StudentsT: Create a Student's T distribution

In distributions3: Probability Distributions as S3 Objects

Create a Student's T distribution

Description

The Student's T distribution is closely related to the Normal() distribution, but has heavier tails. As \nu increases to \infty, the Student's T converges to a Normal. The T distribution appears repeatedly throughout classic frequentist hypothesis testing when comparing group means.

Usage

StudentsT(df)

Arguments

df	Degrees of freedom. Can be any positive number. Often called \nu in textbooks.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Students T random variable with df = \nu.

Support: R, the set of all real numbers

Mean: Undefined unless \nu \ge 2, in which case the mean is zero.

Variance:

\frac{\nu}{\nu - 2}

Undefined if \nu < 1, infinite when 1 < \nu \le 2.

Probability density function (p.d.f):

f(x) = \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu} )^{- \frac{\nu + 1}{2}}

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

Value

A StudentsT object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.