The Studentized Range distribution

Source: R/dist_studentized_range.R

[Stable]

Tukey's studentized range distribution, used for Tukey's honestly significant differences test in ANOVA.

dist_studentized_range(nmeans, df, nranges)

Arguments

nmeans

sample size for range (same for each group).

df

degrees of freedom for \(s\) (see below).

nranges

number of groups whose maximum range is considered.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_studentized_range.html

In the following, let \(Q\) be a Studentized Range random variable with parameters nmeans = \(k\) (number of groups), df = \(\nu\) (degrees of freedom), and nranges = \(n\) (number of ranges).

Support: \(R^+\), the set of positive real numbers.

Mean: Approximated numerically.

Variance: Approximated numerically.

Probability density function (p.d.f): The density does not have a closed-form expression and is computed numerically.

Cumulative distribution function (c.d.f): The c.d.f does not have a simple closed-form expression. For \(n = 1\) (single range), it involves integration over the joint distribution of the sample range and an independent chi-square variable. The general form is computed numerically using algorithms described in the references for stats::ptukey().

Moment generating function (m.g.f): Does not exist in closed form.

See also

stats::Tukey

Examples

dist <- dist_studentized_range(nmeans = c(6, 2), df = c(5, 4), nranges = c(1, 1))

dist
#> <distribution[2]>
#> [1] StudentizedRange(6, 5, 1) StudentizedRange(2, 4, 1)

cdf(dist, 4)
#> [1] 0.8017144 0.9525787

quantile(dist, 0.7)
#> [1] 3.414964 1.682303