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help for ^nct^
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Non-central t programs
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^nctprob^ t' delta df /* yields p
^nctinv^ p delta df /* yields t'
^nctncp^ t' p df /* yields delta
^nctn^ t' delta p /* yields df + 1
^nct2^ t' delta df /* yields two-sided p
^nct2inv^ p delta df /* yields two-sided t'
where
^t'^ is the observed t-value
^delta^ is the noncentrality parameter
^df^ is the degrees of freedom (^df^ is a positive integer)
^p^ is the probability (0 < ^p^ < 1)
For each program, entering the program name with no parameters displays
the command syntax.
Description
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Let Y and Z be independent random variables, where
Z ~ N(delta,1) (Z is distributed Normal with mean delta and variance 1)
Y ~ ChiSq(n) (Y is distributed chi-square with degrees of freedom n).
Then, X = Z / sqrt(Y/n) is said to have a noncentral t distribution with
noncentrality delta and degrees of freedom n.
That is, X ~ t(delta, n).
^nct^xxxx is a family of immediate programs, all related to the noncentral t
distribution. Each one-sided program computes a missing parameter, given
the other parameters, such that P(t<=^t'^| ^delta^, ^df^) = ^p^.
for | use
------+--------------------------------------------------------
^p^ | ^nctprob^ -- Cumulative non-central t probabilities
^t'^ | ^nctinv^ -- Inverse cumulative non-central t values
^delta^ | ^nctncp^ -- Noncentrality parameter of the non-central t
^df^ | ^nctn^ -- Sample size for the cumulative non-central t
There are also two programs that yield two-sided values from a noncentral t,
defined such that P(|t|<=^t'^| ^delta^, ^df^) = 1 - ^p^.
for | use
------+--------------------------------------------------------
^p^ | ^nct2^ -- 2-sided non-central t probabilities
^t'^ | ^nct2inv^ -- Inverse 2-sided non-central t values
The core program, ^nctprob^, computes probabilities from the cumulative
non-central t distribution from negative infinity to ^t'^ for noncentrality
parameter, ^delta^, and positive, integer degrees of freedom, ^df^.
That is, ^nctprob^ computes ^p^ such that P(t<=^t'^| ^delta^, ^df^) = ^p^.
Each program prints its computed value and returns it in global ^S_1^ and in
a result ^r()^. The value computed and the name of the returned parameter
for each program are:
^nctprob^ -- ^r(p)^ -- the probability p
^nctinv^ -- ^r(t)^ -- the critical t'
^nctncp^ -- ^r(delta)^ -- the noncentrality parameter delta
^nctn^ -- ^r(n)^ -- the minimum n
^nct2^ -- ^r(p)^ -- the two-sided tail probability
^nct2inv^ -- ^r(t)^ -- the two-sided critical t'
^nctn^ computes the minimum ^n^ such that, for ^df^ = ^n^ - 1,
when ^p^ < 0.5, P(t<=^t'^|^delta^, ^df^) <= ^p^, and
when ^p^ > 0.5, 1 - P(t<=^t'^|^delta^, ^df^) <= ^p^.
Warning: Convergence time for ^nctn^ is a function of the computed minimum
^n^ and increases greatly when that ^n^ gets large. Because the limit of
the noncentral t (as n gets large) is the noncentral z, a comparison
of the desired probability ^p^ to P(z < ^t'^ - ^delta^) is informative.
If these values are within .005 of each other, convergence time will
be noticably non-negligible.
Notes
-----
These programs call ^integ^. The user is cautioned that the initial
Stata 6.0 release of ^integ^ (version 3.0.4) has a bug. Please install
version 3.0.5 or later.
Some of these programs require installation of programs ^ridder^
(see STB-24, insert ssi5.4 for ^ridder^).
Author
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Thomas J. Steichen
Examples
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. ^nctprob 3.6 3.0 11^ > gives p for t' = 3.6, delta = 3.0, df = 11
. ^nctinv .95 2.0 17^ > gives t' for p = .95, delta = 2.0, df = 17
. ^nctncp 4.045 .95 17^ > gives delta for t' = 4.045, p = .95, df = 17
. ^nctinv 4 2 .95^ > gives n for t' = 4, delta = 2, p = .95
. ^nct2 3.1 2.0 17^ > gives p for |t'| = 3.1, delta = 2.0, df = 17
. ^nct2inv .05 2.0 17^ > gives |t'| for p = .05, delta = 2.0, df = 17
. ^nctprob^ > displays the command syntax