dt(x, df) pt(q, df, ncp=0) qt(p, df) rt(n, df)
x,q
| vector of quantiles. |
p
| vector of probabilities. |
n
| number of observations to generate. |
df
| degrees of freedom. |
ncp
|
non-centrality parameter delta;
currently ncp <= 37.62.
|
df degrees of freedom (and optional noncentrality parameter
ncp). dt gives the density, pt
gives the distribution function, qt gives the quantile function
and rt generates random deviates.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.
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.
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)