1)
Bootstrap p-values for hypothesis tests based on boot
objects can be obtained using the boot.pval
function. The following examples are extensions of those given in the documentation for boot::boot
:
# Hypothesis test for the city data
# H0: ratio = 1.4
library(boot)
ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
city.boot <- boot(city, ratio, R = 999, stype = "w", sim = "ordinary")
boot.pval(city.boot, theta_null = 1.4)
# Studentized test for the two sample difference of means problem
# using the final two series of the gravity data.
diff.means <- function(d, f)
{
n <- nrow(d)
gp1 <- 1:table(as.numeric(d$series))[1]
m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
ss1 <- sum(d[gp1,1]^2 * f[gp1]) - (m1 * m1 * sum(f[gp1]))
ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - (m2 * m2 * sum(f[-gp1]))
c(m1 - m2, (ss1 + ss2)/(sum(f) - 2))
}
grav1 <- gravity[as.numeric(gravity[,2]) >= 7, ]
grav1.boot <- boot(grav1, diff.means, R = 999, stype = "f",
strata = grav1[ ,2])
boot.pval(grav1.boot, type = "stud", theta_null = 0)
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