| TDist {base} | R Documentation |
Density, distribution function, quantile function and random
generation for the t distribution with df degrees of freedom
(and optional noncentrality parameter ncp).
dt(x, df, log = FALSE) pt(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE) qt(p, df, lower.tail = TRUE, log.p = FALSE) rt(n, df)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations to generate. |
df |
degrees of freedom (> 0, maybe non-integer). |
ncp |
non-centrality parameter delta;
currently ncp <= 37.62. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]. |
The t distribution with df = n degrees of
freedom has density
f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)
for all real x. It has mean 0 (for n > 1) and variance n/(n-2) (for n > 2).
The general non-central t
with parameters (df,Del) = (df, ncp)
is defined as a the distribution of
T(df,Del) := (U + Del) / (Chi(df) / sqrt(df))
where U and Chi(df) are independent random
variables, U ~ N(0,1), and
Chi(df)^2
is chi-squared, see pchisq.
The most used applications are power calculations for t-tests:
Let T= (mX - m0) / (S/sqrt(n))
where
mX is the mean and S the sample standard
deviation (sd) of X_1,X_2,...,X_n which are i.i.d.
N(mu,sigma^2).
Then T is distributed as non-centrally t with
df= n-1
degrees of freedom and non-centrality parameter
ncp= mu - m0.
dt gives the density,
pt gives the distribution function,
qt gives the quantile function, and
rt generates random deviates.
Lenth, R. V. (1989). Algorithm AS 243 — Cumulative distribution function of the non-central t distribution, Appl. Statist. 38, 185–189.
df for the F distribution.
1 - pt(1:5, df = 1)
qt(.975, df = c(1:10,20,50,100,1000))
tt <- seq(0,10, len=21)
ncp <- seq(0,6, len=31)
ptn <- outer(tt,ncp, function(t,d) pt(t, df = 3, ncp=d))
image(tt,ncp,ptn, zlim=c(0,1),main=t.tit <- "Non-central t - Probabilities")
persp(tt,ncp,ptn, zlim=0:1, r=2, phi=20, theta=200, main=t.tit,
xlab = "t", ylab = "noncentrality parameter", zlab = "Pr(T <= t)")