Background
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.
This vignette describes how to perform bootstrap t-tests and bootstrap median tests.
Two-sample bootstrap t-tests
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)
#>
#> Welch Two Sample Bootstrap t-test (studentized)
#>
#> data: extra by group
#> t = -1.8608, R = 9999, p-value = 0.08281
#> alternative hypothesis: true difference in means between group 1 and group 2 is not equal to 0
#> 95 percent confidence interval:
#> -3.3910742 0.2349124
#> sample estimates:
#> mean in group 1 mean in group 2
#> 0.75 2.33If 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 intervalYou 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)One-sample bootstrap t-tests
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$extra, mu = 1)
# Using pipes:
sleep |> boot_t_test(extra ~ 1, mu = 1)Bootstrap median tests
Running a bootstrap median test with the
boot_median_test function is completely analogously to
running a bootstrap t-test. The only difference is under the hood -
medians are used instead of means. Because the studentized and BCa
versions of this test use an inner bootstrap to estimate the variance of
the statistic, these takes longer to run than other tests presented
here.
boot_median_test(extra ~ group, data = sleep, type = "perc")
sleep |> boot_median_test(extra ~ group, R = 999, alternative = "less")
boot_median_test(sleep$extra, mu = 1)