boot.pval

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.

Installation

To install the package from CRAN:

install.packages("boot.pval")

To install the development version from Github:

library(devtools)
install_github("mthulin/boot.pval")

Background

p-values can be computed by inverting the corresponding confidence intervals, as described in Section 12.2 of Thulin (2021) 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:

• The p-value of the two-sided test for the parameter theta is the smallest alpha such that theta is not contained in the corresponding 1-alpha confidence interval,
• For a test of the parameter theta with significance level alpha, the set of values of theta that aren’t rejected by the two-sided test (when used as the null hypothesis) is a 1-alpha confidence interval for theta.

Summaries for regression models

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:

• Linear models fitted using lm,
• Generalised linear models fitted using glm or glm.nb,
• Nonlinear models fitted using nls,
• Robust linear models fitted using MASS::rlm,
• Ordered logistic or probit regression models fitted (without weights) using MASS:polr,
• Linear mixed models fitted using lme4::lmer or lmerTest::lmer,
• Generalised linear mixed models fitted using lme4::glmer.
• Cox PH regression models fitted using survival::coxph (using censboot_summary).
• Accelerated failure time models fitted using survival::survreg or rms::psm (using censboot_summary).
• Any regression model such that: 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.

Here is an example 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")

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)

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)
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")

Other hypothesis tests

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)