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 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.

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``,
- 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`

(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")
```

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)
```