| TDist {stats} | 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, ncp=0, 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. If length(n) > 1, the length
is taken to be the number required. |
df |
degrees of freedom (> 0, maybe non-integer). |
ncp |
non-centrality parameter delta;
currently for pt() and dt(), only for 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) * sqrt(n)/sigma.
dt gives the density,
pt gives the distribution function,
qt gives the quantile function, and
rt generates random deviates.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (except non-central versions.)
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)")
op <- par(yaxs="i")
plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32),
main="Non-central t - Density")
par(op)