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, 185189.
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)")